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</head>
<body class="quarto-light">
<div class="reveal">
<div class="slides">
<section id="title-slide" class="quarto-title-block center">
<h1 class="title">Creating data plots for effective decision-making using statistical inference with R</h1>
<div class="quarto-title-authors">
<div class="quarto-title-author">
<div class="quarto-title-author-name">
Dianne Cook <br> Monash University
</div>
</div>
</div>
</section>
<section id="session-2-making-decisions-and-inferential-statements-based-on-data-plots" class="slide level2 center center-align">
<h2>Session 2: Making decisions and inferential statements based on data plots</h2>
</section>
<section id="outline" class="slide level2">
<h2>Outline</h2>
<div class="cell" data-layout-align="center">
<div class="cell-output-display">
<table>
<thead>
<tr class="header">
<th style="text-align: left;">time</th>
<th style="text-align: left;">topic</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: left;">3:00-3:20</td>
<td style="text-align: left;">What is your plot testing?</td>
</tr>
<tr class="even">
<td style="text-align: left;">3:20-3:35</td>
<td style="text-align: left;">Creating null samples</td>
</tr>
<tr class="odd">
<td style="text-align: left;">3:35-4:00</td>
<td style="text-align: left;">Conducting a lineup test</td>
</tr>
<tr class="even">
<td style="text-align: left;">4:00-4:30</td>
<td style="text-align: left;">Testing for best plot design</td>
</tr>
</tbody>
</table>
</div>
</div>
</section>
<section id="what-is-your-plot-testing" class="slide level2">
<h2>What is your plot testing?</h2>
<div class="columns">
<div class="column" style="width:50%;">
<p><br></p>
<div class="cell" data-layout-align="center">
<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>LM_FIT <span class="ot"><-</span> <span class="fu">lm</span>(VAR2 <span class="sc">~</span> VAR1, </span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a> <span class="at">data =</span> DATA)</span>
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a>FIT_ALL <span class="ot"><-</span> <span class="fu">augment</span>(LM_FIT)</span>
<span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplot</span>(FIT_ALL, <span class="fu">aes</span>(<span class="at">x=</span>.FITTED, </span>
<span id="cb1-5"><a href="#cb1-5" aria-hidden="true" tabindex="-1"></a> <span class="at">y=</span>.RESID)) <span class="sc">+</span> </span>
<span id="cb1-6"><a href="#cb1-6" 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>
<p><br> What will we be assessing using this plot?</p>
</div><div class="column" style="width:10%;">
</div><div class="column" style="width:40%;">
<div class="fragment">
<p>Is the model misspecified?</p>
<div style="font-size: 90%;">
<ul>
<li>non-linearity</li>
<li>heteroskedasticity</li>
<li>outliers/anomalies</li>
<li>non-normality</li>
<li>fitted value distribution</li>
</ul>
</div>
</div>
</div>
</div>
</section>
<section id="what-is-your-plot-testing-1" class="slide level2">
<h2>What is your plot testing?</h2>
<div class="columns">
<div class="column" style="width:60%;">
<div class="cell" data-layout-align="center">
<div class="cell-output-display">
<div class="quarto-figure quarto-figure-center">
<figure>
<p><img data-src="slides2_files/figure-revealjs/unnamed-chunk-6-1.png" style="width:100.0%"></p>
</figure>
</div>
</div>
</div>
</div><div class="column" style="width:40%;">
<p>What do you see?</p>
<div class="fragment">
<p>✗ non-linearity <br> ✓ heteroskedasticity <br> ✗ outliers/anomalies <br> ✓ non-normality <br> ✗ fitted value distribution is uniform</p>
</div>
<div class="fragment">
<p><br> <span style="color: #F5191C;"> Are you sure? </span></p>
</div>
</div>
</div>
</section>
<section id="what-is-your-plot-testing-2" class="slide level2">
<h2>What is your plot testing?</h2>
<div class="columns">
<div class="column" style="width:50%;">
<p><br></p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb2"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplot</span>(DATA, </span>
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">aes</span>(<span class="at">x=</span>VAR1, </span>
<span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a> <span class="at">y=</span>VAR2, </span>
<span id="cb2-4"><a href="#cb2-4" aria-hidden="true" tabindex="-1"></a> <span class="at">color=</span>CLASS)) <span class="sc">+</span> </span>
<span id="cb2-5"><a href="#cb2-5" 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>
<p><br> What will we be assessing using this plot?</p>
</div><div class="column" style="width:10%;">
</div><div class="column" style="width:40%;">
