From c79a986ce7fdee9b88845c58d0a379220d099f56 Mon Sep 17 00:00:00 2001
From: "W. Joel Schneider" <w.joel.schneider@gmail.com>
Date: Thu, 29 Feb 2024 15:01:06 -0500
Subject: [PATCH] anova tutorial update

---
 .../2010-10-07 Overture/index.html            |    4 +-
 .../2016-10-26 RESCA-E/RESCAE.html            |    6 +-
 docs/assessingpsyche.html                     |    2 +-
 docs/assessingpsyche.xml                      |    4 +-
 docs/blog.html                                |    2 +-
 docs/blog.xml                                 |   34 +-
 docs/index.html                               |    4 +-
 docs/listings.json                            |   17 +-
 .../index.html                                |    4 +-
 .../index.html                                |    4 +-
 .../index.html                                |    4 +-
 .../index.html                                |    4 +-
 .../index.html                                |    4 +-
 .../index.html                                |    4 +-
 .../index.html                                |    4 +-
 .../index.html                                |    4 +-
 .../index.html                                |    4 +-
 .../index.html                                |    4 +-
 docs/presentations.html                       |    4 +-
 docs/projects.html                            |    4 +-
 docs/search.json                              |  450 ++--
 docs/site_libs/quarto-listing/list.min.js     |    2 +-
 docs/sitemap.xml                              |   76 +-
 docs/software.html                            |    4 +-
 docs/tutorials.html                           |   31 +-
 docs/tutorials.xml                            | 2343 +++++++++++++++--
 docs/tutorials/ANOVA/ANOVA.html               |  354 ++-
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 docs/tutorials/Regression/ggplotcode.svg      |   77 +
 docs/tutorials/Regression/index.html          | 2331 ++++++++++++++++
 .../figure-html/unnamed-chunk-21-1.svg        | 2268 ++++++++++++++++
 .../figure-html/unnamed-chunk-3-1.svg         |  294 +++
 .../figure-html/unnamed-chunk-5-1.svg         | 1939 ++++++++++++++
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 tutorials/ANOVA/ANOVA.qmd                     |   63 +-
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 tutorials/Regression/index.qmd                |  642 +++++
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 190 files changed, 19453 insertions(+), 655 deletions(-)
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diff --git a/docs/AssessingPsyche/2010-10-07 Overture/index.html b/docs/AssessingPsyche/2010-10-07 Overture/index.html
index bc8446c..0905248 100644
--- a/docs/AssessingPsyche/2010-10-07 Overture/index.html	
+++ b/docs/AssessingPsyche/2010-10-07 Overture/index.html	
@@ -2,7 +2,7 @@
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@@ -241,7 +241,7 @@ <h1 class="title">Overture Redux</h1>
 </nav>
 <div id="quarto-sidebar-glass" class="quarto-sidebar-collapse-item" data-bs-toggle="collapse" data-bs-target=".quarto-sidebar-collapse-item"></div>
 <!-- margin-sidebar -->
-    <div id="quarto-margin-sidebar" class="sidebar margin-sidebar">
+    <div id="quarto-margin-sidebar" class="sidebar margin-sidebar zindex-bottom">
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 <main class="content quarto-banner-title-block" id="quarto-document-content">
diff --git a/docs/AssessingPsyche/2016-10-26 RESCA-E/RESCAE.html b/docs/AssessingPsyche/2016-10-26 RESCA-E/RESCAE.html
index 3a2991d..0e68134 100644
--- a/docs/AssessingPsyche/2016-10-26 RESCA-E/RESCAE.html	
+++ b/docs/AssessingPsyche/2016-10-26 RESCA-E/RESCAE.html	
@@ -2,7 +2,7 @@
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@@ -336,7 +336,7 @@ <h2 id="toc-title">Sections</h2>
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 <!-- margin-sidebar -->
-    <div id="quarto-margin-sidebar" class="sidebar margin-sidebar">
+    <div id="quarto-margin-sidebar" class="sidebar margin-sidebar zindex-bottom">
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@@ -484,6 +484,7 @@ <h1>Summary of Influences on RESCA-E Performances</h1>
 </figcaption>
 <div aria-describedby="tbl-subtests-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
 <div class="cell-output-display">
+<div>
 <div id="wkjcdvfxyc" style="padding-left:0px;padding-right:0px;padding-top:10px;padding-bottom:10px;overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
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@@ -1014,6 +1015,7 @@ <h1>Summary of Influences on RESCA-E Performances</h1>
 </div>
 </div>
 </div>
+</div>
 </figure>
 </div>
 </div>
diff --git a/docs/assessingpsyche.html b/docs/assessingpsyche.html
index 278af01..5970de9 100644
--- a/docs/assessingpsyche.html
+++ b/docs/assessingpsyche.html
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diff --git a/docs/assessingpsyche.xml b/docs/assessingpsyche.xml
index 2a2d02e..51d9f61 100644
--- a/docs/assessingpsyche.xml
+++ b/docs/assessingpsyche.xml
@@ -9,7 +9,7 @@
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 <item>
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@@ -195,6 +195,7 @@ Table&nbsp;1: CHC Constructs Hypothesized to Influence Performance on RESCA-E Su
 </figcaption>
 <div aria-describedby="tbl-subtests-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
 <div class="cell-output-display">
+<div>
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@@ -725,6 +726,7 @@ Table&nbsp;1: CHC Constructs Hypothesized to Influence Performance on RESCA-E Su
 </div>
 </div>
 </div>
+</div>
 </figure>
 </div>
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diff --git a/docs/blog.html b/docs/blog.html
index d1c43a7..c586da4 100644
--- a/docs/blog.html
+++ b/docs/blog.html
@@ -2,7 +2,7 @@
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diff --git a/docs/blog.xml b/docs/blog.xml
index eff6d78..50b5ea3 100644
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 <item>
   <title>Making a custom arrowhead for ggplot2 using ggarrow and arrowheadr</title>
@@ -398,6 +398,7 @@ background-color: null;
 font-style: inherit;">\usetikzlibrary</span>{arrows}</span>
 <span id="cb4-2"><span class="kw" style="color: #003B4F;
 background-color: null;
+font-weight: bold;
 font-style: inherit;">\begin</span>{<span class="ex" style="color: null;
 background-color: null;
 font-style: inherit;">tikzpicture</span>}</span>
@@ -432,6 +433,7 @@ font-style: inherit;">\path</span>[-&gt;,</span>
 <span id="cb4-21">    text = black] (B) -&gt; (C);</span>
 <span id="cb4-22"><span class="kw" style="color: #003B4F;
 background-color: null;
+font-weight: bold;
 font-style: inherit;">\end</span>{<span class="ex" style="color: null;
 background-color: null;
 font-style: inherit;">tikzpicture</span>}</span></code></pre></div>
@@ -822,6 +824,7 @@ Figure&nbsp;9: Grid for custom arrowheads
 background-color: null;
 font-style: inherit;">&lt;-</span> <span class="cf" style="color: #003B4F;
 background-color: null;
+font-weight: bold;
 font-style: inherit;">function</span>(<span class="at" style="color: #657422;
 background-color: null;
 font-style: inherit;">a =</span> <span class="dv" style="color: #AD0000;
@@ -2159,6 +2162,7 @@ background-color: null;
 font-style: inherit;">roboto</span>}</span>
 <span id="cb8-11"><span class="kw" style="color: #003B4F;
 background-color: null;
+font-weight: bold;
 font-style: inherit;">\begin</span>{<span class="ex" style="color: null;
 background-color: null;
 font-style: inherit;">document</span>}</span>
@@ -2406,6 +2410,7 @@ font-style: inherit;">{r_{XX}-r_{XX}^2}$</span>}</span>
 <span id="cb8-65">    </span>
 <span id="cb8-66"><span class="kw" style="color: #003B4F;
 background-color: null;
+font-weight: bold;
 font-style: inherit;">\end</span>{<span class="ex" style="color: null;
 background-color: null;
 font-style: inherit;">document</span>}</span></code></pre></div>
@@ -2462,6 +2467,7 @@ font-style: inherit;"># Find where a line intersects with the normal cdf</span><
 background-color: null;
 font-style: inherit;">&lt;-</span> <span class="cf" style="color: #003B4F;
 background-color: null;
+font-weight: bold;
 font-style: inherit;">function</span>(<span class="at" style="color: #657422;
 background-color: null;
 font-style: inherit;">rxx =</span> .<span class="dv" style="color: #AD0000;
@@ -2519,6 +2525,7 @@ background-color: null;
 font-style: inherit;">0</span></span>
 <span id="cb10-17">  <span class="cf" style="color: #003B4F;
 background-color: null;
+font-weight: bold;
 font-style: inherit;">while</span> (<span class="fu" style="color: #4758AB;
 background-color: null;
 font-style: inherit;">abs</span>(diff_y) <span class="sc" style="color: #5E5E5E;
@@ -3187,10 +3194,12 @@ font-style: inherit;">set</span>(</span>
 background-color: null;
 font-style: inherit;">solutionsetter =</span> <span class="cf" style="color: #003B4F;
 background-color: null;
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 background-color: null;
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 <span id="cb1-7">  <span class="cf" style="color: #003B4F;
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+font-weight: bold;
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+font-weight: bold;
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 font-style: inherit;">0</span>)</span>
 <span id="cb5-3">} <span class="cf" style="color: #003B4F;
 background-color: null;
+font-weight: bold;
 font-style: inherit;">else</span> {</span>
 <span id="cb5-4">  y <span class="ot" style="color: #003B4F;
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+    <div id="quarto-margin-sidebar" class="sidebar margin-sidebar zindex-bottom">
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diff --git a/docs/posts/splitting-sentence-texts-into-roughly-equal-segments/index.html b/docs/posts/splitting-sentence-texts-into-roughly-equal-segments/index.html
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+<meta name="generator" content="quarto-1.5.9">
 
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@@ -291,7 +291,7 @@ <h2 id="toc-title">Sections</h2>
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diff --git a/docs/presentations.html b/docs/presentations.html
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+<meta name="generator" content="quarto-1.5.9">
 