<div class="fragment">
<p><br> Is there a difference between the groups?</p>
<ul>
<li>location</li>
<li>shape</li>
<li>outliers/anomalies</li>
</ul>
</div>
</div>
</div>
</section>
<section id="what-is-your-plot-testing-3" class="slide level2">
<h2>What is your plot testing?</h2>
<div class="columns">
<div class="column" style="width:50%;">
<div class="cell" data-layout-align="center">
<div class="cell-output-display">
<div class="quarto-figure quarto-figure-center">
<figure>
<p><img data-src="slides2_files/figure-revealjs/unnamed-chunk-8-1.png" style="width:100.0%"></p>
</figure>
</div>
</div>
</div>
<p><br></p>
</div><div class="column" style="width:10%;">
</div><div class="column" style="width:40%;">
<p>What do you see?</p>
<div class="fragment">
<p>There a difference between the groups</p>
<p>✓ location <br> ✗ shape <br> ✓ outliers/anomalies</p>
</div>
<div class="fragment">
<p><br> <span style="color: #F5191C;"> Are you sure? </span></p>
</div>
</div>
</div>
<div class="hidden">
</div>
</section>
<section id="statistical-thinking" class="slide level2">
<h2>Statistical thinking</h2>
<ul>
<li>Because the <span style="color: #3B99B1;"> plot </span> is specified using a functional mapping of the variables, it <span style="color: #3B99B1;"> is a statistic</span>.</li>
<li>The null and alternative hypotheses are indicated from the plot description.</li>
<li>Applying the function to a dataset provides the observed value.</li>
</ul>
</section>
<section id="null-hypothesis-example-1" class="slide level2">
<h2>Null hypothesis, example 1</h2>
<div class="columns">
<div class="column" style="width:50%;">
<p><br></p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb3"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a>LM_FIT <span class="ot"><-</span> <span class="fu">lm</span>(VAR2 <span class="sc">~</span> VAR1, </span>
<span id="cb3-2"><a href="#cb3-2" aria-hidden="true" tabindex="-1"></a> <span class="at">data =</span> DATA)</span>
<span id="cb3-3"><a href="#cb3-3" aria-hidden="true" tabindex="-1"></a>FIT_ALL <span class="ot"><-</span> <span class="fu">augment</span>(LM_FIT)</span>
<span id="cb3-4"><a href="#cb3-4" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplot</span>(FIT_ALL, <span class="fu">aes</span>(<span class="at">x=</span>.FITTED, </span>
<span id="cb3-5"><a href="#cb3-5" aria-hidden="true" tabindex="-1"></a> <span class="at">y=</span>.RESID)) <span class="sc">+</span> </span>
<span id="cb3-6"><a href="#cb3-6" 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>
<p><br> What is the null hypothesis?</p>
<div class="fragment">
<p><em>There is no relationship between residuals and fitted values.</em> This is <span class="math inline">\(H_o\)</span>.</p>
</div>
</div><div class="column" style="width:5%;">
</div><div class="column" style="width:45%;">
<div class="fragment f80">
<p><br><br><br></p>
<p><strong>Alternative hypothesis</strong>, <span class="math inline">\(H_a\)</span>:</p>
<p><em>There is some relationship</em>, which might be</p>
<ul>
<li>non-linearity</li>
<li>heteroskedasticity</li>
<li>outliers/anomalies</li>
</ul>
</div>
</div>
</div>
</section>
<section id="null-hypothesis-example-2" class="slide level2">
<h2>Null hypothesis, example 2</h2>
<div class="columns">
<div class="column" style="width:50%;">
<p><br></p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb4"><pre class="sourceCode r 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>(DATA, </span>
<span id="cb4-2"><a href="#cb4-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">aes</span>(<span class="at">x=</span>VAR1, </span>
<span id="cb4-3"><a href="#cb4-3" aria-hidden="true" tabindex="-1"></a> <span class="at">y=</span>VAR2, </span>
<span id="cb4-4"><a href="#cb4-4" aria-hidden="true" tabindex="-1"></a> <span class="at">color=</span>CLASS)) <span class="sc">+</span> </span>
<span id="cb4-5"><a href="#cb4-5" 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>
<p><br> What is the null hypothesis?</p>
<div class="fragment">
<p><em>There is no difference between the classes.</em> This is <span class="math inline">\(H_o\)</span>.</p>
</div>
</div><div class="column" style="width:5%;">
</div><div class="column" style="width:45%;">
<div class="fragment f80">
<p><br><br><br> <strong>Alternative hypothesis</strong>, <span class="math inline">\(H_a\)</span>:</p>
<p><em>There is some difference between the classes</em>, which might be</p>
<ul>
<li>location</li>
<li>shape</li>
<li>outliers/anomalies</li>
</ul>
</div>
</div>
</div>
</section>
<section id="exercise" class="slide level2">
<h2>Exercise</h2>
<p>What is being tested in each of these plot descriptions?</p>
<div class="columns">
<div class="column fragment" style="width:30%;">
<div class="cell" data-layout-align="center">
<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><span class="fu">ggplot</span>(DATA, </span>
<span id="cb5-2"><a href="#cb5-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">aes</span>(<span class="at">x=</span>VAR1)) <span class="sc">+</span></span>
<span id="cb5-3"><a href="#cb5-3" aria-hidden="true" tabindex="-1"></a> <span class="fu">geom_histogram</span>()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
</div><div class="column fragment" style="width:38%;">
<div class="cell" data-layout-align="center">
<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">ggplot</span>(DATA, </span>
<span id="cb6-2"><a href="#cb6-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">aes</span>(<span class="at">x=</span>VAR1, </span>
<span id="cb6-3"><a href="#cb6-3" aria-hidden="true" tabindex="-1"></a> <span class="at">fill=</span>VAR2)) <span class="sc">+</span></span>
<span id="cb6-4"><a href="#cb6-4" aria-hidden="true" tabindex="-1"></a> <span class="fu">geom_bar</span>(<span class="at">position=</span><span class="st">"fill"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
</div><div class="column fragment" style="width:30%;">
<div class="cell" data-layout-align="center">
<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">ggplot</span>(DATA, </span>
<span id="cb7-2"><a href="#cb7-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">aes</span>(<span class="at">x=</span>VAR1, </span>
<span id="cb7-3"><a href="#cb7-3" aria-hidden="true" tabindex="-1"></a> <span class="at">y=</span>VAR2)) <span class="sc">+</span></span>
<span id="cb7-4"><a href="#cb7-4" aria-hidden="true" tabindex="-1"></a> <span class="fu">geom_point</span>() <span class="sc">+</span></span>
<span id="cb7-5"><a href="#cb7-5" aria-hidden="true" tabindex="-1"></a> <span class="fu">geom_smooth</span>()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
</div><div class="column fragment" style="width:30%;">
<p>Distribution of VAR1 is ?</p>
</div><div class="column fragment" style="width:38%;">
<p>There is no relationship between VAR1 and VAR2. More specifically, the proportion of VAR2 in each level of VAR1 is the same.</p>
</div><div class="column fragment f80" style="width:30%;">
<p><br> There is no relationship between VAR1 and VAR2. Particularly, VAR2 is not dependent on VAR1 and there is no trend.</p>
</div>
</div>
</section>
<section id="creating-null-samples" class="slide level2 center">
<h2>Creating null samples</h2>
</section>
<section id="statistical-thinking-1" class="slide level2">
<h2>Statistical thinking</h2>
<div class="columns">
<div class="column f80" style="width:50%;">
<p>Sampling distribution for a t-statistic. Values expected assuming <span class="math inline">\(H_o\)</span> is true. <span style="color: #F5191C;"> Shaded areas </span> indicate extreme values.</p>
<div class="cell" data-layout-align="center">
<div class="cell-output-display">
<div class="quarto-figure quarto-figure-center">
<figure>
<p><img data-src="slides2_files/figure-revealjs/unnamed-chunk-15-1.png" style="width:70.0%"></p>
</figure>
</div>
</div>
</div>
</div><div class="column fragment" style="width:50%;">
<p><br><br><br><br></p>
<p><span style="color: #3B99B1;"> For making comparisons when plotting, draw a number of null samples, and plot them with the same script in the plot description.</span></p>
</div>
</div>
</section>
<section id="creating-null-samples-example-1" class="slide level2">
<h2>Creating null samples, example 1</h2>
<div class="columns">
<div class="column f80" style="width:50%;">
<div class="cell" data-layout-align="center">
<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><span class="fu">ggplot</span>(DATA, </span>
<span id="cb8-2"><a href="#cb8-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">aes</span>(<span class="at">x=</span>VAR1, </span>
<span id="cb8-3"><a href="#cb8-3" aria-hidden="true" tabindex="-1"></a> <span class="at">y=</span>VAR2, </span>
<span id="cb8-4"><a href="#cb8-4" aria-hidden="true" tabindex="-1"></a> <span class="at">color=</span>CLASS)) <span class="sc">+</span> </span>
<span id="cb8-5"><a href="#cb8-5" 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>
<p><br></p>
<p><span class="math inline">\(H_o\)</span>: <em>There is no difference between the classes.</em></p>
<div class="fragment">
<p>How would you generate null samples?</p>
</div>
<div class="fragment">