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@@ -226,7 +226,7 @@ <h2 id="toc-title">On this page</h2>
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diff --git a/docs/projects.html b/docs/projects.html
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@@ -203,7 +203,7 @@ <h2 id="toc-title">On this page</h2>
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diff --git a/docs/search.json b/docs/search.json
index e8a477f..64721cf 100644
--- a/docs/search.json
+++ b/docs/search.json
@@ -147,130 +147,235 @@
     "text": "Professional Memberships\n\nMember, American Psychological Association\nMember, National Association of School Psychologists"
   },
   {
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-    "href": "tutorials/ANOVA/ANOVA.html",
-    "title": "ANOVA in R",
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+    "href": "tutorials/Regression/index.html",
+    "title": "Regression in R",
     "section": "",
-    "text": "library(tidyverse)"
+    "text": "Here we are going to use a small data set to predict people’s height using the height of their parents."
   },
   {
-    "objectID": "tutorials/ANOVA/ANOVA.html#summary",
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-    "section": "Summary",
-    "text": "Summary\nThe base R function summary gives us most of what we might want.\n\nsummary(fit)\n\n             Df Sum Sq Mean Sq F value Pr(&gt;F)    \nAttachment    2  20359   10179      93 &lt;2e-16 ***\nResiduals   297  32517     109                   \n---\nSignif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1"
+    "objectID": "tutorials/Regression/index.html#vector-based-images",
+    "href": "tutorials/Regression/index.html#vector-based-images",
+    "title": "Regression in R",
+    "section": "Vector-based Images",
+    "text": "Vector-based Images\nThe .pdf format gives the best image quality but can only be viewed in a .pdf reader. The .svg format is almost as good and can be incorporated into webpages and Office documents. One downside of their near-perfect image quality is that .pdf and .svg image file sizes can become quite large."
   },
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-    "title": "ANOVA in R",
-    "section": "Other Summary Functions",
-    "text": "Other Summary Functions\nIf you want other summary functions, there are many options. Here are a few:\n\nbroom::glance(fit)\n\n\n\n\n\nlogLik\nAIC\nBIC\ndeviance\nnobs\nr.squared\n\n\n\n\n-1129\n2265\n2280\n32517\n300\n0.39\n\n\n\n\n\nbroom::tidy(fit)\n\n\n\n\n\nterm\ndf\nsumsq\nmeansq\nstatistic\np.value\n\n\n\n\nAttachment\n2\n20359\n10179\n93\n0\n\n\nResiduals\n297\n32517\n109\nNA\nNA\n\n\n\n\n\nsjstats::anova_stats(fit)\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nterm\ndf\nsumsq\nmeansq\nstatistic\np.value\netasq\npartial.etasq\nomegasq\npartial.omegasq\nepsilonsq\ncohens.f\npower\n\n\n\n\nAttachment\nAttachment\n2\n20359\n10179\n93\n0\n0.38\n0.38\n0.38\n0.38\n0.38\n0.79\n1\n\n\n…2\nResiduals\n297\n32517\n109\nNA\nNA\nNA\nNA\nNA\nNA\nNA\nNA\nNA\n\n\n\n\n\nperformance::performance(fit)\n\n\n\n\n\nAIC\nAICc\nBIC\nR2\nR2_adjusted\nRMSE\nSigma\n\n\n\n\n2265\n2265\n2280\n0.39\n0.38\n10\n10\n\n\n\n\n\nparameters::parameters(fit)\n\n\n\n\n\nParameter\nSum_Squares\ndf\nMean_Square\nF\np\n\n\n\n\nAttachment\n20359\n2\n10179\n93\n0\n\n\nResiduals\n32517\n297\n109\nNA\nNA\n\n\n\n\n\nreport::report(fit)\n\nThe ANOVA (formula: Anxiety_Questionnaire ~ Attachment) suggests that:\n\n  - The main effect of Attachment is statistically significant and large (F(2,\n297) = 92.98, p &lt; .001; Eta2 = 0.39, 95% CI [0.32, 1.00])\n\nEffect sizes were labelled following Field's (2013) recommendations."
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+    "title": "Regression in R",
+    "section": "Raster Images",
+    "text": "Raster Images\nThe .png format gives good image quality and renders small file sizes.\nThe primary use of the .gif format is to create animated plots. Otherwise stick with .png.\nAlthough the .jpg format is good for photos, it is terrible for plots—it often renders text and sharp corners with pixelated smudges."
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-    "section": "Diagnostic Plots",
-    "text": "Diagnostic Plots\n\nperformance::check_model(fit)"
+    "objectID": "tutorials/Regression/index.html#model-level-statistics",
+    "href": "tutorials/Regression/index.html#model-level-statistics",
+    "title": "Regression in R",
+    "section": "Model-Level Statistics",
+    "text": "Model-Level Statistics\nSome statistics like the coefficient of determination (R2) or the standard error of the estimate (σe) describe the model as a whole.\nThe model-level statistics can be extracted with broom’s glance function or the performance package’s model_performance function.\n\n# Using broom package\nglance(m1)\n\n\n\n\n\nr.squared\nadj.r.squared\nsigma\nstatistic\np.value\ndf\nlogLik\nAIC\nBIC\ndeviance\ndf.residual\nnobs\n\n\n\n\n0.64\n0.63\n2.68\n67.7\n0\n1\n-95.2\n196\n201\n273\n38\n40\n\n\n\n\n\n\n\nmodel_parameters(m1)\n\n\n\n\n\nParameter\nCoefficient\nSE\nCI\nCI_low\nCI_high\nt\ndf_error\np\n\n\n\n\n(Intercept)\n-12.88\n9.78\n0.95\n-32.7\n6.92\n-1.32\n38\n0.2\n\n\navgphgt\n1.19\n0.14\n0.95\n0.9\n1.49\n8.23\n38\n0.0\n\n\n\n\n\n\n\nparameters(m1)\n\n\n\n\n\nParameter\nCoefficient\nSE\nCI\nCI_low\nCI_high\nt\ndf_error\np\n\n\n\n\n(Intercept)\n-12.88\n9.78\n0.95\n-32.7\n6.92\n-1.32\n38\n0.2\n\n\navgphgt\n1.19\n0.14\n0.95\n0.9\n1.49\n8.23\n38\n0.0\n\n\n\n\n\n\n\nstandardise_parameters(m1)\n\n\n\n\n\nParameter\nStd_Coefficient\nCI\nCI_low\nCI_high\n\n\n\n\n(Intercept)\n0.0\n0.95\n-0.19\n0.19\n\n\navgphgt\n0.8\n0.95\n0.60\n1.00\n\n\n\n\n\n\n\nperformance(m1)\n\n\n\n\n\nAIC\nAICc\nBIC\nR2\nR2_adjusted\nRMSE\nSigma\n\n\n\n\n196\n197\n201\n0.64\n0.63\n2.61\n2.68\n\n\n\n\n\n\n\n\nR2 is in the glance function’s r.squared column, and the standard error of the estimate is in the sigma column. If all you wanted was the R2, you could do this:\n\nglance(m1)$r.squared\n\n[1] 0.641\n\n\nOr using the performance package’s r2 function:\n\nr2(m1)\n\n# R2 for Linear Regression\n       R2: 0.641\n  adj. R2: 0.631\n\n\nFor the standard error of the estimate:\n\nglance(m1)$sigma\n\n[1] 2.68\n\n\nWhy would you want just one number instead of reading it from a table? In reproducible research, we intermingle text and code so that it is clear where every number came from. Thus, “hard-coding” your results like this is considered poor practice:\nThe model explains 64% of the variance.\nUsing rmarkdown, instead of typing the numeric results, we type pull the results using an inline code chunk:\nThe model explains `r round(100 * glance(m1)$r.squared, 0)`% of the variance.\nWhich, when rendered, produces the correct output:\n\nThe model explains 64% of the variance.\n\nThat seems like a lot of extra work, right? Yes, it is—unless there is a possibility that your underlying data might change or that you might copy your numbers incorrectly. If you are imperfect, the extra time and effort is worth it. It makes it easy for other scholars to see exactly where each number came from. Hard-coded results are harder to trust.\nAutomated report of the model’s performance:\n\nreport_performance(m1)\n\nThe model explains a statistically significant and substantial proportion of\nvariance (R2 = 0.64, F(1, 38) = 67.73, p &lt; .001, adj. R2 = 0.63)"
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-    "title": "ANOVA in R",
-    "section": "Post-hoc Tests",
-    "text": "Post-hoc Tests\nTukey Honestly Significant Differences\n\nTukeyHSD(fit)\n\n  Tukey multiple comparisons of means\n    95% family-wise confidence level\n\nFit: aov(formula = Anxiety_Questionnaire ~ Attachment, data = d)\n\n$Attachment\n                    diff  lwr  upr p adj\nAvoidant-Secure     -2.3 -5.8  1.3  0.29\nAmbivalent-Secure   16.2 12.8 19.5  0.00\nAmbivalent-Avoidant 18.4 14.8 22.0  0.00\n\nplot(TukeyHSD(fit))"
+    "objectID": "tutorials/Regression/index.html#coefficient-level-statistics",
+    "href": "tutorials/Regression/index.html#coefficient-level-statistics",
+    "title": "Regression in R",
+    "section": "Coefficient-level statistics",
+    "text": "Coefficient-level statistics\nThe regression coefficients—the intercept (b0) and the slope (b1)—have a number of statistics associated with them, which we will discuss later in the course.\nIf you just wanted the intercept and the slope coefficients, use the coef function:\n\ncoef(m1)\n\n(Intercept)     avgphgt \n     -12.88        1.19 \n\n\nThus, the intercept is -12.88 and the slope is 1.19.\nTo get the coefficients (intercept and slope) along with their p-values and other statistical information, use the tidy function:\n\ntidy(m1)\n\n\n\n\n\nterm\nestimate\nstd.error\nstatistic\np.value\n\n\n\n\n(Intercept)\n-12.88\n9.78\n-1.32\n0.2\n\n\navgphgt\n1.19\n0.14\n8.23\n0.0\n\n\n\n\n\n\n\n\nThe parameters package has a slightly more print-friendly version of tidy:\n\nmodel_parameters(m1) \n\n\n\n\n\nParameter\nCoefficient\nSE\nCI\nCI_low\nCI_high\nt\ndf_error\np\n\n\n\n\n(Intercept)\n-12.88\n9.78\n0.95\n-32.7\n6.92\n-1.32\n38\n0.2\n\n\navgphgt\n1.19\n0.14\n0.95\n0.9\n1.49\n8.23\n38\n0.0\n\n\n\n\n\n\n\n\nIf you want standardized parameters:\n\nmodel_parameters(m1, standardize = \"refit\")\n\n\n\n\n\nParameter\nCoefficient\nSE\nCI\nCI_low\nCI_high\nt\ndf_error\np\n\n\n\n\n(Intercept)\n0.0\n0.1\n0.95\n-0.19\n0.19\n0.00\n38\n1\n\n\navgphgt\n0.8\n0.1\n0.95\n0.60\n1.00\n8.23\n38\n0\n\n\n\n\n\n\n\n\nAutomated report of the parameters:\n\nreport_parameters(m1)\n\n  - The intercept is statistically non-significant and negative (beta = -12.88, 95% CI [-32.67, 6.92], t(38) = -1.32, p = 0.196; Std. beta = 2.65e-16, 95% CI [-0.19, 0.19])\n  - The effect of avgphgt is statistically significant and positive (beta = 1.19, 95% CI [0.90, 1.49], t(38) = 8.23, p &lt; .001; Std. beta = 0.80, 95% CI [0.60, 1.00])"
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+    "text": "Observation-Level Statistics\nWe sometimes want the individual-level data like the predictions (\\hat{Y}) and the errors (technically called residuals).\nTo get observation-level statistics, use broom’s augment function:\n\naugment(m1) \n\n\n\n\n\nheight\navgphgt\n.fitted\n.resid\n.hat\n.sigma\n.cooksd\n.std.resid\n\n\n\n\n66\n67.5\n67.6\n-1.64\n0.03\n2.70\n0.00\n-0.62\n\n\n71\n70.0\n70.6\n0.37\n0.04\n2.72\n0.00\n0.14\n\n\n74\n72.5\n73.6\n0.39\n0.10\n2.72\n0.00\n0.15\n\n\n69\n66.5\n66.5\n2.55\n0.03\n2.68\n0.01\n0.96\n\n\n70\n65.0\n64.7\n5.34\n0.04\n2.56\n0.09\n2.03\n\n\n74\n72.0\n73.0\n0.99\n0.09\n2.71\n0.01\n0.39\n\n\n73\n71.0\n71.8\n1.18\n0.06\n2.71\n0.01\n0.45\n\n\n69\n70.0\n70.6\n-1.63\n0.04\n2.70\n0.01\n-0.62\n\n\n69\n71.0\n71.8\n-2.82\n0.06\n2.67\n0.04\n-1.09\n\n\n75\n69.5\n70.0\n4.97\n0.04\n2.59\n0.07\n1.89\n\n\n69\n69.0\n69.4\n-0.43\n0.03\n2.72\n0.00\n-0.16\n\n\n67\n67.5\n67.6\n-0.64\n0.03\n2.71\n0.00\n-0.24\n\n\n73\n71.5\n72.4\n0.58\n0.07\n2.72\n0.00\n0.23\n\n\n74\n70.0\n70.6\n3.37\n0.04\n2.66\n0.04\n1.29\n\n\n63\n61.5\n60.5\n2.51\n0.13\n2.68\n0.07\n1.00\n\n\n67\n68.0\n68.2\n-1.24\n0.03\n2.71\n0.00\n-0.47\n\n\n64\n62.5\n61.7\n2.32\n0.10\n2.69\n0.04\n0.91\n\n\n61\n63.0\n62.3\n-1.28\n0.08\n2.71\n0.01\n-0.50\n\n\n70\n65.5\n65.3\n4.74\n0.04\n2.60\n0.06\n1.80\n\n\n66\n68.0\n68.2\n-2.24\n0.03\n2.69\n0.01\n-0.85\n\n\n70\n72.0\n73.0\n-3.01\n0.09\n2.67\n0.07\n-1.18\n\n\n60\n65.0\n64.7\n-4.66\n0.04\n2.60\n0.07\n-1.78\n\n\n75\n68.5\n68.8\n6.16\n0.03\n2.52\n0.08\n2.33\n\n\n73\n69.0\n69.4\n3.57\n0.03\n2.65\n0.03\n1.35\n\n\n63\n68.0\n68.2\n-5.24\n0.03\n2.57\n0.05\n-1.98\n\n\n63\n65.0\n64.7\n-1.66\n0.04\n2.70\n0.01\n-0.63\n\n\n69\n67.5\n67.6\n1.36\n0.03\n2.71\n0.00\n0.51\n\n\n65\n64.5\n64.1\n0.93\n0.05\n2.71\n0.00\n0.36\n\n\n64\n66.0\n65.8\n-1.85\n0.03\n2.70\n0.01\n-0.70\n\n\n69\n70.0\n70.6\n-1.63\n0.04\n2.70\n0.01\n-0.62\n\n\n65\n68.0\n68.2\n-3.24\n0.03\n2.66\n0.02\n-1.22\n\n\n63\n65.0\n64.7\n-1.66\n0.04\n2.70\n0.01\n-0.63\n\n\n68\n67.5\n67.6\n0.36\n0.03\n2.72\n0.00\n0.13\n\n\n63\n64.5\n64.1\n-1.07\n0.05\n2.71\n0.00\n-0.41\n\n\n62\n64.0\n63.5\n-1.47\n0.06\n2.71\n0.01\n-0.56\n\n\n61\n63.5\n62.9\n-1.87\n0.07\n2.70\n0.02\n-0.72\n\n\n67\n67.0\n67.0\n-0.05\n0.03\n2.72\n0.00\n-0.02\n\n\n63\n64.5\n64.1\n-1.07\n0.05\n2.71\n0.00\n-0.41\n\n\n62\n63.5\n62.9\n-0.87\n0.07\n2.71\n0.00\n-0.34\n\n\n72\n71.5\n72.4\n-0.42\n0.07\n2.72\n0.00\n-0.16\n\n\n\n\n\n\n\n\nThe predicted values \\hat{Y} are in the .fitted column, and the errors (residuals) are in the .resid column.\nIf you wanted to predict a value not in the data, you use the newdata argument. For example, if I wanted to predict the height of a person whose parents had a height of 65:\n\naugment(m1, newdata = tibble(avgphgt = 65))\n\n\n\n\n\navgphgt\n.fitted\n\n\n\n\n65\n64.7"
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+    "section": "Checking assumptions",
+    "text": "Checking assumptions\n\ncheck_model(m2)\n\n\n\n\n\n\n\n\nAll looks well."
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+    "text": "Summarizing Results\nTo summarize the results, use the summary function:\n\nsummary(m2)\n\n\nCall:\nlm(formula = height ~ avgphgt + gender, data = d)\n\nResiduals:\n   Min     1Q Median     3Q    Max \n-3.826 -1.094 -0.347  1.656  4.503 \n\nCoefficients:\n            Estimate Std. Error t value Pr(&gt;|t|)    \n(Intercept)   29.035     11.600    2.50  0.01686 *  \navgphgt        0.671      0.157    4.26  0.00013 ***\ngender        -4.468      0.920   -4.85  2.2e-05 ***\n---\nSignif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n\nResidual standard error: 2.12 on 37 degrees of freedom\nMultiple R-squared:  0.78,  Adjusted R-squared:  0.769 \nF-statistic: 65.7 on 2 and 37 DF,  p-value: 6.6e-13\n\n\nAn automated report:\n\nreport::report(m2)\n\nWe fitted a linear model (estimated using OLS) to predict height with avgphgt\nand gender (formula: height ~ avgphgt + gender). The model explains a\nstatistically significant and substantial proportion of variance (R2 = 0.78,\nF(2, 37) = 65.75, p &lt; .001, adj. R2 = 0.77). The model's intercept,\ncorresponding to avgphgt = 0 and gender = 0, is at 29.03 (95% CI [5.53, 52.54],\nt(37) = 2.50, p = 0.017). Within this model:\n\n  - The effect of avgphgt is statistically significant and positive (beta = 0.67,\n95% CI [0.35, 0.99], t(37) = 4.26, p &lt; .001; Std. beta = 0.45, 95% CI [0.24,\n0.66])\n  - The effect of gender is statistically significant and negative (beta = -4.47,\n95% CI [-6.33, -2.60], t(37) = -4.85, p &lt; .001; Std. beta = -0.51, 95% CI\n[-0.73, -0.30])\n\nStandardized parameters were obtained by fitting the model on a standardized\nversion of the dataset. 95% Confidence Intervals (CIs) and p-values were\ncomputed using a Wald t-distribution approximation.\n\n\nStandardized coefficients:\n\nparameters::model_parameters(m2, standardize = \"refit\")\n\n\n\n\n\nParameter\nCoefficient\nSE\nCI\nCI_low\nCI_high\nt\ndf_error\np\n\n\n\n\n(Intercept)\n0.00\n0.08\n0.95\n-0.15\n0.15\n0.00\n37\n1\n\n\navgphgt\n0.45\n0.11\n0.95\n0.24\n0.66\n4.26\n37\n0\n\n\ngender\n-0.51\n0.11\n0.95\n-0.73\n-0.30\n-4.85\n37\n0\n\n\n\n\n\n\n\n\nThe oddly-named anova function compares 2 models:\n\nanova(m1, m2)\n\n\n\n\n\nRes.Df\nRSS\nDf\nSum of Sq\nF\nPr(&gt;F)\n\n\n\n\n38\n273\n\n\n\n\n\n\n37\n167\n1\n106\n23.6\n0\n\n\n\n\n\n\n\n\nThe p-value is significant, meaning that m2 explains more variance than m1.\nI like the more full-featured set of model comparison functions from the performance package:\n\ncompare_performance(m1, m2) \n\n\n\n\n\nName\nModel\nAIC\nAIC_wt\nAICc\nAICc_wt\nBIC\nBIC_wt\nR2\nR2_adjusted\nRMSE\nSigma\n\n\n\n\nm1\nlm\n196\n0\n197\n0\n201\n0\n0.64\n0.63\n2.61\n2.68\n\n\nm2\nlm\n179\n1\n180\n1\n185\n1\n0.78\n0.77\n2.04\n2.12\n\n\n\n\n\n\n\n\nThis is equivalent to the anova function but in a more readable format:\n\ntest_wald(m1, m2)\n\n\n\n\n\nName\nModel\ndf\ndf_diff\nF\np\n\n\n\n\nm1\nlm\n38\n\n\n\n\n\nm2\nlm\n37\n1\n23.6\n0\n\n\n\n\n\n\n\n\nThis uses Bayes Factor instead of a p-value:\n\ntest_bf(m1, m2)\n\n\n\n\n\nModel\nlog_BF\nBF\n\n\n\n\navgphgt\n\n\n\n\navgphgt + gender\n8.01\n3010\n\n\n\n\n\n\n\n\nA BF &gt; 1 means that m2 is more strongly supported than m1. A BF &lt; 1 means that m1 is more strongly supported than m2.\n\nreport(test_bf(m1, m2))\n\nBayes factors were computed using the BIC approximation, by which BF10 =\nexp((BIC0 - BIC1)/2). Compared to the avgphgt model, we found extreme evidence\n(BF = 3.01e+03) in favour of the avgphgt + gender model (the least supported\nmodel)."
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+    "text": "Canvas Question 1\nQuestion: Is read_1 a significant predictor of read_2?\n\nHide Hints\n\n\nHint 1\nMake model fit object called m_read using the lm function.\n\nm_read &lt;- lm(read_2 ~ read_1, data = d_learn)\n\n\n\nHint 2\nView coefficent-level statistics with broom’s tidy function.\nYou could also use Base R’s summary function.\n\ntidy(m_read)\n\n# or\n\nsummary(m_read)\n\n\n\nHint 3\nIn the read_1 row, is the p.value column less than 0.05?\n\ntidy(m_read)\n\n\n\n\n\n\nterm\nestimate\nstd.error\nstatistic\np.value\n\n\n\n\n(Intercept)\n83.591\n10.922\n7.65\n0.000\n\n\nread_1\n0.251\n0.104\n2.41\n0.021"
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+    "text": "Canvas Question 2\nQuestion: What is the R2 for the m_read model?\n\nHide Hint\n\n\nHint\nView model-level statistics with broom’s glance function.\nYou could also use Base R’s summary function.\n\nglance(m_read)\n\n# or\n\nsummary(m_read)"
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+    "text": "Canvas Question 3\nQuestion: What does the scatter plot look like when read_1 is on the x-axis and read_2 is on the y-axis? Also plot the regression line. Save your plot using the ggsave function and upload it to Canvas.\n\nHide Hints\n\n\nHint 1\nThis will get you started\n\nggplot(d_learn, aes(read_1, read_2))\n\n\n\nPoints Hint\nHere is how you add points.\n\nggplot(d_learn, aes(read_1, read_2)) +\n  geom_point() \n\n\n\nLine Hint\nHere is how you add a regression line.\n\nggplot(d_learn, aes(read_1, read_2)) +\n  geom_point() + \n  geom_smooth(method = \"lm\") \n\n\n\nPlot Polishing\nThis is overkill for now. But soon you will want to fine tune your plots.\n\n# I want to put the equation at x = 130\nequation_x &lt;- 130\nequation_y &lt;- predict(m_read, newdata = tibble(read_1 = equation_x))\n\n# Extracting the coefficients\nb_read &lt;- round(coef(m_read),2)\n\n# The angle of the regression line is the inverse tangent of the slope (converted to degrees)\neq_angle &lt;- atan(b_read[2]) * 180 / pi\n\n# Equation\neq_read &lt;- paste0(\"italic(Y) == \", \n                  b_read[1], \n                  \" + \", \n                  b_read[2], \n                  \" *  italic(X) + italic(e)\")\n\nggplot(d_learn, aes(read_1, read_2)) +\n  geom_point() +\n  geom_smooth(method = \"lm\") +\n  labs(x = \"Reading at Time 1\",\n       y = \"Reading at Time 2\",\n       title = \"Using Reading at Time 1 to Predict Reading at Time 2\") +\n  coord_fixed() +\n  annotate(\n    geom = \"text\",\n    x = equation_x,\n    y = equation_y,\n    angle = eq_angle,\n    label = eq_read,\n    parse = TRUE,\n    vjust = -0.5)"
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+    "text": "Canvas Question 4\nCreate a regression model in which you predict math at time 2 (math_2) using reading at time 1 (math_1). Call the model fit object m_math1.\nQuestion: Does math_1 predict math_2?.\nQuestion: Does math_1 still predict math_2 after controlling for read_1?\n\nHide Hint\n\n\nHint\n\nm_math &lt;- lm(math_2 ~ math_1 + read_1, data = d_learn)\nsummary(m_math)"
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-    "text": "A free online book about psychometric tools for psychological assessment\n\nI have been working on a book project off and on since 2012. I have ambitious goals for this book, but it is not nearly complete. The preliminary, unfinished version is here.\n\n\n\n\n\nAn introduction to tests and measurements\nRenée Tobin and I helped update Ronald Cohen’s Psychological Testing and Assessment: An Introduction to Tests and Measurements. It is a comprehensive textbook suitable for introductory psychological assessment courses.\n\n\n\n\n\n\n\n\n\nA book about making reader-friendly assessment reports that instill hope and promote change.\nMy co-authors and I worked hard to fill the second edition of Essentials of Assessment Report Writing with non-obvious guidance about how to write reports so that the meet the needs of the reports’ readers."
+    "text": "R package for simulating data using standardized coefficients\nTutorial\n\n\n\n\nIn the model below, the path coefficients are standardized. You would like to simulate the variables in the model, but you do not know the disturbance and residual variances. The simstandard package can help.\n\n\n\nA standardized latent variable model\n\n\n\n\n\n\n\nAn R package for detecting unusual scores in a test profile\nTutorial\n\n\n\n\nThis package estimates how unusual a multivariate normal profile is.\n\n\n\n\n\n\nA ggplot2 extension package for creating normal violin plots\n\n\n\n\nI needed to show confidence intervals and conditional normal distributions with specific means and standard deviations. I wrote the ggnormalviolin package to make this happen.\nIt makes plots like this:\n\n\n\nExample plot made with ggnormalviolin\n\n\n\n\n\n\n\nFunctions useful for psychological evaluations\nThis package is still in a preliminary state, just like Individual Psychometrics, the book it accompanies.\n\n\n\n\n\n\n\nMulivariate Confindence Intervals\n\n\n\n\n\n\n\nA set of functions I find convenient to have readily available to me\n\n\n\n\n\n\n\nWhoa! How did you place those labels so perfectly?\n\n\n\n\n\n\n\nAn R package for making digital spirographs\nTutorial\nMy Gallery\n\n\n\n\nMaking digital spirographs is fun! I made an R package called spiro that can make animated spirographs like this one:\n\n\n\n\n\n\nR package for making a custom arrowheads for ggplot2 using ggarrow\nTutorial\n\n\n\n\nThe arrowheadr package allows one to create custom arrowheads that can be used with the ggarrow package."
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-    "text": "An introduction to tests and measurements\nRenée Tobin and I helped update Ronald Cohen’s Psychological Testing and Assessment: An Introduction to Tests and Measurements. It is a comprehensive textbook suitable for introductory psychological assessment courses."
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+    "text": "A ggplot2 extension package for creating normal violin plots\n\n\n\n\nI needed to show confidence intervals and conditional normal distributions with specific means and standard deviations. I wrote the ggnormalviolin package to make this happen.\nIt makes plots like this:\n\n\n\nExample plot made with ggnormalviolin"
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-    "text": "When analyzing questionnaire data, I sometimes put the text of several questions on the y-axis of a plot. Longer text strings will usually make the plot too narrow.\nSuppose we give an admittedly silly questionnaire about preference for darker food. We want to display the percentages of people answering “Yes” to each question.\nHere are the questions and the (completely made-up) percentages:\n# To install the WJSmisc package:\n# remotes::install_github(\"wjschne/WJSmisc\")\nlibrary(WJSmisc)\nlibrary(tidyverse)\nd &lt;- tibble(Item = c(\"When you drink coffee, do you prefer darker roasts?\",\n                     \"Is pumpernickel the best bread?\",\n                     \"Do you like dark chocolate more than you like milk chocolate?\",\n                     \"Are black olives better than green olives?\",\n                     \"Do you enjoy the taste of prunes more than can be prudently admitted aloud?\",\n                     \"Is black licorice the secret to a happly life?\"),\n            Percentage = c(45,4,37, 84,2, 3)) |&gt; \n  mutate(Item = fct_reorder(Item, Percentage))\nA preliminary plot might look like this:\nmyfont &lt;- \"Roboto Condensed\"\nmycolors &lt;- viridis::viridis(n = nrow(d), begin = .2, end = .8) |&gt; tinter::lighten(.5)\nmytextsize &lt;- 17\nmytext &lt;- \"gray60\"\np &lt;- ggplot(d, aes(Percentage, Item)) + \n  geom_col(aes(fill = factor(Percentage))) + \n  geom_richlabel(aes(label = paste0(Percentage, \"%\"), \n                     color = factor(Percentage)), \n                 hjust = 0, fill = \"black\",\n                 text_size = mytextsize,\n                 family = myfont) +\n  labs(y = NULL,\n       caption = \"Note: These numbers were made up for illustrative purposes.\") +\n  theme_minimal(base_size = mytextsize, base_family = myfont) +\n  scale_x_continuous(\"Percent Yes\", \n                     limits = c(0,100), \n                     breaks = NULL,\n                     expand = expansion(mult = c(0,.2))) +\n  scale_fill_manual(values = mycolors) +\n  scale_color_manual(values = mycolors) +\n  theme(plot.title.position = \"plot\", \n        plot.background = element_rect(\"black\"),\n        plot.title = element_text(colour = mytext),\n        plot.caption = element_text(color = \"gray40\"),\n        legend.position = \"none\",\n        axis.title = element_text(color = mytext),\n        panel.grid.major.y = element_blank(),\n        axis.text = element_text(color = mycolors))\np + ggtitle(\"Y-axis labels with no text wrapping\")\nThe questions are taking up a lot of space. We can fix this by using stringr::str_wrap to make the text lines no longer than a specified width:\np + \n  scale_y_discrete(labels = \\(x) str_wrap(x, width = 30)) + \n  ggtitle(\"Splitting Y-axis labels with str_wrap\")\nThis is a definite improvement, but I wish the lines of text were wrapped more evenly. For example, I do not like the fact that “chocolate” and “bread” are by themselves on their own lines. I have spent a fair amount of time breaking lines up by hand and hoped to create a function to automate the process. To achieve a more balanced style of text splitting, I made the str_wrap_equal function.\np + scale_y_discrete(labels = \\(x) str_wrap_equal(x, max_width = 30)) + \n  ggtitle(\"Splitting Y-axis labels with str_wrap_equal\")\nTo each their own, but this style of text wrapping feels better to me here. It is the sort of thing no one but the analyst will notice if it is right but will feel a bit off if not.\nFor your convenience and mine, I put the str_wrap_equal function in the WJSmisc package."
+    "text": "Functions useful for psychological evaluations\nThis package is still in a preliminary state, just like Individual Psychometrics, the book it accompanies.\n\n\n\n\n\n\n\nMulivariate Confindence Intervals"
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-    "text": "Technical Details\nI hesitate to post a new function because the odds are quite high that someone has already done what I did. However, my initial search did not locate a function that does exactly what I needed. If you know of a function have I duplicated unnecessarily, let me know.\nIn deciding whether to break the line, the function does the following:\n\nCalculate the overall length of the text (i.e., number of characters).\nFind k (the number of lines likely needed) by dividing the overall text length by max_width and round up to the nearest integers.\nCalculate the preferred width of the line by dividing the overall text length by the number of lines likely needed.\nAdd to the line one word at a time as long as doing so makes the current line width closer to the preferred width. If not, a new line is started.\nThe function will not make a line longer than what is specified by max_width unless it has to place a single, lonely word that is longer than max_width.\n\nHere is the code for the function. Any suggestions to improve it are welcome.\n\nstr_wrap_equal &lt;- function(x, max_width = 30L, sep = \"\\n\") {\n  purrr::map_chr(x, \\(xi) {\n    # Find overall text length\n    xlen &lt;- stringr::str_length(xi)\n    # Remove any line breaks and allow lines to break at forward slashes\n    xi &lt;- stringr::str_replace(xi, \"\\n\", \" \") |&gt; \n      stringr::str_replace(\"/\", \"/ \")\n    # Number of lines likely needed\n    k &lt;- ceiling(xlen / max_width)\n    # Optimal line length\n    preferred_width &lt;- xlen / k\n    # Split text into words\n    words &lt;- stringr::str_split(xi, pattern = \" \", simplify = F)[[1]]\n    # Number of words in text\n    k_words &lt;- length(words)\n    # Length of each word in text\n    word_len &lt;- stringr::str_length(words)\n    # Create empty text lines with a few extra, if needed\n    textlines &lt;- rep(\"\", k + 10)\n    # Current text line\n    i &lt;- 1\n    # Decide whether to add a word to the current line or to start a new line\n    for (w in seq(k_words)) {\n      # Width of current line before adding a new word\n      current_width &lt;- stringr::str_length(textlines[i])\n      # Width of current line if a new word is added\n      proposed_width &lt;- current_width + word_len[w] + 1\n      # Difference between current width and preferred width\n      current_difference &lt;- abs(current_width - preferred_width)\n      # Difference between proposed width and preferred width\n      proposed_difference &lt;- abs(proposed_width - preferred_width)\n      # Should we start a new line?\n      if (current_difference &lt; proposed_difference | proposed_width &gt; max_width) {\n        i &lt;- i + 1\n      }\n      # Add word to current line, remove spaces, and rejoin words divided by forward slashes\n      textlines[i] &lt;- stringr::str_trim(paste(textlines[i], words[w])) |&gt;\n        stringr::str_replace(\"/ \", \"/\")\n    }\n    # Collapse non-empty lines by separation character\n    paste0(textlines[stringr::str_length(textlines) &gt; 0], collapse = sep)\n  })\n\n}"
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+    "text": "A set of functions I find convenient to have readily available to me\n\n\n\n\n\n\n\nWhoa! How did you place those labels so perfectly?"
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-    "text": "As explained in this ggplot2 vignette, the size parameter in geom_text and geom_label is in millimeters, and the size parameter in all other text elements in ggplot2 is in points.\nIf I specify the base_size of the plot and the size of a label to 16, you can see that the text label is much bigger than 16.\n\nlibrary(tidyverse)\ntextsize &lt;- 16\nggplot() +\n  stat_function(\n    fun = dnorm,\n    xlim = c(-4, 4),\n    geom = \"area\",\n    alpha = .3\n  ) +\n  theme_minimal(base_size = textsize) +\n  annotate(\n    geom = \"text\",\n    x = 0,\n    y = 0,\n    label = \"Mean = 0\",\n    size = textsize,\n    vjust = -.1\n  )\n\n\n\n\n\n\n\n\nIf you do not mind a little trial and error, you can fiddle with the size parameter until you find a value that looks good to you. However, what if we want the axis text labels and the output of geom_text to be exactly the same size?\nWe are in luck because ggplot2 has a .pt constant that will help us convert point sizes to millimeters.\n\nggplot2::.pt\n\n[1] 2.845276\n\n\nWe know that, by default, axis text is .8 times as large as the base_size of the theme.\nLet’s make a function to automate the conversion:\n\nggtext_size &lt;- function(base_size, ratio = 0.8) {\n  ratio * base_size / ggplot2::.pt\n}\n\nNow we can make the label and axis text exactly the same size:\n\nggplot() + \n  stat_function(fun = dnorm, xlim = c(-4,4), geom = \"area\", alpha = .3) +\n  theme_minimal(base_size = textsize) + \n  annotate(\n    geom = \"text\",\n    x = 0,\n    y = 0,\n    label = \"Mean = 0\",\n    size = ggtext_size(textsize),\n    vjust = -.3\n  )\n\n\n\n\n\n\n\n\nFor my own convenience (and possibly yours), I put the ggtext_size function in the WJSmisc package.\nFor more on ggplot font size matters, I found this post by Christophe Nicault to be informative.\n\n\n\nCitationBibTeX citation:@misc{schneider2021,\n  author = {Schneider, W. Joel},\n  title = {Making Text Labels the Same Size as Axis Labels in Ggplot2},\n  date = {2021-08-10},\n  url = {https://wjschne.github.io/posts/making-text-labels-the-same-size-as-axis-labels-in-ggplot2},\n  langid = {en}\n}\nFor attribution, please cite this work as:\nSchneider, W. J. (2021, August 10). Making text labels the same size as\naxis labels in ggplot2. Schneirographs. https://wjschne.github.io/posts/making-text-labels-the-same-size-as-axis-labels-in-ggplot2"
+    "text": "An R package for making digital spirographs\nTutorial\nMy Gallery\n\n\n\n\nMaking digital spirographs is fun! I made an R package called spiro that can make animated spirographs like this one:"
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-    "text": "I am writing a (still very much unfinished) psychometrics book with suggested exercises using R. I wanted to provide solutions to the questions, but I wanted to hide the answers until the reader clicks them.\nAn easy way to create collapsible content is with html’s &lt;details&gt; tag. For example:\nCreate a variable `x`, and assign it a value of 5.\n\n&lt;details&gt;\n  &lt;summary&gt;Suggested Solution&lt;/summary&gt;\n\n```{r}\nx &lt;- 5\n```\n\n&lt;/details&gt;\nCreate a variable x, and assign it a value of 5.\n\n\nSuggested Solution\n\n\nx &lt;- 5\n\n\n\nHowever, I did not want to worry about all the &lt;details&gt; every time I wrote a question. To automate this process, I used two knitr hooks: a chunk hook and an option hook.\nChunk hooks are for customizing the output of a chunk.\nHere I create a new chunk hook that is run before and after the chunk. When it is run before the chunk, it adds the &lt;details&gt; and &lt;summary&gt; tags. After the chunk, it closes the &lt;details&gt; tag.\n\n# Add the details tag before running the chunk \n# and close the tag after running the chunk.\nknitr::knit_hooks$set(\n  solutionsetter = function(before,options) {\n  \n  if (before) {\n    \n    \"\\n\\n&lt;details&gt;&lt;summary&gt;Suggested Solution&lt;/summary&gt;\\n\\n\"\n    \n  } else {\n    \n    \"\\n\\n&lt;/details&gt;\\n\\n\"\n    \n  }\n})\n\nIf all we needed to do was to enclose the output into a &lt;details&gt; tag, then just the solutionsetter hook would be needed. However, I also wanted to set echo=TRUE so that the output would be visible even if the default for echo was FALSE.\nOption hooks are great for when you have a kind of chunk you will use often, and it requires setting many chunk options at once (e.g., a plot variant with different dimensions than your default plot has).\nHere I create a new option hook called solution. It sets the current chunk’s echo option to TRUE and also triggers the new solutionsetter code chunk below. The updated options list needs to be returned explicitly, or the hook will not work.\n\nknitr::opts_hooks$set(solution = function(options) {\n  options$echo &lt;- TRUE\n  options$solutionsetter &lt;- TRUE\n  return(options)\n})\n\nNow all need to do is to set my chunk option to solution = TRUE and the output will be collapsible:\n\nCreate a variable `x`, and assign it a value of 5.\n\n```{r, solution = TRUE}\nx &lt;- 5\n```\nCreate a variable x, and assign it a value of 5.\n\n\n\nSuggested Solution\n\nx &lt;- 5\n\n\n\n\nThe details package\nIf you want a function to enclose the output of a chunk or inline code in &lt;details&gt; tags, use the details package by Jonathan Sidi. For example, an inline chunk like this:\n`r details::details(\"Answer\", summary = \"Question\")`\nWill output like so:\n\n\n Question \n\n\nAnswer\n\n\nThe package has many other uses, which are explained in the package’s vignettes.\n\n\n\n\nCitationBibTeX citation:@misc{schneider2023,\n  author = {Schneider, W. Joel},\n  title = {Creating Collapsible Output with Knitr Chunk Hooks},\n  date = {2023-06-30},\n  url = {https://wjschne.github.io/posts/creating-collapsible-output-with-knitr-chunk-hooks},\n  langid = {en}\n}\nFor attribution, please cite this work as:\nSchneider, W. J. (2023, June 30). Creating collapsible output with knitr\nchunk hooks. Schneirographs. https://wjschne.github.io/posts/creating-collapsible-output-with-knitr-chunk-hooks"
+    "text": "R package for making a custom arrowheads for ggplot2 using ggarrow\nTutorial\n\n\n\n\nThe arrowheadr package allows one to create custom arrowheads that can be used with the ggarrow package."
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-    "text": "Sometimes I want to put a plot caption in the lower right corner of the plot.\n\nlibrary(tidyverse)\np &lt;- ggplot() +\n  stat_function(xlim = c(-4,4), fun = dnorm, n = 801) + \n  labs(x = \"z-scores\", caption = \"Note: Mean = 0, SD = 1\")\n\np\n\n\n\n\n\n\n\n\nHowever, I want the caption to be a little higher, on the same line as the x-axis title. To do so, set a a negative top margin:\n\np + theme(plot.caption = element_text(margin = margin(t = -10, unit = \"pt\")))\n\n\n\n\n\n\n\n\nHere is a more finished example.\n\nlibrary(ggnormalviolin)\nlibrary(ggtext)\nmy_font_size = 16\nmy_font &lt;- \"Roboto Condensed\"\nupdate_geom_defaults(geom = \"text\",new = list(family = my_font) )\nupdate_geom_defaults(geom = \"label\",new = list(family = my_font) )\nupdate_geom_defaults(\"richtext\", list(family = my_font))\ntheme_set(theme_minimal(base_size = my_font_size, base_family = my_font))\n\nd_rect &lt;- tibble(SS = 100, \n                 width = c(20, 40, 60, 80, 122), \n                 fill = paste0(\"gray\", c(95, 90, 80, 70, 65) - 25)) %&gt;% \n  arrange(-width)\n\ntibble(\n  Scale = c(\n    \"Fluid Reasoning\",\n    \"Verbal Comprehension\",\n    \"Visual-Spatial Processing\",\n    \"Working Memory\",\n    \"Processing Speed\",\n    \"Population\"),\n  y = c(5:1 - 0.5, 0),\n  SS = c(115, 111, 109, 86, 79, 100),\n  rxx = c(.93, .92, .92, .92, .88, 0),\n  width = c(rep(1.4, 5), 10.4),\n  alpha = c(rep(1, 5), .3)\n) %&gt;% \n  mutate(true_hat = rxx * (SS - 100) + 100, \n         see = ifelse(rxx == 0, 15, 15 * sqrt(rxx - rxx ^ 2)),\n         Scale = fct_inorder(Scale) %&gt;% fct_rev()) %&gt;% \n  ggplot(aes(y, SS)) + \n  geom_tile(data = d_rect, aes(width = 5.8, x = 2.9, \n                               fill = fill, \n                               height = width, \n                               y = SS)) + \n  geom_normalviolin(aes(mu = true_hat, \n                                        sigma = see, \n                                        width = width,\n                                        alpha = alpha), \n                                    face_left = F, \n                    fill = \"white\") + \n  geom_richtext(aes(label = ifelse(\n    Scale == \"Population\", \n    \"Population Mean\", \n    paste0(\"&lt;span style='font-size:8.5pt;color:white'&gt;(\", \n           round(100 * pnorm(SS, 100, 15),0),\n           \") &lt;/span&gt;\",\n           SS, \n           \"&lt;span style='font-size:9pt;color:#666666'&gt; (\", \n           round(100 * pnorm(SS, 100, 15),0),\n           \")&lt;/span&gt;\"))), \n    vjust = -0.2, \n    lineheight = .8, \n    fill = NA, \n    color = \"gray20\",\n    label.color = NA,\n    label.padding = unit(0,\"mm\")) +\n  geom_text(aes(y = true_hat, \n                label = ifelse(Scale == \"Population\", \n                               \"\", \n                               as.character(Scale))), \n            vjust = 1.5, \n            color = \"gray15\",\n            lineheight = .8) +\n  geom_linerange(aes(ymin = true_hat - 1.96 * see,\n                      ymax = true_hat + 1.96 * see), \n                 linewidth = .5) +\n  geom_pointrange(aes(ymin = true_hat - see,\n                      ymax = true_hat + see), \n                  size = 1.2, \n                  fatten = 1.5) +\n  geom_text(aes(x = x, y = y, label = label), \n            data = tibble(\n              y = c(49.5, 65, 75, 85, 100, 115, 125, 135, 150.5), \n              x = 5.5, \n              label = c(\"Extremely\\nLow Range\", \n                                  \"Very\\nLow\", \n                                  \"Low\\nRange\", \n                                  \"Low\\nAverage\", \n                                  \"Average\\nRange\", \n                                  \"High\\nAverage\", \n                                  \"High\\nRange\", \n                                  \"Very\\nHigh\", \n                                  \"Extremely\\nHigh Range\")), \n            color = \"white\", lineheight = .8, size = 4.25) +\n  scale_y_continuous(\n    \"Standard Scores &lt;span style='font-size:11.7pt;color:#656565'&gt;&lt;br&gt;\n    (and Percentile Ranks)&lt;/span&gt;\", \n    breaks = seq(40, 160, 10), \n    limits = c(37, 163),\n    labels = \\(x) paste0(x,\n                         \"&lt;br&gt;&lt;span style='font-size:10pt;color:#656565'&gt;(\", \n                         pnorm(x,100, 15) %&gt;% \n                           WJSmisc::proportion2percentile(digits = 2) %&gt;%\n                           str_trim(), \n                         \")&lt;/span&gt;\"),\n    expand = expansion()) +\n  scale_x_continuous(NULL, expand = expansion(), breaks = NULL) +\n  scale_alpha_identity() +\n  scale_fill_identity() +\n  coord_flip(clip = \"off\") + \n  theme(axis.title.x = element_markdown(hjust = 0, \n                                        margin = margin(t = 1.25, \n                                                        l = 2,\n                                                        unit = \"mm\")),\n        axis.text.x = element_markdown(colour = \"gray20\"),\n        plot.caption = element_markdown(hjust = 0, \n                                        size = 10, \n                                        margin = margin(t = -10.25, \n                                                        l = 100, \n                                                        unit = \"mm\"), \n                                        color = \"gray40\"),\n        plot.title = element_text(hjust = .025)) +\n  labs(\n    title = \"Display of Cognitive Test Scores\",\n    caption = \"**Notes:** The white normal curves represent the expected \n       true&lt;br&gt;score distributions for each observed score. The black lines&lt;br&gt;\n       underneath span the 68% and 95% confidence intervals.\")\n\n\n\n\n\n\n\n\n\n\n\nCitationBibTeX citation:@misc{schneider2021,\n  author = {Schneider, W. Joel},\n  title = {Caption on Same Line as Axis Title in Ggplot2},\n  date = {2021-07-21},\n  url = {https://wjschne.github.io/posts/caption-on-same-line-as-axis-title-in-ggplot2},\n  langid = {en}\n}\nFor attribution, please cite this work as:\nSchneider, W. J. (2021, July 21). Caption on same line as axis title in\nggplot2. Schneirographs. https://wjschne.github.io/posts/caption-on-same-line-as-axis-title-in-ggplot2"
+    "text": "Nominal, Ordinal, Interval, and Ratio scales\n\n\n\n\n\n\n\n\nHow to visualize which values are most frequent in a distribution\n\n\n\n\n\n\n\n\nFunctions that tell us which values are more likely to occur\n\n\n\n\n\n\n\n\nAn introduction to expected value and how it is related to the concept of the mean of a variable.\n\n\n\n\n\n\n\n\nHow variance is calculated\n\n\n\n\n\n\n\n\nSum of the Reasons Variables Are Normally Normal\n\n\n\n\n\n\n\n\nAn introduction to skewness and its relationship to variability.\n\n\n\n\n\n\n\n\nAn introduction to kurtosis and how it is not just a measure of peakedness.\n\n\n\n\n\n\n\n\nAn introduction to z-scores, stanines, stens, scaled scores, T scores, and index scores\n\n\n\n\n\n\n\n\nCovariance is rarely useful by itself but is indispensable for many statistics."
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-    "text": "Base plot for demonstrations\nlibrary(tidyverse)\n# install.packages(\"remotes\")\n# remotes::install_github(\"teunbrand/ggarrow\")\nlibrary(ggarrow)\n\nnode_radius &lt;- .2\nresection_length &lt;- .03\n\nd_node &lt;- tibble(node = c(\"A\", \"B\", \"C\"),\n       x = c(0, .5, 1),\n       y = c(0, sqrt(3) / 2, 0))\n\nd_edge &lt;- d_node |&gt; \n  mutate(theta_next = atan((y - lag(y)) / (x - lag(x))),\n         x_from = lag(x) + cos(theta_next) * (node_radius + resection_length),\n         y_from = lag(y) + sin(theta_next) * (node_radius + resection_length),\n         x_to = x + cos(theta_next + pi) * (node_radius + resection_length),\n         y_to = y + sin(theta_next + pi) * (node_radius + resection_length)) |&gt; \n  filter(!is.na(x_from))\n\np &lt;- d_edge |&gt;\n  ggplot(aes(x = x_from, y = y_from, xend = x_to, yend = y_to)) +\n  ggforce::geom_circle(aes(\n    x0 = x,\n    y0 = y,\n    r = .2,\n    fill = node\n  ),\n  data = d_node,\n  color = NA, inherit.aes = F) +\n  geom_text(aes(label = node, x = x, y = y), \n            data = d_node, \n            size = 20,\n            family = \"Roboto Condensed\",\n            inherit.aes = F) +\n  coord_fixed() +\n  theme_void() +\n  theme(legend.position = \"none\") +\n     scale_fill_viridis_d(\n       option = \"D\",\n       begin = .2,\n       end = .8,\n       alpha = .5\n     )\n\np\n\n\n\n\n\n\n\n\n\nThe default arrows in ggplot2 are perfectly serviceable. There is the open variety:\n\n\nOpen arrow code\np + geom_segment(arrow = arrow())\n\n\n\n\n\n\n\n\nFigure 1: The ggplot2 open arrow\n\n\n\n\n\nYou can specify which ends of the segment have arrowheads, whether the arrows are open or closed, the length of the arrow, and how sharp the arrow’s point is with the angle argument.\nI happen to prefer a longer, sharper, closed arrow:\n\n\nClosed arrow code\np + geom_segment(\n  arrow = arrow(\n    angle = 15,\n    length = unit(8, \"mm\"),\n    type = \"closed\"\n))\n\n\n\n\n\n\n\n\nFigure 2: The ggplot2 closed arrow\n\n\n\n\n\nNevertheless, I miss the variety of arrows in available in TikZ. I particularly like the latex' arrow:\n\n\nTikZ code with latex’ arrow\n\\usetikzlibrary{arrows}\n\\begin{tikzpicture}\n\\node[fill=violet!50!white, circle] (A) at (0,0) {A};\n\\node[fill=cyan!80!black, circle] (B) at (1,1.732) {B};\n\\node[fill=green!40!black!40, circle] (C) at (2,0) {C};\n\\path[-&gt;,\n    draw,\n    shorten &gt;=2pt,\n    shorten &lt;=2pt,\n    &gt;=latex',\n    thick,\n    color = black!40,\n    text = black] (A) -&gt; (B);\n    \\path[-&gt;,\n    draw,\n    shorten &gt;=2pt,\n    shorten &lt;=2pt,\n    &gt;=latex',\n    thick,\n    color = black!40,\n    text = black] (B) -&gt; (C);\n\\end{tikzpicture}\n\n\n\n\n\n\n\n\nFigure 3: The TikZ latex’ arrow\n\n\n\n\n\n\nThe ggarrow package\nIf you want more variety in drawing arrows in ggplot2, Teun van den Brand’s ggarrow package expands your limits to whatever your imagination can provide.\nThe default arrowhead is already a nice improvement:\n\n\nCode for wings arrow (i.e, stealth arrow in TikZ)\np + geom_arrow_segment(length_head = 10)\n\n\n\n\n\n\n\n\nFigure 4: The “wings” arrow\n\n\n\n\n\nYou can play around with the sharpness of the point (offset) and the sharpness of the barb (inset). You can also add feathers.\n\n\nCode for sharp barbs and feathers\np +\n  geom_arrow_segment(length_head = 4,\n                     arrow_head = arrow_head_wings(offset = 20, \n                                                   inset = 10), \n                     arrow_fins = arrow_fins_feather(), \n                     length_fins = 11)\n\n\n\n\n\n\n\n\nFigure 5: Sharp barbs and feathers\n\n\n\n\n\n\n\nCode for kite arrowhead\np +\n  geom_arrow_segment(length_head = 15,\n                     arrow_head = arrow_head_wings(offset = 22.5,\n                                                   inset = 115),\n                     arrow_fins = arrow_head_wings(offset = 22.5,\n                                                   inset = 115),\n                     length_fins = 15)\n\n\n\n\n\n\n\n\nFigure 6: Double-headed arrow with kite arrowhead\n\n\n\n\n\n\n\nCode for reverse kite arrowhead\np +\n  geom_arrow_segment(length_head = 20,\n                     arrow_head = arrow_head_wings(offset = 45,\n                                                   inset = 120))\n\n\n\n\n\n\n\n\nFigure 7: Reverse kite arrowhead\n\n\n\n\n\nIf the arrow goes too far, you can pull it back with the resect arguments.\n\n\nCode for resecting an arrowhead\np +\n  geom_arrow_segment(length_head = 6,\n                     arrow_head = arrow_head_wings(offset = 120,\n                                                   inset = 35),\n                     resect_head = 2)\n\n\n\n\n\n\n\n\nFigure 8: Demonstration of resecting arrowheads\n\n\n\n\n\nThere is much, much more that can be done. See ggarrow’s arrow ornament vignette for more options.\n\n\nCustom Arrowheads\nNot only does ggarrow offer great arrow geoms with excellent features like resection, one can create any custom arrowhead or feather that can be made with a single polygon.\nThe polygon generally falls between -1 and 1 on x and y, though you can plot outside those limits. In most cases, the point is at (0,1), and the line ends at (0,0):\n\n\nCode\npar(pty = \"s\")\nx &lt;- c(0,1)\ny &lt;- c(0,0)\nplot(x , y, xlim = c(-1, 1), ylim = c(-1, 1))\nrect(-1,-.1,0,.1)\ntext(-.5,0, labels = \"Line\")\ntext(1,0, labels = \"Arrow Point\", adj = 1.2)\npolygon(ggarrow::arrow_head_wings() |&gt; `colnames&lt;-`(c(\"a\", \"b\")))\n\n\n\n\n\n\n\n\nFigure 9: Grid for custom arrowheads\n\n\n\n\n\nThe polygon you create should be be a 2-column matrix with named columns (e.g., x and y). Here I make a elliptical arrowhead.\n\n\nCode\nmake_ellipse &lt;- function(a = 1, b = .5){\n  t &lt;- seq(0,2*pi, length.out = 361)\n  cbind(x = a * cos(t), y = b * sin(t)) \n}\n\np +\n  geom_arrow_segment(length_head = 5,\n                     arrow_head = make_ellipse())\n\n\n\n\n\n\n\n\nFigure 10: Make ellipse\n\n\n\n\n\n\n\nThe arrowheadr package\nI made the arrowheadr package to make custom arrowheads quickly. Some of them are admittedly silly. However, it is now easy to make my favorite kind of arrow in ggplot2:\n\n\nCode\nlibrary(arrowheadr)\n\np + \n  geom_arrow_segment(length_head = 5,\n                     arrow_head = arrow_head_latex())\n\n\n\n\n\nMimicking the latex prime arrowhead\n\n\n\n\nMore examples:\nA catenary:\n\n\nCode\nstlouis &lt;- arrow_head_catenary(base_width = .25, thickness = .15)\n\np + \n  geom_arrow_segment(length_head = 5,\n                     arrow_head = stlouis)\n\n\n\n\n\n\n\n\nFigure 11: The Gateway Arch in St. Louis is shaped like a catenary, not a parabola.\n\n\n\n\n\nThe Cauchy function:\n\n\nCode\np + \n  geom_arrow_segment(length_head = 5,\n                     arrow_head = arrow_head_function(dt, df = 1))\n\n\n\n\n\n\n\n\nFigure 12: Any function can make an arrowhead.\n\n\n\n\n\nRazors:\n\n\nCode\nrazors &lt;- c(1,0,\n  0,.5,\n  -.35,.25,\n  -.35, .21,\n  0,.35,\n  .90,0\n  ) |&gt; \n  v2matrix() |&gt; \n  reflecter() \n\np + \n  geom_arrow_segment(length_head = 10,\n                     arrow_head = razors)\n\n\n\n\n\n\n\n\nFigure 13: The reflecter function takes a set of points and makes them symmetrical.\n\n\n\n\n\nA candle flame:\n\n\nCode\ncandleflame &lt;- arrow_head_wittgenstein_rod(\n  fixed_point = c(-2.75, 0),\n  rod_length = 3.75,\n  nudge = c(1, 0),\n  rescale = .95\n)\n\np + \n  geom_arrow_segment(length_head = 12,\n                     arrow_head = candleflame)\n\n\n\n\n\n\n\n\nFigure 14: Wittgenstein’s rod makes some nice shapes\n\n\n\n\n\nUsing bezier curve control points:\n\n\nCode\ncurved_arrowhead &lt;- arrow_head_bezier(list(\n  c(1,  0,\n    .5, .5,\n    .2, .5),\n  c(.2, .5,\n    .2, .1,\n    -.1, .25,\n    -.3, .25),\n  c(-.3, .25,\n    0, 0,\n    -.3, -.25),\n  c(-.3, -.25,\n    -.1, -.25,\n    .2,  -.1,\n    .2, -.5),\n  c(.2, -.5,\n    .5, -.5,\n    1,  0)\n))\n\np + \n  geom_arrow_segment(length_head = 8,\n                     arrow_head = curved_arrowhead)\n\n\n\n\n\n\n\n\nFigure 15: Bezier curves can make almost anything.\n\n\n\n\n\n\n\n\n\nCitationBibTeX citation:@misc{schneider2023,\n  author = {Schneider, W. Joel},\n  title = {Making a Custom Arrowhead for Ggplot2 Using Ggarrow and\n    Arrowheadr},\n  date = {2023-08-26},\n  url = {https://wjschne.github.io/posts/2023-08-26-making-a-custom-arrowhead-for-ggplot2-using-ggarrow-and-arrowheadr},\n  langid = {en}\n}\nFor attribution, please cite this work as:\nSchneider, W. J. (2023, August 26). Making a custom arrowhead for\nggplot2 using ggarrow and arrowheadr. Schneirographs. https://wjschne.github.io/posts/2023-08-26-making-a-custom-arrowhead-for-ggplot2-using-ggarrow-and-arrowheadr"
+    "text": "Nominal, Ordinal, Interval, and Ratio scales\n\n\n\n\n\n\n\n\nHow to visualize which values are most frequent in a distribution\n\n\n\n\n\n\n\n\nFunctions that tell us which values are more likely to occur\n\n\n\n\n\n\n\n\nAn introduction to expected value and how it is related to the concept of the mean of a variable.\n\n\n\n\n\n\n\n\nHow variance is calculated\n\n\n\n\n\n\n\n\nSum of the Reasons Variables Are Normally Normal\n\n\n\n\n\n\n\n\nAn introduction to skewness and its relationship to variability.\n\n\n\n\n\n\n\n\nAn introduction to kurtosis and how it is not just a measure of peakedness.\n\n\n\n\n\n\n\n\nAn introduction to z-scores, stanines, stens, scaled scores, T scores, and index scores\n\n\n\n\n\n\n\n\nCovariance is rarely useful by itself but is indispensable for many statistics."
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+    "text": "Other Assessment Videos\n\nEmotional Intelligence and CHC Theory\nHow might emotional intelligence relate to psychometric models of intelligence?\n\n\n\n\n\n\n\nMisunderstanding Regression to the Mean\nWhat regression to the mean is and how it is often misunderstood. Examples are provided as to how it is applied to IQ and the death penalty.\n\n\n\n\n\n\n\nTaking latent variable models seriously\nApplying latent score estimates to individuals\n\n\n\n\n\n\n\nSpecific cognitive processing weaknesses are rarely full explanations for academic deficits\nIn which I confess to having had a longstanding misunderstanding about diagnosing learning disorders.\n\n\n\n\n\n\n\nTwo kinds of cognitive ability hierarchies\nSome latent variables represent “hierarchical abstractions.”\n\n\n\n\n\n\n\nA Taxonomy of Influences on Ability Tests\nSome influences are test-specific whereas others are shared across tests. Some influences are transient whereas others are stable. Some influences are relevant to the construct of interest whereas others are irrelevant.\n\n\n\n\n\n\n\nWithin-composite differences: Why measures of the same ability differ?\nThere are a number of lesser-known reasons that two tests that are intended to measure the same ability might differ substantially.\n\n\n\n\n\n\n\nWithin-composite differences: Do large subtest score differences invalidate composite scores?\nIt is often asserted that composite scores should not be interpreted when the scores that make up that composite are discrepant. I show here why this is not typically true, depending on what the composite score is used for.\n\n\n\n\n\n\n\nWriting Assessment Reports People Will Read, Understand, and Remember\nMy appearance on the School Psyched Podcast on October 20, 2019"
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+    "title": "Presentations",
+    "section": "Audio Presentations",
+    "text": "Audio Presentations\n\nThe evolution of cognitive assessment\nMy appearance on the Testing Psychologist Podcast on March 29, 2021"
+  },
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+    "title": "Point labels perpendicular to a curve in ggplot2",
     "section": "",
-    "text": "W. Joel Schneider, Ph.D.\nProfessor\nTemple University\nPsychological Studies in Education\n\n\n\nCounseling Psychology\n\n\nSchool Psychology\n\n\n\n Curriculum Vitae\n Contact Me\n\n\n\n\nAbout Me\nI am a professor at Temple University in the College of Education and Human Development in the Psychological Studies in Education Department. I belong to both the School Psychology and Counseling Psychology programs.\nAlthough I have diverse interests, my primary focus is trying to understand and improve the validity of psychological assessment practices. I teach courses in assessment, counseling, statistics, and research methods.\nI grew up in Southern California (1970–1990) and lived in Córdoba, Argentina (1990–1992). I completed my undergraduate degree in Psychology at the University of California at Berkeley in 1995 and my doctoral studies in clinical psychology at Texas A&M University in 2003. My internship year (2001–2002) was spent in the Dutchess County Department of Behavioral & Community Health in Poughkeepsie, NY. Since then I have been a faculty member at Illinois State University (2002–2017) and Temple University (2017–Present)."
+    "text": "I would like to label points on a sine function so that the labels are always legible. In a sine wave plot in which θ ranges from 0 to 2π, sin(θ) ranges from −1 to +1. Thus, the plot’s xy ratio is\n\\text{plot ratio}= \\frac{2\\pi-0}{1- (-1)}=\\pi\nThe first derivative of the sine function is the cosine function. In my plot, the slope of the tangent line at each point is\n\n\\text{tangent slope} = \\text{plot ratio}\\times \\cos(\\theta)\n The angle of the tangent line’s slope is the arctan of the slope. I would like to place the label perpendicular to the tangent line so I will add 90 degrees (i.e., π/2 radians).\n\n\\text{text angle}=\\tan^{-1}(\\text{tangent slope})+\\pi/2\n\nNow I need a pair of functions that will convert this angle into the right values for ggplot2’s hjust and vjust arguments. I have added the angle2vjust and angle2hjust functions to the WJSmisc package, but I have defined them here as well:\n\nangle2vjust &lt;- function(theta, multiplier = 1, as_degrees = FALSE) {\n  if (as_degrees) theta &lt;- theta * pi / 180\n  (((sin(theta + pi) + 1) / 2) - 0.5) * multiplier + 0.5\n}\n\nangle2hjust &lt;- function(theta, multiplier = 1, as_degrees = FALSE) {\n  if (as_degrees) theta &lt;- theta * pi / 180\n  (((cos(theta + pi) + 1) / 2) - 0.5) * multiplier + 0.5\n}\n\nNow we plot the sine function with labels. I have used geom_richtext from the ggtext package because it allows me to set a white background along with padding and margins.\n\nlibrary(tidyverse)\nlibrary(ggtext)\n\nplot_ratio &lt;- pi\n\ntibble(theta = seq(0, 2 * pi, length.out = 13),\n            y =  sin(theta),\n            tangent_slope = cos(theta) * plot_ratio,\n            text_angle = atan(tangent_slope) + pi / 2) %&gt;%\n  ggplot(aes(theta, y)) +\n  geom_richtext(aes(label = formatC(y, digits = 2, format = \"f\"),\n                    vjust = angle2vjust(text_angle),\n                    hjust = angle2hjust(text_angle)),\n                label.color = NA,\n                label.padding = unit(1, \"pt\"),\n                label.margin = unit(5, \"pt\"),\n                size = 4) +\n  geom_point() +\n  stat_function(fun = sin) +\n  scale_x_continuous(\"&theta;\",\n                     breaks = seq(0, 2 * pi, \n                                  length.out = 13),\n                     minor_breaks = NULL,\n                     labels = function(x) round(x * 180 / pi)) +\n  scale_y_continuous(\"sin(&theta;)\") +\n  coord_fixed(ratio = plot_ratio, clip = \"off\") +\n  theme_minimal(base_size = 16) +\n  theme(axis.title.x = element_markdown(),\n        axis.title.y = element_markdown())\n\n\n\n\n\n\n\n\nIf you do not mind turning your head to one side or the other, a somewhat easier method is to set the label’s vjust to a negative value and rotate the labels by the angle of the tangent line:\n\ntibble(theta = seq(0, 2 * pi, length.out = 13),\n            y = sin(theta),\n            tangent_slope = cos(theta) * plot_ratio,\n            text_angle = atan(tangent_slope)) %&gt;%\n  ggplot(aes(theta, y)) +\n  geom_richtext(aes(label = round(y, 2),\n                    angle = text_angle * 180 / pi),\n                vjust = 0,\n                label.color = NA,\n                label.padding = unit(1, \"pt\"),\n                label.margin = unit(2, \"pt\"),\n                size = 4) +\n  geom_point() +\n  stat_function(fun = sin) +\n  scale_x_continuous(\"&theta;\",\n                     breaks = seq(0, 2 * pi, \n                                  length.out = 13),\n                     minor_breaks = NULL,\n                     labels = function(x) round(x * 180 / pi)) +\n  scale_y_continuous(\"sin(&theta;)\") +\n  coord_fixed(ratio = plot_ratio, clip = \"off\") +\n  theme_minimal(base_size = 16) +\n  theme(axis.title.x = element_markdown(),\n        axis.title.y = element_markdown())\n\n\n\n\n\n\n\n\nWhat if you do not know the function’s first derivative? You can approximate the slope of the tangent line by comparing a function’s output of each point with the output of a slightly deviated point. Here is a plot of the normal cumulative distribution function.\n\n# Small change in x\ndx &lt;- .00000001\nplot_ratio &lt;- 6\n\ntibble(x = seq(-3,3,.5),\n       y = pnorm(x),\n       tangent_slope = plot_ratio * (pnorm(x + dx) - y) / dx,\n       text_angle = atan(tangent_slope) + pi / 2,\n       degrees = text_angle * 180 / pi) %&gt;% \n  ggplot(aes(x,y)) + \n  geom_point() +\n  geom_richtext(aes(label = formatC(y, 2, format = \"f\"),\n                    vjust = angle2vjust(text_angle),\n                    hjust = angle2hjust(text_angle)),\n                label.color = NA,\n                label.padding = unit(1, \"pt\"),\n                label.margin = unit(5, \"pt\"),\n                size = 4) +\n  stat_function(fun = pnorm) + \n  theme_minimal(base_size = 16) + \n  theme(axis.title.x = element_markdown(),\n        axis.title.y = element_markdown()) +\n  coord_fixed(ratio = plot_ratio) + \n  scale_x_continuous(\"*x*\", breaks = -3:3) +\n  scale_y_continuous(\"&Phi;(*x*)\")\n\n\n\n\n\n\n\n\n\n\n\nCitationBibTeX citation:@misc{schneider2021,\n  author = {Schneider, W. Joel},\n  title = {Point Labels Perpendicular to a Curve in Ggplot2},\n  date = {2021-07-27},\n  url = {https://wjschne.github.io/posts/point-labels-perpendicular-to-a-curve-in-ggplot2},\n  langid = {en}\n}\nFor attribution, please cite this work as:\nSchneider, W. J. (2021, July 27). Point labels perpendicular to a curve\nin ggplot2. Schneirographs. https://wjschne.github.io/posts/point-labels-perpendicular-to-a-curve-in-ggplot2"
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-    "text": "Overture Redux\n\n\n\n\n\n\n\n\n\n\n\nSep 12, 2023\n\n\nW. Joel Schneider\n\n\n\n\n\n\n\n\n\n\n\n\nThe RESCA-E Subtests Are Thoughtfully Designed and Highly Refined Measures of CHC Constructs\n\n\nA Review of the Receptive, Expressive & Social Communication Assessment–Elementary\n\n\n\n\n\n\n\n\nOct 26, 2016\n\n\nW. Joel Schneider, W. Joel Schneider\n\n\n\n\n\n\nNo matching items"
+    "text": "While assembling a reply letter for a revise-and-resubmit document, I was documenting for reviewers how we addressed their concerns. The reply letter quoted numerous sentences and paragraphs that had been changed. Unfortunately, each time my co-authors and I edited the paper, I lost track of which sentences and paragraphs I had previously copied into the reply letter. I did not want to misquote my own paper.\nCopying-and-pasting with each edit was getting tedious, and it was an error-prone process. I decided to write a function that would retrieve a named div or span from the paper and print it as a quote in my reply letter. Here it is:\nget_quote &lt;- function(id, file, blockquote = TRUE) {\n  blocktext &lt;- ifelse(blockquote, \"\\n&gt; \", \"\")\n\n  refreplace &lt;- function(x) {\n    # List of sequence types\n    crossrefs &lt;- c(`@fig-` = \"Figure\",\n                   `@fig-` = \"Table\",\n                   `@eq-` = \"Equation\",\n                   `@thm-` = \"Lemma\",\n                   `@lem-` = \"Corollary\",\n                   `@cor-` = \"Proposition\",\n                   `@prp-` = \"Conjecture\",\n                   `@cnj-` = \"Definition\",\n                   `@def-` = \"Example\",\n                   `@exm-` = \"Example\",\n                   `@exr-` = \"Exercise\",\n                   `apafg-` = \"Figure\",\n                   `apatb-` = \"Figure\"\n    )\n\n    # Find all crossreference types and number them\n    make_replacements &lt;- function(x, reftype, prefix) {\n      regstring &lt;- ifelse(stringr::str_starts(reftype, \"\\\\@\"),\n                          paste0(\"\\\\\", reftype, \"(.*?)(?=[\\\\.\\\\?\\\\!\\\\]\\\\}\\\\s,])\"),\n                          paste0(\"\\\\{\", reftype, \"(.*?)\\\\}\"))\n      patterns &lt;- stringr::str_extract_all(string = x, pattern = regstring) |&gt;\n        unlist() |&gt;\n        unique() |&gt;\n        stringr::str_replace(\"\\\\{\", \"\\\\\\\\{\") |&gt;\n        stringr::str_replace(\"\\\\}\", \"\\\\\\\\}\")\n\n      if (all(is.na(patterns))) return(NULL)\n      replacements &lt;- paste(prefix, seq_along(patterns))\n      names(replacements) &lt;- patterns\n      replacements\n    }\n    allreplacements &lt;- purrr::map2(\n      names(crossrefs),\n      crossrefs,\n      \\(rt, pf) make_replacements(\n        x = x,\n        reftype = rt,\n        prefix = pf)) |&gt;\n      unlist()\n\n    stringr::str_replace_all(x, allreplacements)\n\n  }\n\n  filetext &lt;- readLines(file) |&gt;\n    refreplace()\n\n  idcount &lt;- sum(stringr::str_count(filetext, paste0(\"#\", id)))\n  if (idcount &gt; 1)\n    stop(paste0(\n      \"The id (\",\n      id ,\n      \") is not unique. There are \",\n      idcount,\n      \" instances of id = \",\n      id,\n      \".\"\n    ))\n\n  s &lt;- filetext |&gt;\n    paste0(collapse = \"\\n\") |&gt;\n    stringr::str_match(pattern = paste0(\"(?&lt;=\\\\[).+(?=\\\\]\\\\{\\\\#\",\n                               id,\n                               \"\\\\})\"))  |&gt;\n    getElement(1)\n\n  if (is.na(s)) {\n    s &lt;- filetext |&gt;\n      paste0(collapse = \"|||\") |&gt;\n      stringr::str_match(pattern = paste0(\":::\\\\{\\\\#\",\n                                 id,\n                                 \"\\\\}(.*?):::\"))  |&gt;\n      getElement(2) |&gt;\n      stringr::str_replace_all(\"\\\\|\\\\|\\\\|\", blocktext)\n  } else {\n    s &lt;- paste0(blocktext, s)\n  }\n\n  if (is.na(s)) stop(\"Could not find a div or span with id = \", id)\n\n  s\n}\nFor your convenience and mine, I have added this function into the WJSmisc package."
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+    "title": "Insert Text from One Markdown Document into Another",
+    "section": "Remove blockquote formatting",
+    "text": "Remove blockquote formatting\nBy default, the function adds a &gt; before each line in the quoted markdown so that it appears as a block quote. You can turn off the block quote formatting like so:\n`r get_quote(\"id1\", \"index.qmd\", blockquote = FALSE)`\nText in a span"
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+    "title": "Insert Text from One Markdown Document into Another",
+    "section": "Curly braces with additional information",
+    "text": "Curly braces with additional information\nWhat if you need to put additional information in the curly braces of the span or div (e.g., a css class)?\n[Here is more text with extra stuff in the curly braces.]{#id3 .myclass}\nYou can trick the function into thinking that the extra stuff is part of the id like so:\n`r get_quote(\"id3 .myclass\", \"index.qmd\")`\n\nHere is more text with extra stuff in the curly braces."
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-    "text": "The original title of this blog (Assessing Psyche, Engaging Gauss, Seeking Sophia) was more of an aspiration than a description. To keep things simple, going forward the blog’s title will be just AssessingPsyche.\nThe purpose of this blog remains the same from the original post in 2010:\n\nWhen we are at our best, clinicians skilled in psychological assessment see beyond the obvious and communicate something useful and true to a person who needs our help. Uncovering something true often requires a bit of science, sometimes a little math, and always a lot of empathy. Communicating what is true so that it is useful requires still more empathy and a bit of art. We facilitate in our clients a deeper understanding of what is happening to them and present to them a different vision of what they can be.\nIn this blog, I hope to communicate something useful and true to other professionals about psychological assessment. I hope that in communicating what I know, I will come to know more than I do now.\n\n\n\n\nCitationBibTeX citation:@misc{schneider2023,\n  author = {Schneider, W. Joel},\n  title = {Overture {Redux}},\n  date = {2023-09-12},\n  url = {https://wjschne.github.io/AssessingPsyche/2010-10-07 Overture},\n  langid = {en}\n}\nFor attribution, please cite this work as:\nSchneider, W. J. (2023, September 12). Overture Redux.\nAssessingPsyche. https://wjschne.github.io/AssessingPsyche/2010-10-07\nOverture"
+    "text": "I often need to add a prefix or suffix to a character vector:\n\nlibrary(tidyverse)\nx &lt;- c(\"A\", \"B\", \"C\")\npaste0(x, \"_1\")\n\n[1] \"A_1\" \"B_1\" \"C_1\"\n\n\nHowever, if the vector is empty, I do not want bare prefixes or suffixes like this:\n\nx &lt;- character(0)\npaste0(x, \"_1\")\n\n[1] \"_1\"\n\n\nI used to write a lot of tedious if statements like this:\n\nif (length(x) == 0) {\n  y &lt;- character(0)\n} else {\n  y &lt;- paste0(x, \"_1\")\n}\n\nFor R 4.0.1 and later versions, the paste and paste0 functions acquired the recycle0 argument. Setting recycle0 to TRUE returns an empty vector if at least one string is empty:\n\npaste0(x, \"_1\", recycle0 = TRUE)\n\ncharacter(0)\n\n\nIf you need to work with an earlier version of R, you can use the sprintf function instead:\n\n# Non-empty vector\nx &lt;- c(\"A\", \"B\", \"C\")\nsprintf(\"%s_1\", x)\n\n[1] \"A_1\" \"B_1\" \"C_1\"\n\n# Empty vector\nx &lt;- character(0)\nsprintf(\"%s_1\", x)\n\ncharacter(0)\n\n\nThe glue function from the glue package also works nicely:\n\n# Non-empty vector\nx &lt;- c(\"A\", \"B\", \"C\")\nglue::glue(\"{x}_1\")\n\nA_1\nB_1\nC_1\n\n# Empty vector\nx &lt;- character(0)\nglue::glue(\"{x}_1\")\n\n\n\n\nCitationBibTeX citation:@misc{schneider2021,\n  author = {Schneider, W. Joel},\n  title = {Concatenating Vectors Unless a Vector Is Empty},\n  date = {2021-05-13},\n  url = {https://wjschne.github.io/posts/concatenating-vectors-unless-a-vector-is-empty},\n  langid = {en}\n}\nFor attribution, please cite this work as:\nSchneider, W. J. (2021, May 13). Concatenating vectors unless a vector\nis empty. Schneirographs. https://wjschne.github.io/posts/concatenating-vectors-unless-a-vector-is-empty"
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+    "text": "Setup\nThis trick takes a lot of work. I only use it when I need a plot to look its best.\nSuppose I have a Likert questionnaire item with responses ranging from Strongly Disagree to Strongly Agree. Let’s say I have many groups in my study (e.g., college majors), and I want to compare their responses to the item.\nFirst I will create fake data for 26 groups, A–Z.\n\nlibrary(extrafont)\nlibrary(tidyverse)\nmyfont &lt;- \"Roboto Condensed\"\n\nd &lt;- tibble(Group = LETTERS[1:26],\n            mu = rnorm(26, 0, .5),\n            sigma = 1) %&gt;%\n  mutate(x = pmap(list(mean = mu, sd = sigma), rnorm, n = 500)) %&gt;%\n  unnest(x) %&gt;%\n  mutate(Agree = cut_number(x, 6) %&gt;%\n           factor(\n             labels = c(\n               \"Strongly\\nDisagree\",\n               \"Disagree\",\n               \"Slightly\\nDisagree\",\n               \"Slightly\\nAgree\",\n               \"Agree\",\n               \"Strongly\\nAgree\"\n             )\n           )) %&gt;%\n  group_by(Group, Agree) %&gt;%\n  summarise(n = n(), .groups = \"drop\") %&gt;%\n  group_by(Group) %&gt;%\n  arrange(Group, Agree) %&gt;%\n  mutate(p = n / sum(n)) %&gt;%\n  ungroup()\n\n\n\nFirst Attempts at Plotting\n\nd %&gt;%\n  ggplot(mapping = aes(p, Group)) +\n  geom_col(aes(fill = fct_rev(Agree))) +\n  scale_fill_viridis_d(NULL,\n                       begin = .15,\n                       end = 0.8,\n                       direction = -1) +\n  scale_x_continuous(\"Percent\", expand = expansion()) +\n  scale_y_discrete(\"Group\", expand = expansion()) +\n  coord_fixed(1 / 20.5, clip = \"off\") +\n  geom_text(\n    aes(x = p, label = round(100 * p, 0)),\n    color = \"white\",\n    family = myfont,\n    position = position_stack(vjust = .5)\n  )\n\n\n\n\n\n\n\n\nNot bad, but it would look better if we sorted the groups. We have many sorting options. One is that we can convert the Likert scale to numeric and then sort by the group with the highest mean.\n\nd %&gt;%\n  mutate(Group = fct_reorder(Group, .x = as.numeric(Agree) * p, \n                             .fun = mean)) %&gt;%\n  ggplot(mapping = aes(p, Group)) +\n  geom_col(aes(fill = fct_rev(Agree))) +\n  scale_fill_viridis_d(NULL,\n                       begin = .15,\n                       end = 0.8,\n                       direction = -1) +\n  scale_x_continuous(\"Percent\", expand = expansion()) +\n  scale_y_discrete(\"Group\", expand = expansion()) +\n  coord_fixed(1 / 20.5, clip = \"off\") +\n  geom_text(\n    aes(x = p, label = round(100 * p, 0)),\n    color = \"white\",\n    family = myfont,\n    position = position_stack(vjust = .5)\n  )\n\n\n\n\n\n\n\n\nIf I wanted to sort by the “Strongly Agree” category (or any other category):\n\nd %&gt;%\n  mutate(Group = fct_reorder(\n    Group,\n    .x = (Agree == \"Strongly\\nAgree\") * p,\n    .fun = mean\n  )) %&gt;%\n  ggplot(mapping = aes(p, Group)) +\n  geom_col(aes(fill = fct_rev(Agree))) +\n  scale_fill_viridis_d(NULL,\n                       begin = .15,\n                       end = 0.8,\n                       direction = -1) +\n  scale_x_continuous(\"Percent\", expand = expansion()) +\n  scale_y_discrete(\"Group\", expand = expansion()) +\n  coord_fixed(1 / 20.5, clip = \"off\") +\n  geom_text(\n    aes(x = p, label = round(100 * p, 0)),\n    color = \"white\",\n    family = myfont,\n    position = position_stack(vjust = .5)\n  )\n\n\n\n\n\n\n\n\n\n\nFinal Plot\nSo far, these plots look pretty good. However, I wish that the percentage labels were placed in a more aesthetically pleasing way. I am going to group by each of the response categories and then create a loess regression equation to smooth out the placement. It will mean that the labels are no longer centered in the stacked bars, but I think the sacrifice is worth it. I think that the percentage values are much easier to compare because the eye can follow the smooth line of labels.\n\nd &lt;- tibble(Group = LETTERS[1:26],\n            mu = rnorm(26, 0, .5),\n            sigma = 1) %&gt;%\n  mutate(x = pmap(list(mean = mu, sd = sigma), rnorm, n = 500)) %&gt;%\n  unnest(x) %&gt;%\n  mutate(Agree = cut_number(x, 6) %&gt;%\n           factor(\n             labels = c(\n               \"Strongly\\nDisagree\",\n               \"Disagree\",\n               \"Slightly\\nDisagree\",\n               \"Slightly\\nAgree\",\n               \"Agree\",\n               \"Strongly\\nAgree\"\n             )\n           )) %&gt;%\n  mutate(Group = fct_reorder(Group, as.numeric(Agree), .fun = mean)) %&gt;%\n  group_by(Group, Agree) %&gt;%\n  summarise(n = n(), .groups = \"drop\") %&gt;%\n  mutate(Group_position = as.numeric(Group)) %&gt;%\n  group_by(Group) %&gt;%\n  arrange(Group, Agree) %&gt;%\n  mutate(p = n / sum(n),\n         # proportion in each response category by group\n         cp = cumsum(p),\n         # cumulative proportion\n         xpos = cp - p / 2 # center of each stacked bar\n         ) %&gt;%\n         group_by(Agree) %&gt;%\n           nest() %&gt;%\n           mutate(\n             fit = map(data,\n                       loess,\n                       formula = \"xpos ~ Group_position\",\n                       span = 0.45),\n             # loess regression for each response category\n             xhat = map(fit, predict) # x-axis position on smooth line\n           ) %&gt;%\n           select(-fit) %&gt;%\n           unnest(c(data, xhat)) %&gt;%\n           mutate(Group = factor(Group, labels = rev(LETTERS[1:26])))\n         \n         d %&gt;%\n           ggplot(mapping = aes(p, Group)) +\n           geom_col(aes(fill = fct_rev(Agree))) +\n           geom_text(aes(x = xhat,\n                         label = ifelse(p &gt; .01, # No labels on small bars\n                                        round(100 * p, 0),\n                                        \"\")),\n                     color = \"white\",\n                     family = myfont) +\n           scale_fill_viridis_d(begin = .15,\n                                end = 0.8,\n                                direction = -1) +\n           scale_x_continuous(\"Percent\", expand = expansion()) +\n           scale_y_discrete(\"Group\", expand = expansion()) +\n           coord_fixed(1 / 20.5, clip = \"off\") +\n           theme_minimal(base_size = 13, base_family = myfont) +\n           theme(\n             legend.position = \"top\",\n             legend.box.spacing = unit(0.5, \"mm\"),\n             legend.text = element_text(\n               color = \"white\",\n               size = 13,\n               margin = margin(b = -40),\n               # Lower legend text into box\n               vjust = 0.5\n             ),\n             legend.spacing.x = unit(0, \"mm\"),\n             axis.text.y = element_text(hjust = 0.5)\n           ) +\n           guides(\n             fill = guide_legend(\n               title = NULL,\n               nrow = 1,\n               # Put legend on a single row\n               reverse = T,\n               # Reverse order of legend\n               label.position = \"top\",\n               # Put text atop keys\n               keyheight = unit(15, \"mm\"),\n               # Size of legend rectangles\n               keywidth = unit(22.56, \"mm\")\n             )\n           ) \n\n\n\n\n\n\n\n\nThis feels right to me.\n\n\n\n\nCitationBibTeX citation:@misc{schneider2021,\n  author = {Schneider, W. Joel},\n  title = {Bar Chart Labels on Smooth Paths in Ggplot2},\n  date = {2021-07-31},\n  url = {https://wjschne.github.io/posts/bar-chart-labels-on-smooth-paths-in-ggplot2},\n  langid = {en}\n}\nFor attribution, please cite this work as:\nSchneider, W. J. (2021, July 31). Bar chart labels on smooth paths in\nggplot2. Schneirographs. https://wjschne.github.io/posts/bar-chart-labels-on-smooth-paths-in-ggplot2"
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+    "text": "R has a mathematical annotation system via plotmath, but I like the look of true \\LaTeX equations better.\nGetting \\LaTeX equations into ggplot2 plots has never been easy. The tikzdevice package is great if you are generating a .pdf document. If you are not, then you might want to consider other options.\nThe easiest, hassle-free option that I know of is to create the equation in a \\TeX editor and then import the resulting .pdf using ggimage. If I need the best possible image quality, I convert the .pdf to .svg using dvisvgm (see example below).\nFor annotated equations, I like using the aptly-named annotate-equations \\LaTeX package. It uses tikz to remember where parts of equations are on the page. The \\eqnmark and \\eqnmarkbox functions work like so:\n\\eqnmark[color]{node_name}{latex equation terms}\nThen use the \\annotate function like so:\n\\annotate[color]{above,left}{node_name}{annotation text}\n\nMy Annotated Equation\nI am going to refer to the same file name with various endings (e.g., .tex, .pdf, .svg), so I will define it here.\n\nmyfile &lt;- \"annotatedequationsimple\"\n\nHere is the code for the annotated equation. It is saved in annotatedequationsimple.tex.\n\n\nLaTeX code\n```{tikz zscorecode}\n#| code-fold: show\n#| code-summary: \"LaTeX code\"\n#| eval: false\n\n\\documentclass[border={25pt 50pt -35pt 52pt}]{standalone}\n%\\documentclass{article}\n\\usepackage{annotate-equations}\n\\usepackage{xcolor}\n\\definecolor{myviolet}{HTML}{440154}\n\\definecolor{myblue}{HTML}{3B528B}\n\\definecolor{myindigo}{HTML}{21908C}\n\\definecolor{mygreen}{HTML}{5DC863}\n\\usepackage[sfdefault,condensed]{roboto}\n\\begin{document}\n    \n\\renewcommand{\\eqnhighlightheight}{\\mathstrut}\n    \n\\huge$\n\\eqnmark[myviolet]{z}{z} = \n\\frac{\n    \\eqnmark[myblue]{x}{X}-\n    \\eqnmark[myindigo]{mu}{\\mu}}{\n    \\eqnmark[mygreen]{sigma}{\\sigma}}\n$\n\n\n\\annotate[\n    yshift=1em, \n    myviolet,\n    align=right]\n    {above, left}\n    {z}\n    {$z$-score}\n    \n\\annotate[\n    yshift=1em, \n    myblue, \n    align=right]\n    {above,left}\n    {x}\n    {Observed\\\\ Score}\n    \n\n\\annotate[\n    yshift=1em, \n    myindigo]\n    {above,right}\n    {mu}\n    {Population\\\\ Mean\\\\ $\\mu = 100$}\n    \n\n    \n\\annotate[\n    yshift=-.4em, \n    mygreen, \n    align=right]\n    {below,left}\n    {sigma}\n    {Population\\\\ Standard\\\\ Deviation\\\\ $\\sigma = 15$}\n    \n\\end{document}\n\n```\n\n\nNow convert the .tex file to .pdf:\n\npaste0('pdflatex -interaction=nonstopmode ', myfile,'.tex') |&gt; \n  shell()\n\nNow we can convert the .pdf to .svg:\n\npaste0(\"dvisvgm --pdf --output=\", myfile,\".svg \", myfile,\".pdf\") |&gt; \n  shell()\n\nFor best image quality, import the .svg file with svgparser. The read_svg function will create a grid grob that can plotted directly using ggplot2::annotation_custom. If you want fine control of the imported grob, you can plot it using the gggrid package. See here for an example.\nIn the simplest case, we can do this:\n\nlibrary(svgparser)\nmy_svg &lt;- svgparser::read_svg(paste0(myfile, \".svg\"))\nggplot() +\n  theme_void() +\n  annotation_custom(my_svg, xmin = 0, xmax = 1, ymin = 0, ymax = 1) \n\n\n\n\n\n\n\nFigure 1: A simple plot with an annotated equation.\n\n\n\n\n\nHowever, this is no better than just displaying the .svg directly. You probably want to embed the equation in a plot. For example:\n\n\nCode\nmu &lt;- 100\nsigma &lt;- 15\nplot_height &lt;- dnorm(mu, mu, sigma)\nlb &lt;- -4 * sigma + mu\nub &lt;- 4 * sigma + mu\n\nggplot() +\n  annotation_custom(my_svg,\n                    xmin = 112,\n                    xmax = 164,\n                    ymin = .33 * plot_height) +\n  stat_function(\n    fun = \\(x) dnorm(x, mean = mu, sd = sigma),\n    geom = \"area\",\n    n = 1000,\n    fill = \"dodgerblue\",\n    alpha = .5\n  ) +\n  theme_classic(base_family = \"Roboto Condensed\",\n                base_size = 18) +\n  theme(\n    axis.text.x = element_markdown(),\n    axis.title.x = element_markdown(),\n    axis.line = element_blank()\n  )  +\n  scale_x_continuous(\n    \"Observed Score *X*&lt;br&gt;*z*\",\n    breaks = seq(lb, ub, sigma),\n    limits = c(lb, ub),\n    labels = \\(x) paste0(\n      signs::signs(x),\n      \"&lt;br&gt;\",\n      ifelse(\n        x == mu,\n        \"&lt;em&gt;&mu;&lt;/em&gt;\",\n        paste0(\n          signs::signs((x - mu) / sigma,\n                       add_plusses = T,\n                       label_at_zero = \"none\"\n          ),\n          \"&lt;em&gt;&sigma;&lt;/em&gt;\"\n        )\n      )\n    )\n  ) +\n  scale_y_continuous(\n    NULL,\n    limits = c(0, plot_height),\n    expand = expansion(),\n    breaks = NULL\n  ) \n\n\n\n\n\n\n\n\nFigure 2: Importing an .svg file via svgparser and annotation_custom.\n\n\n\n\n\nIf you can live with just a little pixelation, the ggimage package can import a .pdf directly with good results and less hassle, provided you render the plot with the ragg package.\n\n\nCode\nggplot() +\n  geom_image(\n    data = tibble(\n      x = 140,\n      y = .65 * plot_height,\n      image = \"annotatedequationsimple.pdf\"\n    ),\n    aes(x, y, image = image),\n    size = .70\n  ) +\n  stat_function(\n    fun = \\(x) dnorm(x, mean = mu, sd = sigma),\n    geom = \"area\",\n    n = 1000,\n    fill = \"dodgerblue\",\n    alpha = .5\n  ) +\n  theme_classic(base_family = \"Roboto Condensed\", \n                base_size = 18) +\n  theme(\n    axis.text.x = element_markdown(),\n    axis.title.x = element_markdown(),\n    axis.line = element_blank()\n  )  +\n  scale_x_continuous(\n    \"Observed Score *X*&lt;br&gt;*z*\",\n    breaks = seq(lb, ub, sigma),\n    limits = c(lb, ub),\n    labels = \\(x) paste0(\n      signs::signs(x),\n      \"&lt;br&gt;\",\n      ifelse(\n        x == mu,\n        \"&lt;em&gt;&mu;&lt;/em&gt;\",\n        paste0(\n          signs::signs((x - mu) / sigma,\n                       add_plusses = T,\n                       label_at_zero = \"none\"\n          ),\n          \"&lt;em&gt;&sigma;&lt;/em&gt;\"\n        )\n      )\n    )\n  ) +\n  scale_y_continuous(\n    NULL,\n    limits = c(0, plot_height),\n    expand = expansion(),\n    breaks = NULL\n  )\n\n\n\n\n\n\n\n\nFigure 3: Importing a .pdf file via ggimage::geom_image.\n\n\n\n\n\n\n\nA more complex example\nIn this example, I used the \\eqnmarkbox function for greater clarity. The .tex file is saved in a file called annotatedequation.tex.\n\n\nLaTeX code\n\\documentclass[border={10pt 48pt -45pt 62pt}]{standalone}\n%\\documentclass{article}\n\\usepackage{annotate-equations}\n\\usepackage{xcolor}\n\\definecolor{myviolet}{HTML}{414487}\n\\definecolor{myblue}{HTML}{2F6C8E}\n\\definecolor{myblue2}{HTML}{21908C}\n\\definecolor{mygreen}{HTML}{2FB47C}\n\\definecolor{mygreen2}{HTML}{7AD151}\n\\usepackage[sfdefault,condensed]{roboto}\n\\begin{document}\n    \n\\renewcommand{\\eqnhighlightheight}{\\mathstrut}\n    \n$\\LARGE\n\\eqnmarkbox[myviolet]{nodeP}{P\\left(T \\le \\tau \\right)} = \n\\eqnmarkbox[myblue]{phi}{\\Phi}\n\\left(\\frac{\n    \\eqnmarkbox[myblue2]{tau}{\\tau}-\n    \\eqnmarkbox[mygreen]{esttrue}{\\hat{T}}}{\n    \\eqnmarkbox[mygreen2]{sigma}{\\sigma_{T - \\hat{T}}}}\n\\right)$\n\n\n\\annotate[\n    yshift=1em, \n    xshift=11mm, \n    myviolet]\n    {above, left}\n    {nodeP}\n    {Probability true score $T$\\\\ is less than threshold $\\tau$ }\n    \n\\annotate[\n    yshift=-.6em,\n    myblue]\n    {below,left}\n    {phi}\n    {Standard Normal Cumulative\\\\ Distribution Function $\\Phi()$}\n    \n\\annotate[\n    yshift=1.4em, \n    xshift=4mm, \n    myblue2]\n    {above,left}\n    {tau}\n    {Threshold $\\tau=70$}\n    \n\\annotate[\n    yshift=3em, \n    xshift=7mm, \n    mygreen]\n    {above,left}\n    {esttrue}\n    {Estimated True Score $\\hat{T}=r_{XX}(X-\\mu)+\\mu$\\\\ \n    Observed Score $X\\sim \\mathcal{N}\\left(\\mu = 100, \\sigma=15\\right)$\\\\ \n    Reliability Coefficient $r_{XX}=\\{.80,.85,.90,.95,.98\\}$}\n    \n\\annotate[\n    yshift=-2em, \n    mygreen2]\n    {below,left}\n    {sigma}\n    {Standard Error of the Estimate\\\\ \n    $\\sigma_{T-\\hat{T}}=\\sigma\\sqrt{r_{XX}-r_{XX}^2}$}\n    \n\\end{document}\n\n\nNow convert .tex to .pdf:\n\n\nCode\nmyfile &lt;- \"annotatedequation\"\n\npaste0('pdflatex -interaction=nonstopmode ', myfile,'.tex') |&gt; \n  shell()\n\n\nAnd we are ready to plot. This plot shows the probability that an observed score will have a true score less than a specific threshold, given a reliability coefficient.\n\n\nCode\nviridis_start &lt;- .2\nviridis_end &lt;- .8\nthreshold &lt;- 70\n\n# Find where a line intersects with the normal cdf\nfind_x &lt;- function(rxx = .8,\n                   slope = 0.0048,\n                   intercept = .66,\n                   mu = 100,\n                   sigma = 15,\n                   start_x = 60,\n                   threshold = 70) {\n  x &lt;- start_x\n  precision &lt;- .00001\n  diff_y &lt;- precision * 10\n  reps &lt;- 0\n  while (abs(diff_y) &gt; precision) {\n    x &lt;- x - diff_y \n    line_y &lt;- x * slope + intercept\n    curve_y &lt;- pnorm(threshold,\n                     mean = rxx * (x - mu) + mu,\n                     sd = sigma * sqrt(rxx - rxx ^ 2))\n    \n    line_y &lt;- x * slope + intercept\n    curve_y &lt;- pnorm(threshold,\n                     mean = rxx * (x - mu) + mu,\n                     sd = sigma * sqrt(rxx - rxx ^ 2))\n    diff_y &lt;- line_y - curve_y\n    reps &lt;- reps + 1\n  }\n  tibble(x = x, p = line_y, mu = mu, sigma = sigma, threshold = threshold, reps = reps)\n}\n\n\n\n\ndimage = data.frame(x = 115, \n                    y = .62, \n                    image = paste0(myfile, \".pdf\"))\nv_rxx &lt;- round(c(seq(0.80, 0.95, 0.05), 0.98), 2)\n\nd_threshold &lt;-\n  crossing(x = round(seq(40, 160, 0.1), 1),\n           rxx = v_rxx,\n           threshold = 70) %&gt;%\n  mutate(\n    see = 15 * sqrt(rxx - rxx ^ 2),\n    mu = (x - 100) * rxx + 100,\n    p = pnorm(threshold, mu, see)\n  ) %&gt;%\n  group_by(rxx) %&gt;%\n  mutate(acceleration = p - lag(p)) %&gt;%\n  ungroup\n\n\n\nd_labels &lt;- tibble(rxx = v_rxx) %&gt;%\n  mutate(x = map_df(rxx, find_x)) |&gt; \n  unnest(x)\n\nd_threshold %&gt;%\n  ggplot(aes(x, p)) +\n  geom_line(aes(color = factor(rxx)), lwd = 1) +\n  geom_vline(\n    aes(xintercept = threshold),\n    lty = 2,\n    lwd = 1,\n    color = \"gray30\"\n  ) +\n  geom_image(data = dimage,\n             aes(x = x,\n                 y = y,\n                 image = image),\n             size = .87) +\n  geom_richtext(\n    aes(label = rxx_label,\n        color = factor(rxx)),\n    data = d_labels %&gt;%\n      mutate(rxx_label = prob_label(rxx)),\n    angle = -67,\n    size = WJSmisc::ggtext_size(13),\n    label.colour = NA,\n    fill = \"#FFFFFF\",\n    family = \"Roboto Condensed\",\n    label.margin = unit(0, \"mm\"),\n    label.r = unit(2, \"mm\"),\n    label.padding = unit(c(0, 0.75, 0, .5), \"mm\")\n  ) +\n  scale_x_continuous(\n    \"Observed Score\",\n    breaks = seq(40, 160, 15),\n    minor_breaks = seq(40, 160, 5),\n    expand = expansion()\n  ) +\n  scale_y_continuous(\n    paste0(\"Probability True Score &lt; \", threshold),\n    expand = expansion(),\n    breaks = seq(0, 1, 0.1),\n    labels = prob_label,\n    limits = c(0, 1)\n  ) +\n  scale_color_viridis_d(begin = viridis_start,\n                        end = viridis_end) +\n  theme_minimal(base_family = \"Roboto Condensed\", \n                base_size = 16) +\n  theme(legend.position = \"none\", \n        plot.margin = unit(c(3, 5, 2, 2), \"mm\")) +\n  coord_fixed(ratio = 100,\n              clip = \"off\",\n              xlim = c(40, 160)) \n\n\n\n\n\n\n\n\nFigure 4: The probability that a true score is less than 70 depends on the observed score’s position and reliability coefficient.\n\n\n\n\n\n\n\n\n\nCitationBibTeX citation:@misc{schneider2023,\n  author = {Schneider, W. Joel},\n  title = {Annotated Equations in Ggplot2},\n  date = {2023-07-24},\n  url = {https://wjschne.github.io/posts/2023-07-23-latex-equation-in-ggplot2},\n  langid = {en}\n}\nFor attribution, please cite this work as:\nSchneider, W. J. (2023, July 24). Annotated equations in ggplot2.\nSchneirographs. https://wjschne.github.io/posts/2023-07-23-latex-equation-in-ggplot2"
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+    "text": "Making a custom arrowhead for ggplot2 using ggarrow and arrowheadr\n\n\n\n\n\n\nggplot2\n\n\n\nThe ggarrow package allows any polygon to be an arrowhead. \n\n\n\n\n\nAug 26, 2023\n\n\nW. Joel Schneider\n\n\n\n\n\n\n\n\n\n\n\n\nAnnotated equations in ggplot2\n\n\n\n\n\n\nggplot2\n\n\n\nImporting latex into ggplot2\n\n\n\n\n\nJul 24, 2023\n\n\nW. Joel Schneider\n\n\n\n\n\n\n\n\n\n\n\n\nCreating collapsible output with knitr chunk hooks\n\n\n\n\n\n\nknitr\n\n\nmarkdown\n\n\n\nHow to make the output of a knitr code chunk collapsible using knitr hooks.\n\n\n\n\n\nJun 30, 2023\n\n\nW. Joel Schneider\n\n\n\n\n\n\n\n\n\n\n\n\nSplitting sentence texts into roughly equal segments\n\n\n\n\n\n\nggplot2\n\n\n\nFor long text labels on the y-axis, splitting them into roughly equal segments can look better.\n\n\n\n\n\nMay 31, 2023\n\n\nW. Joel Schneider\n\n\n\n\n\n\n\n\n\n\n\n\nInsert Text from One Markdown Document into Another\n\n\n\n\n\n\nmarkdown\n\n\n\nA function for retrieving a div or span from another markdown document\n\n\n\n\n\nMar 30, 2023\n\n\nW. Joel Schneider\n\n\n\n\n\n\n\n\n\n\n\n\nMaking text labels the same size as axis labels in ggplot2\n\n\n\n\n\n\nggplot2\n\n\n\nThe geom_text and geom_label functions do not specifiy text size the same way as the rest of ggplot2 elements do. For consistent text sizes, we can apply a simple conversion.\n\n\n\n\n\nAug 10, 2021\n\n\nW. Joel Schneider\n\n\n\n\n\n\n\n\n\n\n\n\nBar chart labels on smooth paths in ggplot2\n\n\n\n\n\n\nggplot2\n\n\n\nPlacing stacked bar chart labels on a smooth path makes them easier to compare.\n\n\n\n\n\nJul 31, 2021\n\n\nW. Joel Schneider\n\n\n\n\n\n\n\n\n\n\n\n\nPoint labels perpendicular to a curve in ggplot2\n\n\n\n\n\n\nggplot2\n\n\n\nHow to place legible labels for points on a curve in ggplot2\n\n\n\n\n\nJul 27, 2021\n\n\nW. Joel Schneider\n\n\n\n\n\n\n\n\n\n\n\n\nCaption on same line as axis title in ggplot2\n\n\n\n\n\n\nggplot2\n\n\n\nYou can put the caption on the line as the x-axis title\n\n\n\n\n\nJul 21, 2021\n\n\nW. Joel Schneider\n\n\n\n\n\n\n\n\n\n\n\n\nConcatenating vectors unless a vector is empty\n\n\n\n\n\n\nR\n\n\n\nThe paste and paste0 functions have a recycle0 argument that returns an empty vector when any part of the string is empty.\n\n\n\n\n\nMay 13, 2021\n\n\nW. Joel Schneider\n\n\n\n\n\n\nNo matching items"
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     "text": "Alternative Structure\nA cluster analysis can sometimes detect patterns that exploratory factor analyses miss. First, we see the correlation plot (Figure 8), with rectangles around 4 clusters (Ward’s method applied to the correlations subtracted from 1). Again, the pattern is not consistent with the 3 groupings.\n\n\n\n\n\n\n\n\nFigure 8: Correlation plot of the RESCA-E Subtests. Blue = Receptive, Red = Expressive, Green = Social Communication\n\n\n\n\n\nInstead, in the dendrogram in Figure 9, four clusters are highlighted plus a very narrow vocabulary cluster is singled out.\n\n\n\n\n\n\n\n\nFigure 9: Hierarchical Cluster Analysis of the RESCA-E Subtest Correlation Matrix (Ward’s Method). Suggested intepretations of clusters in blue.\n\n\n\n\n\nInstead of a three-part division between receptive, expressive, and social communication, this analysis suggests a 2 × 2 matrix (See Figure 10). There are two expressive clusters, one of which emphasizes social communication whereas the other emphasizes narrative communication (not that narrative is non-social). There are two “receptive” clusters, one of which is more social in that it is about understanding others’ intentions, sometimes via understanding English syntax. The other comprehension cluster consists of tests measuring semantic knowledge (e.g., vocabulary) and narratives.\n\n\n\n\n\n\n\n\nFigure 10: Matrix of Concpetual Groupings\n\n\n\n\n\nFigure 11 shows that the 2 × 2 structure suggested by hierarchical cluster analysis is also present in the the 3D MDS model. Grab the picture with your mouse and move it around to see the structure better.\n\n\n\n\n\n\n\n\nFigure 11: A suggested regrouping of the RESCA-E Subtests.\n\n\n\n\nWhy might there be a division between narrative/semantic vs. social/syntactic abilities? I can only speculate. The clumping of narrative and semantic abilities evokes the classic division of declarative memory into episodic memory and semantic memory. Why would social and syntactic subtests clump together? This is deeply speculative on my part, but there is a developing body of evidence that Broca’s Area not only is vital for the comprehension of syntax, but also the intentions of others (Gentilucci et al., 2006; Hamzei et al., 2003). To oversimplify, perhaps the narrative/semantic vs. social/syntactic abilities distinction reflects to some degree the relative health and functioning of Wernicke’s and Broca’s area, respectively."
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     "section": "",
-    "text": "Making a custom arrowhead for ggplot2 using ggarrow and arrowheadr\n\n\n\n\n\n\nggplot2\n\n\n\nThe ggarrow package allows any polygon to be an arrowhead. \n\n\n\n\n\nAug 26, 2023\n\n\nW. Joel Schneider\n\n\n\n\n\n\n\n\n\n\n\n\nAnnotated equations in ggplot2\n\n\n\n\n\n\nggplot2\n\n\n\nImporting latex into ggplot2\n\n\n\n\n\nJul 24, 2023\n\n\nW. Joel Schneider\n\n\n\n\n\n\n\n\n\n\n\n\nCreating collapsible output with knitr chunk hooks\n\n\n\n\n\n\nknitr\n\n\nmarkdown\n\n\n\nHow to make the output of a knitr code chunk collapsible using knitr hooks.\n\n\n\n\n\nJun 30, 2023\n\n\nW. Joel Schneider\n\n\n\n\n\n\n\n\n\n\n\n\nSplitting sentence texts into roughly equal segments\n\n\n\n\n\n\nggplot2\n\n\n\nFor long text labels on the y-axis, splitting them into roughly equal segments can look better.\n\n\n\n\n\nMay 31, 2023\n\n\nW. Joel Schneider\n\n\n\n\n\n\n\n\n\n\n\n\nInsert Text from One Markdown Document into Another\n\n\n\n\n\n\nmarkdown\n\n\n\nA function for retrieving a div or span from another markdown document\n\n\n\n\n\nMar 30, 2023\n\n\nW. Joel Schneider\n\n\n\n\n\n\n\n\n\n\n\n\nMaking text labels the same size as axis labels in ggplot2\n\n\n\n\n\n\nggplot2\n\n\n\nThe geom_text and geom_label functions do not specifiy text size the same way as the rest of ggplot2 elements do. For consistent text sizes, we can apply a simple conversion.\n\n\n\n\n\nAug 10, 2021\n\n\nW. Joel Schneider\n\n\n\n\n\n\n\n\n\n\n\n\nBar chart labels on smooth paths in ggplot2\n\n\n\n\n\n\nggplot2\n\n\n\nPlacing stacked bar chart labels on a smooth path makes them easier to compare.\n\n\n\n\n\nJul 31, 2021\n\n\nW. Joel Schneider\n\n\n\n\n\n\n\n\n\n\n\n\nPoint labels perpendicular to a curve in ggplot2\n\n\n\n\n\n\nggplot2\n\n\n\nHow to place legible labels for points on a curve in ggplot2\n\n\n\n\n\nJul 27, 2021\n\n\nW. Joel Schneider\n\n\n\n\n\n\n\n\n\n\n\n\nCaption on same line as axis title in ggplot2\n\n\n\n\n\n\nggplot2\n\n\n\nYou can put the caption on the line as the x-axis title\n\n\n\n\n\nJul 21, 2021\n\n\nW. Joel Schneider\n\n\n\n\n\n\n\n\n\n\n\n\nConcatenating vectors unless a vector is empty\n\n\n\n\n\n\nR\n\n\n\nThe paste and paste0 functions have a recycle0 argument that returns an empty vector when any part of the string is empty.\n\n\n\n\n\nMay 13, 2021\n\n\nW. Joel Schneider\n\n\n\n\n\n\nNo matching items"
+    "text": "The original title of this blog (Assessing Psyche, Engaging Gauss, Seeking Sophia) was more of an aspiration than a description. To keep things simple, going forward the blog’s title will be just AssessingPsyche.\nThe purpose of this blog remains the same from the original post in 2010:\n\nWhen we are at our best, clinicians skilled in psychological assessment see beyond the obvious and communicate something useful and true to a person who needs our help. Uncovering something true often requires a bit of science, sometimes a little math, and always a lot of empathy. Communicating what is true so that it is useful requires still more empathy and a bit of art. We facilitate in our clients a deeper understanding of what is happening to them and present to them a different vision of what they can be.\nIn this blog, I hope to communicate something useful and true to other professionals about psychological assessment. I hope that in communicating what I know, I will come to know more than I do now.\n\n\n\n\nCitationBibTeX citation:@misc{schneider2023,\n  author = {Schneider, W. Joel},\n  title = {Overture {Redux}},\n  date = {2023-09-12},\n  url = {https://wjschne.github.io/AssessingPsyche/2010-10-07 Overture},\n  langid = {en}\n}\nFor attribution, please cite this work as:\nSchneider, W. J. (2023, September 12). Overture Redux.\nAssessingPsyche. https://wjschne.github.io/AssessingPsyche/2010-10-07\nOverture"