<p><br> Break any association by permuting (scrambling/shuffling/re-sampling) the CLASS variable.</p>
</div>
</div><div class="column" style="width:50%;">
<div class="fragment">
<div class="cell" data-layout-align="center">
<div class="cell-output-display">
<div class="quarto-figure quarto-figure-center">
<figure>
<p><img data-src="slides2_files/figure-revealjs/unnamed-chunk-17-1.png" style="width:60.0%"></p>
</figure>
</div>
</div>
</div>
</div>
</div>
</div>
</section>
<section id="creating-null-samples-example-2" class="slide level2">
<h2>Creating null samples, example 2</h2>
<div class="columns">
<div class="column f80" style="width:50%;">
<div class="cell" data-layout-align="center">
<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>LM_FIT <span class="ot"><-</span> <span class="fu">lm</span>(VAR2 <span class="sc">~</span> VAR1, </span>
<span id="cb9-2"><a href="#cb9-2" aria-hidden="true" tabindex="-1"></a> <span class="at">data =</span> DATA)</span>
<span id="cb9-3"><a href="#cb9-3" aria-hidden="true" tabindex="-1"></a>FIT_ALL <span class="ot"><-</span> <span class="fu">augment</span>(LM_FIT)</span>
<span id="cb9-4"><a href="#cb9-4" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplot</span>(FIT_ALL, <span class="fu">aes</span>(<span class="at">x=</span>.FITTED, </span>
<span id="cb9-5"><a href="#cb9-5" aria-hidden="true" tabindex="-1"></a> <span class="at">y=</span>.RESID)) <span class="sc">+</span> </span>
<span id="cb9-6"><a href="#cb9-6" 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>
<div class="fragment">
<p><span class="math inline">\(H_o\)</span>: <em>There is no relationship between residuals and fitted values.</em></p>
</div>
<div class="fragment">
<p>How would you generate null samples?</p>
</div>
<div class="fragment">
<p><br> Break any association by</p>
<ul>
<li>permuting residuals,</li>
<li>or residual rotation,</li>
<li>or simulate residuals from a normal distribution.</li>
</ul>
</div>
</div><div class="column" style="width:50%;">
<div class="fragment">
<div class="cell" data-layout-align="center">
<div class="cell-output-display">
<div class="quarto-figure quarto-figure-center">
<figure>
<p><img data-src="slides2_files/figure-revealjs/unnamed-chunk-19-1.png" style="width:60.0%"></p>
</figure>
</div>
</div>
</div>
</div>
</div>
</div>
</section>
<section id="conducting-a-lineup-test" class="slide level2 center">
<h2>Conducting a lineup test</h2>
</section>
<section id="steps" class="slide level2">
<h2>Steps</h2>
<ol type="1">
<li>Create a lineup of <span class="math inline">\(m-1\)</span> null plots + 1 data plot, where the data plot is randomly placed among nulls. Remove any distracting information, like tick labels, titles.</li>
<li>Ask uninvolved observer(s) to pick the plot that is most different. (May need to use a crowd-sourcing service.)</li>
<li>Compute the probability that the data plot was chosen, assuming it is no different from the null plots. This is the <span class="math inline">\(p\)</span>-value.</li>
<li>Decide to reject or fail to reject the null.</li>
</ol>
</section>
<section id="lineup-example-1-12" class="slide level2">
<h2>Lineup example 1 <span style="font-size: 70%;"> (1/2) </span></h2>
<div class="f60">
<div class="cell" data-layout-align="center">
<details>
<summary>Code</summary>
<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">set.seed</span>(<span class="dv">241</span>)</span>
<span id="cb10-2"><a href="#cb10-2" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplot</span>(<span class="fu">lineup</span>(<span class="fu">null_permute</span>(<span class="st">"species"</span>), penguins, <span class="at">n=</span><span class="dv">15</span>), </span>
<span id="cb10-3"><a href="#cb10-3" aria-hidden="true" tabindex="-1"></a> <span class="fu">aes</span>(<span class="at">x=</span>flipper_length_mm, </span>
<span id="cb10-4"><a href="#cb10-4" aria-hidden="true" tabindex="-1"></a> <span class="at">y=</span>bill_length_mm, </span>
<span id="cb10-5"><a href="#cb10-5" aria-hidden="true" tabindex="-1"></a> <span class="at">color=</span>species)) <span class="sc">+</span> </span>
<span id="cb10-6"><a href="#cb10-6" aria-hidden="true" tabindex="-1"></a> <span class="fu">geom_point</span>(<span class="at">alpha=</span><span class="fl">0.8</span>) <span class="sc">+</span></span>
<span id="cb10-7"><a href="#cb10-7" aria-hidden="true" tabindex="-1"></a> <span class="fu">facet_wrap</span>(<span class="sc">~</span>.sample, <span class="at">ncol=</span><span class="dv">5</span>) <span class="sc">+</span></span>
<span id="cb10-8"><a href="#cb10-8" aria-hidden="true" tabindex="-1"></a> <span class="fu">scale_color_discrete_divergingx</span>(<span class="at">palette=</span><span class="st">"Zissou 1"</span>) <span class="sc">+</span></span>
<span id="cb10-9"><a href="#cb10-9" aria-hidden="true" tabindex="-1"></a> <span class="fu">theme</span>(<span class="at">legend.position =</span> <span class="st">"none"</span>,</span>
<span id="cb10-10"><a href="#cb10-10" aria-hidden="true" tabindex="-1"></a> <span class="at">axis.title =</span> <span class="fu">element_blank</span>(),</span>
<span id="cb10-11"><a href="#cb10-11" aria-hidden="true" tabindex="-1"></a> <span class="at">axis.text =</span> <span class="fu">element_blank</span>(),</span>
<span id="cb10-12"><a href="#cb10-12" aria-hidden="true" tabindex="-1"></a> <span class="at">panel.grid.major =</span> <span class="fu">element_blank</span>())</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
<div class="cell-output-display">
<div class="quarto-figure quarto-figure-center">
<figure>
<p><img data-src="slides2_files/figure-revealjs/unnamed-chunk-20-1.png" style="width:80.0%"></p>
</figure>
</div>
</div>
</div>
</div>
</section>
<section id="lineup-example-1-22" class="slide level2">
<h2>Lineup example 1 <span style="font-size: 70%;"> (2/2) </span></h2>
<p>If 10 people are shown this lineup and all 10 pick plot 2, which is the data plot, the <span class="math inline">\(p\)</span>-value will be 0.</p>
<p>Generally, we can compute the probability that the data plot is chosen by <span class="math inline">\(x\)</span> out of <span class="math inline">\(K\)</span> observers, shown a lineup of <span class="math inline">\(m\)</span> plots, using a simulation approach that extends from a binomial distribution, with <span class="math inline">\(p=1/m\)</span>.</p>
<div class="cell" data-layout-align="center">
<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">pvisual</span>(<span class="dv">10</span>, <span class="dv">10</span>, <span class="dv">15</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-stdout">
<pre><code> x simulated binom
[1,] 10 0 1.734168e-12</code></pre>
</div>
</div>
<p>This means we would reject <span class="math inline">\(H_o\)</span> and conclude that there is a difference in the distribution of bill length and flipper length between the species of penguins.</p>
</section>
<section id="lineup-example-2-12" class="slide level2">
<h2>Lineup example 2 <span style="font-size: 70%;"> (1/2) </span></h2>
<div class="f60">
<div class="cell" data-layout-align="center">
<details>
<summary>Code</summary>
<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">data</span>(wasps)</span>
<span id="cb13-2"><a href="#cb13-2" aria-hidden="true" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">258</span>)</span>
<span id="cb13-3"><a href="#cb13-3" aria-hidden="true" tabindex="-1"></a>wasps_l <span class="ot"><-</span> <span class="fu">lineup</span>(<span class="fu">null_permute</span>(<span class="st">"Group"</span>), wasps[,<span class="sc">-</span><span class="dv">1</span>], <span class="at">n=</span><span class="dv">15</span>)</span>
<span id="cb13-4"><a href="#cb13-4" aria-hidden="true" tabindex="-1"></a>wasps_l <span class="ot"><-</span> wasps_l <span class="sc">%>%</span></span>
<span id="cb13-5"><a href="#cb13-5" aria-hidden="true" tabindex="-1"></a> <span class="fu">mutate</span>(<span class="at">LD1 =</span> <span class="cn">NA</span>, <span class="at">LD2 =</span> <span class="cn">NA</span>)</span>
<span id="cb13-6"><a href="#cb13-6" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> (i <span class="cf">in</span> <span class="fu">unique</span>(wasps_l<span class="sc">$</span>.sample)) {</span>
<span id="cb13-7"><a href="#cb13-7" aria-hidden="true" tabindex="-1"></a> x <span class="ot"><-</span> <span class="fu">filter</span>(wasps_l, .sample <span class="sc">==</span> i)</span>
<span id="cb13-8"><a href="#cb13-8" aria-hidden="true" tabindex="-1"></a> xlda <span class="ot"><-</span> MASS<span class="sc">::</span><span class="fu">lda</span>(Group<span class="sc">~</span>., <span class="at">data=</span>x[,<span class="dv">1</span><span class="sc">:</span><span class="dv">42</span>])</span>
<span id="cb13-9"><a href="#cb13-9" aria-hidden="true" tabindex="-1"></a> xp <span class="ot"><-</span> MASS<span class="sc">:::</span><span class="fu">predict.lda</span>(xlda, x, <span class="at">dimen=</span><span class="dv">2</span>)<span class="sc">$</span>x</span>
<span id="cb13-10"><a href="#cb13-10" aria-hidden="true" tabindex="-1"></a> wasps_l<span class="sc">$</span>LD1[wasps_l<span class="sc">$</span>.sample <span class="sc">==</span> i] <span class="ot"><-</span> xp[,<span class="dv">1</span>]</span>