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-    "text": "R has a mathematical annotation system via plotmath, but I like the look of true \\LaTeX equations better.\nGetting \\LaTeX equations into ggplot2 plots has never been easy. The tikzdevice package is great if you are generating a .pdf document. If you are not, then you might want to consider other options.\nThe easiest, hassle-free option that I know of is to create the equation in a \\TeX editor and then import the resulting .pdf using ggimage. If I need the best possible image quality, I convert the .pdf to .svg using dvisvgm (see example below).\nFor annotated equations, I like using the aptly-named annotate-equations \\LaTeX package. It uses tikz to remember where parts of equations are on the page. The \\eqnmark and \\eqnmarkbox functions work like so:\n\\eqnmark[color]{node_name}{latex equation terms}\nThen use the \\annotate function like so:\n\\annotate[color]{above,left}{node_name}{annotation text}\n\nMy Annotated Equation\nI am going to refer to the same file name with various endings (e.g., .tex, .pdf, .svg), so I will define it here.\n\nmyfile &lt;- \"annotatedequationsimple\"\n\nHere is the code for the annotated equation. It is saved in annotatedequationsimple.tex.\n\n\nLaTeX code\n```{tikz zscorecode}\n#| code-fold: show\n#| code-summary: \"LaTeX code\"\n#| eval: false\n\n\\documentclass[border={25pt 50pt -35pt 52pt}]{standalone}\n%\\documentclass{article}\n\\usepackage{annotate-equations}\n\\usepackage{xcolor}\n\\definecolor{myviolet}{HTML}{440154}\n\\definecolor{myblue}{HTML}{3B528B}\n\\definecolor{myindigo}{HTML}{21908C}\n\\definecolor{mygreen}{HTML}{5DC863}\n\\usepackage[sfdefault,condensed]{roboto}\n\\begin{document}\n    \n\\renewcommand{\\eqnhighlightheight}{\\mathstrut}\n    \n\\huge$\n\\eqnmark[myviolet]{z}{z} = \n\\frac{\n    \\eqnmark[myblue]{x}{X}-\n    \\eqnmark[myindigo]{mu}{\\mu}}{\n    \\eqnmark[mygreen]{sigma}{\\sigma}}\n$\n\n\n\\annotate[\n    yshift=1em, \n    myviolet,\n    align=right]\n    {above, left}\n    {z}\n    {$z$-score}\n    \n\\annotate[\n    yshift=1em, \n    myblue, \n    align=right]\n    {above,left}\n    {x}\n    {Observed\\\\ Score}\n    \n\n\\annotate[\n    yshift=1em, \n    myindigo]\n    {above,right}\n    {mu}\n    {Population\\\\ Mean\\\\ $\\mu = 100$}\n    \n\n    \n\\annotate[\n    yshift=-.4em, \n    mygreen, \n    align=right]\n    {below,left}\n    {sigma}\n    {Population\\\\ Standard\\\\ Deviation\\\\ $\\sigma = 15$}\n    \n\\end{document}\n\n```\n\n\nNow convert the .tex file to .pdf:\n\npaste0('pdflatex -interaction=nonstopmode ', myfile,'.tex') |&gt; \n  shell()\n\nNow we can convert the .pdf to .svg:\n\npaste0(\"dvisvgm --pdf --output=\", myfile,\".svg \", myfile,\".pdf\") |&gt; \n  shell()\n\nFor best image quality, import the .svg file with svgparser. The read_svg function will create a grid grob that can plotted directly using ggplot2::annotation_custom. If you want fine control of the imported grob, you can plot it using the gggrid package. See here for an example.\nIn the simplest case, we can do this:\n\nlibrary(svgparser)\nmy_svg &lt;- svgparser::read_svg(paste0(myfile, \".svg\"))\nggplot() +\n  theme_void() +\n  annotation_custom(my_svg, xmin = 0, xmax = 1, ymin = 0, ymax = 1) \n\n\n\n\n\n\n\nFigure 1: A simple plot with an annotated equation.\n\n\n\n\n\nHowever, this is no better than just displaying the .svg directly. You probably want to embed the equation in a plot. For example:\n\n\nCode\nmu &lt;- 100\nsigma &lt;- 15\nplot_height &lt;- dnorm(mu, mu, sigma)\nlb &lt;- -4 * sigma + mu\nub &lt;- 4 * sigma + mu\n\nggplot() +\n  annotation_custom(my_svg,\n                    xmin = 112,\n                    xmax = 164,\n                    ymin = .33 * plot_height) +\n  stat_function(\n    fun = \\(x) dnorm(x, mean = mu, sd = sigma),\n    geom = \"area\",\n    n = 1000,\n    fill = \"dodgerblue\",\n    alpha = .5\n  ) +\n  theme_classic(base_family = \"Roboto Condensed\",\n                base_size = 18) +\n  theme(\n    axis.text.x = element_markdown(),\n    axis.title.x = element_markdown(),\n    axis.line = element_blank()\n  )  +\n  scale_x_continuous(\n    \"Observed Score *X*&lt;br&gt;*z*\",\n    breaks = seq(lb, ub, sigma),\n    limits = c(lb, ub),\n    labels = \\(x) paste0(\n      signs::signs(x),\n      \"&lt;br&gt;\",\n      ifelse(\n        x == mu,\n        \"&lt;em&gt;&mu;&lt;/em&gt;\",\n        paste0(\n          signs::signs((x - mu) / sigma,\n                       add_plusses = T,\n                       label_at_zero = \"none\"\n          ),\n          \"&lt;em&gt;&sigma;&lt;/em&gt;\"\n        )\n      )\n    )\n  ) +\n  scale_y_continuous(\n    NULL,\n    limits = c(0, plot_height),\n    expand = expansion(),\n    breaks = NULL\n  ) \n\n\n\n\n\n\n\n\nFigure 2: Importing an .svg file via svgparser and annotation_custom.\n\n\n\n\n\nIf you can live with just a little pixelation, the ggimage package can import a .pdf directly with good results and less hassle, provided you render the plot with the ragg package.\n\n\nCode\nggplot() +\n  geom_image(\n    data = tibble(\n      x = 140,\n      y = .65 * plot_height,\n      image = \"annotatedequationsimple.pdf\"\n    ),\n    aes(x, y, image = image),\n    size = .70\n  ) +\n  stat_function(\n    fun = \\(x) dnorm(x, mean = mu, sd = sigma),\n    geom = \"area\",\n    n = 1000,\n    fill = \"dodgerblue\",\n    alpha = .5\n  ) +\n  theme_classic(base_family = \"Roboto Condensed\", \n                base_size = 18) +\n  theme(\n    axis.text.x = element_markdown(),\n    axis.title.x = element_markdown(),\n    axis.line = element_blank()\n  )  +\n  scale_x_continuous(\n    \"Observed Score *X*&lt;br&gt;*z*\",\n    breaks = seq(lb, ub, sigma),\n    limits = c(lb, ub),\n    labels = \\(x) paste0(\n      signs::signs(x),\n      \"&lt;br&gt;\",\n      ifelse(\n        x == mu,\n        \"&lt;em&gt;&mu;&lt;/em&gt;\",\n        paste0(\n          signs::signs((x - mu) / sigma,\n                       add_plusses = T,\n                       label_at_zero = \"none\"\n          ),\n          \"&lt;em&gt;&sigma;&lt;/em&gt;\"\n        )\n      )\n    )\n  ) +\n  scale_y_continuous(\n    NULL,\n    limits = c(0, plot_height),\n    expand = expansion(),\n    breaks = NULL\n  )\n\n\n\n\n\n\n\n\nFigure 3: Importing a .pdf file via ggimage::geom_image.\n\n\n\n\n\n\n\nA more complex example\nIn this example, I used the \\eqnmarkbox function for greater clarity. The .tex file is saved in a file called annotatedequation.tex.\n\n\nLaTeX code\n\\documentclass[border={10pt 48pt -45pt 62pt}]{standalone}\n%\\documentclass{article}\n\\usepackage{annotate-equations}\n\\usepackage{xcolor}\n\\definecolor{myviolet}{HTML}{414487}\n\\definecolor{myblue}{HTML}{2F6C8E}\n\\definecolor{myblue2}{HTML}{21908C}\n\\definecolor{mygreen}{HTML}{2FB47C}\n\\definecolor{mygreen2}{HTML}{7AD151}\n\\usepackage[sfdefault,condensed]{roboto}\n\\begin{document}\n    \n\\renewcommand{\\eqnhighlightheight}{\\mathstrut}\n    \n$\\LARGE\n\\eqnmarkbox[myviolet]{nodeP}{P\\left(T \\le \\tau \\right)} = \n\\eqnmarkbox[myblue]{phi}{\\Phi}\n\\left(\\frac{\n    \\eqnmarkbox[myblue2]{tau}{\\tau}-\n    \\eqnmarkbox[mygreen]{esttrue}{\\hat{T}}}{\n    \\eqnmarkbox[mygreen2]{sigma}{\\sigma_{T - \\hat{T}}}}\n\\right)$\n\n\n\\annotate[\n    yshift=1em, \n    xshift=11mm, \n    myviolet]\n    {above, left}\n    {nodeP}\n    {Probability true score $T$\\\\ is less than threshold $\\tau$ }\n    \n\\annotate[\n    yshift=-.6em,\n    myblue]\n    {below,left}\n    {phi}\n    {Standard Normal Cumulative\\\\ Distribution Function $\\Phi()$}\n    \n\\annotate[\n    yshift=1.4em, \n    xshift=4mm, \n    myblue2]\n    {above,left}\n    {tau}\n    {Threshold $\\tau=70$}\n    \n\\annotate[\n    yshift=3em, \n    xshift=7mm, \n    mygreen]\n    {above,left}\n    {esttrue}\n    {Estimated True Score $\\hat{T}=r_{XX}(X-\\mu)+\\mu$\\\\ \n    Observed Score $X\\sim \\mathcal{N}\\left(\\mu = 100, \\sigma=15\\right)$\\\\ \n    Reliability Coefficient $r_{XX}=\\{.80,.85,.90,.95,.98\\}$}\n    \n\\annotate[\n    yshift=-2em, \n    mygreen2]\n    {below,left}\n    {sigma}\n    {Standard Error of the Estimate\\\\ \n    $\\sigma_{T-\\hat{T}}=\\sigma\\sqrt{r_{XX}-r_{XX}^2}$}\n    \n\\end{document}\n\n\nNow convert .tex to .pdf:\n\n\nCode\nmyfile &lt;- \"annotatedequation\"\n\npaste0('pdflatex -interaction=nonstopmode ', myfile,'.tex') |&gt; \n  shell()\n\n\nAnd we are ready to plot. This plot shows the probability that an observed score will have a true score less than a specific threshold, given a reliability coefficient.\n\n\nCode\nviridis_start &lt;- .2\nviridis_end &lt;- .8\nthreshold &lt;- 70\n\n# Find where a line intersects with the normal cdf\nfind_x &lt;- function(rxx = .8,\n                   slope = 0.0048,\n                   intercept = .66,\n                   mu = 100,\n                   sigma = 15,\n                   start_x = 60,\n                   threshold = 70) {\n  x &lt;- start_x\n  precision &lt;- .00001\n  diff_y &lt;- precision * 10\n  reps &lt;- 0\n  while (abs(diff_y) &gt; precision) {\n    x &lt;- x - diff_y \n    line_y &lt;- x * slope + intercept\n    curve_y &lt;- pnorm(threshold,\n                     mean = rxx * (x - mu) + mu,\n                     sd = sigma * sqrt(rxx - rxx ^ 2))\n    \n    line_y &lt;- x * slope + intercept\n    curve_y &lt;- pnorm(threshold,\n                     mean = rxx * (x - mu) + mu,\n                     sd = sigma * sqrt(rxx - rxx ^ 2))\n    diff_y &lt;- line_y - curve_y\n    reps &lt;- reps + 1\n  }\n  tibble(x = x, p = line_y, mu = mu, sigma = sigma, threshold = threshold, reps = reps)\n}\n\n\n\n\ndimage = data.frame(x = 115, \n                    y = .62, \n                    image = paste0(myfile, \".pdf\"))\nv_rxx &lt;- round(c(seq(0.80, 0.95, 0.05), 0.98), 2)\n\nd_threshold &lt;-\n  crossing(x = round(seq(40, 160, 0.1), 1),\n           rxx = v_rxx,\n           threshold = 70) %&gt;%\n  mutate(\n    see = 15 * sqrt(rxx - rxx ^ 2),\n    mu = (x - 100) * rxx + 100,\n    p = pnorm(threshold, mu, see)\n  ) %&gt;%\n  group_by(rxx) %&gt;%\n  mutate(acceleration = p - lag(p)) %&gt;%\n  ungroup\n\n\n\nd_labels &lt;- tibble(rxx = v_rxx) %&gt;%\n  mutate(x = map_df(rxx, find_x)) |&gt; \n  unnest(x)\n\nd_threshold %&gt;%\n  ggplot(aes(x, p)) +\n  geom_line(aes(color = factor(rxx)), lwd = 1) +\n  geom_vline(\n    aes(xintercept = threshold),\n    lty = 2,\n    lwd = 1,\n    color = \"gray30\"\n  ) +\n  geom_image(data = dimage,\n             aes(x = x,\n                 y = y,\n                 image = image),\n             size = .87) +\n  geom_richtext(\n    aes(label = rxx_label,\n        color = factor(rxx)),\n    data = d_labels %&gt;%\n      mutate(rxx_label = prob_label(rxx)),\n    angle = -67,\n    size = WJSmisc::ggtext_size(13),\n    label.colour = NA,\n    fill = \"#FFFFFF\",\n    family = \"Roboto Condensed\",\n    label.margin = unit(0, \"mm\"),\n    label.r = unit(2, \"mm\"),\n    label.padding = unit(c(0, 0.75, 0, .5), \"mm\")\n  ) +\n  scale_x_continuous(\n    \"Observed Score\",\n    breaks = seq(40, 160, 15),\n    minor_breaks = seq(40, 160, 5),\n    expand = expansion()\n  ) +\n  scale_y_continuous(\n    paste0(\"Probability True Score &lt; \", threshold),\n    expand = expansion(),\n    breaks = seq(0, 1, 0.1),\n    labels = prob_label,\n    limits = c(0, 1)\n  ) +\n  scale_color_viridis_d(begin = viridis_start,\n                        end = viridis_end) +\n  theme_minimal(base_family = \"Roboto Condensed\", \n                base_size = 16) +\n  theme(legend.position = \"none\", \n        plot.margin = unit(c(3, 5, 2, 2), \"mm\")) +\n  coord_fixed(ratio = 100,\n              clip = \"off\",\n              xlim = c(40, 160)) \n\n\n\n\n\n\n\n\nFigure 4: The probability that a true score is less than 70 depends on the observed score’s position and reliability coefficient.\n\n\n\n\n\n\n\n\n\nCitationBibTeX citation:@misc{schneider2023,\n  author = {Schneider, W. Joel},\n  title = {Annotated Equations in Ggplot2},\n  date = {2023-07-24},\n  url = {https://wjschne.github.io/posts/2023-07-23-latex-equation-in-ggplot2},\n  langid = {en}\n}\nFor attribution, please cite this work as:\nSchneider, W. J. (2023, July 24). Annotated equations in ggplot2.\nSchneirographs. https://wjschne.github.io/posts/2023-07-23-latex-equation-in-ggplot2"
+    "text": "Overture Redux\n\n\n\n\n\n\n\n\n\n\n\nSep 12, 2023\n\n\nW. Joel Schneider\n\n\n\n\n\n\n\n\n\n\n\n\nThe RESCA-E Subtests Are Thoughtfully Designed and Highly Refined Measures of CHC Constructs\n\n\nA Review of the Receptive, Expressive & Social Communication Assessment–Elementary\n\n\n\n\n\n\n\n\nOct 26, 2016\n\n\nW. Joel Schneider, W. Joel Schneider\n\n\n\n\n\n\nNo matching items"
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-    "text": "Setup\nThis trick takes a lot of work. I only use it when I need a plot to look its best.\nSuppose I have a Likert questionnaire item with responses ranging from Strongly Disagree to Strongly Agree. Let’s say I have many groups in my study (e.g., college majors), and I want to compare their responses to the item.\nFirst I will create fake data for 26 groups, A–Z.\n\nlibrary(extrafont)\nlibrary(tidyverse)\nmyfont &lt;- \"Roboto Condensed\"\n\nd &lt;- tibble(Group = LETTERS[1:26],\n            mu = rnorm(26, 0, .5),\n            sigma = 1) %&gt;%\n  mutate(x = pmap(list(mean = mu, sd = sigma), rnorm, n = 500)) %&gt;%\n  unnest(x) %&gt;%\n  mutate(Agree = cut_number(x, 6) %&gt;%\n           factor(\n             labels = c(\n               \"Strongly\\nDisagree\",\n               \"Disagree\",\n               \"Slightly\\nDisagree\",\n               \"Slightly\\nAgree\",\n               \"Agree\",\n               \"Strongly\\nAgree\"\n             )\n           )) %&gt;%\n  group_by(Group, Agree) %&gt;%\n  summarise(n = n(), .groups = \"drop\") %&gt;%\n  group_by(Group) %&gt;%\n  arrange(Group, Agree) %&gt;%\n  mutate(p = n / sum(n)) %&gt;%\n  ungroup()\n\n\n\nFirst Attempts at Plotting\n\nd %&gt;%\n  ggplot(mapping = aes(p, Group)) +\n  geom_col(aes(fill = fct_rev(Agree))) +\n  scale_fill_viridis_d(NULL,\n                       begin = .15,\n                       end = 0.8,\n                       direction = -1) +\n  scale_x_continuous(\"Percent\", expand = expansion()) +\n  scale_y_discrete(\"Group\", expand = expansion()) +\n  coord_fixed(1 / 20.5, clip = \"off\") +\n  geom_text(\n    aes(x = p, label = round(100 * p, 0)),\n    color = \"white\",\n    family = myfont,\n    position = position_stack(vjust = .5)\n  )\n\n\n\n\n\n\n\n\nNot bad, but it would look better if we sorted the groups. We have many sorting options. One is that we can convert the Likert scale to numeric and then sort by the group with the highest mean.\n\nd %&gt;%\n  mutate(Group = fct_reorder(Group, .x = as.numeric(Agree) * p, \n                             .fun = mean)) %&gt;%\n  ggplot(mapping = aes(p, Group)) +\n  geom_col(aes(fill = fct_rev(Agree))) +\n  scale_fill_viridis_d(NULL,\n                       begin = .15,\n                       end = 0.8,\n                       direction = -1) +\n  scale_x_continuous(\"Percent\", expand = expansion()) +\n  scale_y_discrete(\"Group\", expand = expansion()) +\n  coord_fixed(1 / 20.5, clip = \"off\") +\n  geom_text(\n    aes(x = p, label = round(100 * p, 0)),\n    color = \"white\",\n    family = myfont,\n    position = position_stack(vjust = .5)\n  )\n\n\n\n\n\n\n\n\nIf I wanted to sort by the “Strongly Agree” category (or any other category):\n\nd %&gt;%\n  mutate(Group = fct_reorder(\n    Group,\n    .x = (Agree == \"Strongly\\nAgree\") * p,\n    .fun = mean\n  )) %&gt;%\n  ggplot(mapping = aes(p, Group)) +\n  geom_col(aes(fill = fct_rev(Agree))) +\n  scale_fill_viridis_d(NULL,\n                       begin = .15,\n                       end = 0.8,\n                       direction = -1) +\n  scale_x_continuous(\"Percent\", expand = expansion()) +\n  scale_y_discrete(\"Group\", expand = expansion()) +\n  coord_fixed(1 / 20.5, clip = \"off\") +\n  geom_text(\n    aes(x = p, label = round(100 * p, 0)),\n    color = \"white\",\n    family = myfont,\n    position = position_stack(vjust = .5)\n  )\n\n\n\n\n\n\n\n\n\n\nFinal Plot\nSo far, these plots look pretty good. However, I wish that the percentage labels were placed in a more aesthetically pleasing way. I am going to group by each of the response categories and then create a loess regression equation to smooth out the placement. It will mean that the labels are no longer centered in the stacked bars, but I think the sacrifice is worth it. I think that the percentage values are much easier to compare because the eye can follow the smooth line of labels.\n\nd &lt;- tibble(Group = LETTERS[1:26],\n            mu = rnorm(26, 0, .5),\n            sigma = 1) %&gt;%\n  mutate(x = pmap(list(mean = mu, sd = sigma), rnorm, n = 500)) %&gt;%\n  unnest(x) %&gt;%\n  mutate(Agree = cut_number(x, 6) %&gt;%\n           factor(\n             labels = c(\n               \"Strongly\\nDisagree\",\n               \"Disagree\",\n               \"Slightly\\nDisagree\",\n               \"Slightly\\nAgree\",\n               \"Agree\",\n               \"Strongly\\nAgree\"\n             )\n           )) %&gt;%\n  mutate(Group = fct_reorder(Group, as.numeric(Agree), .fun = mean)) %&gt;%\n  group_by(Group, Agree) %&gt;%\n  summarise(n = n(), .groups = \"drop\") %&gt;%\n  mutate(Group_position = as.numeric(Group)) %&gt;%\n  group_by(Group) %&gt;%\n  arrange(Group, Agree) %&gt;%\n  mutate(p = n / sum(n),\n         # proportion in each response category by group\n         cp = cumsum(p),\n         # cumulative proportion\n         xpos = cp - p / 2 # center of each stacked bar\n         ) %&gt;%\n         group_by(Agree) %&gt;%\n           nest() %&gt;%\n           mutate(\n             fit = map(data,\n                       loess,\n                       formula = \"xpos ~ Group_position\",\n                       span = 0.45),\n             # loess regression for each response category\n             xhat = map(fit, predict) # x-axis position on smooth line\n           ) %&gt;%\n           select(-fit) %&gt;%\n           unnest(c(data, xhat)) %&gt;%\n           mutate(Group = factor(Group, labels = rev(LETTERS[1:26])))\n         \n         d %&gt;%\n           ggplot(mapping = aes(p, Group)) +\n           geom_col(aes(fill = fct_rev(Agree))) +\n           geom_text(aes(x = xhat,\n                         label = ifelse(p &gt; .01, # No labels on small bars\n                                        round(100 * p, 0),\n                                        \"\")),\n                     color = \"white\",\n                     family = myfont) +\n           scale_fill_viridis_d(begin = .15,\n                                end = 0.8,\n                                direction = -1) +\n           scale_x_continuous(\"Percent\", expand = expansion()) +\n           scale_y_discrete(\"Group\", expand = expansion()) +\n           coord_fixed(1 / 20.5, clip = \"off\") +\n           theme_minimal(base_size = 13, base_family = myfont) +\n           theme(\n             legend.position = \"top\",\n             legend.box.spacing = unit(0.5, \"mm\"),\n             legend.text = element_text(\n               color = \"white\",\n               size = 13,\n               margin = margin(b = -40),\n               # Lower legend text into box\n               vjust = 0.5\n             ),\n             legend.spacing.x = unit(0, \"mm\"),\n             axis.text.y = element_text(hjust = 0.5)\n           ) +\n           guides(\n             fill = guide_legend(\n               title = NULL,\n               nrow = 1,\n               # Put legend on a single row\n               reverse = T,\n               # Reverse order of legend\n               label.position = \"top\",\n               # Put text atop keys\n               keyheight = unit(15, \"mm\"),\n               # Size of legend rectangles\n               keywidth = unit(22.56, \"mm\")\n             )\n           ) \n\n\n\n\n\n\n\n\nThis feels right to me.\n\n\n\n\nCitationBibTeX citation:@misc{schneider2021,\n  author = {Schneider, W. Joel},\n  title = {Bar Chart Labels on Smooth Paths in Ggplot2},\n  date = {2021-07-31},\n  url = {https://wjschne.github.io/posts/bar-chart-labels-on-smooth-paths-in-ggplot2},\n  langid = {en}\n}\nFor attribution, please cite this work as:\nSchneider, W. J. (2021, July 31). Bar chart labels on smooth paths in\nggplot2. Schneirographs. https://wjschne.github.io/posts/bar-chart-labels-on-smooth-paths-in-ggplot2"
+    "text": "W. Joel Schneider, Ph.D.\nProfessor\nTemple University\nPsychological Studies in Education\n\n\n\nCounseling Psychology\n\n\nSchool Psychology\n\n\n\n Curriculum Vitae\n Contact Me\n\n\n\n\nAbout Me\nI am a professor at Temple University in the College of Education and Human Development in the Psychological Studies in Education Department. I belong to both the School Psychology and Counseling Psychology programs.\nAlthough I have diverse interests, my primary focus is trying to understand and improve the validity of psychological assessment practices. I teach courses in assessment, counseling, statistics, and research methods.\nI grew up in Southern California (1970–1990) and lived in Córdoba, Argentina (1990–1992). I completed my undergraduate degree in Psychology at the University of California at Berkeley in 1995 and my doctoral studies in clinical psychology at Texas A&M University in 2003. My internship year (2001–2002) was spent in the Dutchess County Department of Behavioral & Community Health in Poughkeepsie, NY. Since then I have been a faculty member at Illinois State University (2002–2017) and Temple University (2017–Present)."
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-    "text": "I often need to add a prefix or suffix to a character vector:\n\nlibrary(tidyverse)\nx &lt;- c(\"A\", \"B\", \"C\")\npaste0(x, \"_1\")\n\n[1] \"A_1\" \"B_1\" \"C_1\"\n\n\nHowever, if the vector is empty, I do not want bare prefixes or suffixes like this:\n\nx &lt;- character(0)\npaste0(x, \"_1\")\n\n[1] \"_1\"\n\n\nI used to write a lot of tedious if statements like this:\n\nif (length(x) == 0) {\n  y &lt;- character(0)\n} else {\n  y &lt;- paste0(x, \"_1\")\n}\n\nFor R 4.0.1 and later versions, the paste and paste0 functions acquired the recycle0 argument. Setting recycle0 to TRUE returns an empty vector if at least one string is empty:\n\npaste0(x, \"_1\", recycle0 = TRUE)\n\ncharacter(0)\n\n\nIf you need to work with an earlier version of R, you can use the sprintf function instead:\n\n# Non-empty vector\nx &lt;- c(\"A\", \"B\", \"C\")\nsprintf(\"%s_1\", x)\n\n[1] \"A_1\" \"B_1\" \"C_1\"\n\n# Empty vector\nx &lt;- character(0)\nsprintf(\"%s_1\", x)\n\ncharacter(0)\n\n\nThe glue function from the glue package also works nicely:\n\n# Non-empty vector\nx &lt;- c(\"A\", \"B\", \"C\")\nglue::glue(\"{x}_1\")\n\nA_1\nB_1\nC_1\n\n# Empty vector\nx &lt;- character(0)\nglue::glue(\"{x}_1\")\n\n\n\n\nCitationBibTeX citation:@misc{schneider2021,\n  author = {Schneider, W. Joel},\n  title = {Concatenating Vectors Unless a Vector Is Empty},\n  date = {2021-05-13},\n  url = {https://wjschne.github.io/posts/concatenating-vectors-unless-a-vector-is-empty},\n  langid = {en}\n}\nFor attribution, please cite this work as:\nSchneider, W. J. (2021, May 13). Concatenating vectors unless a vector\nis empty. Schneirographs. https://wjschne.github.io/posts/concatenating-vectors-unless-a-vector-is-empty"
+    "text": "Base plot for demonstrations\nlibrary(tidyverse)\n# install.packages(\"remotes\")\n# remotes::install_github(\"teunbrand/ggarrow\")\nlibrary(ggarrow)\n\nnode_radius &lt;- .2\nresection_length &lt;- .03\n\nd_node &lt;- tibble(node = c(\"A\", \"B\", \"C\"),\n       x = c(0, .5, 1),\n       y = c(0, sqrt(3) / 2, 0))\n\nd_edge &lt;- d_node |&gt; \n  mutate(theta_next = atan((y - lag(y)) / (x - lag(x))),\n         x_from = lag(x) + cos(theta_next) * (node_radius + resection_length),\n         y_from = lag(y) + sin(theta_next) * (node_radius + resection_length),\n         x_to = x + cos(theta_next + pi) * (node_radius + resection_length),\n         y_to = y + sin(theta_next + pi) * (node_radius + resection_length)) |&gt; \n  filter(!is.na(x_from))\n\np &lt;- d_edge |&gt;\n  ggplot(aes(x = x_from, y = y_from, xend = x_to, yend = y_to)) +\n  ggforce::geom_circle(aes(\n    x0 = x,\n    y0 = y,\n    r = .2,\n    fill = node\n  ),\n  data = d_node,\n  color = NA, inherit.aes = F) +\n  geom_text(aes(label = node, x = x, y = y), \n            data = d_node, \n            size = 20,\n            family = \"Roboto Condensed\",\n            inherit.aes = F) +\n  coord_fixed() +\n  theme_void() +\n  theme(legend.position = \"none\") +\n     scale_fill_viridis_d(\n       option = \"D\",\n       begin = .2,\n       end = .8,\n       alpha = .5\n     )\n\np\n\n\n\n\n\n\n\n\n\nThe default arrows in ggplot2 are perfectly serviceable. There is the open variety:\n\n\nOpen arrow code\np + geom_segment(arrow = arrow())\n\n\n\n\n\n\n\n\nFigure 1: The ggplot2 open arrow\n\n\n\n\n\nYou can specify which ends of the segment have arrowheads, whether the arrows are open or closed, the length of the arrow, and how sharp the arrow’s point is with the angle argument.\nI happen to prefer a longer, sharper, closed arrow:\n\n\nClosed arrow code\np + geom_segment(\n  arrow = arrow(\n    angle = 15,\n    length = unit(8, \"mm\"),\n    type = \"closed\"\n))\n\n\n\n\n\n\n\n\nFigure 2: The ggplot2 closed arrow\n\n\n\n\n\nNevertheless, I miss the variety of arrows in available in TikZ. I particularly like the latex' arrow:\n\n\nTikZ code with latex’ arrow\n\\usetikzlibrary{arrows}\n\\begin{tikzpicture}\n\\node[fill=violet!50!white, circle] (A) at (0,0) {A};\n\\node[fill=cyan!80!black, circle] (B) at (1,1.732) {B};\n\\node[fill=green!40!black!40, circle] (C) at (2,0) {C};\n\\path[-&gt;,\n    draw,\n    shorten &gt;=2pt,\n    shorten &lt;=2pt,\n    &gt;=latex',\n    thick,\n    color = black!40,\n    text = black] (A) -&gt; (B);\n    \\path[-&gt;,\n    draw,\n    shorten &gt;=2pt,\n    shorten &lt;=2pt,\n    &gt;=latex',\n    thick,\n    color = black!40,\n    text = black] (B) -&gt; (C);\n\\end{tikzpicture}\n\n\n\n\n\n\n\n\nFigure 3: The TikZ latex’ arrow\n\n\n\n\n\n\nThe ggarrow package\nIf you want more variety in drawing arrows in ggplot2, Teun van den Brand’s ggarrow package expands your limits to whatever your imagination can provide.\nThe default arrowhead is already a nice improvement:\n\n\nCode for wings arrow (i.e, stealth arrow in TikZ)\np + geom_arrow_segment(length_head = 10)\n\n\n\n\n\n\n\n\nFigure 4: The “wings” arrow\n\n\n\n\n\nYou can play around with the sharpness of the point (offset) and the sharpness of the barb (inset). You can also add feathers.\n\n\nCode for sharp barbs and feathers\np +\n  geom_arrow_segment(length_head = 4,\n                     arrow_head = arrow_head_wings(offset = 20, \n                                                   inset = 10), \n                     arrow_fins = arrow_fins_feather(), \n                     length_fins = 11)\n\n\n\n\n\n\n\n\nFigure 5: Sharp barbs and feathers\n\n\n\n\n\n\n\nCode for kite arrowhead\np +\n  geom_arrow_segment(length_head = 15,\n                     arrow_head = arrow_head_wings(offset = 22.5,\n                                                   inset = 115),\n                     arrow_fins = arrow_head_wings(offset = 22.5,\n                                                   inset = 115),\n                     length_fins = 15)\n\n\n\n\n\n\n\n\nFigure 6: Double-headed arrow with kite arrowhead\n\n\n\n\n\n\n\nCode for reverse kite arrowhead\np +\n  geom_arrow_segment(length_head = 20,\n                     arrow_head = arrow_head_wings(offset = 45,\n                                                   inset = 120))\n\n\n\n\n\n\n\n\nFigure 7: Reverse kite arrowhead\n\n\n\n\n\nIf the arrow goes too far, you can pull it back with the resect arguments.\n\n\nCode for resecting an arrowhead\np +\n  geom_arrow_segment(length_head = 6,\n                     arrow_head = arrow_head_wings(offset = 120,\n                                                   inset = 35),\n                     resect_head = 2)\n\n\n\n\n\n\n\n\nFigure 8: Demonstration of resecting arrowheads\n\n\n\n\n\nThere is much, much more that can be done. See ggarrow’s arrow ornament vignette for more options.\n\n\nCustom Arrowheads\nNot only does ggarrow offer great arrow geoms with excellent features like resection, one can create any custom arrowhead or feather that can be made with a single polygon.\nThe polygon generally falls between -1 and 1 on x and y, though you can plot outside those limits. In most cases, the point is at (0,1), and the line ends at (0,0):\n\n\nCode\npar(pty = \"s\")\nx &lt;- c(0,1)\ny &lt;- c(0,0)\nplot(x , y, xlim = c(-1, 1), ylim = c(-1, 1))\nrect(-1,-.1,0,.1)\ntext(-.5,0, labels = \"Line\")\ntext(1,0, labels = \"Arrow Point\", adj = 1.2)\npolygon(ggarrow::arrow_head_wings() |&gt; `colnames&lt;-`(c(\"a\", \"b\")))\n\n\n\n\n\n\n\n\nFigure 9: Grid for custom arrowheads\n\n\n\n\n\nThe polygon you create should be be a 2-column matrix with named columns (e.g., x and y). Here I make a elliptical arrowhead.\n\n\nCode\nmake_ellipse &lt;- function(a = 1, b = .5){\n  t &lt;- seq(0,2*pi, length.out = 361)\n  cbind(x = a * cos(t), y = b * sin(t)) \n}\n\np +\n  geom_arrow_segment(length_head = 5,\n                     arrow_head = make_ellipse())\n\n\n\n\n\n\n\n\nFigure 10: Make ellipse\n\n\n\n\n\n\n\nThe arrowheadr package\nI made the arrowheadr package to make custom arrowheads quickly. Some of them are admittedly silly. However, it is now easy to make my favorite kind of arrow in ggplot2:\n\n\nCode\nlibrary(arrowheadr)\n\np + \n  geom_arrow_segment(length_head = 5,\n                     arrow_head = arrow_head_latex())\n\n\n\n\n\nMimicking the latex prime arrowhead\n\n\n\n\nMore examples:\nA catenary:\n\n\nCode\nstlouis &lt;- arrow_head_catenary(base_width = .25, thickness = .15)\n\np + \n  geom_arrow_segment(length_head = 5,\n                     arrow_head = stlouis)\n\n\n\n\n\n\n\n\nFigure 11: The Gateway Arch in St. Louis is shaped like a catenary, not a parabola.\n\n\n\n\n\nThe Cauchy function:\n\n\nCode\np + \n  geom_arrow_segment(length_head = 5,\n                     arrow_head = arrow_head_function(dt, df = 1))\n\n\n\n\n\n\n\n\nFigure 12: Any function can make an arrowhead.\n\n\n\n\n\nRazors:\n\n\nCode\nrazors &lt;- c(1,0,\n  0,.5,\n  -.35,.25,\n  -.35, .21,\n  0,.35,\n  .90,0\n  ) |&gt; \n  v2matrix() |&gt; \n  reflecter() \n\np + \n  geom_arrow_segment(length_head = 10,\n                     arrow_head = razors)\n\n\n\n\n\n\n\n\nFigure 13: The reflecter function takes a set of points and makes them symmetrical.\n\n\n\n\n\nA candle flame:\n\n\nCode\ncandleflame &lt;- arrow_head_wittgenstein_rod(\n  fixed_point = c(-2.75, 0),\n  rod_length = 3.75,\n  nudge = c(1, 0),\n  rescale = .95\n)\n\np + \n  geom_arrow_segment(length_head = 12,\n                     arrow_head = candleflame)\n\n\n\n\n\n\n\n\nFigure 14: Wittgenstein’s rod makes some nice shapes\n\n\n\n\n\nUsing bezier curve control points:\n\n\nCode\ncurved_arrowhead &lt;- arrow_head_bezier(list(\n  c(1,  0,\n    .5, .5,\n    .2, .5),\n  c(.2, .5,\n    .2, .1,\n    -.1, .25,\n    -.3, .25),\n  c(-.3, .25,\n    0, 0,\n    -.3, -.25),\n  c(-.3, -.25,\n    -.1, -.25,\n    .2,  -.1,\n    .2, -.5),\n  c(.2, -.5,\n    .5, -.5,\n    1,  0)\n))\n\np + \n  geom_arrow_segment(length_head = 8,\n                     arrow_head = curved_arrowhead)\n\n\n\n\n\n\n\n\nFigure 15: Bezier curves can make almost anything.\n\n\n\n\n\n\n\n\n\nCitationBibTeX citation:@misc{schneider2023,\n  author = {Schneider, W. Joel},\n  title = {Making a Custom Arrowhead for Ggplot2 Using Ggarrow and\n    Arrowheadr},\n  date = {2023-08-26},\n  url = {https://wjschne.github.io/posts/2023-08-26-making-a-custom-arrowhead-for-ggplot2-using-ggarrow-and-arrowheadr},\n  langid = {en}\n}\nFor attribution, please cite this work as:\nSchneider, W. J. (2023, August 26). Making a custom arrowhead for\nggplot2 using ggarrow and arrowheadr. Schneirographs. https://wjschne.github.io/posts/2023-08-26-making-a-custom-arrowhead-for-ggplot2-using-ggarrow-and-arrowheadr"
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-    "text": "While assembling a reply letter for a revise-and-resubmit document, I was documenting for reviewers how we addressed their concerns. The reply letter quoted numerous sentences and paragraphs that had been changed. Unfortunately, each time my co-authors and I edited the paper, I lost track of which sentences and paragraphs I had previously copied into the reply letter. I did not want to misquote my own paper.\nCopying-and-pasting with each edit was getting tedious, and it was an error-prone process. I decided to write a function that would retrieve a named div or span from the paper and print it as a quote in my reply letter. Here it is:\nget_quote &lt;- function(id, file, blockquote = TRUE) {\n  blocktext &lt;- ifelse(blockquote, \"\\n&gt; \", \"\")\n\n  refreplace &lt;- function(x) {\n    # List of sequence types\n    crossrefs &lt;- c(`@fig-` = \"Figure\",\n                   `@fig-` = \"Table\",\n                   `@eq-` = \"Equation\",\n                   `@thm-` = \"Lemma\",\n                   `@lem-` = \"Corollary\",\n                   `@cor-` = \"Proposition\",\n                   `@prp-` = \"Conjecture\",\n                   `@cnj-` = \"Definition\",\n                   `@def-` = \"Example\",\n                   `@exm-` = \"Example\",\n                   `@exr-` = \"Exercise\",\n                   `apafg-` = \"Figure\",\n                   `apatb-` = \"Figure\"\n    )\n\n    # Find all crossreference types and number them\n    make_replacements &lt;- function(x, reftype, prefix) {\n      regstring &lt;- ifelse(stringr::str_starts(reftype, \"\\\\@\"),\n                          paste0(\"\\\\\", reftype, \"(.*?)(?=[\\\\.\\\\?\\\\!\\\\]\\\\}\\\\s,])\"),\n                          paste0(\"\\\\{\", reftype, \"(.*?)\\\\}\"))\n      patterns &lt;- stringr::str_extract_all(string = x, pattern = regstring) |&gt;\n        unlist() |&gt;\n        unique() |&gt;\n        stringr::str_replace(\"\\\\{\", \"\\\\\\\\{\") |&gt;\n        stringr::str_replace(\"\\\\}\", \"\\\\\\\\}\")\n\n      if (all(is.na(patterns))) return(NULL)\n      replacements &lt;- paste(prefix, seq_along(patterns))\n      names(replacements) &lt;- patterns\n      replacements\n    }\n    allreplacements &lt;- purrr::map2(\n      names(crossrefs),\n      crossrefs,\n      \\(rt, pf) make_replacements(\n        x = x,\n        reftype = rt,\n        prefix = pf)) |&gt;\n      unlist()\n\n    stringr::str_replace_all(x, allreplacements)\n\n  }\n\n  filetext &lt;- readLines(file) |&gt;\n    refreplace()\n\n  idcount &lt;- sum(stringr::str_count(filetext, paste0(\"#\", id)))\n  if (idcount &gt; 1)\n    stop(paste0(\n      \"The id (\",\n      id ,\n      \") is not unique. There are \",\n      idcount,\n      \" instances of id = \",\n      id,\n      \".\"\n    ))\n\n  s &lt;- filetext |&gt;\n    paste0(collapse = \"\\n\") |&gt;\n    stringr::str_match(pattern = paste0(\"(?&lt;=\\\\[).+(?=\\\\]\\\\{\\\\#\",\n                               id,\n                               \"\\\\})\"))  |&gt;\n    getElement(1)\n\n  if (is.na(s)) {\n    s &lt;- filetext |&gt;\n      paste0(collapse = \"|||\") |&gt;\n      stringr::str_match(pattern = paste0(\":::\\\\{\\\\#\",\n                                 id,\n                                 \"\\\\}(.*?):::\"))  |&gt;\n      getElement(2) |&gt;\n      stringr::str_replace_all(\"\\\\|\\\\|\\\\|\", blocktext)\n  } else {\n    s &lt;- paste0(blocktext, s)\n  }\n\n  if (is.na(s)) stop(\"Could not find a div or span with id = \", id)\n\n  s\n}\nFor your convenience and mine, I have added this function into the WJSmisc package."
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-    "title": "Insert Text from One Markdown Document into Another",
-    "section": "Remove blockquote formatting",
-    "text": "Remove blockquote formatting\nBy default, the function adds a &gt; before each line in the quoted markdown so that it appears as a block quote. You can turn off the block quote formatting like so:\n`r get_quote(\"id1\", \"index.qmd\", blockquote = FALSE)`\nText in a span"
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-    "title": "Insert Text from One Markdown Document into Another",
-    "section": "Curly braces with additional information",
-    "text": "Curly braces with additional information\nWhat if you need to put additional information in the curly braces of the span or div (e.g., a css class)?\n[Here is more text with extra stuff in the curly braces.]{#id3 .myclass}\nYou can trick the function into thinking that the extra stuff is part of the id like so:\n`r get_quote(\"id3 .myclass\", \"index.qmd\")`\n\nHere is more text with extra stuff in the curly braces."
+    "text": "Sometimes I want to put a plot caption in the lower right corner of the plot.\n\nlibrary(tidyverse)\np &lt;- ggplot() +\n  stat_function(xlim = c(-4,4), fun = dnorm, n = 801) + \n  labs(x = \"z-scores\", caption = \"Note: Mean = 0, SD = 1\")\n\np\n\n\n\n\n\n\n\n\nHowever, I want the caption to be a little higher, on the same line as the x-axis title. To do so, set a a negative top margin:\n\np + theme(plot.caption = element_text(margin = margin(t = -10, unit = \"pt\")))\n\n\n\n\n\n\n\n\nHere is a more finished example.\n\nlibrary(ggnormalviolin)\nlibrary(ggtext)\nmy_font_size = 16\nmy_font &lt;- \"Roboto Condensed\"\nupdate_geom_defaults(geom = \"text\",new = list(family = my_font) )\nupdate_geom_defaults(geom = \"label\",new = list(family = my_font) )\nupdate_geom_defaults(\"richtext\", list(family = my_font))\ntheme_set(theme_minimal(base_size = my_font_size, base_family = my_font))\n\nd_rect &lt;- tibble(SS = 100, \n                 width = c(20, 40, 60, 80, 122), \n                 fill = paste0(\"gray\", c(95, 90, 80, 70, 65) - 25)) %&gt;% \n  arrange(-width)\n\ntibble(\n  Scale = c(\n    \"Fluid Reasoning\",\n    \"Verbal Comprehension\",\n    \"Visual-Spatial Processing\",\n    \"Working Memory\",\n    \"Processing Speed\",\n    \"Population\"),\n  y = c(5:1 - 0.5, 0),\n  SS = c(115, 111, 109, 86, 79, 100),\n  rxx = c(.93, .92, .92, .92, .88, 0),\n  width = c(rep(1.4, 5), 10.4),\n  alpha = c(rep(1, 5), .3)\n) %&gt;% \n  mutate(true_hat = rxx * (SS - 100) + 100, \n         see = ifelse(rxx == 0, 15, 15 * sqrt(rxx - rxx ^ 2)),\n         Scale = fct_inorder(Scale) %&gt;% fct_rev()) %&gt;% \n  ggplot(aes(y, SS)) + \n  geom_tile(data = d_rect, aes(width = 5.8, x = 2.9, \n                               fill = fill, \n                               height = width, \n                               y = SS)) + \n  geom_normalviolin(aes(mu = true_hat, \n                                        sigma = see, \n                                        width = width,\n                                        alpha = alpha), \n                                    face_left = F, \n                    fill = \"white\") + \n  geom_richtext(aes(label = ifelse(\n    Scale == \"Population\", \n    \"Population Mean\", \n    paste0(\"&lt;span style='font-size:8.5pt;color:white'&gt;(\", \n           round(100 * pnorm(SS, 100, 15),0),\n           \") &lt;/span&gt;\",\n           SS, \n           \"&lt;span style='font-size:9pt;color:#666666'&gt; (\", \n           round(100 * pnorm(SS, 100, 15),0),\n           \")&lt;/span&gt;\"))), \n    vjust = -0.2, \n    lineheight = .8, \n    fill = NA, \n    color = \"gray20\",\n    label.color = NA,\n    label.padding = unit(0,\"mm\")) +\n  geom_text(aes(y = true_hat, \n                label = ifelse(Scale == \"Population\", \n                               \"\", \n                               as.character(Scale))), \n            vjust = 1.5, \n            color = \"gray15\",\n            lineheight = .8) +\n  geom_linerange(aes(ymin = true_hat - 1.96 * see,\n                      ymax = true_hat + 1.96 * see), \n                 linewidth = .5) +\n  geom_pointrange(aes(ymin = true_hat - see,\n                      ymax = true_hat + see), \n                  size = 1.2, \n                  fatten = 1.5) +\n  geom_text(aes(x = x, y = y, label = label), \n            data = tibble(\n              y = c(49.5, 65, 75, 85, 100, 115, 125, 135, 150.5), \n              x = 5.5, \n              label = c(\"Extremely\\nLow Range\", \n                                  \"Very\\nLow\", \n                                  \"Low\\nRange\", \n                                  \"Low\\nAverage\", \n                                  \"Average\\nRange\", \n                                  \"High\\nAverage\", \n                                  \"High\\nRange\", \n                                  \"Very\\nHigh\", \n                                  \"Extremely\\nHigh Range\")), \n            color = \"white\", lineheight = .8, size = 4.25) +\n  scale_y_continuous(\n    \"Standard Scores &lt;span style='font-size:11.7pt;color:#656565'&gt;&lt;br&gt;\n    (and Percentile Ranks)&lt;/span&gt;\", \n    breaks = seq(40, 160, 10), \n    limits = c(37, 163),\n    labels = \\(x) paste0(x,\n                         \"&lt;br&gt;&lt;span style='font-size:10pt;color:#656565'&gt;(\", \n                         pnorm(x,100, 15) %&gt;% \n                           WJSmisc::proportion2percentile(digits = 2) %&gt;%\n                           str_trim(), \n                         \")&lt;/span&gt;\"),\n    expand = expansion()) +\n  scale_x_continuous(NULL, expand = expansion(), breaks = NULL) +\n  scale_alpha_identity() +\n  scale_fill_identity() +\n  coord_flip(clip = \"off\") + \n  theme(axis.title.x = element_markdown(hjust = 0, \n                                        margin = margin(t = 1.25, \n                                                        l = 2,\n                                                        unit = \"mm\")),\n        axis.text.x = element_markdown(colour = \"gray20\"),\n        plot.caption = element_markdown(hjust = 0, \n                                        size = 10, \n                                        margin = margin(t = -10.25, \n                                                        l = 100, \n                                                        unit = \"mm\"), \n                                        color = \"gray40\"),\n        plot.title = element_text(hjust = .025)) +\n  labs(\n    title = \"Display of Cognitive Test Scores\",\n    caption = \"**Notes:** The white normal curves represent the expected \n       true&lt;br&gt;score distributions for each observed score. The black lines&lt;br&gt;\n       underneath span the 68% and 95% confidence intervals.\")\n\n\n\n\n\n\n\n\n\n\n\nCitationBibTeX citation:@misc{schneider2021,\n  author = {Schneider, W. Joel},\n  title = {Caption on Same Line as Axis Title in Ggplot2},\n  date = {2021-07-21},\n  url = {https://wjschne.github.io/posts/caption-on-same-line-as-axis-title-in-ggplot2},\n  langid = {en}\n}\nFor attribution, please cite this work as:\nSchneider, W. J. (2021, July 21). Caption on same line as axis title in\nggplot2. Schneirographs. https://wjschne.github.io/posts/caption-on-same-line-as-axis-title-in-ggplot2"
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-    "href": "posts/point-labels-perpendicular-to-a-curve-in-ggplot2/index.html",
-    "title": "Point labels perpendicular to a curve in ggplot2",
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+    "title": "Creating collapsible output with knitr chunk hooks",
     "section": "",
-    "text": "I would like to label points on a sine function so that the labels are always legible. In a sine wave plot in which θ ranges from 0 to 2π, sin(θ) ranges from −1 to +1. Thus, the plot’s xy ratio is\n\\text{plot ratio}= \\frac{2\\pi-0}{1- (-1)}=\\pi\nThe first derivative of the sine function is the cosine function. In my plot, the slope of the tangent line at each point is\n\n\\text{tangent slope} = \\text{plot ratio}\\times \\cos(\\theta)\n The angle of the tangent line’s slope is the arctan of the slope. I would like to place the label perpendicular to the tangent line so I will add 90 degrees (i.e., π/2 radians).\n\n\\text{text angle}=\\tan^{-1}(\\text{tangent slope})+\\pi/2\n\nNow I need a pair of functions that will convert this angle into the right values for ggplot2’s hjust and vjust arguments. I have added the angle2vjust and angle2hjust functions to the WJSmisc package, but I have defined them here as well:\n\nangle2vjust &lt;- function(theta, multiplier = 1, as_degrees = FALSE) {\n  if (as_degrees) theta &lt;- theta * pi / 180\n  (((sin(theta + pi) + 1) / 2) - 0.5) * multiplier + 0.5\n}\n\nangle2hjust &lt;- function(theta, multiplier = 1, as_degrees = FALSE) {\n  if (as_degrees) theta &lt;- theta * pi / 180\n  (((cos(theta + pi) + 1) / 2) - 0.5) * multiplier + 0.5\n}\n\nNow we plot the sine function with labels. I have used geom_richtext from the ggtext package because it allows me to set a white background along with padding and margins.\n\nlibrary(tidyverse)\nlibrary(ggtext)\n\nplot_ratio &lt;- pi\n\ntibble(theta = seq(0, 2 * pi, length.out = 13),\n            y =  sin(theta),\n            tangent_slope = cos(theta) * plot_ratio,\n            text_angle = atan(tangent_slope) + pi / 2) %&gt;%\n  ggplot(aes(theta, y)) +\n  geom_richtext(aes(label = formatC(y, digits = 2, format = \"f\"),\n                    vjust = angle2vjust(text_angle),\n                    hjust = angle2hjust(text_angle)),\n                label.color = NA,\n                label.padding = unit(1, \"pt\"),\n                label.margin = unit(5, \"pt\"),\n                size = 4) +\n  geom_point() +\n  stat_function(fun = sin) +\n  scale_x_continuous(\"&theta;\",\n                     breaks = seq(0, 2 * pi, \n                                  length.out = 13),\n                     minor_breaks = NULL,\n                     labels = function(x) round(x * 180 / pi)) +\n  scale_y_continuous(\"sin(&theta;)\") +\n  coord_fixed(ratio = plot_ratio, clip = \"off\") +\n  theme_minimal(base_size = 16) +\n  theme(axis.title.x = element_markdown(),\n        axis.title.y = element_markdown())\n\n\n\n\n\n\n\n\nIf you do not mind turning your head to one side or the other, a somewhat easier method is to set the label’s vjust to a negative value and rotate the labels by the angle of the tangent line:\n\ntibble(theta = seq(0, 2 * pi, length.out = 13),\n            y = sin(theta),\n            tangent_slope = cos(theta) * plot_ratio,\n            text_angle = atan(tangent_slope)) %&gt;%\n  ggplot(aes(theta, y)) +\n  geom_richtext(aes(label = round(y, 2),\n                    angle = text_angle * 180 / pi),\n                vjust = 0,\n                label.color = NA,\n                label.padding = unit(1, \"pt\"),\n                label.margin = unit(2, \"pt\"),\n                size = 4) +\n  geom_point() +\n  stat_function(fun = sin) +\n  scale_x_continuous(\"&theta;\",\n                     breaks = seq(0, 2 * pi, \n                                  length.out = 13),\n                     minor_breaks = NULL,\n                     labels = function(x) round(x * 180 / pi)) +\n  scale_y_continuous(\"sin(&theta;)\") +\n  coord_fixed(ratio = plot_ratio, clip = \"off\") +\n  theme_minimal(base_size = 16) +\n  theme(axis.title.x = element_markdown(),\n        axis.title.y = element_markdown())\n\n\n\n\n\n\n\n\nWhat if you do not know the function’s first derivative? You can approximate the slope of the tangent line by comparing a function’s output of each point with the output of a slightly deviated point. Here is a plot of the normal cumulative distribution function.\n\n# Small change in x\ndx &lt;- .00000001\nplot_ratio &lt;- 6\n\ntibble(x = seq(-3,3,.5),\n       y = pnorm(x),\n       tangent_slope = plot_ratio * (pnorm(x + dx) - y) / dx,\n       text_angle = atan(tangent_slope) + pi / 2,\n       degrees = text_angle * 180 / pi) %&gt;% \n  ggplot(aes(x,y)) + \n  geom_point() +\n  geom_richtext(aes(label = formatC(y, 2, format = \"f\"),\n                    vjust = angle2vjust(text_angle),\n                    hjust = angle2hjust(text_angle)),\n                label.color = NA,\n                label.padding = unit(1, \"pt\"),\n                label.margin = unit(5, \"pt\"),\n                size = 4) +\n  stat_function(fun = pnorm) + \n  theme_minimal(base_size = 16) + \n  theme(axis.title.x = element_markdown(),\n        axis.title.y = element_markdown()) +\n  coord_fixed(ratio = plot_ratio) + \n  scale_x_continuous(\"*x*\", breaks = -3:3) +\n  scale_y_continuous(\"&Phi;(*x*)\")\n\n\n\n\n\n\n\n\n\n\n\nCitationBibTeX citation:@misc{schneider2021,\n  author = {Schneider, W. Joel},\n  title = {Point Labels Perpendicular to a Curve in Ggplot2},\n  date = {2021-07-27},\n  url = {https://wjschne.github.io/posts/point-labels-perpendicular-to-a-curve-in-ggplot2},\n  langid = {en}\n}\nFor attribution, please cite this work as:\nSchneider, W. J. (2021, July 27). Point labels perpendicular to a curve\nin ggplot2. Schneirographs. https://wjschne.github.io/posts/point-labels-perpendicular-to-a-curve-in-ggplot2"
+    "text": "I am writing a (still very much unfinished) psychometrics book with suggested exercises using R. I wanted to provide solutions to the questions, but I wanted to hide the answers until the reader clicks them.\nAn easy way to create collapsible content is with html’s &lt;details&gt; tag. For example:\nCreate a variable `x`, and assign it a value of 5.\n\n&lt;details&gt;\n  &lt;summary&gt;Suggested Solution&lt;/summary&gt;\n\n```{r}\nx &lt;- 5\n```\n\n&lt;/details&gt;\nCreate a variable x, and assign it a value of 5.\n\n\nSuggested Solution\n\n\nx &lt;- 5\n\n\n\nHowever, I did not want to worry about all the &lt;details&gt; every time I wrote a question. To automate this process, I used two knitr hooks: a chunk hook and an option hook.\nChunk hooks are for customizing the output of a chunk.\nHere I create a new chunk hook that is run before and after the chunk. When it is run before the chunk, it adds the &lt;details&gt; and &lt;summary&gt; tags. After the chunk, it closes the &lt;details&gt; tag.\n\n# Add the details tag before running the chunk \n# and close the tag after running the chunk.\nknitr::knit_hooks$set(\n  solutionsetter = function(before,options) {\n  \n  if (before) {\n    \n    \"\\n\\n&lt;details&gt;&lt;summary&gt;Suggested Solution&lt;/summary&gt;\\n\\n\"\n    \n  } else {\n    \n    \"\\n\\n&lt;/details&gt;\\n\\n\"\n    \n  }\n})\n\nIf all we needed to do was to enclose the output into a &lt;details&gt; tag, then just the solutionsetter hook would be needed. However, I also wanted to set echo=TRUE so that the output would be visible even if the default for echo was FALSE.\nOption hooks are great for when you have a kind of chunk you will use often, and it requires setting many chunk options at once (e.g., a plot variant with different dimensions than your default plot has).\nHere I create a new option hook called solution. It sets the current chunk’s echo option to TRUE and also triggers the new solutionsetter code chunk below. The updated options list needs to be returned explicitly, or the hook will not work.\n\nknitr::opts_hooks$set(solution = function(options) {\n  options$echo &lt;- TRUE\n  options$solutionsetter &lt;- TRUE\n  return(options)\n})\n\nNow all need to do is to set my chunk option to solution = TRUE and the output will be collapsible:\n\nCreate a variable `x`, and assign it a value of 5.\n\n```{r, solution = TRUE}\nx &lt;- 5\n```\nCreate a variable x, and assign it a value of 5.\n\n\n\nSuggested Solution\n\nx &lt;- 5\n\n\n\n\nThe details package\nIf you want a function to enclose the output of a chunk or inline code in &lt;details&gt; tags, use the details package by Jonathan Sidi. For example, an inline chunk like this:\n`r details::details(\"Answer\", summary = \"Question\")`\nWill output like so:\n\n\n Question \n\n\nAnswer\n\n\nThe package has many other uses, which are explained in the package’s vignettes.\n\n\n\n\nCitationBibTeX citation:@misc{schneider2023,\n  author = {Schneider, W. Joel},\n  title = {Creating Collapsible Output with Knitr Chunk Hooks},\n  date = {2023-06-30},\n  url = {https://wjschne.github.io/posts/creating-collapsible-output-with-knitr-chunk-hooks},\n  langid = {en}\n}\nFor attribution, please cite this work as:\nSchneider, W. J. (2023, June 30). Creating collapsible output with knitr\nchunk hooks. Schneirographs. https://wjschne.github.io/posts/creating-collapsible-output-with-knitr-chunk-hooks"
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+    "title": "Making text labels the same size as axis labels in ggplot2",
     "section": "",
-    "text": "Nominal, Ordinal, Interval, and Ratio scales\n\n\n\n\n\n\n\n\nHow to visualize which values are most frequent in a distribution\n\n\n\n\n\n\n\n\nFunctions that tell us which values are more likely to occur\n\n\n\n\n\n\n\n\nAn introduction to expected value and how it is related to the concept of the mean of a variable.\n\n\n\n\n\n\n\n\nHow variance is calculated\n\n\n\n\n\n\n\n\nSum of the Reasons Variables Are Normally Normal\n\n\n\n\n\n\n\n\nAn introduction to skewness and its relationship to variability.\n\n\n\n\n\n\n\n\nAn introduction to kurtosis and how it is not just a measure of peakedness.\n\n\n\n\n\n\n\n\nAn introduction to z-scores, stanines, stens, scaled scores, T scores, and index scores\n\n\n\n\n\n\n\n\nCovariance is rarely useful by itself but is indispensable for many statistics."
+    "text": "As explained in this ggplot2 vignette, the size parameter in geom_text and geom_label is in millimeters, and the size parameter in all other text elements in ggplot2 is in points.\nIf I specify the base_size of the plot and the size of a label to 16, you can see that the text label is much bigger than 16.\n\nlibrary(tidyverse)\ntextsize &lt;- 16\nggplot() +\n  stat_function(\n    fun = dnorm,\n    xlim = c(-4, 4),\n    geom = \"area\",\n    alpha = .3\n  ) +\n  theme_minimal(base_size = textsize) +\n  annotate(\n    geom = \"text\",\n    x = 0,\n    y = 0,\n    label = \"Mean = 0\",\n    size = textsize,\n    vjust = -.1\n  )\n\n\n\n\n\n\n\n\nIf you do not mind a little trial and error, you can fiddle with the size parameter until you find a value that looks good to you. However, what if we want the axis text labels and the output of geom_text to be exactly the same size?\nWe are in luck because ggplot2 has a .pt constant that will help us convert point sizes to millimeters.\n\nggplot2::.pt\n\n[1] 2.845276\n\n\nWe know that, by default, axis text is .8 times as large as the base_size of the theme.\nLet’s make a function to automate the conversion:\n\nggtext_size &lt;- function(base_size, ratio = 0.8) {\n  ratio * base_size / ggplot2::.pt\n}\n\nNow we can make the label and axis text exactly the same size:\n\nggplot() + \n  stat_function(fun = dnorm, xlim = c(-4,4), geom = \"area\", alpha = .3) +\n  theme_minimal(base_size = textsize) + \n  annotate(\n    geom = \"text\",\n    x = 0,\n    y = 0,\n    label = \"Mean = 0\",\n    size = ggtext_size(textsize),\n    vjust = -.3\n  )\n\n\n\n\n\n\n\n\nFor my own convenience (and possibly yours), I put the ggtext_size function in the WJSmisc package.\nFor more on ggplot font size matters, I found this post by Christophe Nicault to be informative.\n\n\n\nCitationBibTeX citation:@misc{schneider2021,\n  author = {Schneider, W. Joel},\n  title = {Making Text Labels the Same Size as Axis Labels in Ggplot2},\n  date = {2021-08-10},\n  url = {https://wjschne.github.io/posts/making-text-labels-the-same-size-as-axis-labels-in-ggplot2},\n  langid = {en}\n}\nFor attribution, please cite this work as:\nSchneider, W. J. (2021, August 10). Making text labels the same size as\naxis labels in ggplot2. Schneirographs. https://wjschne.github.io/posts/making-text-labels-the-same-size-as-axis-labels-in-ggplot2"
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-    "text": "Nominal, Ordinal, Interval, and Ratio scales\n\n\n\n\n\n\n\n\nHow to visualize which values are most frequent in a distribution\n\n\n\n\n\n\n\n\nFunctions that tell us which values are more likely to occur\n\n\n\n\n\n\n\n\nAn introduction to expected value and how it is related to the concept of the mean of a variable.\n\n\n\n\n\n\n\n\nHow variance is calculated\n\n\n\n\n\n\n\n\nSum of the Reasons Variables Are Normally Normal\n\n\n\n\n\n\n\n\nAn introduction to skewness and its relationship to variability.\n\n\n\n\n\n\n\n\nAn introduction to kurtosis and how it is not just a measure of peakedness.\n\n\n\n\n\n\n\n\nAn introduction to z-scores, stanines, stens, scaled scores, T scores, and index scores\n\n\n\n\n\n\n\n\nCovariance is rarely useful by itself but is indispensable for many statistics."
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-    "text": "Other Assessment Videos\n\nEmotional Intelligence and CHC Theory\nHow might emotional intelligence relate to psychometric models of intelligence?\n\n\n\n\n\n\n\nMisunderstanding Regression to the Mean\nWhat regression to the mean is and how it is often misunderstood. Examples are provided as to how it is applied to IQ and the death penalty.\n\n\n\n\n\n\n\nTaking latent variable models seriously\nApplying latent score estimates to individuals\n\n\n\n\n\n\n\nSpecific cognitive processing weaknesses are rarely full explanations for academic deficits\nIn which I confess to having had a longstanding misunderstanding about diagnosing learning disorders.\n\n\n\n\n\n\n\nTwo kinds of cognitive ability hierarchies\nSome latent variables represent “hierarchical abstractions.”\n\n\n\n\n\n\n\nA Taxonomy of Influences on Ability Tests\nSome influences are test-specific whereas others are shared across tests. Some influences are transient whereas others are stable. Some influences are relevant to the construct of interest whereas others are irrelevant.\n\n\n\n\n\n\n\nWithin-composite differences: Why measures of the same ability differ?\nThere are a number of lesser-known reasons that two tests that are intended to measure the same ability might differ substantially.\n\n\n\n\n\n\n\nWithin-composite differences: Do large subtest score differences invalidate composite scores?\nIt is often asserted that composite scores should not be interpreted when the scores that make up that composite are discrepant. I show here why this is not typically true, depending on what the composite score is used for.\n\n\n\n\n\n\n\nWriting Assessment Reports People Will Read, Understand, and Remember\nMy appearance on the School Psyched Podcast on October 20, 2019"
+    "text": "When analyzing questionnaire data, I sometimes put the text of several questions on the y-axis of a plot. Longer text strings will usually make the plot too narrow.\nSuppose we give an admittedly silly questionnaire about preference for darker food. We want to display the percentages of people answering “Yes” to each question.\nHere are the questions and the (completely made-up) percentages:\n# To install the WJSmisc package:\n# remotes::install_github(\"wjschne/WJSmisc\")\nlibrary(WJSmisc)\nlibrary(tidyverse)\nd &lt;- tibble(Item = c(\"When you drink coffee, do you prefer darker roasts?\",\n                     \"Is pumpernickel the best bread?\",\n                     \"Do you like dark chocolate more than you like milk chocolate?\",\n                     \"Are black olives better than green olives?\",\n                     \"Do you enjoy the taste of prunes more than can be prudently admitted aloud?\",\n                     \"Is black licorice the secret to a happly life?\"),\n            Percentage = c(45,4,37, 84,2, 3)) |&gt; \n  mutate(Item = fct_reorder(Item, Percentage))\nA preliminary plot might look like this:\nmyfont &lt;- \"Roboto Condensed\"\nmycolors &lt;- viridis::viridis(n = nrow(d), begin = .2, end = .8) |&gt; tinter::lighten(.5)\nmytextsize &lt;- 17\nmytext &lt;- \"gray60\"\np &lt;- ggplot(d, aes(Percentage, Item)) + \n  geom_col(aes(fill = factor(Percentage))) + \n  geom_richlabel(aes(label = paste0(Percentage, \"%\"), \n                     color = factor(Percentage)), \n                 hjust = 0, fill = \"black\",\n                 text_size = mytextsize,\n                 family = myfont) +\n  labs(y = NULL,\n       caption = \"Note: These numbers were made up for illustrative purposes.\") +\n  theme_minimal(base_size = mytextsize, base_family = myfont) +\n  scale_x_continuous(\"Percent Yes\", \n                     limits = c(0,100), \n                     breaks = NULL,\n                     expand = expansion(mult = c(0,.2))) +\n  scale_fill_manual(values = mycolors) +\n  scale_color_manual(values = mycolors) +\n  theme(plot.title.position = \"plot\", \n        plot.background = element_rect(\"black\"),\n        plot.title = element_text(colour = mytext),\n        plot.caption = element_text(color = \"gray40\"),\n        legend.position = \"none\",\n        axis.title = element_text(color = mytext),\n        panel.grid.major.y = element_blank(),\n        axis.text = element_text(color = mycolors))\np + ggtitle(\"Y-axis labels with no text wrapping\")\nThe questions are taking up a lot of space. We can fix this by using stringr::str_wrap to make the text lines no longer than a specified width:\np + \n  scale_y_discrete(labels = \\(x) str_wrap(x, width = 30)) + \n  ggtitle(\"Splitting Y-axis labels with str_wrap\")\nThis is a definite improvement, but I wish the lines of text were wrapped more evenly. For example, I do not like the fact that “chocolate” and “bread” are by themselves on their own lines. I have spent a fair amount of time breaking lines up by hand and hoped to create a function to automate the process. To achieve a more balanced style of text splitting, I made the str_wrap_equal function.\np + scale_y_discrete(labels = \\(x) str_wrap_equal(x, max_width = 30)) + \n  ggtitle(\"Splitting Y-axis labels with str_wrap_equal\")\nTo each their own, but this style of text wrapping feels better to me here. It is the sort of thing no one but the analyst will notice if it is right but will feel a bit off if not.\nFor your convenience and mine, I put the str_wrap_equal function in the WJSmisc package."
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+    "text": "Technical Details\nI hesitate to post a new function because the odds are quite high that someone has already done what I did. However, my initial search did not locate a function that does exactly what I needed. If you know of a function have I duplicated unnecessarily, let me know.\nIn deciding whether to break the line, the function does the following:\n\nCalculate the overall length of the text (i.e., number of characters).\nFind k (the number of lines likely needed) by dividing the overall text length by max_width and round up to the nearest integers.\nCalculate the preferred width of the line by dividing the overall text length by the number of lines likely needed.\nAdd to the line one word at a time as long as doing so makes the current line width closer to the preferred width. If not, a new line is started.\nThe function will not make a line longer than what is specified by max_width unless it has to place a single, lonely word that is longer than max_width.\n\nHere is the code for the function. Any suggestions to improve it are welcome.\n\nstr_wrap_equal &lt;- function(x, max_width = 30L, sep = \"\\n\") {\n  purrr::map_chr(x, \\(xi) {\n    # Find overall text length\n    xlen &lt;- stringr::str_length(xi)\n    # Remove any line breaks and allow lines to break at forward slashes\n    xi &lt;- stringr::str_replace(xi, \"\\n\", \" \") |&gt; \n      stringr::str_replace(\"/\", \"/ \")\n    # Number of lines likely needed\n    k &lt;- ceiling(xlen / max_width)\n    # Optimal line length\n    preferred_width &lt;- xlen / k\n    # Split text into words\n    words &lt;- stringr::str_split(xi, pattern = \" \", simplify = F)[[1]]\n    # Number of words in text\n    k_words &lt;- length(words)\n    # Length of each word in text\n    word_len &lt;- stringr::str_length(words)\n    # Create empty text lines with a few extra, if needed\n    textlines &lt;- rep(\"\", k + 10)\n    # Current text line\n    i &lt;- 1\n    # Decide whether to add a word to the current line or to start a new line\n    for (w in seq(k_words)) {\n      # Width of current line before adding a new word\n      current_width &lt;- stringr::str_length(textlines[i])\n      # Width of current line if a new word is added\n      proposed_width &lt;- current_width + word_len[w] + 1\n      # Difference between current width and preferred width\n      current_difference &lt;- abs(current_width - preferred_width)\n      # Difference between proposed width and preferred width\n      proposed_difference &lt;- abs(proposed_width - preferred_width)\n      # Should we start a new line?\n      if (current_difference &lt; proposed_difference | proposed_width &gt; max_width) {\n        i &lt;- i + 1\n      }\n      # Add word to current line, remove spaces, and rejoin words divided by forward slashes\n      textlines[i] &lt;- stringr::str_trim(paste(textlines[i], words[w])) |&gt;\n        stringr::str_replace(\"/ \", \"/\")\n    }\n    # Collapse non-empty lines by separation character\n    paste0(textlines[stringr::str_length(textlines) &gt; 0], collapse = sep)\n  })\n\n}"
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+    "text": "A free online book about psychometric tools for psychological assessment\n\nI have been working on a book project off and on since 2012. I have ambitious goals for this book, but it is not nearly complete. The preliminary, unfinished version is here.\n\n\n\n\n\nAn introduction to tests and measurements\nRenée Tobin and I helped update Ronald Cohen’s Psychological Testing and Assessment: An Introduction to Tests and Measurements. It is a comprehensive textbook suitable for introductory psychological assessment courses.\n\n\n\n\n\n\n\n\n\nA book about making reader-friendly assessment reports that instill hope and promote change.\nMy co-authors and I worked hard to fill the second edition of Essentials of Assessment Report Writing with non-obvious guidance about how to write reports so that the meet the needs of the reports’ readers."
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-    "text": "R package for simulating data using standardized coefficients\nTutorial\n\n\n\n\nIn the model below, the path coefficients are standardized. You would like to simulate the variables in the model, but you do not know the disturbance and residual variances. The simstandard package can help.\n\n\n\nA standardized latent variable model"
+    "text": "A free online book about psychometric tools for psychological assessment\n\nI have been working on a book project off and on since 2012. I have ambitious goals for this book, but it is not nearly complete. The preliminary, unfinished version is here."
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-    "text": "An R package for detecting unusual scores in a test profile\nTutorial\n\n\n\n\nThis package estimates how unusual a multivariate normal profile is."
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+    "text": "A book about making reader-friendly assessment reports that instill hope and promote change.\nMy co-authors and I worked hard to fill the second edition of Essentials of Assessment Report Writing with non-obvious guidance about how to write reports so that the meet the needs of the reports’ readers."
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+    "text": "Summary\nThe base R function summary gives us most of what we might want to know.\n\nsummary(fit)\n\n             Df Sum Sq Mean Sq F value Pr(&gt;F)    \nAttachment    2  20359   10179      93 &lt;2e-16 ***\nResiduals   297  32517     109                   \n---\nSignif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n\n\nWe can see that overall model is significant, meaning that at least two groups have reliably different means. At this point we do not know which means differ, but from the plot we have a good guess that the Ambivalent group scores higher on the Anxiety Questionnaire than the other two groups.\nA similar, but tidier display can be found with the parameters function:\n\nparameters(fit)\n\n\n\n\n\nParameter\nSum_Squares\ndf\nMean_Square\nF\np\n\n\n\n\nAttachment\n20359\n2\n10179\n93\n0\n\n\nResiduals\n32517\n297\n109\nNA\nNA"
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-    "text": "An R package for making digital spirographs\nTutorial\nMy Gallery\n\n\n\n\nMaking digital spirographs is fun! I made an R package called spiro that can make animated spirographs like this one:"
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+    "text": "Other Summary Functions\nThe overall “fit” or “performance” of the model:\n\nperformance(fit)\n\n\n\n\n\nAIC\nAICc\nBIC\nR2\nR2_adjusted\nRMSE\nSigma\n\n\n\n\n2265\n2265\n2280\n0.39\n0.38\n10\n10\n\n\n\n\n\n\n\nAIC: Akaike’s Information Criterion\nAICc: AIC with a correction for small sample sizes\nBIC: Bayesian Information Criterion\nR2: R-squared: Coefficient of Determination\nR2_adj: Adjusted r-squared\nRMSE: Root mean squared error =\\sqrt{\\frac{\\sum_{i = 1}^{n}{e_i}}{n}}\nSIGMA: Residual standard deviation (The SD of the residuals =\\sqrt{\\frac{\\sum_{i = 1}^{n}{e_i}}{n-k-1}})"
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-    "text": "R package for making a custom arrowheads for ggplot2 using ggarrow\nTutorial\n\n\n\n\nThe arrowheadr package allows one to create custom arrowheads that can be used with the ggarrow package."
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+    "text": "Diagnostic Plots\n\ncheck_model(fit)\n\n\n\n\n\n\n\n\nNone of the diagnostic checks raise any alarms. Specifically, the residuals are reasonably normal and have consistent variance acrross groups (i.e., the homogeneeity of variance assumption is reasonable)."
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+    "text": "Post-hoc Tests\nWe would like to compare all means but control for family-wise error.\n\nestimate_contrasts(fit)\n\nNo variable was specified for contrast estimation. Selecting `contrast = \"Attachment\"`.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nLevel1\nLevel2\nDifference\nCI_low\nCI_high\nSE\ndf\nt\np\n\n\n\n\nAvoidant\nAmbivalent\n-18.4\n-22.1\n-14.8\n1.5\n297\n-12.1\n0.00\n\n\nSecure\nAmbivalent\n-16.2\n-19.6\n-12.7\n1.4\n297\n-11.3\n0.00\n\n\nSecure\nAvoidant\n2.3\n-1.4\n5.9\n1.5\n297\n1.5\n0.13\n\n\n\n\n\nestimate_contrasts(fit) |&gt; \n  report()\n\nNo variable was specified for contrast estimation. Selecting `contrast = \"Attachment\"`.\n\n\nThe marginal contrasts analysis suggests the following. The difference between\nAvoidant and Ambivalent is negative and statistically significant (difference =\n-18.44, 95% CI [-22.12, -14.77], t(297) = -12.08, p &lt; .001). The difference\nbetween Secure and Ambivalent is negative and statistically significant\n(difference = -16.18, 95% CI [-19.62, -12.74], t(297) = -11.33, p &lt; .001). The\ndifference between Secure and Avoidant is positive and statistically\nnon-significant (difference = 2.27, 95% CI [ -1.37, 5.90], t(297) = 1.50, p =\n0.135)"
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+    "text": "ANOVA in R\n\n\n\n\n\n\n\n\n\n\n\nW. Joel Schneider\n\n\n\n\n\n\n\n\n\n\n\n\nRegression in R\n\n\n\n\n\n\n\n\n\n\n\nW. Joel Schneider\n\n\n\n\n\n\nNo matching items"
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+    "text": "Model Paramters\n\nreport_table(fit)\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nParameter\nSum_Squares\ndf\nMean_Square\nF\np\nEta2\nEta2_CI_low\nEta2_CI_high\n\n\n\n\nAttachment\n20359\n2\n10179\n93\n0\n0.39\n0.32\n1\n\n\nResiduals\n32517\n297\n109\nNA\nNA\nNA\nNA\nNA"
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+    "text": "Automated Interpretation\n\nreport(fit)\n\nThe ANOVA (formula: Anxiety_Questionnaire ~ Attachment) suggests that:\n\n  - The main effect of Attachment is statistically significant and large (F(2,\n297) = 92.98, p &lt; .001; Eta2 = 0.39, 95% CI [0.32, 1.00])\n\nEffect sizes were labelled following Field's (2013) recommendations."
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index 8131881..511346f 100644
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@@ -1,2 +1,2 @@
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n=r("./src/utils/events.js"),s=r("./src/utils/extend.js"),i=r("./src/utils/to-string.js"),a=r("./src/utils/get-by-class.js"),o=r("./src/utils/fuzzy.js");t.exports=function(t,e){e=s({location:0,distance:100,threshold:.4,multiSearch:!0,searchClass:"fuzzy-search"},e=e||{});var r={search:function(n,s){for(var i=e.multiSearch?n.replace(/ +$/,"").split(/ +/):[n],a=0,o=t.items.length;a<o;a++)r.item(t.items[a],s,i)},item:function(t,e,n){for(var s=!0,i=0;i<n.length;i++){for(var a=!1,o=0,l=e.length;o<l;o++)r.values(t.values(),e[o],n[i])&&(a=!0);a||(s=!1)}t.found=s},values:function(t,r,n){if(t.hasOwnProperty(r)){var s=i(t[r]).toLowerCase();if(o(s,n,e))return!0}return!1}};return n.bind(a(t.listContainer,e.searchClass),"keyup",t.utils.events.debounce((function(e){var n=e.target||e.srcElement;t.search(n.value,r.search)}),t.searchDelay)),function(e,n){t.search(e,n,r.search)}}},"./src/index.js":function(t,e,r){var n=r("./node_modules/string-natural-compare/natural-compare.js"),s=r("./src/utils/get-by-class.js"),i=r("./src/utils/extend.js"),a=r("./src/utils/index-of.js"),o=r("./src/utils/events.js"),l=r("./src/utils/to-string.js"),u=r("./src/utils/classes.js"),c=r("./src/utils/get-attribute.js"),f=r("./src/utils/to-array.js");t.exports=function(t,e,h){var d,v=this,g=r("./src/item.js")(v),m=r("./src/add-async.js")(v),p=r("./src/pagination.js")(v);d={start:function(){v.listClass="list",v.searchClass="search",v.sortClass="sort",v.page=1e4,v.i=1,v.items=[],v.visibleItems=[],v.matchingItems=[],v.searched=!1,v.filtered=!1,v.searchColumns=void 0,v.searchDelay=0,v.handlers={updated:[]},v.valueNames=[],v.utils={getByClass:s,extend:i,indexOf:a,events:o,toString:l,naturalSort:n,classes:u,getAttribute:c,toArray:f},v.utils.extend(v,e),v.listContainer="string"==typeof t?document.getElementById(t):t,v.listContainer&&(v.list=s(v.listContainer,v.listClass,!0),v.parse=r("./src/parse.js")(v),v.templater=r("./src/templater.js")(v),v.search=r("./src/search.js")(v),v.filter=r("./src/filter.js")(v),v.sort=r("./src/sort.js")(v),v.fuzzySearch=r("./src/fuzzy-search.js")(v,e.fuzzySearch),this.handlers(),this.items(),this.pagination(),v.update())},handlers:function(){for(var t in v.handlers)v[t]&&v.handlers.hasOwnProperty(t)&&v.on(t,v[t])},items:function(){v.parse(v.list),void 0!==h&&v.add(h)},pagination:function(){if(void 0!==e.pagination){!0===e.pagination&&(e.pagination=[{}]),void 0===e.pagination[0]&&(e.pagination=[e.pagination]);for(var t=0,r=e.pagination.length;t<r;t++)p(e.pagination[t])}}},this.reIndex=function(){v.items=[],v.visibleItems=[],v.matchingItems=[],v.searched=!1,v.filtered=!1,v.parse(v.list)},this.toJSON=function(){for(var t=[],e=0,r=v.items.length;e<r;e++)t.push(v.items[e].values());return t},this.add=function(t,e){if(0!==t.length){if(!e){var r=[],n=!1;void 0===t[0]&&(t=[t]);for(var s=0,i=t.length;s<i;s++){var a;n=v.items.length>v.page,a=new g(t[s],void 0,n),v.items.push(a),r.push(a)}return v.update(),r}m(t.slice(0),e)}},this.show=function(t,e){return this.i=t,this.page=e,v.update(),v},this.remove=function(t,e,r){for(var n=0,s=0,i=v.items.length;s<i;s++)v.items[s].values()[t]==e&&(v.templater.remove(v.items[s],r),v.items.splice(s,1),i--,s--,n++);return v.update(),n},this.get=function(t,e){for(var r=[],n=0,s=v.items.length;n<s;n++){var i=v.items[n];i.values()[t]==e&&r.push(i)}return r},this.size=function(){return v.items.length},this.clear=function(){return v.templater.clear(),v.items=[],v},this.on=function(t,e){return v.handlers[t].push(e),v},this.off=function(t,e){var r=v.handlers[t],n=a(r,e);return n>-1&&r.splice(n,1),v},this.trigger=function(t){for(var e=v.handlers[t].length;e--;)v.handlers[t][e](v);return v},this.reset={filter:function(){for(var t=v.items,e=t.length;e--;)t[e].filtered=!1;return v},search:function(){for(var t=v.items,e=t.length;e--;)t[e].found=!1;return v}},this.update=function(){var t=v.items,e=t.length;v.visibleItems=[],v.matchingItems=[],v.templater.clear();for(var r=0;r<e;r++)t[r].matching()&&v.matchingItems.length+1>=v.i&&v.visibleItems.length<v.page?(t[r].show(),v.visibleItems.push(t[r]),v.matchingItems.push(t[r])):t[r].matching()?(v.matchingItems.push(t[r]),t[r].hide()):t[r].hide();return v.trigger("updated"),v},d.start()}},"./src/item.js":function(t){t.exports=function(t){return function(e,r,n){var s=this;this._values={},this.found=!1,this.filtered=!1;this.values=function(e,r){if(void 0===e)return s._values;for(var n in e)s._values[n]=e[n];!0!==r&&t.templater.set(s,s.values())},this.show=function(){t.templater.show(s)},this.hide=function(){t.templater.hide(s)},this.matching=function(){return t.filtered&&t.searched&&s.found&&s.filtered||t.filtered&&!t.searched&&s.filtered||!t.filtered&&t.searched&&s.found||!t.filtered&&!t.searched},this.visible=function(){return!(!s.elm||s.elm.parentNode!=t.list)},function(e,r,n){if(void 0===r)n?s.values(e,n):s.values(e);else{s.elm=r;var i=t.templater.get(s,e);s.values(i)}}(e,r,n)}}},"./src/pagination.js":function(t,e,r){var n=r("./src/utils/classes.js"),s=r("./src/utils/events.js"),i=r("./src/index.js");t.exports=function(t){var e=!1,r=function(r,s){if(t.page<1)return t.listContainer.style.display="none",void(e=!0);e&&(t.listContainer.style.display="block");var i,o=t.matchingItems.length,l=t.i,u=t.page,c=Math.ceil(o/u),f=Math.ceil(l/u),h=s.innerWindow||2,d=s.left||s.outerWindow||0,v=s.right||s.outerWindow||0;v=c-v,r.clear();for(var g=1;g<=c;g++){var m=f===g?"active":"";a.number(g,d,v,f,h)?(i=r.add({page:g,dotted:!1})[0],m&&n(i.elm).add(m),i.elm.firstChild.setAttribute("data-i",g),i.elm.firstChild.setAttribute("data-page",u)):a.dotted(r,g,d,v,f,h,r.size())&&(i=r.add({page:"...",dotted:!0})[0],n(i.elm).add("disabled"))}},a={number:function(t,e,r,n,s){return this.left(t,e)||this.right(t,r)||this.innerWindow(t,n,s)},left:function(t,e){return t<=e},right:function(t,e){return t>e},innerWindow:function(t,e,r){return t>=e-r&&t<=e+r},dotted:function(t,e,r,n,s,i,a){return this.dottedLeft(t,e,r,n,s,i)||this.dottedRight(t,e,r,n,s,i,a)},dottedLeft:function(t,e,r,n,s,i){return e==r+1&&!this.innerWindow(e,s,i)&&!this.right(e,n)},dottedRight:function(t,e,r,n,s,i,a){return!t.items[a-1].values().dotted&&(e==n&&!this.innerWindow(e,s,i)&&!this.right(e,n))}};return function(e){var n=new i(t.listContainer.id,{listClass:e.paginationClass||"pagination",item:e.item||"<li><a class='page' href='#'></a></li>",valueNames:["page","dotted"],searchClass:"pagination-search-that-is-not-supposed-to-exist",sortClass:"pagination-sort-that-is-not-supposed-to-exist"});s.bind(n.listContainer,"click",(function(e){var r=e.target||e.srcElement,n=t.utils.getAttribute(r,"data-page"),s=t.utils.getAttribute(r,"data-i");s&&t.show((s-1)*n+1,n)})),t.on("updated",(function(){r(n,e)})),r(n,e)}}},"./src/parse.js":function(t,e,r){t.exports=function(t){var e=r("./src/item.js")(t),n=function(r,n){for(var s=0,i=r.length;s<i;s++)t.items.push(new e(n,r[s]))},s=function e(r,s){var i=r.splice(0,50);n(i,s),r.length>0?setTimeout((function(){e(r,s)}),1):(t.update(),t.trigger("parseComplete"))};return t.handlers.parseComplete=t.handlers.parseComplete||[],function(){var e=function(t){for(var e=t.childNodes,r=[],n=0,s=e.length;n<s;n++)void 0===e[n].data&&r.push(e[n]);return r}(t.list),r=t.valueNames;t.indexAsync?s(e,r):n(e,r)}}},"./src/search.js":function(t){t.exports=function(t){var e,r,n,s={resetList:function(){t.i=1,t.templater.clear(),n=void 0},setOptions:function(t){2==t.length&&t[1]instanceof Array?e=t[1]:2==t.length&&"function"==typeof t[1]?(e=void 0,n=t[1]):3==t.length?(e=t[1],n=t[2]):e=void 0},setColumns:function(){0!==t.items.length&&void 0===e&&(e=void 0===t.searchColumns?s.toArray(t.items[0].values()):t.searchColumns)},setSearchString:function(e){e=(e=t.utils.toString(e).toLowerCase()).replace(/[-[\]{}()*+?.,\\^$|#]/g,"\\$&"),r=e},toArray:function(t){var e=[];for(var r in t)e.push(r);return e}},i=function(){for(var n,s=[],i=r;null!==(n=i.match(/"([^"]+)"/));)s.push(n[1]),i=i.substring(0,n.index)+i.substring(n.index+n[0].length);(i=i.trim()).length&&(s=s.concat(i.split(/\s+/)));for(var a=0,o=t.items.length;a<o;a++){var l=t.items[a];if(l.found=!1,s.length){for(var u=0,c=s.length;u<c;u++){for(var f=!1,h=0,d=e.length;h<d;h++){var v=l.values(),g=e[h];if(v.hasOwnProperty(g)&&void 0!==v[g]&&null!==v[g])if(-1!==("string"!=typeof v[g]?v[g].toString():v[g]).toLowerCase().indexOf(s[u])){f=!0;break}}if(!f)break}l.found=f}}},a=function(){t.reset.search(),t.searched=!1},o=function(o){return t.trigger("searchStart"),s.resetList(),s.setSearchString(o),s.setOptions(arguments),s.setColumns(),""===r?a():(t.searched=!0,n?n(r,e):i()),t.update(),t.trigger("searchComplete"),t.visibleItems};return t.handlers.searchStart=t.handlers.searchStart||[],t.handlers.searchComplete=t.handlers.searchComplete||[],t.utils.events.bind(t.utils.getByClass(t.listContainer,t.searchClass),"keyup",t.utils.events.debounce((function(e){var r=e.target||e.srcElement;""===r.value&&!t.searched||o(r.value)}),t.searchDelay)),t.utils.events.bind(t.utils.getByClass(t.listContainer,t.searchClass),"input",(function(t){""===(t.target||t.srcElement).value&&o("")})),o}},"./src/sort.js":function(t){t.exports=function(t){var e={els:void 0,clear:function(){for(var r=0,n=e.els.length;r<n;r++)t.utils.classes(e.els[r]).remove("asc"),t.utils.classes(e.els[r]).remove("desc")},getOrder:function(e){var r=t.utils.getAttribute(e,"data-order");return"asc"==r||"desc"==r?r:t.utils.classes(e).has("desc")?"asc":t.utils.classes(e).has("asc")?"desc":"asc"},getInSensitive:function(e,r){var n=t.utils.getAttribute(e,"data-insensitive");r.insensitive="false"!==n},setOrder:function(r){for(var n=0,s=e.els.length;n<s;n++){var i=e.els[n];if(t.utils.getAttribute(i,"data-sort")===r.valueName){var a=t.utils.getAttribute(i,"data-order");"asc"==a||"desc"==a?a==r.order&&t.utils.classes(i).add(r.order):t.utils.classes(i).add(r.order)}}}},r=function(){t.trigger("sortStart");var r={},n=arguments[0].currentTarget||arguments[0].srcElement||void 0;n?(r.valueName=t.utils.getAttribute(n,"data-sort"),e.getInSensitive(n,r),r.order=e.getOrder(n)):((r=arguments[1]||r).valueName=arguments[0],r.order=r.order||"asc",r.insensitive=void 0===r.insensitive||r.insensitive),e.clear(),e.setOrder(r);var s,i=r.sortFunction||t.sortFunction||null,a="desc"===r.order?-1:1;s=i?function(t,e){return i(t,e,r)*a}:function(e,n){var s=t.utils.naturalSort;return s.alphabet=t.alphabet||r.alphabet||void 0,!s.alphabet&&r.insensitive&&(s=t.utils.naturalSort.caseInsensitive),s(e.values()[r.valueName],n.values()[r.valueName])*a},t.items.sort(s),t.update(),t.trigger("sortComplete")};return t.handlers.sortStart=t.handlers.sortStart||[],t.handlers.sortComplete=t.handlers.sortComplete||[],e.els=t.utils.getByClass(t.listContainer,t.sortClass),t.utils.events.bind(e.els,"click",r),t.on("searchStart",e.clear),t.on("filterStart",e.clear),r}},"./src/templater.js":function(t){var e=function(t){var e,r=this,n=function(e,r){var n=e.cloneNode(!0);n.removeAttribute("id");for(var s=0,i=r.length;s<i;s++){var a=void 0,o=r[s];if(o.data)for(var l=0,u=o.data.length;l<u;l++)n.setAttribute("data-"+o.data[l],"");else o.attr&&o.name?(a=t.utils.getByClass(n,o.name,!0))&&a.setAttribute(o.attr,""):(a=t.utils.getByClass(n,o,!0))&&(a.innerHTML="")}return n},s=function(){for(var e=t.list.childNodes,r=0,n=e.length;r<n;r++)if(void 0===e[r].data)return e[r].cloneNode(!0)},i=function(t){if("string"==typeof t){if(/<tr[\s>]/g.exec(t)){var e=document.createElement("tbody");return e.innerHTML=t,e.firstElementChild}if(-1!==t.indexOf("<")){var r=document.createElement("div");return r.innerHTML=t,r.firstElementChild}}},a=function(e,r,n){var s=void 0,i=function(e){for(var r=0,n=t.valueNames.length;r<n;r++){var s=t.valueNames[r];if(s.data){for(var i=s.data,a=0,o=i.length;a<o;a++)if(i[a]===e)return{data:e}}else{if(s.attr&&s.name&&s.name==e)return s;if(s===e)return e}}}(r);i&&(i.data?e.elm.setAttribute("data-"+i.data,n):i.attr&&i.name?(s=t.utils.getByClass(e.elm,i.name,!0))&&s.setAttribute(i.attr,n):(s=t.utils.getByClass(e.elm,i,!0))&&(s.innerHTML=n))};this.get=function(e,n){r.create(e);for(var s={},i=0,a=n.length;i<a;i++){var o=void 0,l=n[i];if(l.data)for(var u=0,c=l.data.length;u<c;u++)s[l.data[u]]=t.utils.getAttribute(e.elm,"data-"+l.data[u]);else l.attr&&l.name?(o=t.utils.getByClass(e.elm,l.name,!0),s[l.name]=o?t.utils.getAttribute(o,l.attr):""):(o=t.utils.getByClass(e.elm,l,!0),s[l]=o?o.innerHTML:"")}return s},this.set=function(t,e){if(!r.create(t))for(var n in e)e.hasOwnProperty(n)&&a(t,n,e[n])},this.create=function(t){return void 0===t.elm&&(t.elm=e(t.values()),r.set(t,t.values()),!0)},this.remove=function(e){e.elm.parentNode===t.list&&t.list.removeChild(e.elm)},this.show=function(e){r.create(e),t.list.appendChild(e.elm)},this.hide=function(e){void 0!==e.elm&&e.elm.parentNode===t.list&&t.list.removeChild(e.elm)},this.clear=function(){if(t.list.hasChildNodes())for(;t.list.childNodes.length>=1;)t.list.removeChild(t.list.firstChild)},function(){var r;if("function"!=typeof t.item){if(!(r="string"==typeof t.item?-1===t.item.indexOf("<")?document.getElementById(t.item):i(t.item):s()))throw new Error("The list needs to have at least one item on init otherwise you'll have to add a template.");r=n(r,t.valueNames),e=function(){return r.cloneNode(!0)}}else e=function(e){var r=t.item(e);return i(r)}}()};t.exports=function(t){return new e(t)}},"./src/utils/classes.js":function(t,e,r){var n=r("./src/utils/index-of.js"),s=/\s+/;Object.prototype.toString;function i(t){if(!t||!t.nodeType)throw new Error("A DOM element reference is required");this.el=t,this.list=t.classList}t.exports=function(t){return new i(t)},i.prototype.add=function(t){if(this.list)return this.list.add(t),this;var e=this.array();return~n(e,t)||e.push(t),this.el.className=e.join(" "),this},i.prototype.remove=function(t){if(this.list)return this.list.remove(t),this;var e=this.array(),r=n(e,t);return~r&&e.splice(r,1),this.el.className=e.join(" "),this},i.prototype.toggle=function(t,e){return this.list?(void 0!==e?e!==this.list.toggle(t,e)&&this.list.toggle(t):this.list.toggle(t),this):(void 0!==e?e?this.add(t):this.remove(t):this.has(t)?this.remove(t):this.add(t),this)},i.prototype.array=function(){var t=(this.el.getAttribute("class")||"").replace(/^\s+|\s+$/g,"").split(s);return""===t[0]&&t.shift(),t},i.prototype.has=i.prototype.contains=function(t){return this.list?this.list.contains(t):!!~n(this.array(),t)}},"./src/utils/events.js":function(t,e,r){var n=window.addEventListener?"addEventListener":"attachEvent",s=window.removeEventListener?"removeEventListener":"detachEvent",i="addEventListener"!==n?"on":"",a=r("./src/utils/to-array.js");e.bind=function(t,e,r,s){for(var o=0,l=(t=a(t)).length;o<l;o++)t[o][n](i+e,r,s||!1)},e.unbind=function(t,e,r,n){for(var o=0,l=(t=a(t)).length;o<l;o++)t[o][s](i+e,r,n||!1)},e.debounce=function(t,e,r){var n;return e?function(){var 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p=Math.max(1,a-h+1),y=Math.min(a+h,t.length)+e.length,C=Array(y+2);C[y+1]=(1<<m)-1;for(var b=y;b>=p;b--){var j=o[t.charAt(b-1)];if(C[b]=0===m?(C[b+1]<<1|1)&j:(C[b+1]<<1|1)&j|(v[b+1]|v[b])<<1|1|v[b+1],C[b]&d){var x=l(m,b-1);if(x<=u){if(u=x,!((c=b-1)>a))break;p=Math.max(1,2*a-c)}}}if(l(m+1,a)>u)break;v=C}return!(c<0)}},"./src/utils/get-attribute.js":function(t){t.exports=function(t,e){var r=t.getAttribute&&t.getAttribute(e)||null;if(!r)for(var n=t.attributes,s=n.length,i=0;i<s;i++)void 0!==n[i]&&n[i].nodeName===e&&(r=n[i].nodeValue);return r}},"./src/utils/get-by-class.js":function(t){t.exports=function(t,e,r,n){return(n=n||{}).test&&n.getElementsByClassName||!n.test&&document.getElementsByClassName?function(t,e,r){return r?t.getElementsByClassName(e)[0]:t.getElementsByClassName(e)}(t,e,r):n.test&&n.querySelector||!n.test&&document.querySelector?function(t,e,r){return e="."+e,r?t.querySelector(e):t.querySelectorAll(e)}(t,e,r):function(t,e,r){for(var 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n=r("./node_modules/string-natural-compare/natural-compare.js"),s=r("./src/utils/get-by-class.js"),i=r("./src/utils/extend.js"),a=r("./src/utils/index-of.js"),o=r("./src/utils/events.js"),l=r("./src/utils/to-string.js"),u=r("./src/utils/classes.js"),c=r("./src/utils/get-attribute.js"),f=r("./src/utils/to-array.js");t.exports=function(t,e,h){var d,v=this,g=r("./src/item.js")(v),m=r("./src/add-async.js")(v),p=r("./src/pagination.js")(v);d={start:function(){v.listClass="list",v.searchClass="search",v.sortClass="sort",v.page=1e4,v.i=1,v.items=[],v.visibleItems=[],v.matchingItems=[],v.searched=!1,v.filtered=!1,v.searchColumns=void 0,v.searchDelay=0,v.handlers={updated:[]},v.valueNames=[],v.utils={getByClass:s,extend:i,indexOf:a,events:o,toString:l,naturalSort:n,classes:u,getAttribute:c,toArray:f},v.utils.extend(v,e),v.listContainer="string"==typeof 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m=f===g?"active":"";a.number(g,d,v,f,h)?(i=r.add({page:g,dotted:!1})[0],m&&n(i.elm).add(m),i.elm.firstChild.setAttribute("data-i",g),i.elm.firstChild.setAttribute("data-page",u)):a.dotted(r,g,d,v,f,h,r.size())&&(i=r.add({page:"...",dotted:!0})[0],n(i.elm).add("disabled"))}},a={number:function(t,e,r,n,s){return this.left(t,e)||this.right(t,r)||this.innerWindow(t,n,s)},left:function(t,e){return t<=e},right:function(t,e){return t>e},innerWindow:function(t,e,r){return t>=e-r&&t<=e+r},dotted:function(t,e,r,n,s,i,a){return this.dottedLeft(t,e,r,n,s,i)||this.dottedRight(t,e,r,n,s,i,a)},dottedLeft:function(t,e,r,n,s,i){return e==r+1&&!this.innerWindow(e,s,i)&&!this.right(e,n)},dottedRight:function(t,e,r,n,s,i,a){return!t.items[a-1].values().dotted&&(e==n&&!this.innerWindow(e,s,i)&&!this.right(e,n))}};return function(e){var n=new i(t.listContainer.id,{listClass:e.paginationClass||"pagination",item:e.item||"<li><a class='page' 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e,r,n,s={resetList:function(){t.i=1,t.templater.clear(),n=void 0},setOptions:function(t){2==t.length&&t[1]instanceof Array?e=t[1]:2==t.length&&"function"==typeof t[1]?(e=void 0,n=t[1]):3==t.length?(e=t[1],n=t[2]):e=void 0},setColumns:function(){0!==t.items.length&&void 0===e&&(e=void 0===t.searchColumns?s.toArray(t.items[0].values()):t.searchColumns)},setSearchString:function(e){r=e},toArray:function(t){var e=[];for(var r in t)e.push(r);return e}},i=function(){for(var n,s=[],i=r;null!==(n=i.match(/"([^"]+)"/));)s.push(n[1]),i=i.substring(0,n.index)+i.substring(n.index+n[0].length);(i=i.trim()).length&&(s=s.concat(i.split(/\s+/)));for(var a=0,o=t.items.length;a<o;a++){var l=t.items[a];if(l.found=!1,s.length){for(var u=0,c=s.length;u<c;u++){for(var f=!1,h=0,d=e.length;h<d;h++){var v=l.values(),g=e[h];if(v.hasOwnProperty(g)&&void 0!==v[g]&&null!==v[g])if(-1!==("string"!=typeof 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diff --git a/docs/tutorials.xml b/docs/tutorials.xml
index 50adef5..4c27dd1 100644
--- a/docs/tutorials.xml
+++ b/docs/tutorials.xml
@@ -9,8 +9,8 @@
 <link>https://wjschne.github.io/tutorials.html</link>
 <atom:link href="https://wjschne.github.io/tutorials.xml" rel="self" type="application/rss+xml"/>
 <description>{{&lt; meta description-meta &gt;}}</description>
-<generator>quarto-1.4.548</generator>
-<lastBuildDate>Wed, 24 Jan 2024 03:19:31 GMT</lastBuildDate>
+<generator>quarto-1.5.9</generator>
+<lastBuildDate>Thu, 29 Feb 2024 18:24:37 GMT</lastBuildDate>
 <item>
   <title>ANOVA in R</title>
   <dc:creator>W. Joel Schneider</dc:creator>
@@ -25,7 +25,10 @@
 <div class="cell">
 <div class="sourceCode cell-code" id="cb1" style="background: #f1f3f5;"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb1-1"><span class="fu" style="color: #4758AB;
 background-color: null;
-font-style: inherit;">library</span>(tidyverse)</span></code></pre></div>
+font-style: inherit;">library</span>(tidyverse)</span>
+<span id="cb1-2"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">library</span>(easystats)</span></code></pre></div>
 </div>
 </section>
 <section id="import-data-and-create-labelled-factors" class="level1">
@@ -169,11 +172,27 @@ font-style: inherit;">~</span> Attachment, <span class="at" style="color: #65742
 background-color: null;
 font-style: inherit;">data =</span> d)</span></code></pre></div>
 </div>
+<section id="diagnostic-plots" class="level2">
+<h2 class="anchored" data-anchor-id="diagnostic-plots">Diagnostic Plots</h2>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb6" style="background: #f1f3f5;"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb6-1"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">check_model</span>(fit)</span></code></pre></div>
+<div class="cell-output-display">
+<div>
+<figure class="figure">
+<p><img src="https://wjschne.github.io/tutorials/ANOVA/ANOVA_files/figure-html/assumptions-1.png" class="img-fluid figure-img" width="672"></p>
+</figure>
+</div>
+</div>
+</div>
+<p>None of the diagnostic checks raise any alarms. Specifically, the residuals are reasonably normal and have consistent variance acrross groups (i.e., the homogeneeity of variance assumption is reasonable).</p>
+</section>
 <section id="summary" class="level2">
 <h2 class="anchored" data-anchor-id="summary">Summary</h2>
-<p>The base R function <code>summary</code> gives us most of what we might want.</p>
+<p>The base R function <code>summary</code> gives us most of what we might want to know.</p>
 <div class="cell">
-<div class="sourceCode cell-code" id="cb6" style="background: #f1f3f5;"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb6-1"><span class="fu" style="color: #4758AB;
+<div class="sourceCode cell-code" id="cb7" style="background: #f1f3f5;"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb7-1"><span class="fu" style="color: #4758AB;
 background-color: null;
 font-style: inherit;">summary</span>(fit)</span></code></pre></div>
 <div class="cell-output cell-output-stdout">
@@ -184,163 +203,53 @@ Residuals   297  32517     109
 Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1</code></pre>
 </div>
 </div>
-</section>
-<section id="other-summary-functions" class="level2">
-<h2 class="anchored" data-anchor-id="other-summary-functions">Other Summary Functions</h2>
-<p>If you want other summary functions, there are many options. Here are a few:</p>
+<p>We can see that overall model is significant, meaning that at least two groups have reliably different means. At this point we do not know which means differ, but from the plot we have a good guess that the Ambivalent group scores higher on the Anxiety Questionnaire than the other two groups.</p>
+<p>A similar, but tidier display can be found with the <code>parameters</code> function:</p>
 <div class="cell">
-<div class="sourceCode cell-code" id="cb8" style="background: #f1f3f5;"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb8-1">broom<span class="sc" style="color: #5E5E5E;
-background-color: null;
-font-style: inherit;">::</span><span class="fu" style="color: #4758AB;
-background-color: null;
-font-style: inherit;">glance</span>(fit)</span></code></pre></div>
-<div class="cell-output-display">
-<div class="kable-table">
-<table class="table table-sm table-striped small">
-<thead>
-<tr class="header">
-<th style="text-align: right;">logLik</th>
-<th style="text-align: right;">AIC</th>
-<th style="text-align: right;">BIC</th>
-<th style="text-align: right;">deviance</th>
-<th style="text-align: right;">nobs</th>
-<th style="text-align: right;">r.squared</th>
-</tr>
-</thead>
-<tbody>
-<tr class="odd">
-<td style="text-align: right;">-1129</td>
-<td style="text-align: right;">2265</td>
-<td style="text-align: right;">2280</td>
-<td style="text-align: right;">32517</td>
-<td style="text-align: right;">300</td>
-<td style="text-align: right;">0.39</td>
-</tr>
-</tbody>
-</table>
-</div>
-</div>
-<div class="sourceCode cell-code" id="cb9" style="background: #f1f3f5;"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb9-1">broom<span class="sc" style="color: #5E5E5E;
-background-color: null;
-font-style: inherit;">::</span><span class="fu" style="color: #4758AB;
+<div class="sourceCode cell-code" id="cb9" style="background: #f1f3f5;"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb9-1"><span class="fu" style="color: #4758AB;
 background-color: null;
-font-style: inherit;">tidy</span>(fit)</span></code></pre></div>
+font-style: inherit;">parameters</span>(fit)</span></code></pre></div>
 <div class="cell-output-display">
 <div class="kable-table">
 <table class="table table-sm table-striped small">
 <thead>
 <tr class="header">
-<th style="text-align: left;">term</th>
+<th style="text-align: left;">Parameter</th>
+<th style="text-align: right;">Sum_Squares</th>
 <th style="text-align: right;">df</th>
-<th style="text-align: right;">sumsq</th>
-<th style="text-align: right;">meansq</th>
-<th style="text-align: right;">statistic</th>
-<th style="text-align: right;">p.value</th>
+<th style="text-align: right;">Mean_Square</th>
+<th style="text-align: right;">F</th>
+<th style="text-align: right;">p</th>
 </tr>
 </thead>
 <tbody>
 <tr class="odd">
 <td style="text-align: left;">Attachment</td>
-<td style="text-align: right;">2</td>
 <td style="text-align: right;">20359</td>
-<td style="text-align: right;">10179</td>
-<td style="text-align: right;">93</td>
-<td style="text-align: right;">0</td>
-</tr>
-<tr class="even">
-<td style="text-align: left;">Residuals</td>
-<td style="text-align: right;">297</td>
-<td style="text-align: right;">32517</td>
-<td style="text-align: right;">109</td>
-<td style="text-align: right;">NA</td>
-<td style="text-align: right;">NA</td>
-</tr>
-</tbody>
-</table>
-</div>
-</div>
-<div class="sourceCode cell-code" id="cb10" style="background: #f1f3f5;"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb10-1">sjstats<span class="sc" style="color: #5E5E5E;
-background-color: null;
-font-style: inherit;">::</span><span class="fu" style="color: #4758AB;
-background-color: null;
-font-style: inherit;">anova_stats</span>(fit)</span></code></pre></div>
-<div class="cell-output-display">
-<div class="kable-table">
-<table class="table table-sm table-striped small">
-<colgroup>
-<col style="width: 8%">
-<col style="width: 8%">
-<col style="width: 3%">
-<col style="width: 4%">
-<col style="width: 5%">
-<col style="width: 7%">
-<col style="width: 6%">
-<col style="width: 4%">
-<col style="width: 11%">
-<col style="width: 6%">
-<col style="width: 12%">
-<col style="width: 7%">
-<col style="width: 7%">
-<col style="width: 4%">
-</colgroup>
-<thead>
-<tr class="header">
-<th style="text-align: left;"></th>
-<th style="text-align: left;">term</th>
-<th style="text-align: right;">df</th>
-<th style="text-align: right;">sumsq</th>
-<th style="text-align: right;">meansq</th>
-<th style="text-align: right;">statistic</th>
-<th style="text-align: right;">p.value</th>
-<th style="text-align: right;">etasq</th>
-<th style="text-align: right;">partial.etasq</th>
-<th style="text-align: right;">omegasq</th>
-<th style="text-align: right;">partial.omegasq</th>
-<th style="text-align: right;">epsilonsq</th>
-<th style="text-align: right;">cohens.f</th>
-<th style="text-align: right;">power</th>
-</tr>
-</thead>
-<tbody>
-<tr class="odd">
-<td style="text-align: left;">Attachment</td>
-<td style="text-align: left;">Attachment</td>
 <td style="text-align: right;">2</td>
-<td style="text-align: right;">20359</td>
 <td style="text-align: right;">10179</td>
 <td style="text-align: right;">93</td>
 <td style="text-align: right;">0</td>
-<td style="text-align: right;">0.38</td>
-<td style="text-align: right;">0.38</td>
-<td style="text-align: right;">0.38</td>
-<td style="text-align: right;">0.38</td>
-<td style="text-align: right;">0.38</td>
-<td style="text-align: right;">0.79</td>
-<td style="text-align: right;">1</td>
 </tr>
 <tr class="even">
-<td style="text-align: left;">…2</td>
 <td style="text-align: left;">Residuals</td>
-<td style="text-align: right;">297</td>
 <td style="text-align: right;">32517</td>
+<td style="text-align: right;">297</td>
 <td style="text-align: right;">109</td>
 <td style="text-align: right;">NA</td>
 <td style="text-align: right;">NA</td>
-<td style="text-align: right;">NA</td>
-<td style="text-align: right;">NA</td>
-<td style="text-align: right;">NA</td>
-<td style="text-align: right;">NA</td>
-<td style="text-align: right;">NA</td>
-<td style="text-align: right;">NA</td>
-<td style="text-align: right;">NA</td>
 </tr>
 </tbody>
 </table>
 </div>
 </div>
-<div class="sourceCode cell-code" id="cb11" style="background: #f1f3f5;"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb11-1">performance<span class="sc" style="color: #5E5E5E;
-background-color: null;
-font-style: inherit;">::</span><span class="fu" style="color: #4758AB;
+</div>
+</section>
+<section id="other-summary-functions" class="level2">
+<h2 class="anchored" data-anchor-id="other-summary-functions">Other Summary Functions</h2>
+<p>The overall “fit” or “performance” of the model:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb10" style="background: #f1f3f5;"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb10-1"><span class="fu" style="color: #4758AB;
 background-color: null;
 font-style: inherit;">performance</span>(fit)</span></code></pre></div>
 <div class="cell-output-display">
@@ -371,117 +280,2161 @@ font-style: inherit;">performance</span>(fit)</span></code></pre></div>
 </table>
 </div>
 </div>
-<div class="sourceCode cell-code" id="cb12" style="background: #f1f3f5;"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb12-1">parameters<span class="sc" style="color: #5E5E5E;
+</div>
+<ul>
+<li><code>AIC</code>: <a href="https://en.wikipedia.org/wiki/Akaike_information_criterion">Akaike’s Information Criterion</a></li>
+<li><code>AICc</code>: AIC with a correction for small sample sizes</li>
+<li><code>BIC</code>: <a href="https://en.wikipedia.org/wiki/Bayesian_information_criterion">Bayesian Information Criterion</a></li>
+<li><code>R2</code>: R-squared: <a href="https://en.wikipedia.org/wiki/Coefficient_of_determination">Coefficient of Determination</a></li>
+<li><code>R2_adj</code>: <a href="https://en.wikipedia.org/wiki/Coefficient_of_determination#Adjusted_R2">Adjusted r-squared</a></li>
+<li><code>RMSE</code>: Root mean squared error <img src="https://latex.codecogs.com/png.latex?=%5Csqrt%7B%5Cfrac%7B%5Csum_%7Bi%20=%201%7D%5E%7Bn%7D%7Be_i%7D%7D%7Bn%7D%7D"></li>
+<li><code>SIGMA</code>: Residual standard deviation (The SD of the residuals <img src="https://latex.codecogs.com/png.latex?=%5Csqrt%7B%5Cfrac%7B%5Csum_%7Bi%20=%201%7D%5E%7Bn%7D%7Be_i%7D%7D%7Bn-k-1%7D%7D">)</li>
+</ul>
+</section>
+<section id="automated-interpretation" class="level2">
+<h2 class="anchored" data-anchor-id="automated-interpretation">Automated Interpretation</h2>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb11" style="background: #f1f3f5;"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb11-1"><span class="fu" style="color: #4758AB;
 background-color: null;
-font-style: inherit;">::</span><span class="fu" style="color: #4758AB;
+font-style: inherit;">report</span>(fit)</span></code></pre></div>
+<div class="cell-output cell-output-stdout">
+<pre><code>The ANOVA (formula: Anxiety_Questionnaire ~ Attachment) suggests that:
+
+  - The main effect of Attachment is statistically significant and large (F(2,
+297) = 92.98, p &lt; .001; Eta2 = 0.39, 95% CI [0.32, 1.00])
+
+Effect sizes were labelled following Field's (2013) recommendations.</code></pre>
+</div>
+</div>
+</section>
+<section id="post-hoc-tests" class="level2">
+<h2 class="anchored" data-anchor-id="post-hoc-tests">Post-hoc Tests</h2>
+<p>We would like to compare all means but control for family-wise error.</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb13" style="background: #f1f3f5;"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb13-1"><span class="fu" style="color: #4758AB;
 background-color: null;
-font-style: inherit;">parameters</span>(fit)</span></code></pre></div>
+font-style: inherit;">estimate_contrasts</span>(fit)</span></code></pre></div>
+<div class="cell-output cell-output-stderr">
+<pre><code>No variable was specified for contrast estimation. Selecting `contrast = "Attachment"`.</code></pre>
+</div>
 <div class="cell-output-display">
 <div class="kable-table">
 <table class="table table-sm table-striped small">
+<colgroup>
+<col style="width: 13%">
+<col style="width: 16%">
+<col style="width: 16%">
+<col style="width: 10%">
+<col style="width: 12%">
+<col style="width: 6%">
+<col style="width: 6%">
+<col style="width: 9%">
+<col style="width: 7%">
+</colgroup>
 <thead>
 <tr class="header">
-<th style="text-align: left;">Parameter</th>
-<th style="text-align: right;">Sum_Squares</th>
+<th style="text-align: left;">Level1</th>
+<th style="text-align: left;">Level2</th>
+<th style="text-align: right;">Difference</th>
+<th style="text-align: right;">CI_low</th>
+<th style="text-align: right;">CI_high</th>
+<th style="text-align: right;">SE</th>
 <th style="text-align: right;">df</th>
-<th style="text-align: right;">Mean_Square</th>
-<th style="text-align: right;">F</th>
+<th style="text-align: right;">t</th>
 <th style="text-align: right;">p</th>
 </tr>
 </thead>
 <tbody>
 <tr class="odd">
-<td style="text-align: left;">Attachment</td>
-<td style="text-align: right;">20359</td>
-<td style="text-align: right;">2</td>
-<td style="text-align: right;">10179</td>
-<td style="text-align: right;">93</td>
-<td style="text-align: right;">0</td>
+<td style="text-align: left;">Avoidant</td>
+<td style="text-align: left;">Ambivalent</td>
+<td style="text-align: right;">-18.4</td>
+<td style="text-align: right;">-22.1</td>
+<td style="text-align: right;">-14.8</td>
+<td style="text-align: right;">1.5</td>
+<td style="text-align: right;">297</td>
+<td style="text-align: right;">-12.1</td>
+<td style="text-align: right;">0.00</td>
 </tr>
 <tr class="even">
-<td style="text-align: left;">Residuals</td>
-<td style="text-align: right;">32517</td>
+<td style="text-align: left;">Secure</td>
+<td style="text-align: left;">Ambivalent</td>
+<td style="text-align: right;">-16.2</td>
+<td style="text-align: right;">-19.6</td>
+<td style="text-align: right;">-12.7</td>
+<td style="text-align: right;">1.4</td>
 <td style="text-align: right;">297</td>
-<td style="text-align: right;">109</td>
-<td style="text-align: right;">NA</td>
-<td style="text-align: right;">NA</td>
+<td style="text-align: right;">-11.3</td>
+<td style="text-align: right;">0.00</td>
+</tr>
+<tr class="odd">
+<td style="text-align: left;">Secure</td>
+<td style="text-align: left;">Avoidant</td>
+<td style="text-align: right;">2.3</td>
+<td style="text-align: right;">-1.4</td>
+<td style="text-align: right;">5.9</td>
+<td style="text-align: right;">1.5</td>
+<td style="text-align: right;">297</td>
+<td style="text-align: right;">1.5</td>
+<td style="text-align: right;">0.13</td>
 </tr>
 </tbody>
 </table>
 </div>
 </div>
-<div class="sourceCode cell-code" id="cb13" style="background: #f1f3f5;"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb13-1">report<span class="sc" style="color: #5E5E5E;
+<div class="sourceCode cell-code" id="cb15" style="background: #f1f3f5;"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb15-1"><span class="fu" style="color: #4758AB;
 background-color: null;
-font-style: inherit;">::</span><span class="fu" style="color: #4758AB;
+font-style: inherit;">estimate_contrasts</span>(fit) <span class="sc" style="color: #5E5E5E;
 background-color: null;
-font-style: inherit;">report</span>(fit)</span></code></pre></div>
+font-style: inherit;">|&gt;</span> </span>
+<span id="cb15-2">  <span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">report</span>()</span></code></pre></div>
+<div class="cell-output cell-output-stderr">
+<pre><code>No variable was specified for contrast estimation. Selecting `contrast = "Attachment"`.</code></pre>
+</div>
 <div class="cell-output cell-output-stdout">
-<pre><code>The ANOVA (formula: Anxiety_Questionnaire ~ Attachment) suggests that:
+<pre><code>The marginal contrasts analysis suggests the following. The difference between
+Avoidant and Ambivalent is negative and statistically significant (difference =
+-18.44, 95% CI [-22.12, -14.77], t(297) = -12.08, p &lt; .001). The difference
+between Secure and Ambivalent is negative and statistically significant
+(difference = -16.18, 95% CI [-19.62, -12.74], t(297) = -11.33, p &lt; .001). The
+difference between Secure and Avoidant is positive and statistically
+non-significant (difference = 2.27, 95% CI [ -1.37, 5.90], t(297) = 1.50, p =
+0.135)</code></pre>
+</div>
+</div>
 
-  - The main effect of Attachment is statistically significant and large (F(2,
-297) = 92.98, p &lt; .001; Eta2 = 0.39, 95% CI [0.32, 1.00])
 
-Effect sizes were labelled following Field's (2013) recommendations.</code></pre>
+</section>
+</section>
+
+ ]]></description>
+  <guid>https://wjschne.github.io/tutorials/ANOVA/ANOVA.html</guid>
+  <pubDate>Thu, 29 Feb 2024 18:24:37 GMT</pubDate>
+</item>
+<item>
+  <title>Regression in R</title>
+  <dc:creator>W. Joel Schneider</dc:creator>
+  <link>https://wjschne.github.io/tutorials/Regression/</link>
+  <description><![CDATA[ 
+
+
+
+
+<p>Here we are going to use a small data set to predict people’s height using the height of their parents.</p>
+<section id="loading-packages" class="level1 page-columns page-full">
+<h1>Loading packages</h1>
+
+<div class="no-row-height column-margin column-container"><div class="callout callout-style-default callout-note callout-titled" title="R Package">
+<div class="callout-header d-flex align-content-center">
+<div class="callout-icon-container">
+<i class="callout-icon"></i>
 </div>
+<div class="callout-title-container flex-fill">
+R Package
 </div>
-</section>
-<section id="diagnostic-plots" class="level2">
-<h2 class="anchored" data-anchor-id="diagnostic-plots">Diagnostic Plots</h2>
+</div>
+<div class="callout-body-container callout-body">
+<p>A <strong>R package</strong> is software that extends the capabilities of R. To use a package, you must first install it on your machine. You only need to install it once. By default, the package tab is in the lower right pane in RStudio. It has two buttons, one for installing new packages and one for updating packages you have previously installed.</p>
+<div id="fig-packages" class="quarto-figure quarto-figure-center quarto-float anchored">
+<figure class="quarto-float quarto-float-fig figure">
+<div aria-describedby="fig-packages-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
+<img src="https://wjschne.github.io/tutorials/Regression/updatepackages.png" class="img-fluid figure-img">
+</div>
+<figcaption class="quarto-float-caption-bottom quarto-float-caption quarto-float-fig" id="fig-packages-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
+Figure&nbsp;1: RStudio’s “Install” and “Update” buttons in the “Package” tab in the lower right pane
+</figcaption>
+</figure>
+</div>
+</div>
+</div></div><p>First we load packages to enhance R’s abilities. Some packages are designed to work with several other packages as a system. The <a href="https://www.tidyverse.org/">tidyverse package</a> is a “meta-package” that installs and loads a coherent set of packages designed to help you import, manipulate, visualize, and interpret data. If you do not have a recent version of tidyverse already installed, you can install it with this code:</p>
 <div class="cell">
-<div class="sourceCode cell-code" id="cb15" style="background: #f1f3f5;"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb15-1">performance<span class="sc" style="color: #5E5E5E;
+<div class="sourceCode cell-code" id="cb1" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb1-1"><span class="fu" style="color: #4758AB;
 background-color: null;
-font-style: inherit;">::</span><span class="fu" style="color: #4758AB;
+font-style: inherit;">install.packages</span>(<span class="st" style="color: #20794D;
 background-color: null;
-font-style: inherit;">check_model</span>(fit)</span></code></pre></div>
-<div class="cell-output-display">
-<div>
-<figure class="figure">
-<p><img src="https://wjschne.github.io/tutorials/ANOVA/ANOVA_files/figure-html/assumptions-1.png" class="img-fluid figure-img" width="672"></p>
-</figure>
+font-style: inherit;">"tidyverse"</span>)</span></code></pre></div>
 </div>
+<div class="callout callout-style-default callout-tip callout-titled">
+<div class="callout-header d-flex align-content-center">
+<div class="callout-icon-container">
+<i class="callout-icon"></i>
 </div>
+<div class="callout-title-container flex-fill">
+Tip
 </div>
-</section>
-<section id="post-hoc-tests" class="level2">
-<h2 class="anchored" data-anchor-id="post-hoc-tests">Post-hoc Tests</h2>
-<p>Tukey Honestly Significant Differences</p>
+</div>
+<div class="callout-body-container callout-body">
+<p>You only have to install the tidyverse once. Thereafter, from time to time, you can let RStudio’s “Update” button in the “Packages” pane help you keep your packages current.</p>
+</div>
+</div>
+<p>When you “load” the tidyvert</p>
+<p>The <code>tidyverse</code> allows us to do many things, including plotting with <code>ggplot2</code>. The <code>haven</code> package imports SPSS data. The <code>broom</code> package tidies up our regression results with the <code>tidy</code>, <code>glance</code>, and <code>augment</code> functions.</p>
+<p>If any of these packages are not installed, click the <strong>Packages</strong> tab, click the <strong>Install</strong> button, enter the names of the packages separated by commas, and click <strong>Install</strong>.</p>
+<p>To load the packages, run this code:</p>
 <div class="cell">
-<div class="sourceCode cell-code" id="cb16" style="background: #f1f3f5;"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb16-1"><span class="fu" style="color: #4758AB;
+<div class="sourceCode cell-code" id="cb2" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb2-1"><span class="fu" style="color: #4758AB;
 background-color: null;
-font-style: inherit;">TukeyHSD</span>(fit)</span></code></pre></div>
-<div class="cell-output cell-output-stdout">
-<pre><code>  Tukey multiple comparisons of means
-    95% family-wise confidence level
-
-Fit: aov(formula = Anxiety_Questionnaire ~ Attachment, data = d)
+font-style: inherit;">library</span>(tidyverse) <span class="co" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;"># Loads primary packages for data wrangling and plotting</span></span>
+<span id="cb2-2"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">library</span>(haven) <span class="co" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;"># Imports data saved in SPSS, SAS, or Stata format</span></span>
+<span id="cb2-3"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">library</span>(broom) <span class="co" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;"># Displays model information</span></span>
+<span id="cb2-4"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">library</span>(performance) <span class="co" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;"># Displays model information </span></span>
+<span id="cb2-5"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">library</span>(parameters) <span class="co" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;"># Displays model parameter information</span></span>
+<span id="cb2-6"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">library</span>(report) <span class="co" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;"># Automated interpretations</span></span></code></pre></div>
+</div>
+</section>
+<section id="import-data" class="level1 page-columns page-full">
+<h1>Import data</h1>
 