<span id="cb13-11"><a href="#cb13-11" aria-hidden="true" tabindex="-1"></a> wasps_l<span class="sc">$</span>LD2[wasps_l<span class="sc">$</span>.sample <span class="sc">==</span> i] <span class="ot"><-</span> xp[,<span class="dv">2</span>]</span>
<span id="cb13-12"><a href="#cb13-12" aria-hidden="true" tabindex="-1"></a>}</span>
<span id="cb13-13"><a href="#cb13-13" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplot</span>(wasps_l, </span>
<span id="cb13-14"><a href="#cb13-14" aria-hidden="true" tabindex="-1"></a> <span class="fu">aes</span>(<span class="at">x=</span>LD1, </span>
<span id="cb13-15"><a href="#cb13-15" aria-hidden="true" tabindex="-1"></a> <span class="at">y=</span>LD2, </span>
<span id="cb13-16"><a href="#cb13-16" aria-hidden="true" tabindex="-1"></a> <span class="at">color=</span>Group)) <span class="sc">+</span> </span>
<span id="cb13-17"><a href="#cb13-17" aria-hidden="true" tabindex="-1"></a> <span class="fu">geom_point</span>(<span class="at">alpha=</span><span class="fl">0.8</span>) <span class="sc">+</span></span>
<span id="cb13-18"><a href="#cb13-18" aria-hidden="true" tabindex="-1"></a> <span class="fu">facet_wrap</span>(<span class="sc">~</span>.sample, <span class="at">ncol=</span><span class="dv">5</span>) <span class="sc">+</span></span>
<span id="cb13-19"><a href="#cb13-19" aria-hidden="true" tabindex="-1"></a> <span class="fu">scale_color_discrete_divergingx</span>(<span class="at">palette=</span><span class="st">"Zissou 1"</span>) <span class="sc">+</span></span>
<span id="cb13-20"><a href="#cb13-20" aria-hidden="true" tabindex="-1"></a> <span class="fu">theme</span>(<span class="at">legend.position =</span> <span class="st">"none"</span>,</span>
<span id="cb13-21"><a href="#cb13-21" aria-hidden="true" tabindex="-1"></a> <span class="at">axis.title =</span> <span class="fu">element_blank</span>(),</span>
<span id="cb13-22"><a href="#cb13-22" aria-hidden="true" tabindex="-1"></a> <span class="at">axis.text =</span> <span class="fu">element_blank</span>(),</span>
<span id="cb13-23"><a href="#cb13-23" aria-hidden="true" tabindex="-1"></a> <span class="at">panel.grid.major =</span> <span class="fu">element_blank</span>())</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
<div class="cell-output-display">
<div class="quarto-figure quarto-figure-center">
<figure>
<p><img data-src="slides2_files/figure-revealjs/unnamed-chunk-22-1.png" style="width:80.0%"></p>
</figure>
</div>
</div>
</div>
</div>
</section>
<section id="lineup-example-2-22" class="slide level2">
<h2>Lineup example 2 <span style="font-size: 70%;"> (2/2) </span></h2>
<p>If 10 people are shown this lineup and 1 picked the data plot (position 6), which is the data plot, the <span class="math inline">\(p\)</span>-value will be large.</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb14"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb14-1"><a href="#cb14-1" aria-hidden="true" tabindex="-1"></a><span class="fu">pvisual</span>(<span class="dv">1</span>, <span class="dv">10</span>, <span class="dv">15</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-stdout">
<pre><code> x simulated binom
[1,] 1 0.4639 0.4983882</code></pre>
</div>
</div>
<p>This means we would NOT reject <span class="math inline">\(H_o\)</span> and conclude that there is NO difference in the distribution of groups.</p>
</section>
<section id="what-is-the-p-value" class="slide level2">
<h2>What is the <span class="math inline">\(p\)</span>-value?</h2>
<ul>
<li>Suppose <span class="math inline">\(X\)</span> is the number of nandus out of <span class="math inline">\(n\)</span> independent tosses.</li>
<li>Let <span class="math inline">\(p\)</span> be the probability of getting a <img src="images/heads.png" height="70px" style="vertical-align:middle;"> for this coin.</li>
<li><strong>Hypotheses</strong>: <span class="math inline">\(H_0: p = 0.5\)</span> vs. <span class="math inline">\(H_a: p > 0.5\)</span>. <br> Alternative <span class="math inline">\(H_a\)</span> is saying we believe that the coin is biased to nandus. <br> Alternative needs to be decided before seeing data.</li>
<li><strong>Assumption</strong>: Each toss is independent with equal chance of getting a nandu.</li>
</ul>
<!--
**Test statistic**: $X \sim B(n, p)$. Recall $E(X\mid H_0) = np_0$.<br> We observe $n, x, \widehat{p}$. Test statistic is $\widehat{p} - p_0$.
- **P-value**: $P(X ~ \geq ~ x\mid H_0)$
- Conclusion**: Reject null hypothesis when the $p$-value is less than<br> some significance level $\alpha$. Usually $\alpha = 0.05$.