-$Attachment
-                    diff  lwr  upr p adj
-Avoidant-Secure     -2.3 -5.8  1.3  0.29
-Ambivalent-Secure   16.2 12.8 19.5  0.00
-Ambivalent-Avoidant 18.4 14.8 22.0  0.00</code></pre>
+<div class="no-row-height column-margin column-container"><div class="">
+<p>If I use only one data set in an analysis, I call it <code>d</code>. If I need multiple data sets, I use a <code>d_</code> prefix to differentiate them. For example, if I have separate data sets for juniors and seniors, I might call them, <code>d_juniors</code> and <code>d_seniors</code>, respectively. This kind of consistency seems like extra work, but it pays off in making it easy for <em>future-you</em> to read and understand your own code.</p>
+</div></div><p>You can import the <code>height.sav</code> data set directly from my github repository for this course. There is no need to save the data anywhere. This code loads the data into variable <code>d</code> using the <a href="https://haven.tidyverse.org/reference/read_spss.html"><code>read_spss</code></a> function from the <a href="https://haven.tidyverse.org/">haven package</a>.</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb3" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb3-1">d <span class="ot" style="color: #003B4F;
+background-color: null;
+font-style: inherit;">&lt;-</span> <span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">read_spss</span>(<span class="st" style="color: #20794D;
+background-color: null;
+font-style: inherit;">"https://github.com/wjschne/EDUC5325/raw/master/height.sav"</span>)</span></code></pre></div>
 </div>
-<div class="sourceCode cell-code" id="cb18" style="background: #f1f3f5;"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb18-1"><span class="fu" style="color: #4758AB;
+</section>
+<section id="make-a-plot" class="level1">
+<h1>Make a plot</h1>
+<p>Use the <a href="https://ggplot2.tidyverse.org/reference/ggplot.html"><code>ggplot</code></a> function from the <a href="https://ggplot2.tidyverse.org/">ggplot2</a> package to make a scatterplot with points and a regression line:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb4" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb4-1"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">ggplot</span>(d, <span class="fu" style="color: #4758AB;
 background-color: null;
-font-style: inherit;">plot</span>(<span class="fu" style="color: #4758AB;
+font-style: inherit;">aes</span>(avgphgt,height)) <span class="sc" style="color: #5E5E5E;
 background-color: null;
-font-style: inherit;">TukeyHSD</span>(fit))</span></code></pre></div>
+font-style: inherit;">+</span></span>
+<span id="cb4-2">  <span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">geom_point</span>() <span class="sc" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;">+</span></span>
+<span id="cb4-3">  <span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">geom_smooth</span>(<span class="at" style="color: #657422;
+background-color: null;
+font-style: inherit;">method =</span> <span class="st" style="color: #20794D;
+background-color: null;
+font-style: inherit;">"lm"</span>)</span></code></pre></div>
 <div class="cell-output-display">
 <div>
 <figure class="figure">
-<p><img src="https://wjschne.github.io/tutorials/ANOVA/ANOVA_files/figure-html/tukey-1.png" class="img-fluid figure-img" width="672"></p>
+<p><img src="https://wjschne.github.io/tutorials/Regression/index_files/figure-html/unnamed-chunk-3-1.svg" style="width:100.0%;height:100.0%" class="figure-img"></p>
 </figure>
 </div>
 </div>
 </div>
-
-
-</section>
+<p>Annotated, here is what the code does:</p>
+<div class="quarto-figure quarto-figure-center">
+<figure class="figure">
+<p><img src="https://wjschne.github.io/tutorials/Regression/ggplotcode.svg" class="img-fluid figure-img"></p>
+<figcaption>Annotated ggplot2 code</figcaption>
+</figure>
+</div>
 </section>
-
- ]]></description>
-  <guid>https://wjschne.github.io/tutorials/ANOVA/ANOVA.html</guid>
-  <pubDate>Wed, 24 Jan 2024 03:19:31 GMT</pubDate>
+<section id="save-the-plot" class="level1">
+<h1>Save the plot!</h1>
+<p>You can save to your hard drive a high-quality plot with the <a href="https://ggplot2.tidyverse.org/reference/ggsave.html"><code>ggsave</code></a> function. It will save whatever the last plot you created with <a href="https://ggplot2.tidyverse.org/reference/ggplot.html"><code>ggplot</code></a>. It will guess the file format from the file extension.</p>
+<p>Here I save a .pdf file of the plot:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb5" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb5-1"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">ggsave</span>(<span class="st" style="color: #20794D;
+background-color: null;
+font-style: inherit;">"my_plot.pdf"</span>)</span></code></pre></div>
+</div>
+<section id="vector-based-images" class="level2">
+<h2 class="anchored" data-anchor-id="vector-based-images">Vector-based Images</h2>
+<p>The .pdf format gives the best image quality but can only be viewed in a .pdf reader. The .svg format is almost as good and can be incorporated into webpages and Office documents. One downside of their near-perfect image quality is that .pdf and .svg image file sizes can become quite large.</p>
+</section>
+<section id="raster-images" class="level2">
+<h2 class="anchored" data-anchor-id="raster-images">Raster Images</h2>
+<p>The .png format gives good image quality and renders small file sizes.</p>
+<p>The primary use of the .gif format is to create animated plots. Otherwise stick with .png.</p>
+<p>Although the .jpg format is good for photos, it is terrible for plots—it often renders text and sharp corners with pixelated smudges.</p>
+</section>
+</section>
+<section id="creating-the-model" class="level1">
+<h1>Creating the model</h1>
+<p>To run regression models, use the <a href="https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/lm"><code>lm</code></a> function. The <code>lm</code> stands for “linear model.” The general pattern is <code>lm(Y~X, data = d)</code>, which means “<code>Y</code> is predicted by <code>X</code>, using the data set <code>d</code>.”</p>
+<p>Here we predict height from Average of Parent Height (<code>avgphgt</code>):</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb6" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb6-1">m1 <span class="ot" style="color: #003B4F;
+background-color: null;
+font-style: inherit;">&lt;-</span> <span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">lm</span>(height<span class="sc" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;">~</span>avgphgt, <span class="at" style="color: #657422;
+background-color: null;
+font-style: inherit;">data =</span> d)</span></code></pre></div>
+</div>
+<p>Notice that we did not get any results. Instead, we created the model fit object <code>m1</code>, which contains information about the model. There are a variety of functions we can use to extract information from <code>m1</code>.</p>
+</section>
+<section id="checking-assumptions" class="level1">
+<h1>Checking Assumptions</h1>
+<p>Regression assumes that the observations should independent, and the residuals should be normal and homoscedastic. The <a href="https://easystats.github.io/performance/index.html">performance package</a> has a great function for checking model assumptions: <a href="https://easystats.github.io/performance/reference/check_model.html"><code>check_model</code></a></p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb7" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb7-1"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">check_model</span>(m1)</span></code></pre></div>
+<div class="cell-output-display">
+<div>
+<figure class="figure">
+<p><img src="https://wjschne.github.io/tutorials/Regression/index_files/figure-html/unnamed-chunk-5-1.svg" style="width:100.0%;height:100.0%" class="figure-img"></p>
+</figure>
+</div>
+</div>
+</div>
+<p>Here we see that none of the assumptions have been severely violated.</p>
+</section>
+<section id="summarising-results" class="level1">
+<h1>Summarising Results</h1>
+<p>Base R gives you most of what you would want to know about the regression results with the <code>summary</code> function:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb8" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb8-1"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">summary</span>(m1)</span></code></pre></div>
+<div class="cell-output cell-output-stdout">
+<pre><code>
+Call:
+lm(formula = height ~ avgphgt, data = d)
+
+Residuals:
+   Min     1Q Median     3Q    Max 
+-5.241 -1.649 -0.539  1.224  6.163 
+
+Coefficients:
+            Estimate Std. Error t value Pr(&gt;|t|)    
+(Intercept)  -12.878      9.779   -1.32      0.2    
+avgphgt        1.193      0.145    8.23  5.7e-10 ***
+---
+Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
+
+Residual standard error: 2.68 on 38 degrees of freedom
+Multiple R-squared:  0.641, Adjusted R-squared:  0.631 
+F-statistic: 67.7 on 1 and 38 DF,  p-value: 5.7e-10</code></pre>
+</div>
+</div>
+<p>This output is not pretty, nor was it intended to be. It is designed for you, the analyst. The <code>summary</code> function’s print method is optimized for reading results in the console, not in a document. Presentation-worthy results need a lot more care and attention.</p>
+<p>An automated report from the <a href="https://easystats.github.io/report/articles/report.html">report</a> package:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb10" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb10-1">report<span class="sc" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;">::</span><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">report</span>(m1)</span></code></pre></div>
+<div class="cell-output cell-output-stdout">
+<pre><code>We fitted a linear model (estimated using OLS) to predict height with avgphgt
+(formula: height ~ avgphgt). The model explains a statistically significant and
+substantial proportion of variance (R2 = 0.64, F(1, 38) = 67.73, p &lt; .001, adj.
+R2 = 0.63). The model's intercept, corresponding to avgphgt = 0, is at -12.88
+(95% CI [-32.67, 6.92], t(38) = -1.32, p = 0.196). Within this model:
+
+  - The effect of avgphgt is statistically significant and positive (beta = 1.19,
+95% CI [0.90, 1.49], t(38) = 8.23, p &lt; .001; Std. beta = 0.80, 95% CI [0.60,
+1.00])
+
+Standardized parameters were obtained by fitting the model on a standardized
+version of the dataset. 95% Confidence Intervals (CIs) and p-values were
+computed using a Wald t-distribution approximation.</code></pre>
+</div>
+</div>
+<p>The <code>glance</code>, <code>tidy</code>, and <code>augment</code> functions from the <a href="https://broom.tidymodels.org/">broom</a> package make the output more amenable for further analysis or for presentation.</p>
+<section id="model-level-statistics" class="level2">
+<h2 class="anchored" data-anchor-id="model-level-statistics">Model-Level Statistics</h2>
+<p>Some statistics like the <em>coefficient of determination</em> (<em>R</em><sup>2</sup>) or the <em>standard error of the estimate</em> (<em>σ<sub>e</sub></em>) describe the model as a whole.</p>
+<p>The model-level statistics can be extracted with broom’s <a href="https://broom.tidymodels.org/reference/data.frame_tidiers.html"><code>glance</code></a> function or the performance package’s <a href="https://easystats.github.io/performance/reference/model_performance.lm.html"><code>model_performance</code></a> function.</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb12" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb12-1"><span class="co" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;"># Using broom package</span></span>
+<span id="cb12-2"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">glance</span>(m1)</span></code></pre></div>
+<div class="cell-output-display">
+<div>
+<table class="lightable-paper table table-sm table-striped small" data-quarto-postprocess="true">
+<thead>
+<tr class="header">
+<th style="text-align: right;" data-quarto-table-cell-role="th">r.squared</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">adj.r.squared</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">sigma</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">statistic</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">p.value</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">df</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">logLik</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">AIC</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">BIC</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">deviance</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">df.residual</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">nobs</th>
+</tr>
+</thead>
+<tbody>
+<tr class="odd">
+<td style="text-align: right;">0.64</td>
+<td style="text-align: right;">0.63</td>
+<td style="text-align: right;">2.68</td>
+<td style="text-align: right;">67.7</td>
+<td style="text-align: right;">0</td>
+<td style="text-align: right;">1</td>
+<td style="text-align: right;">-95.2</td>
+<td style="text-align: right;">196</td>
+<td style="text-align: right;">201</td>
+<td style="text-align: right;">273</td>
+<td style="text-align: right;">38</td>
+<td style="text-align: right;">40</td>
+</tr>
+</tbody>
+</table>
+
+
+</div>
+</div>
+<div class="sourceCode cell-code" id="cb13" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb13-1"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">model_parameters</span>(m1)</span></code></pre></div>
+<div class="cell-output-display">
+<div>
+<table class="lightable-paper table table-sm table-striped small" data-quarto-postprocess="true">
+<thead>
+<tr class="header">
+<th style="text-align: left;" data-quarto-table-cell-role="th">Parameter</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">Coefficient</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">SE</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">CI</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">CI_low</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">CI_high</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">t</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">df_error</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">p</th>
+</tr>
+</thead>
+<tbody>
+<tr class="odd">
+<td style="text-align: left;">(Intercept)</td>
+<td style="text-align: right;">-12.88</td>
+<td style="text-align: right;">9.78</td>
+<td style="text-align: right;">0.95</td>
+<td style="text-align: right;">-32.7</td>
+<td style="text-align: right;">6.92</td>
+<td style="text-align: right;">-1.32</td>
+<td style="text-align: right;">38</td>
+<td style="text-align: right;">0.2</td>
+</tr>
+<tr class="even">
+<td style="text-align: left;">avgphgt</td>
+<td style="text-align: right;">1.19</td>
+<td style="text-align: right;">0.14</td>
+<td style="text-align: right;">0.95</td>
+<td style="text-align: right;">0.9</td>
+<td style="text-align: right;">1.49</td>
+<td style="text-align: right;">8.23</td>
+<td style="text-align: right;">38</td>
+<td style="text-align: right;">0.0</td>
+</tr>
+</tbody>
+</table>
+
+
+</div>
+</div>
+<div class="sourceCode cell-code" id="cb14" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb14-1"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">parameters</span>(m1)</span></code></pre></div>
+<div class="cell-output-display">
+<div>
+<table class="lightable-paper table table-sm table-striped small" data-quarto-postprocess="true">
+<thead>
+<tr class="header">
+<th style="text-align: left;" data-quarto-table-cell-role="th">Parameter</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">Coefficient</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">SE</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">CI</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">CI_low</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">CI_high</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">t</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">df_error</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">p</th>
+</tr>
+</thead>
+<tbody>
+<tr class="odd">
+<td style="text-align: left;">(Intercept)</td>
+<td style="text-align: right;">-12.88</td>
+<td style="text-align: right;">9.78</td>
+<td style="text-align: right;">0.95</td>
+<td style="text-align: right;">-32.7</td>
+<td style="text-align: right;">6.92</td>
+<td style="text-align: right;">-1.32</td>
+<td style="text-align: right;">38</td>
+<td style="text-align: right;">0.2</td>
+</tr>
+<tr class="even">
+<td style="text-align: left;">avgphgt</td>
+<td style="text-align: right;">1.19</td>
+<td style="text-align: right;">0.14</td>
+<td style="text-align: right;">0.95</td>
+<td style="text-align: right;">0.9</td>
+<td style="text-align: right;">1.49</td>
+<td style="text-align: right;">8.23</td>
+<td style="text-align: right;">38</td>
+<td style="text-align: right;">0.0</td>
+</tr>
+</tbody>
+</table>
+
+
+</div>
+</div>
+<div class="sourceCode cell-code" id="cb15" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb15-1"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">standardise_parameters</span>(m1)</span></code></pre></div>
+<div class="cell-output-display">
+<div>
+<table class="lightable-paper table table-sm table-striped small" data-quarto-postprocess="true">
+<thead>
+<tr class="header">
+<th style="text-align: left;" data-quarto-table-cell-role="th">Parameter</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">Std_Coefficient</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">CI</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">CI_low</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">CI_high</th>
+</tr>
+</thead>
+<tbody>
+<tr class="odd">
+<td style="text-align: left;">(Intercept)</td>
+<td style="text-align: right;">0.0</td>
+<td style="text-align: right;">0.95</td>
+<td style="text-align: right;">-0.19</td>
+<td style="text-align: right;">0.19</td>
+</tr>
+<tr class="even">
+<td style="text-align: left;">avgphgt</td>
+<td style="text-align: right;">0.8</td>
+<td style="text-align: right;">0.95</td>
+<td style="text-align: right;">0.60</td>
+<td style="text-align: right;">1.00</td>
+</tr>
+</tbody>
+</table>
+
+
+</div>
+</div>
+<div class="sourceCode cell-code" id="cb16" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb16-1"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">performance</span>(m1)</span></code></pre></div>
+<div class="cell-output-display">
+<div>
+<table class="lightable-paper table table-sm table-striped small" data-quarto-postprocess="true">
+<thead>
+<tr class="header">
+<th style="text-align: right;" data-quarto-table-cell-role="th">AIC</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">AICc</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">BIC</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">R2</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">R2_adjusted</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">RMSE</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">Sigma</th>
+</tr>
+</thead>
+<tbody>
+<tr class="odd">
+<td style="text-align: right;">196</td>
+<td style="text-align: right;">197</td>
+<td style="text-align: right;">201</td>
+<td style="text-align: right;">0.64</td>
+<td style="text-align: right;">0.63</td>
+<td style="text-align: right;">2.61</td>
+<td style="text-align: right;">2.68</td>
+</tr>
+</tbody>
+</table>
+
+
+</div>
+</div>
+</div>
+<p><em>R</em><sup>2</sup> is in the <code>glance</code> function’s <code>r.squared</code> column, and the <em>standard error of the estimate</em> is in the <code>sigma</code> column. If all you wanted was the <em>R</em><sup>2</sup>, you could do this:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb17" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb17-1"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">glance</span>(m1)<span class="sc" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;">$</span>r.squared</span></code></pre></div>
+<div class="cell-output cell-output-stdout">
+<pre><code>[1] 0.641</code></pre>
+</div>
+</div>
+<p>Or using the performance package’s <a href="https://easystats.github.io/performance/reference/r2.html"><code>r2</code></a> function:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb19" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb19-1"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">r2</span>(m1)</span></code></pre></div>
+<div class="cell-output cell-output-stdout">
+<pre><code># R2 for Linear Regression
+       R2: 0.641
+  adj. R2: 0.631</code></pre>
+</div>
+</div>
+<p>For the <em>standard error of the estimate</em>:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb21" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb21-1"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">glance</span>(m1)<span class="sc" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;">$</span>sigma</span></code></pre></div>
+<div class="cell-output cell-output-stdout">
+<pre><code>[1] 2.68</code></pre>
+</div>
+</div>
+<p>Why would you want just one number instead of reading it from a table? In reproducible research, we intermingle text and code so that it is clear where every number came from. Thus, “hard-coding” your results like this is considered poor practice:</p>
+<pre><code>The model explains 64% of the variance.</code></pre>
+<p>Using rmarkdown, instead of typing the numeric results, we type pull the results using an <a href="https://rmarkdown.rstudio.com/lesson-4.html">inline code chunk</a>:</p>
+<div class="sourceCode" id="cb24" style="background: #f1f3f5;"><pre class="sourceCode default fold-show code-with-copy"><code class="sourceCode default"><span id="cb24-1">The model explains `r round(100 * glance(m1)$r.squared, 0)`% of the variance.</span></code></pre></div>
+<p>Which, when rendered, produces the correct output:</p>
+<blockquote class="blockquote">
+<p>The model explains 64% of the variance.</p>
+</blockquote>
+<p>That seems like a lot of extra work, right? Yes, it is—unless there is a possibility that your underlying data might change or that you might copy your numbers incorrectly. If you are imperfect, the extra time and effort is worth it. It makes it easy for other scholars to see exactly where each number came from. Hard-coded results are harder to trust.</p>
+<p>Automated report of the model’s performance:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb25" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb25-1"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">report_performance</span>(m1)</span></code></pre></div>
+<div class="cell-output cell-output-stdout">
+<pre><code>The model explains a statistically significant and substantial proportion of
+variance (R2 = 0.64, F(1, 38) = 67.73, p &lt; .001, adj. R2 = 0.63)</code></pre>
+</div>
+</div>
+</section>
+<section id="coefficient-level-statistics" class="level2">
+<h2 class="anchored" data-anchor-id="coefficient-level-statistics">Coefficient-level statistics</h2>
+<p>The regression coefficients—the intercept (<em>b</em><sub>0</sub>) and the slope (<em>b</em><sub>1</sub>)—have a number of statistics associated with them, which we will discuss later in the course.</p>
+<p>If you just wanted the intercept and the slope coefficients, use the <code>coef</code> function:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb27" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb27-1"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">coef</span>(m1)</span></code></pre></div>
+<div class="cell-output cell-output-stdout">
+<pre><code>(Intercept)     avgphgt 
+     -12.88        1.19 </code></pre>
+</div>
+</div>
+<p>Thus, the intercept is -12.88 and the slope is 1.19.</p>
+<p>To get the coefficients (intercept and slope) along with their p-values and other statistical information, use the <a href="https://broom.tidymodels.org/reference/tidy.lm.html"><code>tidy</code></a> function:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb29" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb29-1"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">tidy</span>(m1)</span></code></pre></div>
+<div class="cell-output-display">
+<div>
+<table class="lightable-paper table table-sm table-striped small" data-quarto-postprocess="true">
+<thead>
+<tr class="header">
+<th style="text-align: left;" data-quarto-table-cell-role="th">term</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">estimate</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">std.error</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">statistic</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">p.value</th>
+</tr>
+</thead>
+<tbody>
+<tr class="odd">
+<td style="text-align: left;">(Intercept)</td>
+<td style="text-align: right;">-12.88</td>
+<td style="text-align: right;">9.78</td>
+<td style="text-align: right;">-1.32</td>
+<td style="text-align: right;">0.2</td>
+</tr>
+<tr class="even">
+<td style="text-align: left;">avgphgt</td>
+<td style="text-align: right;">1.19</td>
+<td style="text-align: right;">0.14</td>
+<td style="text-align: right;">8.23</td>
+<td style="text-align: right;">0.0</td>
+</tr>
+</tbody>
+</table>
+
+
+</div>
+</div>
+</div>
+<p>The parameters package has a slightly more print-friendly version of <code>tidy</code>:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb30" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb30-1"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">model_parameters</span>(m1) </span></code></pre></div>
+<div class="cell-output-display">
+<div>
+<table class="lightable-paper table table-sm table-striped small" data-quarto-postprocess="true">
+<thead>
+<tr class="header">
+<th style="text-align: left;" data-quarto-table-cell-role="th">Parameter</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">Coefficient</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">SE</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">CI</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">CI_low</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">CI_high</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">t</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">df_error</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">p</th>
+</tr>
+</thead>
+<tbody>
+<tr class="odd">
+<td style="text-align: left;">(Intercept)</td>
+<td style="text-align: right;">-12.88</td>
+<td style="text-align: right;">9.78</td>
+<td style="text-align: right;">0.95</td>
+<td style="text-align: right;">-32.7</td>
+<td style="text-align: right;">6.92</td>
+<td style="text-align: right;">-1.32</td>
+<td style="text-align: right;">38</td>
+<td style="text-align: right;">0.2</td>
+</tr>
+<tr class="even">
+<td style="text-align: left;">avgphgt</td>
+<td style="text-align: right;">1.19</td>
+<td style="text-align: right;">0.14</td>
+<td style="text-align: right;">0.95</td>
+<td style="text-align: right;">0.9</td>
+<td style="text-align: right;">1.49</td>
+<td style="text-align: right;">8.23</td>
+<td style="text-align: right;">38</td>
+<td style="text-align: right;">0.0</td>
+</tr>
+</tbody>
+</table>
+
+
+</div>
+</div>
+</div>
+<p>If you want standardized parameters:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb31" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb31-1"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">model_parameters</span>(m1, <span class="at" style="color: #657422;
+background-color: null;
+font-style: inherit;">standardize =</span> <span class="st" style="color: #20794D;
+background-color: null;
+font-style: inherit;">"refit"</span>)</span></code></pre></div>
+<div class="cell-output-display">
+<div>
+<table class="lightable-paper table table-sm table-striped small" data-quarto-postprocess="true">
+<thead>
+<tr class="header">
+<th style="text-align: left;" data-quarto-table-cell-role="th">Parameter</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">Coefficient</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">SE</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">CI</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">CI_low</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">CI_high</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">t</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">df_error</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">p</th>
+</tr>
+</thead>
+<tbody>
+<tr class="odd">
+<td style="text-align: left;">(Intercept)</td>
+<td style="text-align: right;">0.0</td>
+<td style="text-align: right;">0.1</td>
+<td style="text-align: right;">0.95</td>
+<td style="text-align: right;">-0.19</td>
+<td style="text-align: right;">0.19</td>
+<td style="text-align: right;">0.00</td>
+<td style="text-align: right;">38</td>
+<td style="text-align: right;">1</td>
+</tr>
+<tr class="even">
+<td style="text-align: left;">avgphgt</td>
+<td style="text-align: right;">0.8</td>
+<td style="text-align: right;">0.1</td>
+<td style="text-align: right;">0.95</td>
+<td style="text-align: right;">0.60</td>
+<td style="text-align: right;">1.00</td>
+<td style="text-align: right;">8.23</td>
+<td style="text-align: right;">38</td>
+<td style="text-align: right;">0</td>
+</tr>
+</tbody>
+</table>
+
+
+</div>
+</div>
+</div>
+<p>Automated report of the parameters:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb32" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb32-1"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">report_parameters</span>(m1)</span></code></pre></div>
+<div class="cell-output cell-output-stdout">
+<pre><code>  - The intercept is statistically non-significant and negative (beta = -12.88, 95% CI [-32.67, 6.92], t(38) = -1.32, p = 0.196; Std. beta = 2.65e-16, 95% CI [-0.19, 0.19])
+  - The effect of avgphgt is statistically significant and positive (beta = 1.19, 95% CI [0.90, 1.49], t(38) = 8.23, p &lt; .001; Std. beta = 0.80, 95% CI [0.60, 1.00])</code></pre>
+</div>
+</div>
+</section>
+<section id="observation-level-statistics" class="level2">
+<h2 class="anchored" data-anchor-id="observation-level-statistics">Observation-Level Statistics</h2>
+<p>We sometimes want the individual-level data like the predictions (<img src="https://latex.codecogs.com/png.latex?%5Chat%7BY%7D">) and the errors (technically called <em>residuals</em>).</p>
+<p>To get observation-level statistics, use broom’s <a href="https://broom.tidymodels.org/reference/augment.lm.html"><code>augment</code></a> function:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb34" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb34-1"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">augment</span>(m1) </span></code></pre></div>
+<div class="cell-output-display">
+<div>
+<table class="lightable-paper table table-sm table-striped small" data-quarto-postprocess="true">
+<thead>
+<tr class="header">
+<th style="text-align: right;" data-quarto-table-cell-role="th">height</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">avgphgt</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">.fitted</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">.resid</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">.hat</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">.sigma</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">.cooksd</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">.std.resid</th>
+</tr>
+</thead>
+<tbody>
+<tr class="odd">
+<td style="text-align: right;">66</td>
+<td style="text-align: right;">67.5</td>
+<td style="text-align: right;">67.6</td>
+<td style="text-align: right;">-1.64</td>
+<td style="text-align: right;">0.03</td>
+<td style="text-align: right;">2.70</td>
+<td style="text-align: right;">0.00</td>
+<td style="text-align: right;">-0.62</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">71</td>
+<td style="text-align: right;">70.0</td>
+<td style="text-align: right;">70.6</td>
+<td style="text-align: right;">0.37</td>
+<td style="text-align: right;">0.04</td>
+<td style="text-align: right;">2.72</td>
+<td style="text-align: right;">0.00</td>
+<td style="text-align: right;">0.14</td>
+</tr>
+<tr class="odd">
+<td style="text-align: right;">74</td>
+<td style="text-align: right;">72.5</td>
+<td style="text-align: right;">73.6</td>
+<td style="text-align: right;">0.39</td>
+<td style="text-align: right;">0.10</td>
+<td style="text-align: right;">2.72</td>
+<td style="text-align: right;">0.00</td>
+<td style="text-align: right;">0.15</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">69</td>
+<td style="text-align: right;">66.5</td>
+<td style="text-align: right;">66.5</td>
+<td style="text-align: right;">2.55</td>
+<td style="text-align: right;">0.03</td>
+<td style="text-align: right;">2.68</td>
+<td style="text-align: right;">0.01</td>
+<td style="text-align: right;">0.96</td>
+</tr>
+<tr class="odd">
+<td style="text-align: right;">70</td>
+<td style="text-align: right;">65.0</td>
+<td style="text-align: right;">64.7</td>
+<td style="text-align: right;">5.34</td>
+<td style="text-align: right;">0.04</td>
+<td style="text-align: right;">2.56</td>
+<td style="text-align: right;">0.09</td>
+<td style="text-align: right;">2.03</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">74</td>
+<td style="text-align: right;">72.0</td>
+<td style="text-align: right;">73.0</td>
+<td style="text-align: right;">0.99</td>
+<td style="text-align: right;">0.09</td>
+<td style="text-align: right;">2.71</td>
+<td style="text-align: right;">0.01</td>
+<td style="text-align: right;">0.39</td>
+</tr>
+<tr class="odd">
+<td style="text-align: right;">73</td>
+<td style="text-align: right;">71.0</td>
+<td style="text-align: right;">71.8</td>
+<td style="text-align: right;">1.18</td>
+<td style="text-align: right;">0.06</td>
+<td style="text-align: right;">2.71</td>
+<td style="text-align: right;">0.01</td>
+<td style="text-align: right;">0.45</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">69</td>
+<td style="text-align: right;">70.0</td>
+<td style="text-align: right;">70.6</td>
+<td style="text-align: right;">-1.63</td>
+<td style="text-align: right;">0.04</td>
+<td style="text-align: right;">2.70</td>
+<td style="text-align: right;">0.01</td>
+<td style="text-align: right;">-0.62</td>
+</tr>
+<tr class="odd">
+<td style="text-align: right;">69</td>
+<td style="text-align: right;">71.0</td>
+<td style="text-align: right;">71.8</td>
+<td style="text-align: right;">-2.82</td>
+<td style="text-align: right;">0.06</td>
+<td style="text-align: right;">2.67</td>
+<td style="text-align: right;">0.04</td>
+<td style="text-align: right;">-1.09</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">75</td>
+<td style="text-align: right;">69.5</td>
+<td style="text-align: right;">70.0</td>
+<td style="text-align: right;">4.97</td>
+<td style="text-align: right;">0.04</td>
+<td style="text-align: right;">2.59</td>
+<td style="text-align: right;">0.07</td>
+<td style="text-align: right;">1.89</td>
+</tr>
+<tr class="odd">
+<td style="text-align: right;">69</td>
+<td style="text-align: right;">69.0</td>
+<td style="text-align: right;">69.4</td>
+<td style="text-align: right;">-0.43</td>
+<td style="text-align: right;">0.03</td>
+<td style="text-align: right;">2.72</td>
+<td style="text-align: right;">0.00</td>
+<td style="text-align: right;">-0.16</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">67</td>
+<td style="text-align: right;">67.5</td>
+<td style="text-align: right;">67.6</td>
+<td style="text-align: right;">-0.64</td>
+<td style="text-align: right;">0.03</td>
+<td style="text-align: right;">2.71</td>
+<td style="text-align: right;">0.00</td>
+<td style="text-align: right;">-0.24</td>
+</tr>
+<tr class="odd">
+<td style="text-align: right;">73</td>
+<td style="text-align: right;">71.5</td>
+<td style="text-align: right;">72.4</td>
+<td style="text-align: right;">0.58</td>
+<td style="text-align: right;">0.07</td>
+<td style="text-align: right;">2.72</td>
+<td style="text-align: right;">0.00</td>
+<td style="text-align: right;">0.23</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">74</td>
+<td style="text-align: right;">70.0</td>
+<td style="text-align: right;">70.6</td>
+<td style="text-align: right;">3.37</td>
+<td style="text-align: right;">0.04</td>
+<td style="text-align: right;">2.66</td>
+<td style="text-align: right;">0.04</td>
+<td style="text-align: right;">1.29</td>
+</tr>
+<tr class="odd">
+<td style="text-align: right;">63</td>
+<td style="text-align: right;">61.5</td>
+<td style="text-align: right;">60.5</td>
+<td style="text-align: right;">2.51</td>
+<td style="text-align: right;">0.13</td>
+<td style="text-align: right;">2.68</td>
+<td style="text-align: right;">0.07</td>
+<td style="text-align: right;">1.00</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">67</td>
+<td style="text-align: right;">68.0</td>
+<td style="text-align: right;">68.2</td>
+<td style="text-align: right;">-1.24</td>
+<td style="text-align: right;">0.03</td>
+<td style="text-align: right;">2.71</td>
+<td style="text-align: right;">0.00</td>
+<td style="text-align: right;">-0.47</td>
+</tr>
+<tr class="odd">
+<td style="text-align: right;">64</td>
+<td style="text-align: right;">62.5</td>
+<td style="text-align: right;">61.7</td>
+<td style="text-align: right;">2.32</td>
+<td style="text-align: right;">0.10</td>
+<td style="text-align: right;">2.69</td>
+<td style="text-align: right;">0.04</td>
+<td style="text-align: right;">0.91</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">61</td>
+<td style="text-align: right;">63.0</td>
+<td style="text-align: right;">62.3</td>
+<td style="text-align: right;">-1.28</td>
+<td style="text-align: right;">0.08</td>
+<td style="text-align: right;">2.71</td>
+<td style="text-align: right;">0.01</td>
+<td style="text-align: right;">-0.50</td>
+</tr>
+<tr class="odd">
+<td style="text-align: right;">70</td>
+<td style="text-align: right;">65.5</td>
+<td style="text-align: right;">65.3</td>
+<td style="text-align: right;">4.74</td>
+<td style="text-align: right;">0.04</td>
+<td style="text-align: right;">2.60</td>
+<td style="text-align: right;">0.06</td>
+<td style="text-align: right;">1.80</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">66</td>
+<td style="text-align: right;">68.0</td>
+<td style="text-align: right;">68.2</td>
+<td style="text-align: right;">-2.24</td>
+<td style="text-align: right;">0.03</td>
+<td style="text-align: right;">2.69</td>
+<td style="text-align: right;">0.01</td>
+<td style="text-align: right;">-0.85</td>
+</tr>
+<tr class="odd">
+<td style="text-align: right;">70</td>
+<td style="text-align: right;">72.0</td>
+<td style="text-align: right;">73.0</td>
+<td style="text-align: right;">-3.01</td>
+<td style="text-align: right;">0.09</td>
+<td style="text-align: right;">2.67</td>
+<td style="text-align: right;">0.07</td>
+<td style="text-align: right;">-1.18</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">60</td>
+<td style="text-align: right;">65.0</td>
+<td style="text-align: right;">64.7</td>
+<td style="text-align: right;">-4.66</td>
+<td style="text-align: right;">0.04</td>
+<td style="text-align: right;">2.60</td>
+<td style="text-align: right;">0.07</td>
+<td style="text-align: right;">-1.78</td>
+</tr>
+<tr class="odd">
+<td style="text-align: right;">75</td>
+<td style="text-align: right;">68.5</td>
+<td style="text-align: right;">68.8</td>
+<td style="text-align: right;">6.16</td>
+<td style="text-align: right;">0.03</td>
+<td style="text-align: right;">2.52</td>
+<td style="text-align: right;">0.08</td>
+<td style="text-align: right;">2.33</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">73</td>
+<td style="text-align: right;">69.0</td>
+<td style="text-align: right;">69.4</td>
+<td style="text-align: right;">3.57</td>
+<td style="text-align: right;">0.03</td>
+<td style="text-align: right;">2.65</td>
+<td style="text-align: right;">0.03</td>
+<td style="text-align: right;">1.35</td>
+</tr>
+<tr class="odd">
+<td style="text-align: right;">63</td>
+<td style="text-align: right;">68.0</td>
+<td style="text-align: right;">68.2</td>
+<td style="text-align: right;">-5.24</td>
+<td style="text-align: right;">0.03</td>
+<td style="text-align: right;">2.57</td>
+<td style="text-align: right;">0.05</td>
+<td style="text-align: right;">-1.98</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">63</td>
+<td style="text-align: right;">65.0</td>
+<td style="text-align: right;">64.7</td>
+<td style="text-align: right;">-1.66</td>
+<td style="text-align: right;">0.04</td>
+<td style="text-align: right;">2.70</td>
+<td style="text-align: right;">0.01</td>
+<td style="text-align: right;">-0.63</td>
+</tr>
+<tr class="odd">
+<td style="text-align: right;">69</td>
+<td style="text-align: right;">67.5</td>
+<td style="text-align: right;">67.6</td>
+<td style="text-align: right;">1.36</td>
+<td style="text-align: right;">0.03</td>
+<td style="text-align: right;">2.71</td>
+<td style="text-align: right;">0.00</td>
+<td style="text-align: right;">0.51</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">65</td>
+<td style="text-align: right;">64.5</td>
+<td style="text-align: right;">64.1</td>
+<td style="text-align: right;">0.93</td>
+<td style="text-align: right;">0.05</td>
+<td style="text-align: right;">2.71</td>
+<td style="text-align: right;">0.00</td>
+<td style="text-align: right;">0.36</td>
+</tr>
+<tr class="odd">
+<td style="text-align: right;">64</td>
+<td style="text-align: right;">66.0</td>
+<td style="text-align: right;">65.8</td>
+<td style="text-align: right;">-1.85</td>
+<td style="text-align: right;">0.03</td>
+<td style="text-align: right;">2.70</td>
+<td style="text-align: right;">0.01</td>
+<td style="text-align: right;">-0.70</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">69</td>
+<td style="text-align: right;">70.0</td>
+<td style="text-align: right;">70.6</td>
+<td style="text-align: right;">-1.63</td>
+<td style="text-align: right;">0.04</td>
+<td style="text-align: right;">2.70</td>
+<td style="text-align: right;">0.01</td>
+<td style="text-align: right;">-0.62</td>
+</tr>
+<tr class="odd">
+<td style="text-align: right;">65</td>
+<td style="text-align: right;">68.0</td>
+<td style="text-align: right;">68.2</td>
+<td style="text-align: right;">-3.24</td>
+<td style="text-align: right;">0.03</td>
+<td style="text-align: right;">2.66</td>
+<td style="text-align: right;">0.02</td>
+<td style="text-align: right;">-1.22</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">63</td>
+<td style="text-align: right;">65.0</td>
+<td style="text-align: right;">64.7</td>
+<td style="text-align: right;">-1.66</td>
+<td style="text-align: right;">0.04</td>
+<td style="text-align: right;">2.70</td>
+<td style="text-align: right;">0.01</td>
+<td style="text-align: right;">-0.63</td>
+</tr>
+<tr class="odd">
+<td style="text-align: right;">68</td>
+<td style="text-align: right;">67.5</td>
+<td style="text-align: right;">67.6</td>
+<td style="text-align: right;">0.36</td>
+<td style="text-align: right;">0.03</td>
+<td style="text-align: right;">2.72</td>
+<td style="text-align: right;">0.00</td>
+<td style="text-align: right;">0.13</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">63</td>
+<td style="text-align: right;">64.5</td>
+<td style="text-align: right;">64.1</td>
+<td style="text-align: right;">-1.07</td>
+<td style="text-align: right;">0.05</td>
+<td style="text-align: right;">2.71</td>
+<td style="text-align: right;">0.00</td>
+<td style="text-align: right;">-0.41</td>
+</tr>
+<tr class="odd">
+<td style="text-align: right;">62</td>
+<td style="text-align: right;">64.0</td>
+<td style="text-align: right;">63.5</td>
+<td style="text-align: right;">-1.47</td>
+<td style="text-align: right;">0.06</td>
+<td style="text-align: right;">2.71</td>
+<td style="text-align: right;">0.01</td>
+<td style="text-align: right;">-0.56</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">61</td>
+<td style="text-align: right;">63.5</td>
+<td style="text-align: right;">62.9</td>
+<td style="text-align: right;">-1.87</td>
+<td style="text-align: right;">0.07</td>
+<td style="text-align: right;">2.70</td>
+<td style="text-align: right;">0.02</td>
+<td style="text-align: right;">-0.72</td>
+</tr>
+<tr class="odd">
+<td style="text-align: right;">67</td>
+<td style="text-align: right;">67.0</td>
+<td style="text-align: right;">67.0</td>
+<td style="text-align: right;">-0.05</td>
+<td style="text-align: right;">0.03</td>
+<td style="text-align: right;">2.72</td>
+<td style="text-align: right;">0.00</td>
+<td style="text-align: right;">-0.02</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">63</td>
+<td style="text-align: right;">64.5</td>
+<td style="text-align: right;">64.1</td>
+<td style="text-align: right;">-1.07</td>
+<td style="text-align: right;">0.05</td>
+<td style="text-align: right;">2.71</td>
+<td style="text-align: right;">0.00</td>
+<td style="text-align: right;">-0.41</td>
+</tr>
+<tr class="odd">
+<td style="text-align: right;">62</td>
+<td style="text-align: right;">63.5</td>
+<td style="text-align: right;">62.9</td>
+<td style="text-align: right;">-0.87</td>
+<td style="text-align: right;">0.07</td>
+<td style="text-align: right;">2.71</td>
+<td style="text-align: right;">0.00</td>
+<td style="text-align: right;">-0.34</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">72</td>
+<td style="text-align: right;">71.5</td>
+<td style="text-align: right;">72.4</td>
+<td style="text-align: right;">-0.42</td>
+<td style="text-align: right;">0.07</td>
+<td style="text-align: right;">2.72</td>
+<td style="text-align: right;">0.00</td>
+<td style="text-align: right;">-0.16</td>
+</tr>
+</tbody>
+</table>
+
+
+</div>
+</div>
+</div>
+<p>The predicted values <img src="https://latex.codecogs.com/png.latex?%5Chat%7BY%7D"> are in the <code>.fitted</code> column, and the errors (residuals) are in the <code>.resid</code> column.</p>
+<p>If you wanted to predict a value not in the data, you use the <code>newdata</code> argument. For example, if I wanted to predict the height of a person whose parents had a height of 65:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb35" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb35-1"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">augment</span>(m1, <span class="at" style="color: #657422;
+background-color: null;
+font-style: inherit;">newdata =</span> <span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">tibble</span>(<span class="at" style="color: #657422;
+background-color: null;
+font-style: inherit;">avgphgt =</span> <span class="dv" style="color: #AD0000;
+background-color: null;
+font-style: inherit;">65</span>))</span></code></pre></div>
+<div class="cell-output-display">
+<div>
+<table class="lightable-paper table table-sm table-striped small" data-quarto-postprocess="true">
+<thead>
+<tr class="header">
+<th style="text-align: right;" data-quarto-table-cell-role="th">avgphgt</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">.fitted</th>
+</tr>
+</thead>
+<tbody>
+<tr class="odd">
+<td style="text-align: right;">65</td>
+<td style="text-align: right;">64.7</td>
+</tr>
+</tbody>
+</table>
+
+
+</div>
+</div>
+</div>
+</section>
+</section>
+<section id="multiple-regression" class="level1">
+<h1>Multiple Regression</h1>
+<p>In this data set, all participants identified as either male or female. We assume that males are, on average, taller than females. The codes are male = 1 and female = 2, thus our predicted effect is negative.</p>
+<p>We run the regression model with 2 predictors like so:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb36" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb36-1">m2 <span class="ot" style="color: #003B4F;
+background-color: null;
+font-style: inherit;">&lt;-</span> <span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">lm</span>(height <span class="sc" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;">~</span> avgphgt <span class="sc" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;">+</span> gender, <span class="at" style="color: #657422;
+background-color: null;
+font-style: inherit;">data =</span> d)</span></code></pre></div>
+</div>
+<section id="checking-assumptions-1" class="level2">
+<h2 class="anchored" data-anchor-id="checking-assumptions-1">Checking assumptions</h2>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb37" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb37-1"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">check_model</span>(m2)</span></code></pre></div>
+<div class="cell-output-display">
+<div>
+<figure class="figure">
+<p><img src="https://wjschne.github.io/tutorials/Regression/index_files/figure-html/unnamed-chunk-21-1.svg" style="width:100.0%;height:100.0%" class="figure-img"></p>
+</figure>
+</div>
+</div>
+</div>
+<p>All looks well.</p>
+</section>
+<section id="summarizing-results" class="level2">
+<h2 class="anchored" data-anchor-id="summarizing-results">Summarizing Results</h2>
+<p>To summarize the results, use the <code>summary</code> function:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb38" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb38-1"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">summary</span>(m2)</span></code></pre></div>
+<div class="cell-output cell-output-stdout">
+<pre><code>
+Call:
+lm(formula = height ~ avgphgt + gender, data = d)
+
+Residuals:
+   Min     1Q Median     3Q    Max 
+-3.826 -1.094 -0.347  1.656  4.503 
+
+Coefficients:
+            Estimate Std. Error t value Pr(&gt;|t|)    
+(Intercept)   29.035     11.600    2.50  0.01686 *  
+avgphgt        0.671      0.157    4.26  0.00013 ***
+gender        -4.468      0.920   -4.85  2.2e-05 ***
+---
+Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
+
+Residual standard error: 2.12 on 37 degrees of freedom
+Multiple R-squared:  0.78,  Adjusted R-squared:  0.769 
+F-statistic: 65.7 on 2 and 37 DF,  p-value: 6.6e-13</code></pre>
+</div>
+</div>
+<p>An automated report:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb40" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb40-1">report<span class="sc" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;">::</span><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">report</span>(m2)</span></code></pre></div>
+<div class="cell-output cell-output-stdout">
+<pre><code>We fitted a linear model (estimated using OLS) to predict height with avgphgt
+and gender (formula: height ~ avgphgt + gender). The model explains a
+statistically significant and substantial proportion of variance (R2 = 0.78,
+F(2, 37) = 65.75, p &lt; .001, adj. R2 = 0.77). The model's intercept,
+corresponding to avgphgt = 0 and gender = 0, is at 29.03 (95% CI [5.53, 52.54],
+t(37) = 2.50, p = 0.017). Within this model:
+
+  - The effect of avgphgt is statistically significant and positive (beta = 0.67,
+95% CI [0.35, 0.99], t(37) = 4.26, p &lt; .001; Std. beta = 0.45, 95% CI [0.24,
+0.66])
+  - The effect of gender is statistically significant and negative (beta = -4.47,
+95% CI [-6.33, -2.60], t(37) = -4.85, p &lt; .001; Std. beta = -0.51, 95% CI
+[-0.73, -0.30])
+
+Standardized parameters were obtained by fitting the model on a standardized
+version of the dataset. 95% Confidence Intervals (CIs) and p-values were
+computed using a Wald t-distribution approximation.</code></pre>
+</div>
+</div>
+<p>Standardized coefficients:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb42" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb42-1">parameters<span class="sc" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;">::</span><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">model_parameters</span>(m2, <span class="at" style="color: #657422;
+background-color: null;
+font-style: inherit;">standardize =</span> <span class="st" style="color: #20794D;
+background-color: null;
+font-style: inherit;">"refit"</span>)</span></code></pre></div>
+<div class="cell-output-display">
+<div>
+<table class="lightable-paper table table-sm table-striped small" data-quarto-postprocess="true">
+<thead>
+<tr class="header">
+<th style="text-align: left;" data-quarto-table-cell-role="th">Parameter</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">Coefficient</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">SE</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">CI</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">CI_low</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">CI_high</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">t</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">df_error</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">p</th>
+</tr>
+</thead>
+<tbody>
+<tr class="odd">
+<td style="text-align: left;">(Intercept)</td>
+<td style="text-align: right;">0.00</td>
+<td style="text-align: right;">0.08</td>
+<td style="text-align: right;">0.95</td>
+<td style="text-align: right;">-0.15</td>
+<td style="text-align: right;">0.15</td>
+<td style="text-align: right;">0.00</td>
+<td style="text-align: right;">37</td>
+<td style="text-align: right;">1</td>
+</tr>
+<tr class="even">
+<td style="text-align: left;">avgphgt</td>
+<td style="text-align: right;">0.45</td>
+<td style="text-align: right;">0.11</td>
+<td style="text-align: right;">0.95</td>
+<td style="text-align: right;">0.24</td>
+<td style="text-align: right;">0.66</td>
+<td style="text-align: right;">4.26</td>
+<td style="text-align: right;">37</td>
+<td style="text-align: right;">0</td>
+</tr>
+<tr class="odd">
+<td style="text-align: left;">gender</td>
+<td style="text-align: right;">-0.51</td>
+<td style="text-align: right;">0.11</td>
+<td style="text-align: right;">0.95</td>
+<td style="text-align: right;">-0.73</td>
+<td style="text-align: right;">-0.30</td>
+<td style="text-align: right;">-4.85</td>
+<td style="text-align: right;">37</td>
+<td style="text-align: right;">0</td>
+</tr>
+</tbody>
+</table>
+
+
+</div>
+</div>
+</div>
+<p>The oddly-named <code>anova</code> function compares 2 models:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb43" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb43-1"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">anova</span>(m1, m2)</span></code></pre></div>
+<div class="cell-output-display">
+<div>
+<table class="lightable-paper table table-sm table-striped small" data-quarto-postprocess="true">
+<thead>
+<tr class="header">
+<th style="text-align: right;" data-quarto-table-cell-role="th">Res.Df</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">RSS</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">Df</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">Sum of Sq</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">F</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">Pr(&gt;F)</th>
+</tr>
+</thead>
+<tbody>
+<tr class="odd">
+<td style="text-align: right;">38</td>
+<td style="text-align: right;">273</td>
+<td style="text-align: right;"></td>
+<td style="text-align: right;"></td>
+<td style="text-align: right;"></td>
+<td style="text-align: right;"></td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">37</td>
+<td style="text-align: right;">167</td>
+<td style="text-align: right;">1</td>
+<td style="text-align: right;">106</td>
+<td style="text-align: right;">23.6</td>
+<td style="text-align: right;">0</td>
+</tr>
+</tbody>
+</table>
+
+
+</div>
+</div>
+</div>
+<p>The p-value is significant, meaning that <code>m2</code> explains more variance than <code>m1</code>.</p>
+<p>I like the more full-featured set of model comparison functions from the performance package:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb44" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb44-1"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">compare_performance</span>(m1, m2) </span></code></pre></div>
+<div class="cell-output-display">
+<div>
+<table class="lightable-paper table table-sm table-striped small" data-quarto-postprocess="true">
+<thead>
+<tr class="header">
+<th style="text-align: left;" data-quarto-table-cell-role="th">Name</th>
+<th style="text-align: left;" data-quarto-table-cell-role="th">Model</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">AIC</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">AIC_wt</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">AICc</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">AICc_wt</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">BIC</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">BIC_wt</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">R2</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">R2_adjusted</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">RMSE</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">Sigma</th>
+</tr>
+</thead>
+<tbody>
+<tr class="odd">
+<td style="text-align: left;">m1</td>
+<td style="text-align: left;">lm</td>
+<td style="text-align: right;">196</td>
+<td style="text-align: right;">0</td>
+<td style="text-align: right;">197</td>
+<td style="text-align: right;">0</td>
+<td style="text-align: right;">201</td>
+<td style="text-align: right;">0</td>
+<td style="text-align: right;">0.64</td>
+<td style="text-align: right;">0.63</td>
+<td style="text-align: right;">2.61</td>
+<td style="text-align: right;">2.68</td>
+</tr>
+<tr class="even">
+<td style="text-align: left;">m2</td>
+<td style="text-align: left;">lm</td>
+<td style="text-align: right;">179</td>
+<td style="text-align: right;">1</td>
+<td style="text-align: right;">180</td>
+<td style="text-align: right;">1</td>
+<td style="text-align: right;">185</td>
+<td style="text-align: right;">1</td>
+<td style="text-align: right;">0.78</td>
+<td style="text-align: right;">0.77</td>
+<td style="text-align: right;">2.04</td>
+<td style="text-align: right;">2.12</td>
+</tr>
+</tbody>
+</table>
+
+
+</div>
+</div>
+</div>
+<p>This is equivalent to the <code>anova</code> function but in a more readable format:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb45" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb45-1"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">test_wald</span>(m1, m2)</span></code></pre></div>
+<div class="cell-output-display">
+<div>
+<table class="lightable-paper table table-sm table-striped small" data-quarto-postprocess="true">
+<thead>
+<tr class="header">
+<th style="text-align: left;" data-quarto-table-cell-role="th">Name</th>
+<th style="text-align: left;" data-quarto-table-cell-role="th">Model</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">df</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">df_diff</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">F</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">p</th>
+</tr>
+</thead>
+<tbody>
+<tr class="odd">
+<td style="text-align: left;">m1</td>
+<td style="text-align: left;">lm</td>
+<td style="text-align: right;">38</td>
+<td style="text-align: right;"></td>
+<td style="text-align: right;"></td>
+<td style="text-align: right;"></td>
+</tr>
+<tr class="even">
+<td style="text-align: left;">m2</td>
+<td style="text-align: left;">lm</td>
+<td style="text-align: right;">37</td>
+<td style="text-align: right;">1</td>
+<td style="text-align: right;">23.6</td>
+<td style="text-align: right;">0</td>
+</tr>
+</tbody>
+</table>
+
+
+</div>
+</div>
+</div>
+<p>This uses Bayes Factor instead of a p-value:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb46" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb46-1"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">test_bf</span>(m1, m2)</span></code></pre></div>
+<div class="cell-output-display">
+<div>
+<table class="lightable-paper table table-sm table-striped small" data-quarto-postprocess="true">
+<thead>
+<tr class="header">
+<th style="text-align: left;" data-quarto-table-cell-role="th">Model</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">log_BF</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">BF</th>
+</tr>
+</thead>
+<tbody>
+<tr class="odd">
+<td style="text-align: left;">avgphgt</td>
+<td style="text-align: right;"></td>
+<td style="text-align: right;"></td>
+</tr>
+<tr class="even">
+<td style="text-align: left;">avgphgt + gender</td>
+<td style="text-align: right;">8.01</td>
+<td style="text-align: right;">3010</td>
+</tr>
+</tbody>
+</table>
+
+
+</div>
+</div>
+</div>
+<p>A BF &gt; 1 means that <code>m2</code> is more strongly supported than <code>m1</code>. A BF &lt; 1 means that <code>m1</code> is more strongly supported than <code>m2</code>.</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb47" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb47-1"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">report</span>(<span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">test_bf</span>(m1, m2))</span></code></pre></div>
+<div class="cell-output cell-output-stdout">
+<pre><code>Bayes factors were computed using the BIC approximation, by which BF10 =
+exp((BIC0 - BIC1)/2). Compared to the avgphgt model, we found extreme evidence
+(BF = 3.01e+03) in favour of the avgphgt + gender model (the least supported
+model).</code></pre>
+</div>
+</div>
+</section>
+</section>
+<section id="all-the-code-in-one-place" class="level1">
+<h1>All the code in one place:</h1>
+<p>The preceding analyses might seem like a lot, but it is not really so much when you see it all in just a few lines of code. Here are all the main analyses:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb49" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb49-1"><span class="co" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;"># Load packages</span></span>
+<span id="cb49-2"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">library</span>(tidyverse)</span>
+<span id="cb49-3"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">library</span>(haven)</span>
+<span id="cb49-4"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">library</span>(broom)</span>
+<span id="cb49-5"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">library</span>(performance)</span>
+<span id="cb49-6"></span>
+<span id="cb49-7"><span class="co" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;"># Import data</span></span>
+<span id="cb49-8">d <span class="ot" style="color: #003B4F;
+background-color: null;
+font-style: inherit;">&lt;-</span> <span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">read_spss</span>(<span class="st" style="color: #20794D;
+background-color: null;
+font-style: inherit;">"https://github.com/wjschne/EDUC5325/raw/master/height.sav"</span>)</span>
+<span id="cb49-9"></span>
+<span id="cb49-10"><span class="co" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;"># Plot data</span></span>
+<span id="cb49-11"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">ggplot</span>(d, <span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">aes</span>(weight,height)) <span class="sc" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;">+</span></span>
+<span id="cb49-12">  <span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">geom_point</span>() <span class="sc" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;">+</span></span>
+<span id="cb49-13">  <span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">geom_smooth</span>(<span class="at" style="color: #657422;
+background-color: null;
+font-style: inherit;">method =</span> <span class="st" style="color: #20794D;
+background-color: null;
+font-style: inherit;">"lm"</span>)</span>
+<span id="cb49-14"></span>
+<span id="cb49-15"><span class="co" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;"># Save plot</span></span>
+<span id="cb49-16"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">ggsave</span>(<span class="st" style="color: #20794D;
+background-color: null;
+font-style: inherit;">"my_plot.pdf"</span>)</span>
+<span id="cb49-17"></span>
+<span id="cb49-18"><span class="co" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;"># Create regression model</span></span>
+<span id="cb49-19">m1 <span class="ot" style="color: #003B4F;
+background-color: null;
+font-style: inherit;">&lt;-</span> <span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">lm</span>(height<span class="sc" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;">~</span>avgphgt, <span class="at" style="color: #657422;
+background-color: null;
+font-style: inherit;">data =</span> d)</span>
+<span id="cb49-20"></span>
+<span id="cb49-21"><span class="co" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;"># Display results</span></span>
+<span id="cb49-22"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">glance</span>(m1)</span>
+<span id="cb49-23"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">tidy</span>(m1)</span>
+<span id="cb49-24"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">augment</span>(m1)</span>
+<span id="cb49-25"></span>
+<span id="cb49-26"><span class="co" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;"># Multiple regression</span></span>
+<span id="cb49-27">m2 <span class="ot" style="color: #003B4F;
+background-color: null;
+font-style: inherit;">&lt;-</span> <span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">lm</span>(height <span class="sc" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;">~</span> avgphgt <span class="sc" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;">+</span> gender, <span class="at" style="color: #657422;
+background-color: null;
+font-style: inherit;">data =</span> d)</span>
+<span id="cb49-28"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">summary</span>(m2)</span>
+<span id="cb49-29">parameters<span class="sc" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;">::</span><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">model_parameters</span>(m2, <span class="at" style="color: #657422;
+background-color: null;
+font-style: inherit;">standardize =</span> <span class="st" style="color: #20794D;
+background-color: null;
+font-style: inherit;">"refit"</span>)</span>
+<span id="cb49-30"></span>
+<span id="cb49-31"><span class="co" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;"># Compare model m1 and model m2</span></span>
+<span id="cb49-32"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">anova</span>(m1, m2)</span>
+<span id="cb49-33"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">compare_performance</span>(m1, m2)</span></code></pre></div>
+</div>
+</section>
+<section id="canvas-questions" class="level1">
+<h1>Canvas Questions</h1>
+<p>Use the Transfer of Learning data set. A set of 38 inmates participated in a study that tested the Transfer of Learning Hypothesis. Some people believe that studying Latin is particularly beneficial for progress in other academic disciplines. Each inmate was given a reading test and a math test before the study began. Some inmates were randomly assigned to participate in a 48-week Latin course. The control group studied the lives of famous athletes for the same time period. Each inmate took the reading and math test again to see if studying Latin improved their academic skills. Personal information was also collected about each inmate including their annual income before going to prison, whether or not the inmate had a documented learning disorder, and whether or not the inmate had been convicted of a violent offense. Here are the variable names:</p>
+<ul>
+<li><code>read_1</code> = Reading before study</li>
+<li><code>read_2</code> = Reading after study</li>
+<li><code>math_1</code> = Math before study</li>
+<li><code>math_2</code> = Math after study</li>
+<li><code>latin</code> = 1 (Studied Latin), 0 (Studied sports history)</li>
+<li><code>violent</code> = 1 (convicted of a violent offense), 0 (convicted of a non-violent offense)</li>
+<li><code>learning</code> = 1 (Learning disabled), 0 (Not learning disabled)</li>
+<li><code>income</code> = Annual income before incarceration</li>
+</ul>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb50" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb50-1">d_learn <span class="ot" style="color: #003B4F;
+background-color: null;
+font-style: inherit;">&lt;-</span> <span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">read_csv</span>(<span class="st" style="color: #20794D;
+background-color: null;
+font-style: inherit;">"https://github.com/wjschne/EDUC5529/raw/master/transfer_of_learning.csv"</span>) </span></code></pre></div>
+</div>
+<p><span class="instruction">Assume α = 0.05 for all hypothesis tests.</span></p>
+<p><span class="instruction">Create a regression model in which you predict reading at time 2 (<code>read_2</code>) using reading at time 1 (<code>read_1</code>). Call the model fit object <code>m_read</code>.</span></p>
+<section id="canvas-question-1" class="level2 tabset tabset-fade tabset-pills">
+<h2 class="tabset tabset-fade tabset-pills anchored" data-anchor-id="canvas-question-1">Canvas Question 1</h2>
+<p><span class="question"><strong>Question</strong>: Is <code>read_1</code> a significant predictor of <code>read_2</code>?</span></p>
+<section id="hide-hints" class="level3">
+<h3 class="anchored" data-anchor-id="hide-hints">Hide Hints</h3>
+</section>
+<section id="hint-1" class="level3">
+<h3 class="anchored" data-anchor-id="hint-1">Hint 1</h3>
+<p>Make model fit object called <code>m_read</code> using the <code>lm</code> function.</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb51" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb51-1">m_read <span class="ot" style="color: #003B4F;
+background-color: null;
+font-style: inherit;">&lt;-</span> <span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">lm</span>(read_2 <span class="sc" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;">~</span> read_1, <span class="at" style="color: #657422;
+background-color: null;
+font-style: inherit;">data =</span> d_learn)</span></code></pre></div>
+</div>
+</section>
+<section id="hint-2" class="level3">
+<h3 class="anchored" data-anchor-id="hint-2">Hint 2</h3>
+<p>View coefficent-level statistics with broom’s <code>tidy</code> function.</p>
+<p>You could also use Base R’s <code>summary</code> function.</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb52" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb52-1"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">tidy</span>(m_read)</span>
+<span id="cb52-2"></span>
+<span id="cb52-3"><span class="co" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;"># or</span></span>
+<span id="cb52-4"></span>
+<span id="cb52-5"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">summary</span>(m_read)</span></code></pre></div>
+</div>
+</section>
+<section id="hint-3" class="level3">
+<h3 class="anchored" data-anchor-id="hint-3">Hint 3</h3>
+<p>In the read_1 row, is the p.value column less than 0.05?</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb53" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb53-1"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">tidy</span>(m_read)</span></code></pre></div>
+</div>
+<div class="cell">
+<div class="cell-output-display">
+<table class="table table-sm table-striped small">
+<thead>
+<tr class="header">
+<th style="text-align: left;">term</th>
+<th style="text-align: right;">estimate</th>
+<th style="text-align: right;">std.error</th>
+<th style="text-align: right;">statistic</th>
+<th style="text-align: right;">p.value</th>
+</tr>
+</thead>
+<tbody>
+<tr class="odd">
+<td style="text-align: left;">(Intercept)</td>
+<td style="text-align: right;">83.591</td>
+<td style="text-align: right;">10.922</td>
+<td style="text-align: right;">7.65</td>
+<td style="text-align: right;">0.000</td>
+</tr>
+<tr class="even">
+<td style="text-align: left;">read_1</td>
+<td style="text-align: right;">0.251</td>
+<td style="text-align: right;">0.104</td>
+<td style="text-align: right;">2.41</td>
+<td style="text-align: right;">0.021</td>
+</tr>
+</tbody>
+</table>
+</div>
+</div>
+</section>
+</section>
+<section id="canvas-question-2" class="level2 tabset tabset-fade tabset-pills">
+<h2 class="tabset tabset-fade tabset-pills anchored" data-anchor-id="canvas-question-2">Canvas Question 2</h2>
+<p><span class="question"><strong>Question</strong>: What is the <em>R</em><sup>2</sup> for the <code>m_read</code> model?</span></p>
+<section id="hide-hint" class="level3">
+<h3 class="anchored" data-anchor-id="hide-hint">Hide Hint</h3>
+</section>
+<section id="hint" class="level3">
+<h3 class="anchored" data-anchor-id="hint">Hint</h3>
+<p>View model-level statistics with broom’s <code>glance</code> function.</p>
+<p>You could also use Base R’s <code>summary</code> function.</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb54" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb54-1"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">glance</span>(m_read)</span>
+<span id="cb54-2"></span>
+<span id="cb54-3"><span class="co" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;"># or</span></span>
+<span id="cb54-4"></span>
+<span id="cb54-5"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">summary</span>(m_read)</span></code></pre></div>
+</div>
+</section>
+</section>
+<section id="canvas-question-3" class="level2 tabset tabset-fade tabset-pills">
+<h2 class="tabset tabset-fade tabset-pills anchored" data-anchor-id="canvas-question-3">Canvas Question 3</h2>
+<p><span class="question"><strong>Question</strong>: What does the scatter plot look like when <code>read_1</code> is on the <em>x</em>-axis and <code>read_2</code> is on the <em>y</em>-axis? Also plot the regression line. Save your plot using the <code>ggsave</code> function and upload it to Canvas.</span></p>
+<section id="hide-hints-1" class="level3">
+<h3 class="anchored" data-anchor-id="hide-hints-1">Hide Hints</h3>
+</section>
+<section id="hint-1-1" class="level3">
+<h3 class="anchored" data-anchor-id="hint-1-1">Hint 1</h3>
+<p>This will get you started</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb55" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb55-1"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">ggplot</span>(d_learn, <span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">aes</span>(read_1, read_2))</span></code></pre></div>
+</div>
+</section>
+<section id="points-hint" class="level3">
+<h3 class="anchored" data-anchor-id="points-hint">Points Hint</h3>
+<p>Here is how you add points.</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb56" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb56-1"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">ggplot</span>(d_learn, <span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">aes</span>(read_1, read_2)) <span class="sc" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;">+</span></span>
+<span id="cb56-2">  <span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">geom_point</span>() </span></code></pre></div>
+</div>
+</section>
+<section id="line-hint" class="level3">
+<h3 class="anchored" data-anchor-id="line-hint">Line Hint</h3>
+<p>Here is how you add a regression line.</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb57" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb57-1"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">ggplot</span>(d_learn, <span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">aes</span>(read_1, read_2)) <span class="sc" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;">+</span></span>
+<span id="cb57-2">  <span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">geom_point</span>() <span class="sc" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;">+</span> </span>
+<span id="cb57-3">  <span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">geom_smooth</span>(<span class="at" style="color: #657422;
+background-color: null;
+font-style: inherit;">method =</span> <span class="st" style="color: #20794D;
+background-color: null;
+font-style: inherit;">"lm"</span>) </span></code></pre></div>
+</div>
+</section>
+<section id="plot-polishing" class="level3">
+<h3 class="anchored" data-anchor-id="plot-polishing">Plot Polishing</h3>
+<p>This is overkill for now. But soon you will want to fine tune your plots.</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb58" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb58-1"><span class="co" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;"># I want to put the equation at x = 130</span></span>
+<span id="cb58-2">equation_x <span class="ot" style="color: #003B4F;
+background-color: null;
+font-style: inherit;">&lt;-</span> <span class="dv" style="color: #AD0000;
+background-color: null;
+font-style: inherit;">130</span></span>
+<span id="cb58-3">equation_y <span class="ot" style="color: #003B4F;
+background-color: null;
+font-style: inherit;">&lt;-</span> <span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">predict</span>(m_read, <span class="at" style="color: #657422;
+background-color: null;
+font-style: inherit;">newdata =</span> <span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">tibble</span>(<span class="at" style="color: #657422;
+background-color: null;
+font-style: inherit;">read_1 =</span> equation_x))</span>
+<span id="cb58-4"></span>
+<span id="cb58-5"><span class="co" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;"># Extracting the coefficients</span></span>
+<span id="cb58-6">b_read <span class="ot" style="color: #003B4F;
+background-color: null;
+font-style: inherit;">&lt;-</span> <span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">round</span>(<span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">coef</span>(m_read),<span class="dv" style="color: #AD0000;
+background-color: null;
+font-style: inherit;">2</span>)</span>
+<span id="cb58-7"></span>
+<span id="cb58-8"><span class="co" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;"># The angle of the regression line is the inverse tangent of the slope (converted to degrees)</span></span>
+<span id="cb58-9">eq_angle <span class="ot" style="color: #003B4F;
+background-color: null;
+font-style: inherit;">&lt;-</span> <span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">atan</span>(b_read[<span class="dv" style="color: #AD0000;
+background-color: null;
+font-style: inherit;">2</span>]) <span class="sc" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;">*</span> <span class="dv" style="color: #AD0000;
+background-color: null;
+font-style: inherit;">180</span> <span class="sc" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;">/</span> pi</span>
+<span id="cb58-10"></span>
+<span id="cb58-11"><span class="co" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;"># Equation</span></span>
+<span id="cb58-12">eq_read <span class="ot" style="color: #003B4F;
+background-color: null;
+font-style: inherit;">&lt;-</span> <span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">paste0</span>(<span class="st" style="color: #20794D;
+background-color: null;
+font-style: inherit;">"italic(Y) == "</span>, </span>
+<span id="cb58-13">                  b_read[<span class="dv" style="color: #AD0000;
+background-color: null;
+font-style: inherit;">1</span>], </span>
+<span id="cb58-14">                  <span class="st" style="color: #20794D;
+background-color: null;
+font-style: inherit;">" + "</span>, </span>
+<span id="cb58-15">                  b_read[<span class="dv" style="color: #AD0000;
+background-color: null;
+font-style: inherit;">2</span>], </span>
+<span id="cb58-16">                  <span class="st" style="color: #20794D;
+background-color: null;
+font-style: inherit;">" *  italic(X) + italic(e)"</span>)</span>
+<span id="cb58-17"></span>
+<span id="cb58-18"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">ggplot</span>(d_learn, <span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">aes</span>(read_1, read_2)) <span class="sc" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;">+</span></span>
+<span id="cb58-19">  <span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">geom_point</span>() <span class="sc" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;">+</span></span>
+<span id="cb58-20">  <span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">geom_smooth</span>(<span class="at" style="color: #657422;
+background-color: null;
+font-style: inherit;">method =</span> <span class="st" style="color: #20794D;
+background-color: null;
+font-style: inherit;">"lm"</span>) <span class="sc" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;">+</span></span>
+<span id="cb58-21">  <span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">labs</span>(<span class="at" style="color: #657422;
+background-color: null;
+font-style: inherit;">x =</span> <span class="st" style="color: #20794D;
+background-color: null;
+font-style: inherit;">"Reading at Time 1"</span>,</span>
+<span id="cb58-22">       <span class="at" style="color: #657422;
+background-color: null;
+font-style: inherit;">y =</span> <span class="st" style="color: #20794D;
+background-color: null;
+font-style: inherit;">"Reading at Time 2"</span>,</span>
+<span id="cb58-23">       <span class="at" style="color: #657422;
+background-color: null;
+font-style: inherit;">title =</span> <span class="st" style="color: #20794D;
+background-color: null;
+font-style: inherit;">"Using Reading at Time 1 to Predict Reading at Time 2"</span>) <span class="sc" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;">+</span></span>
+<span id="cb58-24">  <span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">coord_fixed</span>() <span class="sc" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;">+</span></span>
+<span id="cb58-25">  <span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">annotate</span>(</span>
+<span id="cb58-26">    <span class="at" style="color: #657422;
+background-color: null;
+font-style: inherit;">geom =</span> <span class="st" style="color: #20794D;
+background-color: null;
+font-style: inherit;">"text"</span>,</span>
+<span id="cb58-27">    <span class="at" style="color: #657422;
+background-color: null;
+font-style: inherit;">x =</span> equation_x,</span>
+<span id="cb58-28">    <span class="at" style="color: #657422;
+background-color: null;
+font-style: inherit;">y =</span> equation_y,</span>
+<span id="cb58-29">    <span class="at" style="color: #657422;
+background-color: null;
+font-style: inherit;">angle =</span> eq_angle,</span>
+<span id="cb58-30">    <span class="at" style="color: #657422;
+background-color: null;
+font-style: inherit;">label =</span> eq_read,</span>
+<span id="cb58-31">    <span class="at" style="color: #657422;
+background-color: null;
+font-style: inherit;">parse =</span> <span class="cn" style="color: #8f5902;
+background-color: null;
+font-style: inherit;">TRUE</span>,</span>
+<span id="cb58-32">    <span class="at" style="color: #657422;
+background-color: null;
+font-style: inherit;">vjust =</span> <span class="sc" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;">-</span><span class="fl" style="color: #AD0000;
+background-color: null;
+font-style: inherit;">0.5</span>)</span></code></pre></div>
+</div>
+</section>
+</section>
+<section id="canvas-question-4" class="level2 tabset tabset-fade tabset-pills">
+<h2 class="tabset tabset-fade tabset-pills anchored" data-anchor-id="canvas-question-4">Canvas Question 4</h2>
+<p><span class="instruction">Create a regression model in which you predict math at time 2 (<code>math_2</code>) using reading at time 1 (<code>math_1</code>). Call the model fit object <code>m_math1</code>.</span></p>
+<p><span class="question"><strong>Question</strong>: Does <code>math_1</code> predict <code>math_2</code>?.</span></p>
+<p><span class="question"><strong>Question</strong>: Does <code>math_1</code> still predict <code>math_2</code> after controlling for <code>read_1</code>?</span></p>
+<section id="hide-hint-1" class="level3">
+<h3 class="anchored" data-anchor-id="hide-hint-1">Hide Hint</h3>
+</section>
+<section id="hint-4" class="level3">
+<h3 class="anchored" data-anchor-id="hint-4">Hint</h3>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb59" style="background: #f1f3f5;"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb59-1">m_math <span class="ot" style="color: #003B4F;
+background-color: null;
+font-style: inherit;">&lt;-</span> <span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">lm</span>(math_2 <span class="sc" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;">~</span> math_1 <span class="sc" style="color: #5E5E5E;
+background-color: null;
+font-style: inherit;">+</span> read_1, <span class="at" style="color: #657422;
+background-color: null;
+font-style: inherit;">data =</span> d_learn)</span>
+<span id="cb59-2"><span class="fu" style="color: #4758AB;
+background-color: null;
+font-style: inherit;">summary</span>(m_math)</span></code></pre></div>
+</div>
+
+
+</section>
+</section>
+</section>
+
+ ]]></description>
+  <guid>https://wjschne.github.io/tutorials/Regression/</guid>
+  <pubDate>Thu, 29 Feb 2024 18:24:37 GMT</pubDate>
 </item>
 </channel>
 </rss>
diff --git a/docs/tutorials/ANOVA/ANOVA.html b/docs/tutorials/ANOVA/ANOVA.html
index 463f927..6d7cfcf 100644
--- a/docs/tutorials/ANOVA/ANOVA.html
+++ b/docs/tutorials/ANOVA/ANOVA.html
@@ -2,7 +2,7 @@
 <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"><head>
 