-->
</section>
<section id="what-is-the-p-value-1" class="slide level2">
<h2>What is the <span class="math inline">\(p\)</span>-value?</h2>
<ul>
<li>Suppose I have a coin that I’m going to flip <img src="images/tails.png" height="70px" style="vertical-align:middle;"> <img src="images/heads.png" height="70px" style="vertical-align:middle;"></li>
<li><strong>Experiment 1</strong>: I flipped the coin 10 times and this is the result:</li>
</ul>
<center>
<div class="cell" data-layout-align="center">
<p><img src="images/heads.png" height="70px" style="vertical-align:middle;"><img src="images/heads.png" height="70px" style="vertical-align:middle;"><img src="images/heads.png" height="70px" style="vertical-align:middle;"><img src="images/tails.png" height="70px" style="vertical-align:middle;"><img src="images/tails.png" height="70px" style="vertical-align:middle;"><img src="images/heads.png" height="70px" style="vertical-align:middle;"><img src="images/heads.png" height="70px" style="vertical-align:middle;"><img src="images/heads.png" height="70px" style="vertical-align:middle;"><img src="images/tails.png" height="70px" style="vertical-align:middle;"><img src="images/heads.png" height="70px" style="vertical-align:middle;"></p>
</div>
</center>
<ul>
<li>The result is 7 nandus and 3 tails. So 70% are nandus.</li>
<li>Do you believe the coin is biased based on this data?</li>
</ul>
</section>
<section id="what-is-the-p-value-2" class="slide level2">
<h2>What is the <span class="math inline">\(p\)</span>-value?</h2>
<ul>
<li><strong>Experiment 2</strong>: Suppose now I flip the coin 100 times and this is the outcome:</li>
</ul>
<div class="cell" data-layout-align="center">
<p><img src="images/heads.png" height="70px" style="vertical-align:middle;"><img src="images/tails.png" height="70px" style="vertical-align:middle;"><img src="images/heads.png" height="70px" style="vertical-align:middle;"><img src="images/heads.png" height="70px" style="vertical-align:middle;"><img src="images/heads.png" height="70px" style="vertical-align:middle;"><img src="images/heads.png" height="70px" style="vertical-align:middle;"><img src="images/heads.png" height="70px" style="vertical-align:middle;"><img src="images/heads.png" height="70px" style="vertical-align:middle;"><img src="images/heads.png" height="70px" style="vertical-align:middle;"><img src="images/heads.png" height="70px" style="vertical-align:middle;"><img src="images/heads.png" height="70px" style="vertical-align:middle;"><img src="images/heads.png" height="70px" style="vertical-align:middle;"><img src="images/heads.png" height="70px" style="vertical-align:middle;"><img src="images/tails.png" height="70px" style="vertical-align:middle;"><img src="images/heads.png" height="70px" style="vertical-align:middle;"><img src="images/heads.png" height="70px" style="vertical-align:middle;"><img src="images/heads.png" height="70px" style="vertical-align:middle;"><img src="images/heads.png" height="70px" style="vertical-align:middle;"><img src="images/heads.png" height="70px" style="vertical-align:middle;"><img src="images/tails.png" height="70px" style="vertical-align:middle;"><img src="images/heads.png" height="70px" style="vertical-align:middle;"><img src="images/heads.png" height="70px" style="vertical-align:middle;"><img src="images/heads.png" height="70px" style="vertical-align:middle;"><img src="images/tails.png" height="70px" style="vertical-align:middle;"><img src="images/heads.png" height="70px" style="vertical-align:middle;"><img src="images/heads.png" height="70px" style="vertical-align:middle;"><img src="images/heads.png" height="70px" style="vertical-align:middle;"><img src="images/heads.png" height="70px" style="vertical-align:middle;"><img src="images/heads.png" height="70px" style="vertical-align:middle;"><img src="images/heads.png" height="70px" style="vertical-align:middle;"><img src="images/heads.png" height="70px" style="vertical-align:middle;"><img src="images/tails.png" height="70px" style="vertical-align:middle;"><img src="images/tails.png" height="70px" style="vertical-align:middle;"><img src="images/tails.png" height="70px" style="vertical-align:middle;"><img src="images/tails.png" height="70px" style="vertical-align:middle;"><img src="images/heads.png" height="70px" style="vertical-align:middle;"><img src="images/heads.png" height="70px" style="vertical-align:middle;"><img src="images/tails.png" height="70px" style="vertical-align:middle;"><img src="images/heads.png" height="70px" style="vertical-align:middle;"><img src="images/heads.png" height="70px" style="vertical-align:middle;"><img src="images/heads.png" height="70px" style="vertical-align:middle;"><img src="images/tails.png" height="70px" style="vertical-align:middle;"></p>
</div>
<ul>
<li>We observe 70 nandus and 30 tails. So again 70% are nandus.</li>
<li>Based on this data, do you think the coin is biased?</li>
</ul>
</section>
<section>
<section id="calculate-the-p-value" class="title-slide slide level1 center">
<h1>Calculate the <span class="math inline">\(p\)</span>-value</h1>
<div class="columns">
<div class="column f70" style="width:50%;">
<p><strong>Experiment 1 (n=10)</strong></p>
<ul>
<li>We observed <span class="math inline">\(x=7\)</span>, or <span class="math inline">\(\widehat{p} = 0.7\)</span>.</li>