 <meta charset="utf-8">
-<meta name="generator" content="quarto-1.4.548">
+<meta name="generator" content="quarto-1.5.9">
 
 <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
 
@@ -105,6 +105,49 @@
   }
 }</script>
 
+  <script>window.backupDefine = window.define; window.define = undefined;</script><script src="https://cdn.jsdelivr.net/npm/katex@0.15.1/dist/katex.min.js"></script>
+  <script>document.addEventListener("DOMContentLoaded", function () {
+ var mathElements = document.getElementsByClassName("math");
+ var macros = [];
+ for (var i = 0; i < mathElements.length; i++) {
+  var texText = mathElements[i].firstChild;
+  if (mathElements[i].tagName == "SPAN") {
+   katex.render(texText.data, mathElements[i], {
+    displayMode: mathElements[i].classList.contains('display'),
+    throwOnError: false,
+    macros: macros,
+    fleqn: false
+   });
+}}});
+  </script>
+  <script>window.define = window.backupDefine; window.backupDefine = undefined;</script><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.15.1/dist/katex.min.css">
+
+<script type="text/javascript">
+const typesetMath = (el) => {
+  if (window.MathJax) {
+    // MathJax Typeset
+    window.MathJax.typeset([el]);
+  } else if (window.katex) {
+    // KaTeX Render
+    var mathElements = el.getElementsByClassName("math");
+    var macros = [];
+    for (var i = 0; i < mathElements.length; i++) {
+      var texText = mathElements[i].firstChild;
+      if (mathElements[i].tagName == "SPAN") {
+        window.katex.render(texText.data, mathElements[i], {
+          displayMode: mathElements[i].classList.contains('display'),
+          throwOnError: false,
+          macros: macros,
+          fleqn: false
+        });
+      }
+    }
+  }
+}
+window.Quarto = {
+  typesetMath
+};
+</script>
 
 <link rel="stylesheet" href="../../styles.css">
 <meta property="og:title" content="W. Joel Schneider">
@@ -233,9 +276,10 @@ <h2 id="toc-title">On this page</h2>
   <li><a href="#plot" id="toc-plot" class="nav-link" data-scroll-target="#plot">Plot</a></li>
   <li><a href="#create-one-way-anova" id="toc-create-one-way-anova" class="nav-link" data-scroll-target="#create-one-way-anova">Create One-Way ANOVA</a>
   <ul class="collapse">
+  <li><a href="#diagnostic-plots" id="toc-diagnostic-plots" class="nav-link" data-scroll-target="#diagnostic-plots">Diagnostic Plots</a></li>
   <li><a href="#summary" id="toc-summary" class="nav-link" data-scroll-target="#summary">Summary</a></li>
   <li><a href="#other-summary-functions" id="toc-other-summary-functions" class="nav-link" data-scroll-target="#other-summary-functions">Other Summary Functions</a></li>
-  <li><a href="#diagnostic-plots" id="toc-diagnostic-plots" class="nav-link" data-scroll-target="#diagnostic-plots">Diagnostic Plots</a></li>
+  <li><a href="#automated-interpretation" id="toc-automated-interpretation" class="nav-link" data-scroll-target="#automated-interpretation">Automated Interpretation</a></li>
   <li><a href="#post-hoc-tests" id="toc-post-hoc-tests" class="nav-link" data-scroll-target="#post-hoc-tests">Post-hoc Tests</a></li>
   </ul></li>
   </ul>
@@ -243,7 +287,7 @@ <h2 id="toc-title">On this page</h2>
 </nav>
 <div id="quarto-sidebar-glass" class="quarto-sidebar-collapse-item" data-bs-toggle="collapse" data-bs-target=".quarto-sidebar-collapse-item"></div>
 <!-- margin-sidebar -->
-    <div id="quarto-margin-sidebar" class="sidebar margin-sidebar">
+    <div id="quarto-margin-sidebar" class="sidebar margin-sidebar zindex-bottom">
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 <!-- main -->
 <main class="content" id="quarto-document-content">
@@ -276,7 +320,8 @@ <h1 class="title">ANOVA in R</h1>
 <section id="load-packages" class="level1">
 <h1>Load Packages</h1>
 <div class="cell">
-<div class="sourceCode cell-code" id="cb1"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(tidyverse)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode cell-code" id="cb1"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(tidyverse)</span>
+<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(easystats)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 </div>
 </section>
 <section id="import-data-and-create-labelled-factors" class="level1">
@@ -356,11 +401,25 @@ <h1>Create One-Way ANOVA</h1>
 <div class="cell">
 <div class="sourceCode cell-code" id="cb5"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a>fit <span class="ot">&lt;-</span> <span class="fu">aov</span>(Anxiety_Questionnaire <span class="sc">~</span> Attachment, <span class="at">data =</span> d)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 </div>
+<section id="diagnostic-plots" class="level2">
+<h2 class="anchored" data-anchor-id="diagnostic-plots">Diagnostic Plots</h2>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb6"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a><span class="fu">check_model</span>(fit)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output-display">
+<div>
+<figure class="figure">
+<p><img src="ANOVA_files/figure-html/assumptions-1.png" class="img-fluid figure-img" width="672"></p>
+</figure>
+</div>
+</div>
+</div>
+<p>None of the diagnostic checks raise any alarms. Specifically, the residuals are reasonably normal and have consistent variance acrross groups (i.e., the homogeneeity of variance assumption is reasonable).</p>
+</section>
 <section id="summary" class="level2">
 <h2 class="anchored" data-anchor-id="summary">Summary</h2>
-<p>The base R function <code>summary</code> gives us most of what we might want.</p>
+<p>The base R function <code>summary</code> gives us most of what we might want to know.</p>
 <div class="cell">
-<div class="sourceCode cell-code" id="cb6"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a><span class="fu">summary</span>(fit)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode cell-code" id="cb7"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a><span class="fu">summary</span>(fit)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <div class="cell-output cell-output-stdout">
 <pre><code>             Df Sum Sq Mean Sq F value Pr(&gt;F)    
 Attachment    2  20359   10179      93 &lt;2e-16 ***
@@ -369,149 +428,51 @@ <h2 class="anchored" data-anchor-id="summary">Summary</h2>
 Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1</code></pre>
 </div>
 </div>
-</section>
-<section id="other-summary-functions" class="level2">
-<h2 class="anchored" data-anchor-id="other-summary-functions">Other Summary Functions</h2>
-<p>If you want other summary functions, there are many options. Here are a few:</p>
+<p>We can see that overall model is significant, meaning that at least two groups have reliably different means. At this point we do not know which means differ, but from the plot we have a good guess that the Ambivalent group scores higher on the Anxiety Questionnaire than the other two groups.</p>
+<p>A similar, but tidier display can be found with the <code>parameters</code> function:</p>
 <div class="cell">
-<div class="sourceCode cell-code" id="cb8"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb8-1"><a href="#cb8-1" aria-hidden="true" tabindex="-1"></a>broom<span class="sc">::</span><span class="fu">glance</span>(fit)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode cell-code" id="cb9"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb9-1"><a href="#cb9-1" aria-hidden="true" tabindex="-1"></a><span class="fu">parameters</span>(fit)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <div class="cell-output-display">
 <div class="kable-table">
 <table class="table table-sm table-striped small">
 <thead>
 <tr class="header">
-<th style="text-align: right;">logLik</th>
-<th style="text-align: right;">AIC</th>
-<th style="text-align: right;">BIC</th>
-<th style="text-align: right;">deviance</th>
-<th style="text-align: right;">nobs</th>
-<th style="text-align: right;">r.squared</th>
-</tr>
-</thead>
-<tbody>
-<tr class="odd">
-<td style="text-align: right;">-1129</td>
-<td style="text-align: right;">2265</td>
-<td style="text-align: right;">2280</td>
-<td style="text-align: right;">32517</td>
-<td style="text-align: right;">300</td>
-<td style="text-align: right;">0.39</td>
-</tr>
-</tbody>
-</table>
-</div>
-</div>
-<div class="sourceCode cell-code" id="cb9"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb9-1"><a href="#cb9-1" aria-hidden="true" tabindex="-1"></a>broom<span class="sc">::</span><span class="fu">tidy</span>(fit)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
-<div class="cell-output-display">
-<div class="kable-table">
-<table class="table table-sm table-striped small">
-<thead>
-<tr class="header">
-<th style="text-align: left;">term</th>
+<th style="text-align: left;">Parameter</th>
+<th style="text-align: right;">Sum_Squares</th>
 <th style="text-align: right;">df</th>
-<th style="text-align: right;">sumsq</th>
-<th style="text-align: right;">meansq</th>
-<th style="text-align: right;">statistic</th>
-<th style="text-align: right;">p.value</th>
+<th style="text-align: right;">Mean_Square</th>
+<th style="text-align: right;">F</th>
+<th style="text-align: right;">p</th>
 </tr>
 </thead>
 <tbody>
 <tr class="odd">
 <td style="text-align: left;">Attachment</td>
-<td style="text-align: right;">2</td>
 <td style="text-align: right;">20359</td>
-<td style="text-align: right;">10179</td>
-<td style="text-align: right;">93</td>
-<td style="text-align: right;">0</td>
-</tr>
-<tr class="even">
-<td style="text-align: left;">Residuals</td>
-<td style="text-align: right;">297</td>
-<td style="text-align: right;">32517</td>
-<td style="text-align: right;">109</td>
-<td style="text-align: right;">NA</td>
-<td style="text-align: right;">NA</td>
-</tr>
-</tbody>
-</table>
-</div>
-</div>
-<div class="sourceCode cell-code" id="cb10"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb10-1"><a href="#cb10-1" aria-hidden="true" tabindex="-1"></a>sjstats<span class="sc">::</span><span class="fu">anova_stats</span>(fit)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
-<div class="cell-output-display">
-<div class="kable-table">
-<table class="table table-sm table-striped small">
-<colgroup>
-<col style="width: 8%">
-<col style="width: 8%">
-<col style="width: 3%">
-<col style="width: 4%">
-<col style="width: 5%">
-<col style="width: 7%">
-<col style="width: 6%">
-<col style="width: 4%">
-<col style="width: 11%">
-<col style="width: 6%">
-<col style="width: 12%">
-<col style="width: 7%">
-<col style="width: 7%">
-<col style="width: 4%">
-</colgroup>
-<thead>
-<tr class="header">
-<th style="text-align: left;"></th>
-<th style="text-align: left;">term</th>
-<th style="text-align: right;">df</th>
-<th style="text-align: right;">sumsq</th>
-<th style="text-align: right;">meansq</th>
-<th style="text-align: right;">statistic</th>
-<th style="text-align: right;">p.value</th>
-<th style="text-align: right;">etasq</th>
-<th style="text-align: right;">partial.etasq</th>
-<th style="text-align: right;">omegasq</th>
-<th style="text-align: right;">partial.omegasq</th>
-<th style="text-align: right;">epsilonsq</th>
-<th style="text-align: right;">cohens.f</th>
-<th style="text-align: right;">power</th>
-</tr>
-</thead>
-<tbody>
-<tr class="odd">
-<td style="text-align: left;">Attachment</td>
-<td style="text-align: left;">Attachment</td>
 <td style="text-align: right;">2</td>
-<td style="text-align: right;">20359</td>
 <td style="text-align: right;">10179</td>
 <td style="text-align: right;">93</td>
 <td style="text-align: right;">0</td>
-<td style="text-align: right;">0.38</td>
-<td style="text-align: right;">0.38</td>
-<td style="text-align: right;">0.38</td>
-<td style="text-align: right;">0.38</td>
-<td style="text-align: right;">0.38</td>
-<td style="text-align: right;">0.79</td>
-<td style="text-align: right;">1</td>
 </tr>
 <tr class="even">
-<td style="text-align: left;">…2</td>
 <td style="text-align: left;">Residuals</td>
-<td style="text-align: right;">297</td>
 <td style="text-align: right;">32517</td>
+<td style="text-align: right;">297</td>
 <td style="text-align: right;">109</td>
 <td style="text-align: right;">NA</td>
 <td style="text-align: right;">NA</td>
-<td style="text-align: right;">NA</td>
-<td style="text-align: right;">NA</td>
-<td style="text-align: right;">NA</td>
-<td style="text-align: right;">NA</td>
-<td style="text-align: right;">NA</td>
-<td style="text-align: right;">NA</td>
-<td style="text-align: right;">NA</td>
 </tr>
 </tbody>
 </table>
 </div>
 </div>
-<div class="sourceCode cell-code" id="cb11"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb11-1"><a href="#cb11-1" aria-hidden="true" tabindex="-1"></a>performance<span class="sc">::</span><span class="fu">performance</span>(fit)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+</div>
+</section>
+<section id="other-summary-functions" class="level2">
+<h2 class="anchored" data-anchor-id="other-summary-functions">Other Summary Functions</h2>
+<p>The overall “fit” or “performance” of the model:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb10"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb10-1"><a href="#cb10-1" aria-hidden="true" tabindex="-1"></a><span class="fu">performance</span>(fit)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <div class="cell-output-display">
 <div class="kable-table">
 <table class="table table-sm table-striped small">
@@ -540,89 +501,118 @@ <h2 class="anchored" data-anchor-id="other-summary-functions">Other Summary Func
 </table>
 </div>
 </div>
-<div class="sourceCode cell-code" id="cb12"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb12-1"><a href="#cb12-1" aria-hidden="true" tabindex="-1"></a>parameters<span class="sc">::</span><span class="fu">parameters</span>(fit)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+</div>
+<ul>
+<li><code>AIC</code>: <a href="https://en.wikipedia.org/wiki/Akaike_information_criterion">Akaike’s Information Criterion</a></li>
+<li><code>AICc</code>: AIC with a correction for small sample sizes</li>
+<li><code>BIC</code>: <a href="https://en.wikipedia.org/wiki/Bayesian_information_criterion">Bayesian Information Criterion</a></li>
+<li><code>R2</code>: R-squared: <a href="https://en.wikipedia.org/wiki/Coefficient_of_determination">Coefficient of Determination</a></li>
+<li><code>R2_adj</code>: <a href="https://en.wikipedia.org/wiki/Coefficient_of_determination#Adjusted_R2">Adjusted r-squared</a></li>
+<li><code>RMSE</code>: Root mean squared error <span class="math inline">=\sqrt{\frac{\sum_{i = 1}^{n}{e_i}}{n}}</span></li>
+<li><code>SIGMA</code>: Residual standard deviation (The SD of the residuals <span class="math inline">=\sqrt{\frac{\sum_{i = 1}^{n}{e_i}}{n-k-1}}</span>)</li>
+</ul>
+</section>
+<section id="automated-interpretation" class="level2">
+<h2 class="anchored" data-anchor-id="automated-interpretation">Automated Interpretation</h2>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb11"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb11-1"><a href="#cb11-1" aria-hidden="true" tabindex="-1"></a><span class="fu">report</span>(fit)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output cell-output-stdout">
+<pre><code>The ANOVA (formula: Anxiety_Questionnaire ~ Attachment) suggests that:
+
+  - The main effect of Attachment is statistically significant and large (F(2,
+297) = 92.98, p &lt; .001; Eta2 = 0.39, 95% CI [0.32, 1.00])
+
+Effect sizes were labelled following Field's (2013) recommendations.</code></pre>
+</div>
+</div>
+</section>
+<section id="post-hoc-tests" class="level2">
+<h2 class="anchored" data-anchor-id="post-hoc-tests">Post-hoc Tests</h2>
+<p>We would like to compare all means but control for family-wise error.</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb13"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb13-1"><a href="#cb13-1" aria-hidden="true" tabindex="-1"></a><span class="fu">estimate_contrasts</span>(fit)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output cell-output-stderr">
+<pre><code>No variable was specified for contrast estimation. Selecting `contrast = "Attachment"`.</code></pre>
+</div>
 <div class="cell-output-display">
 <div class="kable-table">
 <table class="table table-sm table-striped small">
+<colgroup>
+<col style="width: 13%">
+<col style="width: 16%">
+<col style="width: 16%">
+<col style="width: 10%">
+<col style="width: 12%">
+<col style="width: 6%">
+<col style="width: 6%">
+<col style="width: 9%">
+<col style="width: 7%">
+</colgroup>
 <thead>
 <tr class="header">
-<th style="text-align: left;">Parameter</th>
-<th style="text-align: right;">Sum_Squares</th>
+<th style="text-align: left;">Level1</th>
+<th style="text-align: left;">Level2</th>
+<th style="text-align: right;">Difference</th>
+<th style="text-align: right;">CI_low</th>
+<th style="text-align: right;">CI_high</th>
+<th style="text-align: right;">SE</th>
 <th style="text-align: right;">df</th>
-<th style="text-align: right;">Mean_Square</th>
-<th style="text-align: right;">F</th>
+<th style="text-align: right;">t</th>
 <th style="text-align: right;">p</th>
 </tr>
 </thead>
 <tbody>
 <tr class="odd">
-<td style="text-align: left;">Attachment</td>
-<td style="text-align: right;">20359</td>
-<td style="text-align: right;">2</td>
-<td style="text-align: right;">10179</td>
-<td style="text-align: right;">93</td>
-<td style="text-align: right;">0</td>
+<td style="text-align: left;">Avoidant</td>
+<td style="text-align: left;">Ambivalent</td>
+<td style="text-align: right;">-18.4</td>
+<td style="text-align: right;">-22.1</td>
+<td style="text-align: right;">-14.8</td>
+<td style="text-align: right;">1.5</td>
+<td style="text-align: right;">297</td>
+<td style="text-align: right;">-12.1</td>
+<td style="text-align: right;">0.00</td>
 </tr>
 <tr class="even">
-<td style="text-align: left;">Residuals</td>
-<td style="text-align: right;">32517</td>
+<td style="text-align: left;">Secure</td>
+<td style="text-align: left;">Ambivalent</td>
+<td style="text-align: right;">-16.2</td>
+<td style="text-align: right;">-19.6</td>
+<td style="text-align: right;">-12.7</td>
+<td style="text-align: right;">1.4</td>
 <td style="text-align: right;">297</td>
-<td style="text-align: right;">109</td>
-<td style="text-align: right;">NA</td>
-<td style="text-align: right;">NA</td>
+<td style="text-align: right;">-11.3</td>
+<td style="text-align: right;">0.00</td>
+</tr>
+<tr class="odd">
+<td style="text-align: left;">Secure</td>
+<td style="text-align: left;">Avoidant</td>
+<td style="text-align: right;">2.3</td>
+<td style="text-align: right;">-1.4</td>
+<td style="text-align: right;">5.9</td>
+<td style="text-align: right;">1.5</td>
+<td style="text-align: right;">297</td>
+<td style="text-align: right;">1.5</td>
+<td style="text-align: right;">0.13</td>
 </tr>
 </tbody>
 </table>
 </div>
 </div>
-<div class="sourceCode cell-code" id="cb13"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb13-1"><a href="#cb13-1" aria-hidden="true" tabindex="-1"></a>report<span class="sc">::</span><span class="fu">report</span>(fit)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
-<div class="cell-output cell-output-stdout">
-<pre><code>The ANOVA (formula: Anxiety_Questionnaire ~ Attachment) suggests that:
-
-  - The main effect of Attachment is statistically significant and large (F(2,
-297) = 92.98, p &lt; .001; Eta2 = 0.39, 95% CI [0.32, 1.00])
-
-Effect sizes were labelled following Field's (2013) recommendations.</code></pre>
-</div>
-</div>
-</section>
-<section id="diagnostic-plots" class="level2">
-<h2 class="anchored" data-anchor-id="diagnostic-plots">Diagnostic Plots</h2>
-<div class="cell">
-<div class="sourceCode cell-code" id="cb15"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb15-1"><a href="#cb15-1" aria-hidden="true" tabindex="-1"></a>performance<span class="sc">::</span><span class="fu">check_model</span>(fit)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
-<div class="cell-output-display">
-<div>
-<figure class="figure">
-<p><img src="ANOVA_files/figure-html/assumptions-1.png" class="img-fluid figure-img" width="672"></p>
-</figure>
+<div class="sourceCode cell-code" id="cb15"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb15-1"><a href="#cb15-1" aria-hidden="true" tabindex="-1"></a><span class="fu">estimate_contrasts</span>(fit) <span class="sc">|&gt;</span> </span>
+<span id="cb15-2"><a href="#cb15-2" aria-hidden="true" tabindex="-1"></a>  <span class="fu">report</span>()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output cell-output-stderr">
+<pre><code>No variable was specified for contrast estimation. Selecting `contrast = "Attachment"`.</code></pre>
 </div>
-</div>
-</div>
-</section>
-<section id="post-hoc-tests" class="level2">
-<h2 class="anchored" data-anchor-id="post-hoc-tests">Post-hoc Tests</h2>
-<p>Tukey Honestly Significant Differences</p>
-<div class="cell">
-<div class="sourceCode cell-code" id="cb16"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb16-1"><a href="#cb16-1" aria-hidden="true" tabindex="-1"></a><span class="fu">TukeyHSD</span>(fit)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <div class="cell-output cell-output-stdout">
-<pre><code>  Tukey multiple comparisons of means
-    95% family-wise confidence level
-
-Fit: aov(formula = Anxiety_Questionnaire ~ Attachment, data = d)
-
-$Attachment
-                    diff  lwr  upr p adj
-Avoidant-Secure     -2.3 -5.8  1.3  0.29
-Ambivalent-Secure   16.2 12.8 19.5  0.00
-Ambivalent-Avoidant 18.4 14.8 22.0  0.00</code></pre>
-</div>
-<div class="sourceCode cell-code" id="cb18"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb18-1"><a href="#cb18-1" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(<span class="fu">TukeyHSD</span>(fit))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
-<div class="cell-output-display">
-<div>
-<figure class="figure">
-<p><img src="ANOVA_files/figure-html/tukey-1.png" class="img-fluid figure-img" width="672"></p>
-</figure>
-</div>
+<pre><code>The marginal contrasts analysis suggests the following. The difference between
+Avoidant and Ambivalent is negative and statistically significant (difference =
+-18.44, 95% CI [-22.12, -14.77], t(297) = -12.08, p &lt; .001). The difference
+between Secure and Ambivalent is negative and statistically significant
+(difference = -16.18, 95% CI [-19.62, -12.74], t(297) = -11.33, p &lt; .001). The
+difference between Secure and Avoidant is positive and statistically
+non-significant (difference = 2.27, 95% CI [ -1.37, 5.90], t(297) = 1.50, p =
+0.135)</code></pre>
 </div>
 </div>
 
diff --git a/docs/tutorials/ANOVA/ANOVA_files/figure-html/assumptions-1.png b/docs/tutorials/ANOVA/ANOVA_files/figure-html/assumptions-1.png
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+<h1 class="title">Regression in R</h1>
+</div>
+
+
+
+<div class="quarto-title-meta">
+
+    <div>
+    <div class="quarto-title-meta-heading">Author</div>
+    <div class="quarto-title-meta-contents">
+             <p>W. Joel Schneider </p>
+          </div>
+  </div>
+    
+  
+    
+  </div>
+  
+
+
+</header>
+
+
+<p>Here we are going to use a small data set to predict people’s height using the height of their parents.</p>
+<section id="loading-packages" class="level1 page-columns page-full">
+<h1>Loading packages</h1>
+
+<div class="no-row-height column-margin column-container"><div class="callout callout-style-default callout-note callout-titled" title="R Package">
+<div class="callout-header d-flex align-content-center">
+<div class="callout-icon-container">
+<i class="callout-icon"></i>
+</div>
+<div class="callout-title-container flex-fill">
+R Package
+</div>
+</div>
+<div class="callout-body-container callout-body">
+<p>A <strong>R package</strong> is software that extends the capabilities of R. To use a package, you must first install it on your machine. You only need to install it once. By default, the package tab is in the lower right pane in RStudio. It has two buttons, one for installing new packages and one for updating packages you have previously installed.</p>
+<div id="fig-packages" class="quarto-figure quarto-figure-center quarto-float anchored">
+<figure class="quarto-float quarto-float-fig figure">
+<div aria-describedby="fig-packages-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
+<img src="updatepackages.png" class="img-fluid figure-img">
+</div>
+<figcaption class="quarto-float-caption-bottom quarto-float-caption quarto-float-fig" id="fig-packages-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
+Figure&nbsp;1: RStudio’s “Install” and “Update” buttons in the “Package” tab in the lower right pane
+</figcaption>
+</figure>
+</div>
+</div>
+</div></div><p>First we load packages to enhance R’s abilities. Some packages are designed to work with several other packages as a system. The <a href="https://www.tidyverse.org/">tidyverse package</a> is a “meta-package” that installs and loads a coherent set of packages designed to help you import, manipulate, visualize, and interpret data. If you do not have a recent version of tidyverse already installed, you can install it with this code:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb1"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="fu">install.packages</span>(<span class="st">"tidyverse"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+</div>
+<div class="callout callout-style-default callout-tip callout-titled">
+<div class="callout-header d-flex align-content-center">
+<div class="callout-icon-container">
+<i class="callout-icon"></i>
+</div>
+<div class="callout-title-container flex-fill">
+Tip
+</div>
+</div>
+<div class="callout-body-container callout-body">
+<p>You only have to install the tidyverse once. Thereafter, from time to time, you can let RStudio’s “Update” button in the “Packages” pane help you keep your packages current.</p>
+</div>
+</div>
+<p>When you “load” the tidyvert</p>
+<p>The <code>tidyverse</code> allows us to do many things, including plotting with <code>ggplot2</code>. The <code>haven</code> package imports SPSS data. The <code>broom</code> package tidies up our regression results with the <code>tidy</code>, <code>glance</code>, and <code>augment</code> functions.</p>
+<p>If any of these packages are not installed, click the <strong>Packages</strong> tab, click the <strong>Install</strong> button, enter the names of the packages separated by commas, and click <strong>Install</strong>.</p>
+<p>To load the packages, run this code:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb2"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(tidyverse) <span class="co"># Loads primary packages for data wrangling and plotting</span></span>
+<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(haven) <span class="co"># Imports data saved in SPSS, SAS, or Stata format</span></span>
+<span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(broom) <span class="co"># Displays model information</span></span>
+<span id="cb2-4"><a href="#cb2-4" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(performance) <span class="co"># Displays model information </span></span>
+<span id="cb2-5"><a href="#cb2-5" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(parameters) <span class="co"># Displays model parameter information</span></span>
+<span id="cb2-6"><a href="#cb2-6" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(report) <span class="co"># Automated interpretations</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+</div>
+</section>
+<section id="import-data" class="level1 page-columns page-full">
+<h1>Import data</h1>
+
+<div class="no-row-height column-margin column-container"><div class="">
+<p>If I use only one data set in an analysis, I call it <code>d</code>. If I need multiple data sets, I use a <code>d_</code> prefix to differentiate them. For example, if I have separate data sets for juniors and seniors, I might call them, <code>d_juniors</code> and <code>d_seniors</code>, respectively. This kind of consistency seems like extra work, but it pays off in making it easy for <em>future-you</em> to read and understand your own code.</p>
+</div></div><p>You can import the <code>height.sav</code> data set directly from my github repository for this course. There is no need to save the data anywhere. This code loads the data into variable <code>d</code> using the <a href="https://haven.tidyverse.org/reference/read_spss.html"><code>read_spss</code></a> function from the <a href="https://haven.tidyverse.org/">haven package</a>.</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb3"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a>d <span class="ot">&lt;-</span> <span class="fu">read_spss</span>(<span class="st">"https://github.com/wjschne/EDUC5325/raw/master/height.sav"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+</div>
+</section>
+<section id="make-a-plot" class="level1">
+<h1>Make a plot</h1>
+<p>Use the <a href="https://ggplot2.tidyverse.org/reference/ggplot.html"><code>ggplot</code></a> function from the <a href="https://ggplot2.tidyverse.org/">ggplot2</a> package to make a scatterplot with points and a regression line:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb4"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplot</span>(d, <span class="fu">aes</span>(avgphgt,height)) <span class="sc">+</span></span>
+<span id="cb4-2"><a href="#cb4-2" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_point</span>() <span class="sc">+</span></span>
+<span id="cb4-3"><a href="#cb4-3" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_smooth</span>(<span class="at">method =</span> <span class="st">"lm"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output-display">
+<div>
+<figure class="figure">
+<p><img src="index_files/figure-html/unnamed-chunk-3-1.svg" style="width:100.0%;height:100.0%" class="figure-img"></p>
+</figure>
+</div>
+</div>
+</div>
+<p>Annotated, here is what the code does:</p>
+<div class="quarto-figure quarto-figure-center">
+<figure class="figure">
+<p><img src="ggplotcode.svg" class="img-fluid figure-img"></p>
+<figcaption>Annotated ggplot2 code</figcaption>
+</figure>
+</div>
+</section>
+<section id="save-the-plot" class="level1">
+<h1>Save the plot!</h1>
+<p>You can save to your hard drive a high-quality plot with the <a href="https://ggplot2.tidyverse.org/reference/ggsave.html"><code>ggsave</code></a> function. It will save whatever the last plot you created with <a href="https://ggplot2.tidyverse.org/reference/ggplot.html"><code>ggplot</code></a>. It will guess the file format from the file extension.</p>
+<p>Here I save a .pdf file of the plot:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb5"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a><span class="fu">ggsave</span>(<span class="st">"my_plot.pdf"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+</div>
+<section id="vector-based-images" class="level2">
+<h2 class="anchored" data-anchor-id="vector-based-images">Vector-based Images</h2>
+<p>The .pdf format gives the best image quality but can only be viewed in a .pdf reader. The .svg format is almost as good and can be incorporated into webpages and Office documents. One downside of their near-perfect image quality is that .pdf and .svg image file sizes can become quite large.</p>
+</section>
+<section id="raster-images" class="level2">
+<h2 class="anchored" data-anchor-id="raster-images">Raster Images</h2>
+<p>The .png format gives good image quality and renders small file sizes.</p>
+<p>The primary use of the .gif format is to create animated plots. Otherwise stick with .png.</p>
+<p>Although the .jpg format is good for photos, it is terrible for plots—it often renders text and sharp corners with pixelated smudges.</p>
+</section>
+</section>
+<section id="creating-the-model" class="level1">
+<h1>Creating the model</h1>
+<p>To run regression models, use the <a href="https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/lm"><code>lm</code></a> function. The <code>lm</code> stands for “linear model.” The general pattern is <code>lm(Y~X, data = d)</code>, which means “<code>Y</code> is predicted by <code>X</code>, using the data set <code>d</code>.”</p>
+<p>Here we predict height from Average of Parent Height (<code>avgphgt</code>):</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb6"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a>m1 <span class="ot">&lt;-</span> <span class="fu">lm</span>(height<span class="sc">~</span>avgphgt, <span class="at">data =</span> d)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+</div>
+<p>Notice that we did not get any results. Instead, we created the model fit object <code>m1</code>, which contains information about the model. There are a variety of functions we can use to extract information from <code>m1</code>.</p>
+</section>
+<section id="checking-assumptions" class="level1">
+<h1>Checking Assumptions</h1>
+<p>Regression assumes that the observations should independent, and the residuals should be normal and homoscedastic. The <a href="https://easystats.github.io/performance/index.html">performance package</a> has a great function for checking model assumptions: <a href="https://easystats.github.io/performance/reference/check_model.html"><code>check_model</code></a></p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb7"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a><span class="fu">check_model</span>(m1)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output-display">
+<div>
+<figure class="figure">
+<p><img src="index_files/figure-html/unnamed-chunk-5-1.svg" style="width:100.0%;height:100.0%" class="figure-img"></p>
+</figure>
+</div>
+</div>
+</div>
+<p>Here we see that none of the assumptions have been severely violated.</p>
+</section>
+<section id="summarising-results" class="level1">
+<h1>Summarising Results</h1>
+<p>Base R gives you most of what you would want to know about the regression results with the <code>summary</code> function:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb8"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb8-1"><a href="#cb8-1" aria-hidden="true" tabindex="-1"></a><span class="fu">summary</span>(m1)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output cell-output-stdout">
+<pre><code>
+Call:
+lm(formula = height ~ avgphgt, data = d)
+
+Residuals:
+   Min     1Q Median     3Q    Max 
+-5.241 -1.649 -0.539  1.224  6.163 
+
+Coefficients:
+            Estimate Std. Error t value Pr(&gt;|t|)    
+(Intercept)  -12.878      9.779   -1.32      0.2    
+avgphgt        1.193      0.145    8.23  5.7e-10 ***
+---
+Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
+
+Residual standard error: 2.68 on 38 degrees of freedom
+Multiple R-squared:  0.641, Adjusted R-squared:  0.631 
+F-statistic: 67.7 on 1 and 38 DF,  p-value: 5.7e-10</code></pre>
+</div>
+</div>
+<p>This output is not pretty, nor was it intended to be. It is designed for you, the analyst. The <code>summary</code> function’s print method is optimized for reading results in the console, not in a document. Presentation-worthy results need a lot more care and attention.</p>
+<p>An automated report from the <a href="https://easystats.github.io/report/articles/report.html">report</a> package:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb10"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb10-1"><a href="#cb10-1" aria-hidden="true" tabindex="-1"></a>report<span class="sc">::</span><span class="fu">report</span>(m1)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output cell-output-stdout">
+<pre><code>We fitted a linear model (estimated using OLS) to predict height with avgphgt
+(formula: height ~ avgphgt). The model explains a statistically significant and
+substantial proportion of variance (R2 = 0.64, F(1, 38) = 67.73, p &lt; .001, adj.
+R2 = 0.63). The model's intercept, corresponding to avgphgt = 0, is at -12.88
+(95% CI [-32.67, 6.92], t(38) = -1.32, p = 0.196). Within this model:
+
+  - The effect of avgphgt is statistically significant and positive (beta = 1.19,
+95% CI [0.90, 1.49], t(38) = 8.23, p &lt; .001; Std. beta = 0.80, 95% CI [0.60,
+1.00])
+
+Standardized parameters were obtained by fitting the model on a standardized
+version of the dataset. 95% Confidence Intervals (CIs) and p-values were
+computed using a Wald t-distribution approximation.</code></pre>
+</div>
+</div>
+<p>The <code>glance</code>, <code>tidy</code>, and <code>augment</code> functions from the <a href="https://broom.tidymodels.org/">broom</a> package make the output more amenable for further analysis or for presentation.</p>
+<section id="model-level-statistics" class="level2">
+<h2 class="anchored" data-anchor-id="model-level-statistics">Model-Level Statistics</h2>
+<p>Some statistics like the <em>coefficient of determination</em> (<em>R</em><sup>2</sup>) or the <em>standard error of the estimate</em> (<em>σ<sub>e</sub></em>) describe the model as a whole.</p>
+<p>The model-level statistics can be extracted with broom’s <a href="https://broom.tidymodels.org/reference/data.frame_tidiers.html"><code>glance</code></a> function or the performance package’s <a href="https://easystats.github.io/performance/reference/model_performance.lm.html"><code>model_performance</code></a> function.</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb12"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb12-1"><a href="#cb12-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Using broom package</span></span>
+<span id="cb12-2"><a href="#cb12-2" aria-hidden="true" tabindex="-1"></a><span class="fu">glance</span>(m1)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output-display">
+<div>
+<table class="lightable-paper table table-sm table-striped small" data-quarto-postprocess="true">
+<thead>
+<tr class="header">
+<th style="text-align: right;" data-quarto-table-cell-role="th">r.squared</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">adj.r.squared</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">sigma</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">statistic</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">p.value</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">df</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">logLik</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">AIC</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">BIC</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">deviance</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">df.residual</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">nobs</th>
+</tr>
+</thead>
+<tbody>
+<tr class="odd">
+<td style="text-align: right;">0.64</td>
+<td style="text-align: right;">0.63</td>
+<td style="text-align: right;">2.68</td>
+<td style="text-align: right;">67.7</td>
+<td style="text-align: right;">0</td>
+<td style="text-align: right;">1</td>
+<td style="text-align: right;">-95.2</td>
+<td style="text-align: right;">196</td>
+<td style="text-align: right;">201</td>
+<td style="text-align: right;">273</td>
+<td style="text-align: right;">38</td>
+<td style="text-align: right;">40</td>
+</tr>
+</tbody>
+</table>
+
+
+</div>
+</div>
+<div class="sourceCode cell-code" id="cb13"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb13-1"><a href="#cb13-1" aria-hidden="true" tabindex="-1"></a><span class="fu">model_parameters</span>(m1)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output-display">
+<div>
+<table class="lightable-paper table table-sm table-striped small" data-quarto-postprocess="true">
+<thead>
+<tr class="header">
+<th style="text-align: left;" data-quarto-table-cell-role="th">Parameter</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">Coefficient</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">SE</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">CI</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">CI_low</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">CI_high</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">t</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">df_error</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">p</th>
+</tr>
+</thead>
+<tbody>
+<tr class="odd">
+<td style="text-align: left;">(Intercept)</td>
+<td style="text-align: right;">-12.88</td>
+<td style="text-align: right;">9.78</td>
+<td style="text-align: right;">0.95</td>
+<td style="text-align: right;">-32.7</td>
+<td style="text-align: right;">6.92</td>
+<td style="text-align: right;">-1.32</td>
+<td style="text-align: right;">38</td>
+<td style="text-align: right;">0.2</td>
+</tr>
+<tr class="even">
+<td style="text-align: left;">avgphgt</td>
+<td style="text-align: right;">1.19</td>
+<td style="text-align: right;">0.14</td>
+<td style="text-align: right;">0.95</td>
+<td style="text-align: right;">0.9</td>
+<td style="text-align: right;">1.49</td>
+<td style="text-align: right;">8.23</td>
+<td style="text-align: right;">38</td>
+<td style="text-align: right;">0.0</td>
+</tr>
+</tbody>
+</table>
+
+
+</div>
+</div>
+<div class="sourceCode cell-code" id="cb14"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb14-1"><a href="#cb14-1" aria-hidden="true" tabindex="-1"></a><span class="fu">parameters</span>(m1)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output-display">
+<div>
+<table class="lightable-paper table table-sm table-striped small" data-quarto-postprocess="true">
+<thead>
+<tr class="header">
+<th style="text-align: left;" data-quarto-table-cell-role="th">Parameter</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">Coefficient</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">SE</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">CI</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">CI_low</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">CI_high</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">t</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">df_error</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">p</th>
+</tr>
+</thead>
+<tbody>
+<tr class="odd">
+<td style="text-align: left;">(Intercept)</td>
+<td style="text-align: right;">-12.88</td>
+<td style="text-align: right;">9.78</td>
+<td style="text-align: right;">0.95</td>
+<td style="text-align: right;">-32.7</td>
+<td style="text-align: right;">6.92</td>
+<td style="text-align: right;">-1.32</td>
+<td style="text-align: right;">38</td>
+<td style="text-align: right;">0.2</td>
+</tr>
+<tr class="even">
+<td style="text-align: left;">avgphgt</td>
+<td style="text-align: right;">1.19</td>
+<td style="text-align: right;">0.14</td>
+<td style="text-align: right;">0.95</td>
+<td style="text-align: right;">0.9</td>
+<td style="text-align: right;">1.49</td>
+<td style="text-align: right;">8.23</td>
+<td style="text-align: right;">38</td>
+<td style="text-align: right;">0.0</td>
+</tr>
+</tbody>
+</table>
+
+
+</div>
+</div>
+<div class="sourceCode cell-code" id="cb15"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb15-1"><a href="#cb15-1" aria-hidden="true" tabindex="-1"></a><span class="fu">standardise_parameters</span>(m1)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output-display">
+<div>
+<table class="lightable-paper table table-sm table-striped small" data-quarto-postprocess="true">
+<thead>
+<tr class="header">
+<th style="text-align: left;" data-quarto-table-cell-role="th">Parameter</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">Std_Coefficient</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">CI</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">CI_low</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">CI_high</th>
+</tr>
+</thead>
+<tbody>
+<tr class="odd">
+<td style="text-align: left;">(Intercept)</td>
+<td style="text-align: right;">0.0</td>
+<td style="text-align: right;">0.95</td>
+<td style="text-align: right;">-0.19</td>
+<td style="text-align: right;">0.19</td>
+</tr>
+<tr class="even">
+<td style="text-align: left;">avgphgt</td>
+<td style="text-align: right;">0.8</td>
+<td style="text-align: right;">0.95</td>
+<td style="text-align: right;">0.60</td>
+<td style="text-align: right;">1.00</td>
+</tr>
+</tbody>
+</table>
+
+
+</div>
+</div>
+<div class="sourceCode cell-code" id="cb16"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb16-1"><a href="#cb16-1" aria-hidden="true" tabindex="-1"></a><span class="fu">performance</span>(m1)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output-display">
+<div>
+<table class="lightable-paper table table-sm table-striped small" data-quarto-postprocess="true">
+<thead>
+<tr class="header">
+<th style="text-align: right;" data-quarto-table-cell-role="th">AIC</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">AICc</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">BIC</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">R2</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">R2_adjusted</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">RMSE</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">Sigma</th>
+</tr>
+</thead>
+<tbody>
+<tr class="odd">
+<td style="text-align: right;">196</td>
+<td style="text-align: right;">197</td>
+<td style="text-align: right;">201</td>
+<td style="text-align: right;">0.64</td>
+<td style="text-align: right;">0.63</td>
+<td style="text-align: right;">2.61</td>
+<td style="text-align: right;">2.68</td>
+</tr>
+</tbody>
+</table>
+
+
+</div>
+</div>
+</div>
+<p><em>R</em><sup>2</sup> is in the <code>glance</code> function’s <code>r.squared</code> column, and the <em>standard error of the estimate</em> is in the <code>sigma</code> column. If all you wanted was the <em>R</em><sup>2</sup>, you could do this:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb17"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb17-1"><a href="#cb17-1" aria-hidden="true" tabindex="-1"></a><span class="fu">glance</span>(m1)<span class="sc">$</span>r.squared</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output cell-output-stdout">
+<pre><code>[1] 0.641</code></pre>
+</div>
+</div>
+<p>Or using the performance package’s <a href="https://easystats.github.io/performance/reference/r2.html"><code>r2</code></a> function:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb19"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb19-1"><a href="#cb19-1" aria-hidden="true" tabindex="-1"></a><span class="fu">r2</span>(m1)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output cell-output-stdout">
+<pre><code># R2 for Linear Regression
+       R2: 0.641
+  adj. R2: 0.631</code></pre>
+</div>
+</div>
+<p>For the <em>standard error of the estimate</em>:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb21"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb21-1"><a href="#cb21-1" aria-hidden="true" tabindex="-1"></a><span class="fu">glance</span>(m1)<span class="sc">$</span>sigma</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output cell-output-stdout">
+<pre><code>[1] 2.68</code></pre>
+</div>
+</div>
+<p>Why would you want just one number instead of reading it from a table? In reproducible research, we intermingle text and code so that it is clear where every number came from. Thus, “hard-coding” your results like this is considered poor practice:</p>
+<pre><code>The model explains 64% of the variance.</code></pre>
+<p>Using rmarkdown, instead of typing the numeric results, we type pull the results using an <a href="https://rmarkdown.rstudio.com/lesson-4.html">inline code chunk</a>:</p>
+<div class="sourceCode" id="cb24"><pre class="sourceCode default fold-show code-with-copy"><code class="sourceCode default"><span id="cb24-1"><a href="#cb24-1" aria-hidden="true" tabindex="-1"></a>The model explains `r round(100 * glance(m1)$r.squared, 0)`% of the variance.</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<p>Which, when rendered, produces the correct output:</p>
+<blockquote class="blockquote">
+<p>The model explains 64% of the variance.</p>
+</blockquote>
+<p>That seems like a lot of extra work, right? Yes, it is—unless there is a possibility that your underlying data might change or that you might copy your numbers incorrectly. If you are imperfect, the extra time and effort is worth it. It makes it easy for other scholars to see exactly where each number came from. Hard-coded results are harder to trust.</p>
+<p>Automated report of the model’s performance:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb25"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb25-1"><a href="#cb25-1" aria-hidden="true" tabindex="-1"></a><span class="fu">report_performance</span>(m1)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output cell-output-stdout">
+<pre><code>The model explains a statistically significant and substantial proportion of
+variance (R2 = 0.64, F(1, 38) = 67.73, p &lt; .001, adj. R2 = 0.63)</code></pre>
+</div>
+</div>
+</section>
+<section id="coefficient-level-statistics" class="level2">
+<h2 class="anchored" data-anchor-id="coefficient-level-statistics">Coefficient-level statistics</h2>
+<p>The regression coefficients—the intercept (<em>b</em><sub>0</sub>) and the slope (<em>b</em><sub>1</sub>)—have a number of statistics associated with them, which we will discuss later in the course.</p>
+<p>If you just wanted the intercept and the slope coefficients, use the <code>coef</code> function:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb27"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb27-1"><a href="#cb27-1" aria-hidden="true" tabindex="-1"></a><span class="fu">coef</span>(m1)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output cell-output-stdout">
+<pre><code>(Intercept)     avgphgt 
+     -12.88        1.19 </code></pre>
+</div>
+</div>
+<p>Thus, the intercept is -12.88 and the slope is 1.19.</p>
+<p>To get the coefficients (intercept and slope) along with their p-values and other statistical information, use the <a href="https://broom.tidymodels.org/reference/tidy.lm.html"><code>tidy</code></a> function:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb29"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb29-1"><a href="#cb29-1" aria-hidden="true" tabindex="-1"></a><span class="fu">tidy</span>(m1)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output-display">
+<div>
+<table class="lightable-paper table table-sm table-striped small" data-quarto-postprocess="true">
+<thead>
+<tr class="header">
+<th style="text-align: left;" data-quarto-table-cell-role="th">term</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">estimate</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">std.error</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">statistic</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">p.value</th>
+</tr>
+</thead>
+<tbody>
+<tr class="odd">
+<td style="text-align: left;">(Intercept)</td>
+<td style="text-align: right;">-12.88</td>
+<td style="text-align: right;">9.78</td>
+<td style="text-align: right;">-1.32</td>
+<td style="text-align: right;">0.2</td>
+</tr>
+<tr class="even">
+<td style="text-align: left;">avgphgt</td>
+<td style="text-align: right;">1.19</td>
+<td style="text-align: right;">0.14</td>
+<td style="text-align: right;">8.23</td>
+<td style="text-align: right;">0.0</td>
+</tr>
+</tbody>
+</table>
+
+
+</div>
+</div>
+</div>
+<p>The parameters package has a slightly more print-friendly version of <code>tidy</code>:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb30"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb30-1"><a href="#cb30-1" aria-hidden="true" tabindex="-1"></a><span class="fu">model_parameters</span>(m1) </span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output-display">
+<div>
+<table class="lightable-paper table table-sm table-striped small" data-quarto-postprocess="true">
+<thead>
+<tr class="header">
+<th style="text-align: left;" data-quarto-table-cell-role="th">Parameter</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">Coefficient</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">SE</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">CI</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">CI_low</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">CI_high</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">t</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">df_error</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">p</th>
+</tr>
+</thead>
+<tbody>
+<tr class="odd">
+<td style="text-align: left;">(Intercept)</td>
+<td style="text-align: right;">-12.88</td>
+<td style="text-align: right;">9.78</td>
+<td style="text-align: right;">0.95</td>
+<td style="text-align: right;">-32.7</td>
+<td style="text-align: right;">6.92</td>
+<td style="text-align: right;">-1.32</td>
+<td style="text-align: right;">38</td>
+<td style="text-align: right;">0.2</td>
+</tr>
+<tr class="even">
+<td style="text-align: left;">avgphgt</td>
+<td style="text-align: right;">1.19</td>
+<td style="text-align: right;">0.14</td>
+<td style="text-align: right;">0.95</td>
+<td style="text-align: right;">0.9</td>
+<td style="text-align: right;">1.49</td>
+<td style="text-align: right;">8.23</td>
+<td style="text-align: right;">38</td>
+<td style="text-align: right;">0.0</td>
+</tr>
+</tbody>
+</table>
+
+
+</div>
+</div>
+</div>
+<p>If you want standardized parameters:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb31"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb31-1"><a href="#cb31-1" aria-hidden="true" tabindex="-1"></a><span class="fu">model_parameters</span>(m1, <span class="at">standardize =</span> <span class="st">"refit"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output-display">
+<div>
+<table class="lightable-paper table table-sm table-striped small" data-quarto-postprocess="true">
+<thead>
+<tr class="header">
+<th style="text-align: left;" data-quarto-table-cell-role="th">Parameter</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">Coefficient</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">SE</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">CI</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">CI_low</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">CI_high</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">t</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">df_error</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">p</th>
+</tr>
+</thead>
+<tbody>
+<tr class="odd">
+<td style="text-align: left;">(Intercept)</td>
+<td style="text-align: right;">0.0</td>
+<td style="text-align: right;">0.1</td>
+<td style="text-align: right;">0.95</td>
+<td style="text-align: right;">-0.19</td>
+<td style="text-align: right;">0.19</td>
+<td style="text-align: right;">0.00</td>
+<td style="text-align: right;">38</td>
+<td style="text-align: right;">1</td>
+</tr>
+<tr class="even">
+<td style="text-align: left;">avgphgt</td>
+<td style="text-align: right;">0.8</td>
+<td style="text-align: right;">0.1</td>
+<td style="text-align: right;">0.95</td>
+<td style="text-align: right;">0.60</td>
+<td style="text-align: right;">1.00</td>
+<td style="text-align: right;">8.23</td>
+<td style="text-align: right;">38</td>
+<td style="text-align: right;">0</td>
+</tr>
+</tbody>
+</table>
+
+
+</div>
+</div>
+</div>
+<p>Automated report of the parameters:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb32"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb32-1"><a href="#cb32-1" aria-hidden="true" tabindex="-1"></a><span class="fu">report_parameters</span>(m1)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output cell-output-stdout">
+<pre><code>  - The intercept is statistically non-significant and negative (beta = -12.88, 95% CI [-32.67, 6.92], t(38) = -1.32, p = 0.196; Std. beta = 2.65e-16, 95% CI [-0.19, 0.19])
+  - The effect of avgphgt is statistically significant and positive (beta = 1.19, 95% CI [0.90, 1.49], t(38) = 8.23, p &lt; .001; Std. beta = 0.80, 95% CI [0.60, 1.00])</code></pre>
+</div>
+</div>
+</section>
+<section id="observation-level-statistics" class="level2">
+<h2 class="anchored" data-anchor-id="observation-level-statistics">Observation-Level Statistics</h2>
+<p>We sometimes want the individual-level data like the predictions (<span class="math inline">\hat{Y}</span>) and the errors (technically called <em>residuals</em>).</p>
+<p>To get observation-level statistics, use broom’s <a href="https://broom.tidymodels.org/reference/augment.lm.html"><code>augment</code></a> function:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb34"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb34-1"><a href="#cb34-1" aria-hidden="true" tabindex="-1"></a><span class="fu">augment</span>(m1) </span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output-display">
+<div>
+<table class="lightable-paper table table-sm table-striped small" data-quarto-postprocess="true">
+<thead>
+<tr class="header">
+<th style="text-align: right;" data-quarto-table-cell-role="th">height</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">avgphgt</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">.fitted</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">.resid</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">.hat</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">.sigma</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">.cooksd</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">.std.resid</th>
+</tr>
+</thead>
+<tbody>
+<tr class="odd">
+<td style="text-align: right;">66</td>
+<td style="text-align: right;">67.5</td>
+<td style="text-align: right;">67.6</td>
+<td style="text-align: right;">-1.64</td>
+<td style="text-align: right;">0.03</td>
+<td style="text-align: right;">2.70</td>
+<td style="text-align: right;">0.00</td>
+<td style="text-align: right;">-0.62</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">71</td>
+<td style="text-align: right;">70.0</td>
+<td style="text-align: right;">70.6</td>
+<td style="text-align: right;">0.37</td>
+<td style="text-align: right;">0.04</td>
+<td style="text-align: right;">2.72</td>
+<td style="text-align: right;">0.00</td>
+<td style="text-align: right;">0.14</td>
+</tr>
+<tr class="odd">
+<td style="text-align: right;">74</td>
+<td style="text-align: right;">72.5</td>
+<td style="text-align: right;">73.6</td>
+<td style="text-align: right;">0.39</td>
+<td style="text-align: right;">0.10</td>
+<td style="text-align: right;">2.72</td>
+<td style="text-align: right;">0.00</td>
+<td style="text-align: right;">0.15</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">69</td>
+<td style="text-align: right;">66.5</td>
+<td style="text-align: right;">66.5</td>
+<td style="text-align: right;">2.55</td>
+<td style="text-align: right;">0.03</td>
+<td style="text-align: right;">2.68</td>
+<td style="text-align: right;">0.01</td>
+<td style="text-align: right;">0.96</td>
+</tr>
+<tr class="odd">
+<td style="text-align: right;">70</td>
+<td style="text-align: right;">65.0</td>
+<td style="text-align: right;">64.7</td>
+<td style="text-align: right;">5.34</td>
+<td style="text-align: right;">0.04</td>
+<td style="text-align: right;">2.56</td>
+<td style="text-align: right;">0.09</td>
+<td style="text-align: right;">2.03</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">74</td>
+<td style="text-align: right;">72.0</td>
+<td style="text-align: right;">73.0</td>
+<td style="text-align: right;">0.99</td>
+<td style="text-align: right;">0.09</td>
+<td style="text-align: right;">2.71</td>
+<td style="text-align: right;">0.01</td>
+<td style="text-align: right;">0.39</td>
+</tr>
+<tr class="odd">
+<td style="text-align: right;">73</td>
+<td style="text-align: right;">71.0</td>
+<td style="text-align: right;">71.8</td>
+<td style="text-align: right;">1.18</td>
+<td style="text-align: right;">0.06</td>
+<td style="text-align: right;">2.71</td>
+<td style="text-align: right;">0.01</td>
+<td style="text-align: right;">0.45</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">69</td>
+<td style="text-align: right;">70.0</td>
+<td style="text-align: right;">70.6</td>
+<td style="text-align: right;">-1.63</td>
+<td style="text-align: right;">0.04</td>
+<td style="text-align: right;">2.70</td>
+<td style="text-align: right;">0.01</td>
+<td style="text-align: right;">-0.62</td>
+</tr>
+<tr class="odd">
+<td style="text-align: right;">69</td>
+<td style="text-align: right;">71.0</td>
+<td style="text-align: right;">71.8</td>
+<td style="text-align: right;">-2.82</td>
+<td style="text-align: right;">0.06</td>
+<td style="text-align: right;">2.67</td>
+<td style="text-align: right;">0.04</td>
+<td style="text-align: right;">-1.09</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">75</td>
+<td style="text-align: right;">69.5</td>
+<td style="text-align: right;">70.0</td>
+<td style="text-align: right;">4.97</td>
+<td style="text-align: right;">0.04</td>
+<td style="text-align: right;">2.59</td>
+<td style="text-align: right;">0.07</td>
+<td style="text-align: right;">1.89</td>
+</tr>
+<tr class="odd">
+<td style="text-align: right;">69</td>
+<td style="text-align: right;">69.0</td>
+<td style="text-align: right;">69.4</td>
+<td style="text-align: right;">-0.43</td>
+<td style="text-align: right;">0.03</td>
+<td style="text-align: right;">2.72</td>
+<td style="text-align: right;">0.00</td>
+<td style="text-align: right;">-0.16</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">67</td>
+<td style="text-align: right;">67.5</td>
+<td style="text-align: right;">67.6</td>
+<td style="text-align: right;">-0.64</td>
+<td style="text-align: right;">0.03</td>
+<td style="text-align: right;">2.71</td>
+<td style="text-align: right;">0.00</td>
+<td style="text-align: right;">-0.24</td>
+</tr>
+<tr class="odd">
+<td style="text-align: right;">73</td>
+<td style="text-align: right;">71.5</td>
+<td style="text-align: right;">72.4</td>
+<td style="text-align: right;">0.58</td>
+<td style="text-align: right;">0.07</td>
+<td style="text-align: right;">2.72</td>
+<td style="text-align: right;">0.00</td>
+<td style="text-align: right;">0.23</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">74</td>
+<td style="text-align: right;">70.0</td>
+<td style="text-align: right;">70.6</td>
+<td style="text-align: right;">3.37</td>
+<td style="text-align: right;">0.04</td>
+<td style="text-align: right;">2.66</td>
+<td style="text-align: right;">0.04</td>
+<td style="text-align: right;">1.29</td>
+</tr>
+<tr class="odd">
+<td style="text-align: right;">63</td>
+<td style="text-align: right;">61.5</td>
+<td style="text-align: right;">60.5</td>
+<td style="text-align: right;">2.51</td>
+<td style="text-align: right;">0.13</td>
+<td style="text-align: right;">2.68</td>
+<td style="text-align: right;">0.07</td>
+<td style="text-align: right;">1.00</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">67</td>
+<td style="text-align: right;">68.0</td>
+<td style="text-align: right;">68.2</td>
+<td style="text-align: right;">-1.24</td>
+<td style="text-align: right;">0.03</td>
+<td style="text-align: right;">2.71</td>
+<td style="text-align: right;">0.00</td>
+<td style="text-align: right;">-0.47</td>
+</tr>
+<tr class="odd">
+<td style="text-align: right;">64</td>
+<td style="text-align: right;">62.5</td>
+<td style="text-align: right;">61.7</td>
+<td style="text-align: right;">2.32</td>
+<td style="text-align: right;">0.10</td>
+<td style="text-align: right;">2.69</td>
+<td style="text-align: right;">0.04</td>
+<td style="text-align: right;">0.91</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">61</td>
+<td style="text-align: right;">63.0</td>
+<td style="text-align: right;">62.3</td>
+<td style="text-align: right;">-1.28</td>
+<td style="text-align: right;">0.08</td>
+<td style="text-align: right;">2.71</td>
+<td style="text-align: right;">0.01</td>
+<td style="text-align: right;">-0.50</td>
+</tr>
+<tr class="odd">
+<td style="text-align: right;">70</td>
+<td style="text-align: right;">65.5</td>
+<td style="text-align: right;">65.3</td>
+<td style="text-align: right;">4.74</td>
+<td style="text-align: right;">0.04</td>
+<td style="text-align: right;">2.60</td>
+<td style="text-align: right;">0.06</td>
+<td style="text-align: right;">1.80</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">66</td>
+<td style="text-align: right;">68.0</td>
+<td style="text-align: right;">68.2</td>
+<td style="text-align: right;">-2.24</td>
+<td style="text-align: right;">0.03</td>
+<td style="text-align: right;">2.69</td>
+<td style="text-align: right;">0.01</td>
+<td style="text-align: right;">-0.85</td>
+</tr>
+<tr class="odd">
+<td style="text-align: right;">70</td>
+<td style="text-align: right;">72.0</td>
+<td style="text-align: right;">73.0</td>
+<td style="text-align: right;">-3.01</td>
+<td style="text-align: right;">0.09</td>
+<td style="text-align: right;">2.67</td>
+<td style="text-align: right;">0.07</td>
+<td style="text-align: right;">-1.18</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">60</td>
+<td style="text-align: right;">65.0</td>
+<td style="text-align: right;">64.7</td>
+<td style="text-align: right;">-4.66</td>
+<td style="text-align: right;">0.04</td>
+<td style="text-align: right;">2.60</td>
+<td style="text-align: right;">0.07</td>
+<td style="text-align: right;">-1.78</td>
+</tr>
+<tr class="odd">
+<td style="text-align: right;">75</td>
+<td style="text-align: right;">68.5</td>
+<td style="text-align: right;">68.8</td>
+<td style="text-align: right;">6.16</td>
+<td style="text-align: right;">0.03</td>
+<td style="text-align: right;">2.52</td>
+<td style="text-align: right;">0.08</td>
+<td style="text-align: right;">2.33</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">73</td>
+<td style="text-align: right;">69.0</td>
+<td style="text-align: right;">69.4</td>
+<td style="text-align: right;">3.57</td>
+<td style="text-align: right;">0.03</td>
+<td style="text-align: right;">2.65</td>
+<td style="text-align: right;">0.03</td>
+<td style="text-align: right;">1.35</td>
+</tr>
+<tr class="odd">
+<td style="text-align: right;">63</td>
+<td style="text-align: right;">68.0</td>
+<td style="text-align: right;">68.2</td>
+<td style="text-align: right;">-5.24</td>
+<td style="text-align: right;">0.03</td>
+<td style="text-align: right;">2.57</td>
+<td style="text-align: right;">0.05</td>
+<td style="text-align: right;">-1.98</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">63</td>
+<td style="text-align: right;">65.0</td>
+<td style="text-align: right;">64.7</td>
+<td style="text-align: right;">-1.66</td>
+<td style="text-align: right;">0.04</td>
+<td style="text-align: right;">2.70</td>
+<td style="text-align: right;">0.01</td>
+<td style="text-align: right;">-0.63</td>
+</tr>
+<tr class="odd">
+<td style="text-align: right;">69</td>
+<td style="text-align: right;">67.5</td>
+<td style="text-align: right;">67.6</td>
+<td style="text-align: right;">1.36</td>
+<td style="text-align: right;">0.03</td>
+<td style="text-align: right;">2.71</td>
+<td style="text-align: right;">0.00</td>
+<td style="text-align: right;">0.51</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">65</td>
+<td style="text-align: right;">64.5</td>
+<td style="text-align: right;">64.1</td>
+<td style="text-align: right;">0.93</td>
+<td style="text-align: right;">0.05</td>
+<td style="text-align: right;">2.71</td>
+<td style="text-align: right;">0.00</td>
+<td style="text-align: right;">0.36</td>
+</tr>
+<tr class="odd">
+<td style="text-align: right;">64</td>
+<td style="text-align: right;">66.0</td>
+<td style="text-align: right;">65.8</td>
+<td style="text-align: right;">-1.85</td>
+<td style="text-align: right;">0.03</td>
+<td style="text-align: right;">2.70</td>
+<td style="text-align: right;">0.01</td>
+<td style="text-align: right;">-0.70</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">69</td>
+<td style="text-align: right;">70.0</td>
+<td style="text-align: right;">70.6</td>
+<td style="text-align: right;">-1.63</td>
+<td style="text-align: right;">0.04</td>
+<td style="text-align: right;">2.70</td>
+<td style="text-align: right;">0.01</td>
+<td style="text-align: right;">-0.62</td>
+</tr>
+<tr class="odd">
+<td style="text-align: right;">65</td>
+<td style="text-align: right;">68.0</td>
+<td style="text-align: right;">68.2</td>
+<td style="text-align: right;">-3.24</td>
+<td style="text-align: right;">0.03</td>
+<td style="text-align: right;">2.66</td>
+<td style="text-align: right;">0.02</td>
+<td style="text-align: right;">-1.22</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">63</td>
+<td style="text-align: right;">65.0</td>
+<td style="text-align: right;">64.7</td>
+<td style="text-align: right;">-1.66</td>
+<td style="text-align: right;">0.04</td>
+<td style="text-align: right;">2.70</td>
+<td style="text-align: right;">0.01</td>
+<td style="text-align: right;">-0.63</td>
+</tr>
+<tr class="odd">
+<td style="text-align: right;">68</td>
+<td style="text-align: right;">67.5</td>
+<td style="text-align: right;">67.6</td>
+<td style="text-align: right;">0.36</td>
+<td style="text-align: right;">0.03</td>
+<td style="text-align: right;">2.72</td>
+<td style="text-align: right;">0.00</td>
+<td style="text-align: right;">0.13</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">63</td>
+<td style="text-align: right;">64.5</td>
+<td style="text-align: right;">64.1</td>
+<td style="text-align: right;">-1.07</td>
+<td style="text-align: right;">0.05</td>
+<td style="text-align: right;">2.71</td>
+<td style="text-align: right;">0.00</td>
+<td style="text-align: right;">-0.41</td>
+</tr>
+<tr class="odd">
+<td style="text-align: right;">62</td>
+<td style="text-align: right;">64.0</td>
+<td style="text-align: right;">63.5</td>
+<td style="text-align: right;">-1.47</td>
+<td style="text-align: right;">0.06</td>
+<td style="text-align: right;">2.71</td>
+<td style="text-align: right;">0.01</td>
+<td style="text-align: right;">-0.56</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">61</td>
+<td style="text-align: right;">63.5</td>
+<td style="text-align: right;">62.9</td>
+<td style="text-align: right;">-1.87</td>
+<td style="text-align: right;">0.07</td>
+<td style="text-align: right;">2.70</td>
+<td style="text-align: right;">0.02</td>
+<td style="text-align: right;">-0.72</td>
+</tr>
+<tr class="odd">
+<td style="text-align: right;">67</td>
+<td style="text-align: right;">67.0</td>
+<td style="text-align: right;">67.0</td>
+<td style="text-align: right;">-0.05</td>
+<td style="text-align: right;">0.03</td>
+<td style="text-align: right;">2.72</td>
+<td style="text-align: right;">0.00</td>
+<td style="text-align: right;">-0.02</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">63</td>
+<td style="text-align: right;">64.5</td>
+<td style="text-align: right;">64.1</td>
+<td style="text-align: right;">-1.07</td>
+<td style="text-align: right;">0.05</td>
+<td style="text-align: right;">2.71</td>
+<td style="text-align: right;">0.00</td>
+<td style="text-align: right;">-0.41</td>
+</tr>
+<tr class="odd">
+<td style="text-align: right;">62</td>
+<td style="text-align: right;">63.5</td>
+<td style="text-align: right;">62.9</td>
+<td style="text-align: right;">-0.87</td>
+<td style="text-align: right;">0.07</td>
+<td style="text-align: right;">2.71</td>
+<td style="text-align: right;">0.00</td>
+<td style="text-align: right;">-0.34</td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">72</td>
+<td style="text-align: right;">71.5</td>
+<td style="text-align: right;">72.4</td>
+<td style="text-align: right;">-0.42</td>
+<td style="text-align: right;">0.07</td>
+<td style="text-align: right;">2.72</td>
+<td style="text-align: right;">0.00</td>
+<td style="text-align: right;">-0.16</td>
+</tr>
+</tbody>
+</table>
+
+
+</div>
+</div>
+</div>
+<p>The predicted values <span class="math inline">\hat{Y}</span> are in the <code>.fitted</code> column, and the errors (residuals) are in the <code>.resid</code> column.</p>
+<p>If you wanted to predict a value not in the data, you use the <code>newdata</code> argument. For example, if I wanted to predict the height of a person whose parents had a height of 65:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb35"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb35-1"><a href="#cb35-1" aria-hidden="true" tabindex="-1"></a><span class="fu">augment</span>(m1, <span class="at">newdata =</span> <span class="fu">tibble</span>(<span class="at">avgphgt =</span> <span class="dv">65</span>))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output-display">
+<div>
+<table class="lightable-paper table table-sm table-striped small" data-quarto-postprocess="true">
+<thead>
+<tr class="header">
+<th style="text-align: right;" data-quarto-table-cell-role="th">avgphgt</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">.fitted</th>
+</tr>
+</thead>
+<tbody>
+<tr class="odd">
+<td style="text-align: right;">65</td>
+<td style="text-align: right;">64.7</td>
+</tr>
+</tbody>
+</table>
+
+
+</div>
+</div>
+</div>
+</section>
+</section>
+<section id="multiple-regression" class="level1">
+<h1>Multiple Regression</h1>
+<p>In this data set, all participants identified as either male or female. We assume that males are, on average, taller than females. The codes are male = 1 and female = 2, thus our predicted effect is negative.</p>
+<p>We run the regression model with 2 predictors like so:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb36"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb36-1"><a href="#cb36-1" aria-hidden="true" tabindex="-1"></a>m2 <span class="ot">&lt;-</span> <span class="fu">lm</span>(height <span class="sc">~</span> avgphgt <span class="sc">+</span> gender, <span class="at">data =</span> d)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+</div>
+<section id="checking-assumptions-1" class="level2">
+<h2 class="anchored" data-anchor-id="checking-assumptions-1">Checking assumptions</h2>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb37"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb37-1"><a href="#cb37-1" aria-hidden="true" tabindex="-1"></a><span class="fu">check_model</span>(m2)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output-display">
+<div>
+<figure class="figure">
+<p><img src="index_files/figure-html/unnamed-chunk-21-1.svg" style="width:100.0%;height:100.0%" class="figure-img"></p>
+</figure>
+</div>
+</div>
+</div>
+<p>All looks well.</p>
+</section>
+<section id="summarizing-results" class="level2">
+<h2 class="anchored" data-anchor-id="summarizing-results">Summarizing Results</h2>
+<p>To summarize the results, use the <code>summary</code> function:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb38"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb38-1"><a href="#cb38-1" aria-hidden="true" tabindex="-1"></a><span class="fu">summary</span>(m2)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output cell-output-stdout">
+<pre><code>
+Call:
+lm(formula = height ~ avgphgt + gender, data = d)
+
+Residuals:
+   Min     1Q Median     3Q    Max 
+-3.826 -1.094 -0.347  1.656  4.503 
+
+Coefficients:
+            Estimate Std. Error t value Pr(&gt;|t|)    
+(Intercept)   29.035     11.600    2.50  0.01686 *  
+avgphgt        0.671      0.157    4.26  0.00013 ***
+gender        -4.468      0.920   -4.85  2.2e-05 ***
+---
+Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
+
+Residual standard error: 2.12 on 37 degrees of freedom
+Multiple R-squared:  0.78,  Adjusted R-squared:  0.769 
+F-statistic: 65.7 on 2 and 37 DF,  p-value: 6.6e-13</code></pre>
+</div>
+</div>
+<p>An automated report:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb40"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb40-1"><a href="#cb40-1" aria-hidden="true" tabindex="-1"></a>report<span class="sc">::</span><span class="fu">report</span>(m2)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output cell-output-stdout">
+<pre><code>We fitted a linear model (estimated using OLS) to predict height with avgphgt
+and gender (formula: height ~ avgphgt + gender). The model explains a
+statistically significant and substantial proportion of variance (R2 = 0.78,
+F(2, 37) = 65.75, p &lt; .001, adj. R2 = 0.77). The model's intercept,
+corresponding to avgphgt = 0 and gender = 0, is at 29.03 (95% CI [5.53, 52.54],
+t(37) = 2.50, p = 0.017). Within this model:
+
+  - The effect of avgphgt is statistically significant and positive (beta = 0.67,
+95% CI [0.35, 0.99], t(37) = 4.26, p &lt; .001; Std. beta = 0.45, 95% CI [0.24,
+0.66])
+  - The effect of gender is statistically significant and negative (beta = -4.47,
+95% CI [-6.33, -2.60], t(37) = -4.85, p &lt; .001; Std. beta = -0.51, 95% CI
+[-0.73, -0.30])
+
+Standardized parameters were obtained by fitting the model on a standardized
+version of the dataset. 95% Confidence Intervals (CIs) and p-values were
+computed using a Wald t-distribution approximation.</code></pre>
+</div>
+</div>
+<p>Standardized coefficients:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb42"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb42-1"><a href="#cb42-1" aria-hidden="true" tabindex="-1"></a>parameters<span class="sc">::</span><span class="fu">model_parameters</span>(m2, <span class="at">standardize =</span> <span class="st">"refit"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output-display">
+<div>
+<table class="lightable-paper table table-sm table-striped small" data-quarto-postprocess="true">
+<thead>
+<tr class="header">
+<th style="text-align: left;" data-quarto-table-cell-role="th">Parameter</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">Coefficient</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">SE</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">CI</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">CI_low</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">CI_high</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">t</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">df_error</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">p</th>
+</tr>
+</thead>
+<tbody>
+<tr class="odd">
+<td style="text-align: left;">(Intercept)</td>
+<td style="text-align: right;">0.00</td>
+<td style="text-align: right;">0.08</td>
+<td style="text-align: right;">0.95</td>
+<td style="text-align: right;">-0.15</td>
+<td style="text-align: right;">0.15</td>
+<td style="text-align: right;">0.00</td>
+<td style="text-align: right;">37</td>
+<td style="text-align: right;">1</td>
+</tr>
+<tr class="even">
+<td style="text-align: left;">avgphgt</td>
+<td style="text-align: right;">0.45</td>
+<td style="text-align: right;">0.11</td>
+<td style="text-align: right;">0.95</td>
+<td style="text-align: right;">0.24</td>
+<td style="text-align: right;">0.66</td>
+<td style="text-align: right;">4.26</td>
+<td style="text-align: right;">37</td>
+<td style="text-align: right;">0</td>
+</tr>
+<tr class="odd">
+<td style="text-align: left;">gender</td>
+<td style="text-align: right;">-0.51</td>
+<td style="text-align: right;">0.11</td>
+<td style="text-align: right;">0.95</td>
+<td style="text-align: right;">-0.73</td>
+<td style="text-align: right;">-0.30</td>
+<td style="text-align: right;">-4.85</td>
+<td style="text-align: right;">37</td>
+<td style="text-align: right;">0</td>
+</tr>
+</tbody>
+</table>
+
+
+</div>
+</div>
+</div>
+<p>The oddly-named <code>anova</code> function compares 2 models:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb43"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb43-1"><a href="#cb43-1" aria-hidden="true" tabindex="-1"></a><span class="fu">anova</span>(m1, m2)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output-display">
+<div>
+<table class="lightable-paper table table-sm table-striped small" data-quarto-postprocess="true">
+<thead>
+<tr class="header">
+<th style="text-align: right;" data-quarto-table-cell-role="th">Res.Df</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">RSS</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">Df</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">Sum of Sq</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">F</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">Pr(&gt;F)</th>
+</tr>
+</thead>
+<tbody>
+<tr class="odd">
+<td style="text-align: right;">38</td>
+<td style="text-align: right;">273</td>
+<td style="text-align: right;"></td>
+<td style="text-align: right;"></td>
+<td style="text-align: right;"></td>
+<td style="text-align: right;"></td>
+</tr>
+<tr class="even">
+<td style="text-align: right;">37</td>
+<td style="text-align: right;">167</td>
+<td style="text-align: right;">1</td>
+<td style="text-align: right;">106</td>
+<td style="text-align: right;">23.6</td>
+<td style="text-align: right;">0</td>
+</tr>
+</tbody>
+</table>
+
+
+</div>
+</div>
+</div>
+<p>The p-value is significant, meaning that <code>m2</code> explains more variance than <code>m1</code>.</p>
+<p>I like the more full-featured set of model comparison functions from the performance package:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb44"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb44-1"><a href="#cb44-1" aria-hidden="true" tabindex="-1"></a><span class="fu">compare_performance</span>(m1, m2) </span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output-display">
+<div>
+<table class="lightable-paper table table-sm table-striped small" data-quarto-postprocess="true">
+<thead>
+<tr class="header">
+<th style="text-align: left;" data-quarto-table-cell-role="th">Name</th>
+<th style="text-align: left;" data-quarto-table-cell-role="th">Model</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">AIC</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">AIC_wt</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">AICc</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">AICc_wt</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">BIC</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">BIC_wt</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">R2</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">R2_adjusted</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">RMSE</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">Sigma</th>
+</tr>
+</thead>
+<tbody>
+<tr class="odd">
+<td style="text-align: left;">m1</td>
+<td style="text-align: left;">lm</td>
+<td style="text-align: right;">196</td>
+<td style="text-align: right;">0</td>
+<td style="text-align: right;">197</td>
+<td style="text-align: right;">0</td>
+<td style="text-align: right;">201</td>
+<td style="text-align: right;">0</td>
+<td style="text-align: right;">0.64</td>
+<td style="text-align: right;">0.63</td>
+<td style="text-align: right;">2.61</td>
+<td style="text-align: right;">2.68</td>
+</tr>
+<tr class="even">
+<td style="text-align: left;">m2</td>
+<td style="text-align: left;">lm</td>
+<td style="text-align: right;">179</td>
+<td style="text-align: right;">1</td>
+<td style="text-align: right;">180</td>
+<td style="text-align: right;">1</td>
+<td style="text-align: right;">185</td>
+<td style="text-align: right;">1</td>
+<td style="text-align: right;">0.78</td>
+<td style="text-align: right;">0.77</td>
+<td style="text-align: right;">2.04</td>
+<td style="text-align: right;">2.12</td>
+</tr>
+</tbody>
+</table>
+
+
+</div>
+</div>
+</div>
+<p>This is equivalent to the <code>anova</code> function but in a more readable format:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb45"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb45-1"><a href="#cb45-1" aria-hidden="true" tabindex="-1"></a><span class="fu">test_wald</span>(m1, m2)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output-display">
+<div>
+<table class="lightable-paper table table-sm table-striped small" data-quarto-postprocess="true">
+<thead>
+<tr class="header">
+<th style="text-align: left;" data-quarto-table-cell-role="th">Name</th>
+<th style="text-align: left;" data-quarto-table-cell-role="th">Model</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">df</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">df_diff</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">F</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">p</th>
+</tr>
+</thead>
+<tbody>
+<tr class="odd">
+<td style="text-align: left;">m1</td>
+<td style="text-align: left;">lm</td>
+<td style="text-align: right;">38</td>
+<td style="text-align: right;"></td>
+<td style="text-align: right;"></td>
+<td style="text-align: right;"></td>
+</tr>
+<tr class="even">
+<td style="text-align: left;">m2</td>
+<td style="text-align: left;">lm</td>
+<td style="text-align: right;">37</td>
+<td style="text-align: right;">1</td>
+<td style="text-align: right;">23.6</td>
+<td style="text-align: right;">0</td>
+</tr>
+</tbody>
+</table>
+
+
+</div>
+</div>
+</div>
+<p>This uses Bayes Factor instead of a p-value:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb46"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb46-1"><a href="#cb46-1" aria-hidden="true" tabindex="-1"></a><span class="fu">test_bf</span>(m1, m2)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output-display">
+<div>
+<table class="lightable-paper table table-sm table-striped small" data-quarto-postprocess="true">
+<thead>
+<tr class="header">
+<th style="text-align: left;" data-quarto-table-cell-role="th">Model</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">log_BF</th>
+<th style="text-align: right;" data-quarto-table-cell-role="th">BF</th>
+</tr>
+</thead>
+<tbody>
+<tr class="odd">
+<td style="text-align: left;">avgphgt</td>
+<td style="text-align: right;"></td>
+<td style="text-align: right;"></td>
+</tr>
+<tr class="even">
+<td style="text-align: left;">avgphgt + gender</td>
+<td style="text-align: right;">8.01</td>
+<td style="text-align: right;">3010</td>
+</tr>
+</tbody>
+</table>
+
+
+</div>
+</div>
+</div>
+<p>A BF &gt; 1 means that <code>m2</code> is more strongly supported than <code>m1</code>. A BF &lt; 1 means that <code>m1</code> is more strongly supported than <code>m2</code>.</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb47"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb47-1"><a href="#cb47-1" aria-hidden="true" tabindex="-1"></a><span class="fu">report</span>(<span class="fu">test_bf</span>(m1, m2))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output cell-output-stdout">
+<pre><code>Bayes factors were computed using the BIC approximation, by which BF10 =
+exp((BIC0 - BIC1)/2). Compared to the avgphgt model, we found extreme evidence
+(BF = 3.01e+03) in favour of the avgphgt + gender model (the least supported
+model).</code></pre>
+</div>
+</div>
+</section>
+</section>
+<section id="all-the-code-in-one-place" class="level1">
+<h1>All the code in one place:</h1>
+<p>The preceding analyses might seem like a lot, but it is not really so much when you see it all in just a few lines of code. Here are all the main analyses:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb49"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb49-1"><a href="#cb49-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Load packages</span></span>
+<span id="cb49-2"><a href="#cb49-2" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(tidyverse)</span>
+<span id="cb49-3"><a href="#cb49-3" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(haven)</span>
+<span id="cb49-4"><a href="#cb49-4" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(broom)</span>
+<span id="cb49-5"><a href="#cb49-5" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(performance)</span>
+<span id="cb49-6"><a href="#cb49-6" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb49-7"><a href="#cb49-7" aria-hidden="true" tabindex="-1"></a><span class="co"># Import data</span></span>
+<span id="cb49-8"><a href="#cb49-8" aria-hidden="true" tabindex="-1"></a>d <span class="ot">&lt;-</span> <span class="fu">read_spss</span>(<span class="st">"https://github.com/wjschne/EDUC5325/raw/master/height.sav"</span>)</span>
+<span id="cb49-9"><a href="#cb49-9" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb49-10"><a href="#cb49-10" aria-hidden="true" tabindex="-1"></a><span class="co"># Plot data</span></span>
+<span id="cb49-11"><a href="#cb49-11" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplot</span>(d, <span class="fu">aes</span>(weight,height)) <span class="sc">+</span></span>
+<span id="cb49-12"><a href="#cb49-12" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_point</span>() <span class="sc">+</span></span>
+<span id="cb49-13"><a href="#cb49-13" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_smooth</span>(<span class="at">method =</span> <span class="st">"lm"</span>)</span>
+<span id="cb49-14"><a href="#cb49-14" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb49-15"><a href="#cb49-15" aria-hidden="true" tabindex="-1"></a><span class="co"># Save plot</span></span>
+<span id="cb49-16"><a href="#cb49-16" aria-hidden="true" tabindex="-1"></a><span class="fu">ggsave</span>(<span class="st">"my_plot.pdf"</span>)</span>
+<span id="cb49-17"><a href="#cb49-17" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb49-18"><a href="#cb49-18" aria-hidden="true" tabindex="-1"></a><span class="co"># Create regression model</span></span>
+<span id="cb49-19"><a href="#cb49-19" aria-hidden="true" tabindex="-1"></a>m1 <span class="ot">&lt;-</span> <span class="fu">lm</span>(height<span class="sc">~</span>avgphgt, <span class="at">data =</span> d)</span>
+<span id="cb49-20"><a href="#cb49-20" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb49-21"><a href="#cb49-21" aria-hidden="true" tabindex="-1"></a><span class="co"># Display results</span></span>
+<span id="cb49-22"><a href="#cb49-22" aria-hidden="true" tabindex="-1"></a><span class="fu">glance</span>(m1)</span>
+<span id="cb49-23"><a href="#cb49-23" aria-hidden="true" tabindex="-1"></a><span class="fu">tidy</span>(m1)</span>
+<span id="cb49-24"><a href="#cb49-24" aria-hidden="true" tabindex="-1"></a><span class="fu">augment</span>(m1)</span>
+<span id="cb49-25"><a href="#cb49-25" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb49-26"><a href="#cb49-26" aria-hidden="true" tabindex="-1"></a><span class="co"># Multiple regression</span></span>
+<span id="cb49-27"><a href="#cb49-27" aria-hidden="true" tabindex="-1"></a>m2 <span class="ot">&lt;-</span> <span class="fu">lm</span>(height <span class="sc">~</span> avgphgt <span class="sc">+</span> gender, <span class="at">data =</span> d)</span>
+<span id="cb49-28"><a href="#cb49-28" aria-hidden="true" tabindex="-1"></a><span class="fu">summary</span>(m2)</span>
+<span id="cb49-29"><a href="#cb49-29" aria-hidden="true" tabindex="-1"></a>parameters<span class="sc">::</span><span class="fu">model_parameters</span>(m2, <span class="at">standardize =</span> <span class="st">"refit"</span>)</span>
+<span id="cb49-30"><a href="#cb49-30" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb49-31"><a href="#cb49-31" aria-hidden="true" tabindex="-1"></a><span class="co"># Compare model m1 and model m2</span></span>
+<span id="cb49-32"><a href="#cb49-32" aria-hidden="true" tabindex="-1"></a><span class="fu">anova</span>(m1, m2)</span>
+<span id="cb49-33"><a href="#cb49-33" aria-hidden="true" tabindex="-1"></a><span class="fu">compare_performance</span>(m1, m2)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+</div>
+</section>
+<section id="canvas-questions" class="level1">
+<h1>Canvas Questions</h1>
+<p>Use the Transfer of Learning data set. A set of 38 inmates participated in a study that tested the Transfer of Learning Hypothesis. Some people believe that studying Latin is particularly beneficial for progress in other academic disciplines. Each inmate was given a reading test and a math test before the study began. Some inmates were randomly assigned to participate in a 48-week Latin course. The control group studied the lives of famous athletes for the same time period. Each inmate took the reading and math test again to see if studying Latin improved their academic skills. Personal information was also collected about each inmate including their annual income before going to prison, whether or not the inmate had a documented learning disorder, and whether or not the inmate had been convicted of a violent offense. Here are the variable names:</p>
+<ul>
+<li><code>read_1</code> = Reading before study</li>
+<li><code>read_2</code> = Reading after study</li>
+<li><code>math_1</code> = Math before study</li>
+<li><code>math_2</code> = Math after study</li>
+<li><code>latin</code> = 1 (Studied Latin), 0 (Studied sports history)</li>
+<li><code>violent</code> = 1 (convicted of a violent offense), 0 (convicted of a non-violent offense)</li>
+<li><code>learning</code> = 1 (Learning disabled), 0 (Not learning disabled)</li>
+<li><code>income</code> = Annual income before incarceration</li>
+</ul>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb50"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb50-1"><a href="#cb50-1" aria-hidden="true" tabindex="-1"></a>d_learn <span class="ot">&lt;-</span> <span class="fu">read_csv</span>(<span class="st">"https://github.com/wjschne/EDUC5529/raw/master/transfer_of_learning.csv"</span>) </span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+</div>
+<p><span class="instruction">Assume α = 0.05 for all hypothesis tests.</span></p>
+<p><span class="instruction">Create a regression model in which you predict reading at time 2 (<code>read_2</code>) using reading at time 1 (<code>read_1</code>). Call the model fit object <code>m_read</code>.</span></p>
+<section id="canvas-question-1" class="level2 tabset tabset-fade tabset-pills">
+<h2 class="tabset tabset-fade tabset-pills anchored" data-anchor-id="canvas-question-1">Canvas Question 1</h2>
+<p><span class="question"><strong>Question</strong>: Is <code>read_1</code> a significant predictor of <code>read_2</code>?</span></p>
+<section id="hide-hints" class="level3">
+<h3 class="anchored" data-anchor-id="hide-hints">Hide Hints</h3>
+</section>
+<section id="hint-1" class="level3">
+<h3 class="anchored" data-anchor-id="hint-1">Hint 1</h3>
+<p>Make model fit object called <code>m_read</code> using the <code>lm</code> function.</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb51"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb51-1"><a href="#cb51-1" aria-hidden="true" tabindex="-1"></a>m_read <span class="ot">&lt;-</span> <span class="fu">lm</span>(read_2 <span class="sc">~</span> read_1, <span class="at">data =</span> d_learn)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+</div>
+</section>
+<section id="hint-2" class="level3">
+<h3 class="anchored" data-anchor-id="hint-2">Hint 2</h3>
+<p>View coefficent-level statistics with broom’s <code>tidy</code> function.</p>
+<p>You could also use Base R’s <code>summary</code> function.</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb52"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb52-1"><a href="#cb52-1" aria-hidden="true" tabindex="-1"></a><span class="fu">tidy</span>(m_read)</span>
+<span id="cb52-2"><a href="#cb52-2" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb52-3"><a href="#cb52-3" aria-hidden="true" tabindex="-1"></a><span class="co"># or</span></span>
+<span id="cb52-4"><a href="#cb52-4" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb52-5"><a href="#cb52-5" aria-hidden="true" tabindex="-1"></a><span class="fu">summary</span>(m_read)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+</div>
+</section>
+<section id="hint-3" class="level3">
+<h3 class="anchored" data-anchor-id="hint-3">Hint 3</h3>
+<p>In the read_1 row, is the p.value column less than 0.05?</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb53"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb53-1"><a href="#cb53-1" aria-hidden="true" tabindex="-1"></a><span class="fu">tidy</span>(m_read)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+</div>
+<div class="cell">
+<div class="cell-output-display">
+<table class="table table-sm table-striped small">
+<thead>
+<tr class="header">
+<th style="text-align: left;">term</th>
+<th style="text-align: right;">estimate</th>
+<th style="text-align: right;">std.error</th>
+<th style="text-align: right;">statistic</th>
+<th style="text-align: right;">p.value</th>
+</tr>
+</thead>
+<tbody>
+<tr class="odd">
+<td style="text-align: left;">(Intercept)</td>
+<td style="text-align: right;">83.591</td>
+<td style="text-align: right;">10.922</td>
+<td style="text-align: right;">7.65</td>
+<td style="text-align: right;">0.000</td>
+</tr>
+<tr class="even">
+<td style="text-align: left;">read_1</td>
+<td style="text-align: right;">0.251</td>
+<td style="text-align: right;">0.104</td>
+<td style="text-align: right;">2.41</td>
+<td style="text-align: right;">0.021</td>
+</tr>
+</tbody>
+</table>
+</div>
+</div>
+</section>
+</section>
+<section id="canvas-question-2" class="level2 tabset tabset-fade tabset-pills">
+<h2 class="tabset tabset-fade tabset-pills anchored" data-anchor-id="canvas-question-2">Canvas Question 2</h2>
+<p><span class="question"><strong>Question</strong>: What is the <em>R</em><sup>2</sup> for the <code>m_read</code> model?</span></p>
+<section id="hide-hint" class="level3">
+<h3 class="anchored" data-anchor-id="hide-hint">Hide Hint</h3>
+</section>
+<section id="hint" class="level3">
+<h3 class="anchored" data-anchor-id="hint">Hint</h3>
+<p>View model-level statistics with broom’s <code>glance</code> function.</p>
+<p>You could also use Base R’s <code>summary</code> function.</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb54"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb54-1"><a href="#cb54-1" aria-hidden="true" tabindex="-1"></a><span class="fu">glance</span>(m_read)</span>
+<span id="cb54-2"><a href="#cb54-2" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb54-3"><a href="#cb54-3" aria-hidden="true" tabindex="-1"></a><span class="co"># or</span></span>
+<span id="cb54-4"><a href="#cb54-4" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb54-5"><a href="#cb54-5" aria-hidden="true" tabindex="-1"></a><span class="fu">summary</span>(m_read)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+</div>
+</section>
+</section>
+<section id="canvas-question-3" class="level2 tabset tabset-fade tabset-pills">
+<h2 class="tabset tabset-fade tabset-pills anchored" data-anchor-id="canvas-question-3">Canvas Question 3</h2>
+<p><span class="question"><strong>Question</strong>: What does the scatter plot look like when <code>read_1</code> is on the <em>x</em>-axis and <code>read_2</code> is on the <em>y</em>-axis? Also plot the regression line. Save your plot using the <code>ggsave</code> function and upload it to Canvas.</span></p>
+<section id="hide-hints-1" class="level3">
+<h3 class="anchored" data-anchor-id="hide-hints-1">Hide Hints</h3>
+</section>
+<section id="hint-1-1" class="level3">
+<h3 class="anchored" data-anchor-id="hint-1-1">Hint 1</h3>
+<p>This will get you started</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb55"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb55-1"><a href="#cb55-1" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplot</span>(d_learn, <span class="fu">aes</span>(read_1, read_2))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+</div>
+</section>
+<section id="points-hint" class="level3">
+<h3 class="anchored" data-anchor-id="points-hint">Points Hint</h3>
+<p>Here is how you add points.</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb56"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb56-1"><a href="#cb56-1" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplot</span>(d_learn, <span class="fu">aes</span>(read_1, read_2)) <span class="sc">+</span></span>
+<span id="cb56-2"><a href="#cb56-2" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_point</span>() </span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+</div>
+</section>
+<section id="line-hint" class="level3">
+<h3 class="anchored" data-anchor-id="line-hint">Line Hint</h3>
+<p>Here is how you add a regression line.</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb57"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb57-1"><a href="#cb57-1" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplot</span>(d_learn, <span class="fu">aes</span>(read_1, read_2)) <span class="sc">+</span></span>
+<span id="cb57-2"><a href="#cb57-2" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_point</span>() <span class="sc">+</span> </span>
+<span id="cb57-3"><a href="#cb57-3" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_smooth</span>(<span class="at">method =</span> <span class="st">"lm"</span>) </span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+</div>
+</section>
+<section id="plot-polishing" class="level3">
+<h3 class="anchored" data-anchor-id="plot-polishing">Plot Polishing</h3>
+<p>This is overkill for now. But soon you will want to fine tune your plots.</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb58"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb58-1"><a href="#cb58-1" aria-hidden="true" tabindex="-1"></a><span class="co"># I want to put the equation at x = 130</span></span>
+<span id="cb58-2"><a href="#cb58-2" aria-hidden="true" tabindex="-1"></a>equation_x <span class="ot">&lt;-</span> <span class="dv">130</span></span>
+<span id="cb58-3"><a href="#cb58-3" aria-hidden="true" tabindex="-1"></a>equation_y <span class="ot">&lt;-</span> <span class="fu">predict</span>(m_read, <span class="at">newdata =</span> <span class="fu">tibble</span>(<span class="at">read_1 =</span> equation_x))</span>
+<span id="cb58-4"><a href="#cb58-4" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb58-5"><a href="#cb58-5" aria-hidden="true" tabindex="-1"></a><span class="co"># Extracting the coefficients</span></span>
+<span id="cb58-6"><a href="#cb58-6" aria-hidden="true" tabindex="-1"></a>b_read <span class="ot">&lt;-</span> <span class="fu">round</span>(<span class="fu">coef</span>(m_read),<span class="dv">2</span>)</span>
+<span id="cb58-7"><a href="#cb58-7" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb58-8"><a href="#cb58-8" aria-hidden="true" tabindex="-1"></a><span class="co"># The angle of the regression line is the inverse tangent of the slope (converted to degrees)</span></span>
+<span id="cb58-9"><a href="#cb58-9" aria-hidden="true" tabindex="-1"></a>eq_angle <span class="ot">&lt;-</span> <span class="fu">atan</span>(b_read[<span class="dv">2</span>]) <span class="sc">*</span> <span class="dv">180</span> <span class="sc">/</span> pi</span>
+<span id="cb58-10"><a href="#cb58-10" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb58-11"><a href="#cb58-11" aria-hidden="true" tabindex="-1"></a><span class="co"># Equation</span></span>
+<span id="cb58-12"><a href="#cb58-12" aria-hidden="true" tabindex="-1"></a>eq_read <span class="ot">&lt;-</span> <span class="fu">paste0</span>(<span class="st">"italic(Y) == "</span>, </span>
+<span id="cb58-13"><a href="#cb58-13" aria-hidden="true" tabindex="-1"></a>                  b_read[<span class="dv">1</span>], </span>
+<span id="cb58-14"><a href="#cb58-14" aria-hidden="true" tabindex="-1"></a>                  <span class="st">" + "</span>, </span>
+<span id="cb58-15"><a href="#cb58-15" aria-hidden="true" tabindex="-1"></a>                  b_read[<span class="dv">2</span>], </span>
+<span id="cb58-16"><a href="#cb58-16" aria-hidden="true" tabindex="-1"></a>                  <span class="st">" *  italic(X) + italic(e)"</span>)</span>
+<span id="cb58-17"><a href="#cb58-17" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb58-18"><a href="#cb58-18" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplot</span>(d_learn, <span class="fu">aes</span>(read_1, read_2)) <span class="sc">+</span></span>
+<span id="cb58-19"><a href="#cb58-19" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_point</span>() <span class="sc">+</span></span>
+<span id="cb58-20"><a href="#cb58-20" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_smooth</span>(<span class="at">method =</span> <span class="st">"lm"</span>) <span class="sc">+</span></span>
+<span id="cb58-21"><a href="#cb58-21" aria-hidden="true" tabindex="-1"></a>  <span class="fu">labs</span>(<span class="at">x =</span> <span class="st">"Reading at Time 1"</span>,</span>
+<span id="cb58-22"><a href="#cb58-22" aria-hidden="true" tabindex="-1"></a>       <span class="at">y =</span> <span class="st">"Reading at Time 2"</span>,</span>
+<span id="cb58-23"><a href="#cb58-23" aria-hidden="true" tabindex="-1"></a>       <span class="at">title =</span> <span class="st">"Using Reading at Time 1 to Predict Reading at Time 2"</span>) <span class="sc">+</span></span>
+<span id="cb58-24"><a href="#cb58-24" aria-hidden="true" tabindex="-1"></a>  <span class="fu">coord_fixed</span>() <span class="sc">+</span></span>
+<span id="cb58-25"><a href="#cb58-25" aria-hidden="true" tabindex="-1"></a>  <span class="fu">annotate</span>(</span>
+<span id="cb58-26"><a href="#cb58-26" aria-hidden="true" tabindex="-1"></a>    <span class="at">geom =</span> <span class="st">"text"</span>,</span>
+<span id="cb58-27"><a href="#cb58-27" aria-hidden="true" tabindex="-1"></a>    <span class="at">x =</span> equation_x,</span>
+<span id="cb58-28"><a href="#cb58-28" aria-hidden="true" tabindex="-1"></a>    <span class="at">y =</span> equation_y,</span>
+<span id="cb58-29"><a href="#cb58-29" aria-hidden="true" tabindex="-1"></a>    <span class="at">angle =</span> eq_angle,</span>
+<span id="cb58-30"><a href="#cb58-30" aria-hidden="true" tabindex="-1"></a>    <span class="at">label =</span> eq_read,</span>
+<span id="cb58-31"><a href="#cb58-31" aria-hidden="true" tabindex="-1"></a>    <span class="at">parse =</span> <span class="cn">TRUE</span>,</span>
+<span id="cb58-32"><a href="#cb58-32" aria-hidden="true" tabindex="-1"></a>    <span class="at">vjust =</span> <span class="sc">-</span><span class="fl">0.5</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+</div>
+</section>
+</section>
+<section id="canvas-question-4" class="level2 tabset tabset-fade tabset-pills">
+<h2 class="tabset tabset-fade tabset-pills anchored" data-anchor-id="canvas-question-4">Canvas Question 4</h2>
+<p><span class="instruction">Create a regression model in which you predict math at time 2 (<code>math_2</code>) using reading at time 1 (<code>math_1</code>). Call the model fit object <code>m_math1</code>.</span></p>
+<p><span class="question"><strong>Question</strong>: Does <code>math_1</code> predict <code>math_2</code>?.</span></p>
+<p><span class="question"><strong>Question</strong>: Does <code>math_1</code> still predict <code>math_2</code> after controlling for <code>read_1</code>?</span></p>
+<section id="hide-hint-1" class="level3">
+<h3 class="anchored" data-anchor-id="hide-hint-1">Hide Hint</h3>
+</section>
+<section id="hint-4" class="level3">
+<h3 class="anchored" data-anchor-id="hint-4">Hint</h3>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb59"><pre class="sourceCode r fold-show code-with-copy"><code class="sourceCode r"><span id="cb59-1"><a href="#cb59-1" aria-hidden="true" tabindex="-1"></a>m_math <span class="ot">&lt;-</span> <span class="fu">lm</span>(math_2 <span class="sc">~</span> math_1 <span class="sc">+</span> read_1, <span class="at">data =</span> d_learn)</span>
+<span id="cb59-2"><a href="#cb59-2" aria-hidden="true" tabindex="-1"></a><span class="fu">summary</span>(m_math)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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diff --git a/tutorials/ANOVA/ANOVA.qmd b/tutorials/ANOVA/ANOVA.qmd
index 11137f2..95c47db 100644
--- a/tutorials/ANOVA/ANOVA.qmd
+++ b/tutorials/ANOVA/ANOVA.qmd
@@ -9,6 +9,7 @@ format:
 ```{r setup, include=FALSE}
 knitr::opts_chunk$set(echo = TRUE)
 library(tidyverse)
+library(easystats)
 library(haven)
 library(sjstats)
 options(digits = 2)
@@ -18,6 +19,7 @@ options(digits = 2)
 