<li>Assuming <span class="math inline">\(H_0\)</span> is true, we expect <span class="math inline">\(np=10\times 0.5=5\)</span>.</li>
<li>Calculate the <span class="math inline">\(P(X \geq 7)\)</span></li>
</ul>
<p><br> <br></p>
<div class="cell" data-layout-align="center">
<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">sum</span>(<span class="fu">dbinom</span>(<span class="dv">7</span><span class="sc">:</span><span class="dv">10</span>, <span class="dv">10</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 class="cell-output cell-output-stdout">
<pre><code>[1] 0.171875</code></pre>
</div>
</div>
</div><div class="column f70" style="width:50%;">
<p><strong>Experiment 2 (n=100)</strong></p>
<ul>
<li>We observed <span class="math inline">\(x=70\)</span>, or <span class="math inline">\(\widehat{p} = 0.7\)</span>.</li>
<li>Assuming <span class="math inline">\(H_0\)</span> is true, we expect <span class="math inline">\(np=100\times 0.5=50\)</span>.</li>
<li>Calculate the <span class="math inline">\(P(X \geq 70)\)</span></li>
</ul>
<p><br> <br></p>
<div class="cell" data-layout-align="center">
<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">sum</span>(<span class="fu">dbinom</span>(<span class="dv">70</span><span class="sc">:</span><span class="dv">100</span>, <span class="dv">100</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 class="cell-output cell-output-stdout">
<pre><code>[1] 3.92507e-05</code></pre>
</div>
</div>
</div>
</div>
</section>
<section id="lineup-p-value-and-power" class="slide level2">
<h2>Lineup <span class="math inline">\(p\)</span>-value and power</h2>
<p>Suppose <span class="math inline">\(x\)</span> out of <span class="math inline">\(n\)</span> people detected the data plot from a lineup, then the <strong>visual inference p-value</strong> is given as <span class="math inline">\(P(X \geq x)\)</span> where <span class="math inline">\(X \sim B(n, 1/m)\)</span>, but</p>
<div class="fragment">
<p>the assumption of independence is not strictly satisfied, if people are shown the same lineup. So the <span class="math inline">\(p\)</span>-value is computed by simulation with</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb20"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb20-1"><a href="#cb20-1" aria-hidden="true" tabindex="-1"></a>nullabor<span class="sc">::</span><span class="fu">pvisual</span>()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
</div>
<div class="fragment">
<p><br> and the <strong>power of a lineup</strong> is estimated as <span class="math inline">\(x/n\)</span>. We’ll use this to compare the signal strengths for different plot designs. <em>Stay tuned!</em></p>
</div>
</section>
<section id="lineup-example-3-12" class="slide level2">
<h2>Lineup example 3 <span style="font-size: 70%;"> (1/2) </span></h2>
<div class="columns">
<div class="column" style="width:15%;">
<p><br><br>Which plot is the most different?</p>
</div><div class="column" style="width:85%;">
<div class="cell" data-layout-align="center">
<div class="cell-output-display">
<div class="quarto-figure quarto-figure-center">
<figure>
<p><img data-src="slides2_files/figure-revealjs/unnamed-chunk-28-1.png" style="width:90.0%"></p>
</figure>
</div>
</div>
</div>
</div>
</div>
</section>
<section id="lineup-example-3-22" class="slide level2">
<h2>Lineup example 3 <span style="font-size: 70%;"> (2/2) </span></h2>
<div class="columns">
<div class="column" style="width:50%;">
<p>Plot description was:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb21"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb21-1"><a href="#cb21-1" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplot</span>(stars, <span class="fu">aes</span>(<span class="at">x=</span>temp)) <span class="sc">+</span></span>
<span id="cb21-2"><a href="#cb21-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">geom_density</span>()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<div class="fragment">
<p>In particular, the researcher is interested to know if star temperature is a skewed distribution.</p>
</div>
<div class="fragment">
<p><span class="math inline">\(H_o: X\sim exp(\widehat{\lambda})\)</span> <br> <span class="math inline">\(H_a:\)</span> it has a different distribution.</p>
</div>
</div><div class="column fragment" style="width:50%;">
<div class="fragment">
<p>Generate the lineup with:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb22"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb22-1"><a href="#cb22-1" aria-hidden="true" tabindex="-1"></a><span class="fu">lineup</span>(<span class="fu">null_dist</span>(<span class="st">"temp"</span>, <span class="st">"exp"</span>, </span>
<span id="cb22-2"><a href="#cb22-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">list</span>(<span class="at">rate =</span> <span class="dv">1</span> <span class="sc">/</span> </span>
<span id="cb22-3"><a href="#cb22-3" aria-hidden="true" tabindex="-1"></a> <span class="fu">mean</span>(dslabs<span class="sc">::</span>stars<span class="sc">$</span>temp))), </span>
<span id="cb22-4"><a href="#cb22-4" aria-hidden="true" tabindex="-1"></a> stars, <span class="at">n=</span><span class="dv">15</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
</div>
<div class="fragment">