 ```{r load}
 library(tidyverse)
+library(easystats)
 ```
 
 # Import Data and Create Labelled Factors
@@ -68,44 +70,69 @@ The `aov` (Analysis of Variance) function is what you want to use. The `anova` f
 fit <- aov(Anxiety_Questionnaire ~ Attachment, data = d)
 ```
 
+## Diagnostic Plots
+
+```{r assumptions}
+#| fig-height: 9
+check_model(fit)
+```
+
+None of the diagnostic checks raise any alarms. Specifically, the residuals are reasonably normal and have consistent variance acrross groups (i.e., the homogeneeity of variance assumption is reasonable).
+
 ## Summary
 
-The base R function `summary` gives us most of what we might want.
+The base R function `summary` gives us most of what we might want to know.
 
 ```{r summary}
 summary(fit)
 ```
 
-## Other Summary Functions
+We can see that overall model is significant, meaning that at least two groups have reliably different means. At this point we do not know which means differ, but from the plot we have a good guess that the Ambivalent group scores higher on the Anxiety Questionnaire than the other two groups.
+
+A similar, but tidier display can be found with the `parameters` function:
+
+```{r}
+parameters(fit)
+```
 
-If you want other summary functions, there are many options. Here are a few:
 
-```{r othersummary}
-broom::glance(fit)
-broom::tidy(fit)
+## Other Summary Functions
 
-sjstats::anova_stats(fit)
+The overall "fit" or "performance" of the model:
 
-performance::performance(fit)
-parameters::parameters(fit)
-report::report(fit)
+```{r}
+performance(fit)
 ```
 
-## Diagnostic Plots
+* `AIC`: [Akaike's Information Criterion](https://en.wikipedia.org/wiki/Akaike_information_criterion)
+* `AICc`: AIC with a correction for small sample sizes
+* `BIC`: [Bayesian Information Criterion](https://en.wikipedia.org/wiki/Bayesian_information_criterion)
+* `R2`: R-squared: [Coefficient of Determination](https://en.wikipedia.org/wiki/Coefficient_of_determination)
+* `R2_adj`: [Adjusted r-squared](https://en.wikipedia.org/wiki/Coefficient_of_determination#Adjusted_R2)
+* `RMSE`: Root mean squared error $=\sqrt{\frac{\sum_{i = 1}^{n}{e_i}}{n}}$
+* `SIGMA`: Residual standard deviation (The SD of the residuals $=\sqrt{\frac{\sum_{i = 1}^{n}{e_i}}{n-k-1}}$) 
+
+
+## Automated Interpretation
+
+```{r}
+report(fit)
 
-```{r assumptions}
-#| fig-height: 7
-performance::check_model(fit)
 ```
 
 
 ## Post-hoc Tests
 
-Tukey Honestly Significant Differences
+We would like to compare all means but control for family-wise error.
 
-```{r tukey, fig.height=8}
-TukeyHSD(fit)
-plot(TukeyHSD(fit))
+```{r contrasts, fig.height=8}
+estimate_contrasts(fit)
+estimate_contrasts(fit) |> 
+  report()
 ```
 
 
+
+
+
+
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new file mode 100644
index 0000000..467e397
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diff --git a/tutorials/Regression/index.qmd b/tutorials/Regression/index.qmd
new file mode 100644
index 0000000..0e4a448
--- /dev/null
+++ b/tutorials/Regression/index.qmd
@@ -0,0 +1,642 @@
+---
+title: "Regression in R"
+author: "W. Joel Schneider"
+format: 
+  html: 
+    df-print: kable
+---
+
+```{r}
+#| label: setup
+#| include: false
+#| cache: false
+
+knitr::opts_chunk$set(
+  warning = F,
+  message = F,
+  dev = "svg",
+  class.source = 'fold-show',
+  echo = TRUE,
+  dev = "svg",
+  cache = F,
+  out.width = '100%',
+  out.height = '100%'
+)
+
+
+
+
+library(extrafont)
+extrafont::loadfonts(device = "win")
+library(tidyverse)
+library(lubridate)
+library(performance)
+library(haven)
+library(broom)
+library(parameters)
+library(knitr)
+library(report)
+
+options(digits = 3, knitr.kable.NA = '', knitr.kable.digits = 2)
+
+# library(magick)
+
+# dutchmasters::dutchmasters
+options(digits = 3)
+bg_color <- "#ECE5D3"
+fore_color <- "#2D2718"
+line_color <- "#7C4728"
+area_color <- "#624E3E"
+bsize = 16
+tsize = 0.8 * bsize / 2.845276
+myfont <- "Roboto Condensed"
+
+CurrentQuestion <- 0
+
+inc_num <- function() {
+  CurrentQuestion <<- CurrentQuestion + 1
+  CurrentQuestion
+}
+
+CurrentFigure <- 0
+
+inc_fig <- function() {
+  CurrentFigure <<- CurrentFigure + 1
+  CurrentFigure
+  }
+
+show_answer <- FALSE
+
+library(kableExtra)
+library(knitr)
+
+registerS3method("knit_print", "data.frame", function(x, ...) {
+  knitr::knit_print(knitr::kable(x, digits = 2) |> kableExtra::kable_paper(full_width = T, html_font = NULL), ...)
+})
+
+```
+
+Here we are going to use a small data set to predict people's height using the height of their parents.
+
+# Loading packages
+
+:::{.callout-note .column-margin title="R Package"}
+A **R package** is software that extends the capabilities of R. To use a package, you must first install it on your machine. You only need to install it once. By default, the package tab is in the lower right pane in RStudio. It has two buttons, one for installing new packages and one for updating packages you have previously installed.
+
+![RStudio's "Install" and "Update" buttons in the "Package" tab in the lower right pane](updatepackages.png){#fig-packages}
+
+:::
+
+First we load packages to enhance R's abilities. Some packages are designed to work with several other packages as a system. The [tidyverse package](https://www.tidyverse.org/) is a "meta-package" that installs and loads a coherent set of packages designed to help you import, manipulate, visualize, and interpret data. If you do not have a recent version of tidyverse already installed, you can install it with this code:
+
+```{r installtidyverse}
+#| eval: false
+
+install.packages("tidyverse")
+```
+
+:::{.callout-tip}
+
+You only have to install the tidyverse once. Thereafter, from time to time, you can let RStudio's "Update" button in the "Packages" pane help you keep your packages current.
+
+
+:::
+When you "load" the tidyvert
+
+
+The `tidyverse` allows us to do many things, including plotting with `ggplot2`. The `haven` package imports SPSS data. The `broom` package tidies up our regression results with the `tidy`, `glance`, and `augment` functions.
+
+If any of these packages are not installed, click the **Packages** tab, click the **Install** button, enter the names of the packages separated by commas, and click **Install**.
+
+To load the packages, run this code:
+
+```{r loadpacakges}
+library(tidyverse) # Loads primary packages for data wrangling and plotting
+library(haven) # Imports data saved in SPSS, SAS, or Stata format
+library(broom) # Displays model information
+library(performance) # Displays model information 
+library(parameters) # Displays model parameter information
+library(report) # Automated interpretations
+```
+
+# Import data
+
+:::{.column-margin}
+If I use only one data set in an analysis, I call it `d`. If I need multiple data sets, I use a `d_` prefix to differentiate them. For example, if I have separate data sets for juniors and seniors, I might call them, `d_juniors` and `d_seniors`, respectively. This kind of consistency seems like extra work, but it pays off in making it easy for *future-you* to read and understand your own code.
+:::
+
+You can import the `height.sav` data set directly from my github repository for this course. There is no need to save the data anywhere. This code loads the data into variable `d` using the [`read_spss`](https://haven.tidyverse.org/reference/read_spss.html) function from the [haven package](https://haven.tidyverse.org/).
+
+```{r}
+d <- read_spss("https://github.com/wjschne/EDUC5325/raw/master/height.sav")
+```
+
+# Make a plot
+
+Use the [`ggplot`](https://ggplot2.tidyverse.org/reference/ggplot.html) function from the [ggplot2](https://ggplot2.tidyverse.org/) package to make a scatterplot with points and a regression line:
+
+```{r}
+ggplot(d, aes(avgphgt,height)) +
+  geom_point() +
+  geom_smooth(method = "lm")
+```
+
+Annotated, here is what the code does:
+
+![Annotated ggplot2 code](ggplotcode.svg)
+
+# Save the plot!
+
+You can save to your hard drive a high-quality plot with the [`ggsave`](https://ggplot2.tidyverse.org/reference/ggsave.html) function. It will save whatever the last plot you created with [`ggplot`](https://ggplot2.tidyverse.org/reference/ggplot.html). It will guess the file format from the file extension. 
+
+Here I save a .pdf file of the plot:
+
+```{r saveplot}
+#| eval: false
+ggsave("my_plot.pdf")
+```
+
+
+## Vector-based Images
+
+The .pdf format gives the best image quality but can only be viewed in a .pdf reader. The .svg format is almost as good and can be incorporated into webpages and Office documents. One downside of their near-perfect image quality is that .pdf and .svg image file sizes can become quite large. 
+
+## Raster Images
+
+The .png format gives good image quality and renders small file sizes. 
+
+The primary use of the .gif format is to create animated plots. Otherwise stick with .png.
+
+Although the .jpg format is good for photos, it is terrible for plots---it often renders text and sharp corners with pixelated smudges. 
+
+# Creating the model
+
+To run regression models, use the [`lm`](https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/lm) function. The `lm` stands for "linear model." The general pattern is `lm(Y~X, data = d)`, which means "`Y` is predicted by `X`, using the data set `d`."
+
+Here we predict height from Average of Parent Height (`avgphgt`):
+
+```{r}
+m1 <- lm(height~avgphgt, data = d)
+```
+
+Notice that we did not get any results. Instead, we created the model fit object `m1`, which contains information about the model. There are a variety of functions we can use to extract information from `m1`.
+
+# Checking Assumptions
+
+Regression assumes that the observations should independent, and the residuals should be normal and homoscedastic. The [performance package](https://easystats.github.io/performance/index.html) has a great function for checking model assumptions: [`check_model`](https://easystats.github.io/performance/reference/check_model.html)
+
+```{r}
+check_model(m1)
+```
+
+Here we see that none of the assumptions have been severely violated.
+
+# Summarising Results
+
+Base R gives you most of what you would want to know about the regression results with the `summary` function:
+
+```{r}
+summary(m1)
+```
+
+This output is not pretty, nor was it intended to be. It is designed for you, the analyst. The `summary` function's print method is optimized for reading results in the console, not in a document.  Presentation-worthy results need a lot more care and attention.
+
+An automated report from the [report](https://easystats.github.io/report/articles/report.html) package:
+
+```{r report_m1}
+#| warning: false
+report::report(m1)
+```
+
+
+
+The `glance`, `tidy`, and `augment` functions from the [broom](https://broom.tidymodels.org/) package make the output more amenable for further analysis or for presentation.
+
+## Model-Level Statistics
+
+Some statistics like the *coefficient of determination* (*R*^2^) or the *standard error of the estimate* (*&sigma;~e~*) describe the model as a whole.
+
+The model-level statistics can be extracted with broom's [`glance`](https://broom.tidymodels.org/reference/data.frame_tidiers.html) function or the performance package's [`model_performance`](https://easystats.github.io/performance/reference/model_performance.lm.html) function.
+
+```{r}
+# Using broom package
+glance(m1)
+model_parameters(m1)
+parameters(m1)
+standardise_parameters(m1)
+performance(m1)
+```
+
+
+*R*^2^ is in the `glance` function's `r.squared` column, and the *standard error of the estimate* is in the `sigma` column.
+If all you wanted was the *R*^2^, you could do this:
+
+```{r}
+glance(m1)$r.squared
+```
+
+Or using the performance package's [`r2`](https://easystats.github.io/performance/reference/r2.html) function:
+
+```{r}
+r2(m1)
+```
+
+
+For the *standard error of the estimate*:
+
+```{r}
+glance(m1)$sigma
+```
+
+
+Why would you want just one number instead of reading it from a table? In reproducible research, we intermingle text and code so that it is clear where every number came from. Thus, "hard-coding" your results like this is considered poor practice:
+
+```
+The model explains 64% of the variance.
+```
+
+Using rmarkdown, instead of typing the numeric results, we type pull the results using an [inline code chunk](https://rmarkdown.rstudio.com/lesson-4.html):
+
+
+````{verbatim}
+The model explains `r round(100 * glance(m1)$r.squared, 0)`% of the variance.
+````
+
+Which, when rendered, produces the correct output:
+
+> The model explains `r round(100 * glance(m1)$r.squared, 0)`% of the variance.
+
+That seems like a lot of extra work, right? Yes, it is---unless there is a possibility that your underlying data might change or that you might copy your numbers incorrectly. If you are imperfect, the extra time and effort is worth it. It makes it easy for other scholars to see exactly where each number came from. Hard-coded results are harder to trust.
+
+Automated report of the model's performance:
+
+```{r}
+#| warning: false
+report_performance(m1)
+
+
+```
+
+
+## Coefficient-level statistics
+
+The regression coefficients---the intercept (*b*~0~) and the slope (*b*~1~)---have a number of statistics associated with them, which we will discuss later in the course.
+
+If you just wanted the intercept and the slope coefficients, use the `coef` function:
+
+```{r}
+coef(m1)
+```
+
+Thus, the intercept is `r round(coef(m1)[1],2)` and the slope is `r round(coef(m1)[2],2)`.
+
+To get the coefficients (intercept and slope) along with their p-values and other statistical information, use the [`tidy`](https://broom.tidymodels.org/reference/tidy.lm.html) function:
+
+```{r}
+tidy(m1)
+```
+
+The parameters package has a slightly more print-friendly version of `tidy`:
+
+```{r}
+model_parameters(m1) 
+```
+
+If you want standardized parameters:
+
+```{r}
+model_parameters(m1, standardize = "refit")
+```
+
+Automated report of the parameters:
+
+```{r}
+#| warning: false
+report_parameters(m1)
+```
+
+
+## Observation-Level Statistics
+
+We sometimes want the individual-level data like the predictions ($\hat{Y}$) and the errors (technically called *residuals*).
+
+To get observation-level statistics, use broom's [`augment`](https://broom.tidymodels.org/reference/augment.lm.html) function:
+
+```{r}
+augment(m1) 
+```
+
+The predicted values $\hat{Y}$ are in the `.fitted` column, and the errors (residuals) are in the `.resid` column.
+
+If you wanted to predict a value not in the data, you use the `newdata` argument. For example, if I wanted to predict the height of a person whose parents had a height of 65:
+
+```{r}
+augment(m1, newdata = tibble(avgphgt = 65))
+```
+
+
+# Multiple Regression
+
+In this data set, all participants identified as either male or female. We assume that males are, on average, taller than females. The codes are male = 1 and female = 2, thus our predicted effect is negative.
+
+We run the regression model with 2 predictors like so:
+
+```{r}
+m2 <- lm(height ~ avgphgt + gender, data = d)
+```
+
+## Checking assumptions
+
+```{r}
+check_model(m2)
+
+
+```
+
+All looks well.
+
+## Summarizing Results
+
+To summarize the results, use the `summary` function:
+
+```{r}
+summary(m2)
+```
+
+An automated report:
+
+```{r}
+#| warning: false
+report::report(m2)
+```
+
+
+
+Standardized coefficients:
+
+```{r}
+parameters::model_parameters(m2, standardize = "refit")
+```
+
+The oddly-named `anova` function compares 2 models:
+
+```{r}
+anova(m1, m2)
+```
+
+The p-value is significant, meaning that `m2` explains more variance than `m1`.
+
+
+I like the more full-featured set of model comparison functions from the performance package: 
+
+```{r}
+compare_performance(m1, m2) 
+```
+
+This is equivalent to the `anova` function but in a more readable format:
+
+```{r}
+test_wald(m1, m2)
+```
+
+
+This uses Bayes Factor instead of a p-value:
+
+```{r}
+test_bf(m1, m2)
+```
+
+A BF > 1 means that `m2` is more strongly supported than `m1`. A BF < 1 means that `m1` is more strongly supported than `m2`.
+
+```{r}
+report(test_bf(m1, m2))
+```
+
+
+# All the code in one place:
+
+The preceding analyses might seem like a lot, but it is not really so much when you see it all in just a few lines of code. Here are all the main analyses:
+
+```{r}
+#| eval: !expr F
+# Load packages
+library(tidyverse)
+library(haven)
+library(broom)
+library(performance)
+
+# Import data
+d <- read_spss("https://github.com/wjschne/EDUC5325/raw/master/height.sav")
+
+# Plot data
+ggplot(d, aes(weight,height)) +
+  geom_point() +
+  geom_smooth(method = "lm")
+
+# Save plot
+ggsave("my_plot.pdf")
+
+# Create regression model
+m1 <- lm(height~avgphgt, data = d)
+
+# Display results
+glance(m1)
+tidy(m1)
+augment(m1)
+
+# Multiple regression
+m2 <- lm(height ~ avgphgt + gender, data = d)
+summary(m2)
+parameters::model_parameters(m2, standardize = "refit")
+
+# Compare model m1 and model m2
+anova(m1, m2)
+compare_performance(m1, m2)
+```
+
+# Canvas Questions
+
+Use the Transfer of Learning data set. A set of 38 inmates participated in a study that tested the Transfer of Learning Hypothesis. Some people believe that studying Latin is particularly beneficial for progress in other academic disciplines. Each inmate was given a reading test and a math test before the study began. Some inmates were randomly assigned to participate in a 48-week Latin course. The control group studied the lives of famous athletes for the same time period. Each inmate took the reading and math test again to see if studying Latin improved their academic skills. Personal information was also collected about each inmate including their annual income before going to prison, whether or not the inmate had a documented learning disorder, and whether or not the inmate had been convicted of a violent offense. Here are the variable names:
+
+* `read_1` = Reading before study
+* `read_2` = Reading after study
+* `math_1` = Math before study
+* `math_2` = Math after study
+* `latin` = 1 (Studied Latin), 0 (Studied sports history)
+* `violent` = 1 (convicted of a violent offense), 0 (convicted of a non-violent offense)
+* `learning` = 1 (Learning disabled), 0 (Not learning disabled)
+* `income` = Annual income before incarceration
+
+```{r}
+#| message: !expr F
+d_learn <- read_csv("https://github.com/wjschne/EDUC5529/raw/master/transfer_of_learning.csv") 
+```
+
+<span class="instruction">Assume &alpha; = 0.05 for all hypothesis tests.</span>
+
+<span class="instruction">Create a regression model in which you predict reading at time 2 (`read_2`) using reading at time 1 (`read_1`). Call the model fit object `m_read`.</span>
+
+## Canvas Question `r inc_num()` {.tabset .tabset-fade .tabset-pills}
+
+<span class = "question">**Question**: Is `read_1` a significant predictor of `read_2`?</span>
+
+### Hide Hints
+
+
+### Hint 1
+
+Make model fit object called `m_read` using the `lm` function.
+
+```{r}
+m_read <- lm(read_2 ~ read_1, data = d_learn)
+```
+
+
+### Hint 2
+
+View coefficent-level statistics with broom's `tidy` function.
+
+You could also use Base R's `summary` function.
+
+```{r}
+#| eval: !expr show_answer
+tidy(m_read)
+
+# or
+
+summary(m_read)
+```
+
+### Hint 3
+
+In the read_1 row, is the p.value column less than 0.05?
+
+```{r}
+#| eval: !expr F
+tidy(m_read)
+```
+
+```{r}
+#| echo: !expr F
+tidy(m_read) %>% knitr::kable(digit = 3)
+```
+
+## Canvas Question `r inc_num()` {.tabset .tabset-fade .tabset-pills}
+
+<span class = "question">**Question**: What is the *R*^2^ for the `m_read` model?</span>
+
+### Hide Hint
+
+
+### Hint
+
+View model-level statistics with broom's `glance` function.
+
+You could also use Base R's `summary` function.
+
+```{r}
+#| eval: !expr show_answer
+glance(m_read)
+
+# or
+
+summary(m_read)
+```
+
+
+## Canvas Question `r inc_num()` {.tabset .tabset-fade .tabset-pills}
+
+<span class = "question">**Question**: What does the scatter plot look like when `read_1` is on the *x*-axis and `read_2` is on the *y*-axis? Also plot the regression line. Save your plot using the `ggsave` function and upload it to Canvas.</span>
+
+### Hide Hints
+
+### Hint 1
+
+This will get you started
+
+```{r}
+#| eval: !expr show_answer
+ggplot(d_learn, aes(read_1, read_2))
+```
+
+### Points Hint
+
+Here is how you add points.
+
+```{r}
+#| eval: !expr show_answer
+ggplot(d_learn, aes(read_1, read_2)) +
+  geom_point() 
+```
+
+### Line Hint
+
+Here is how you add a regression line.
+
+```{r}
+#| eval: !expr show_answer
+ggplot(d_learn, aes(read_1, read_2)) +
+  geom_point() + 
+  geom_smooth(method = "lm") 
+```
+
+### Plot Polishing
+
+This is overkill for now. But soon you will want to fine tune your plots.
+
+```{r}
+#| eval: !expr show_answer
+# I want to put the equation at x = 130
+equation_x <- 130
+equation_y <- predict(m_read, newdata = tibble(read_1 = equation_x))
+
+# Extracting the coefficients
+b_read <- round(coef(m_read),2)
+
+# The angle of the regression line is the inverse tangent of the slope (converted to degrees)
+eq_angle <- atan(b_read[2]) * 180 / pi
+
+# Equation
+eq_read <- paste0("italic(Y) == ", 
+                  b_read[1], 
+                  " + ", 
+                  b_read[2], 
+                  " *  italic(X) + italic(e)")
+
+ggplot(d_learn, aes(read_1, read_2)) +
+  geom_point() +
+  geom_smooth(method = "lm") +
+  labs(x = "Reading at Time 1",
+       y = "Reading at Time 2",
+       title = "Using Reading at Time 1 to Predict Reading at Time 2") +
+  coord_fixed() +
+  annotate(
+    geom = "text",
+    x = equation_x,
+    y = equation_y,
+    angle = eq_angle,
+    label = eq_read,
+    parse = TRUE,
+    vjust = -0.5)
+```
+
+## Canvas Question `r inc_num()` {.tabset .tabset-fade .tabset-pills}
+
+<span class="instruction">Create a regression model in which you predict math at time 2 (`math_2`) using reading at time 1 (`math_1`). Call the model fit object `m_math1`.</span>
+
+<span class = "question">**Question**: Does `math_1`  predict `math_2`?.</span>
+
+
+<span class = "question">**Question**: Does `math_1` still predict `math_2` after controlling for `read_1`?</span>
+
+### Hide Hint
+
+### Hint
+
+
+```{r}
+#| eval: !expr show_answer
+m_math <- lm(math_2 ~ math_1 + read_1, data = d_learn)
+summary(m_math)
+```
+
diff --git a/tutorials/Regression/index_cache/html/__packages b/tutorials/Regression/index_cache/html/__packages
new file mode 100644
index 0000000..af49bbb
--- /dev/null
+++ b/tutorials/Regression/index_cache/html/__packages
@@ -0,0 +1,18 @@
+extrafont
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+ggplot2
+tibble
+tidyr
+readr
+purrr
+dplyr
+stringr
+forcats
+lubridate
+performance
+haven
+broom
+parameters
+knitr
+report
+kableExtra
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