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<title>The Value of Accepting the Null Hypothesis</title>
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<li><a href="#background" id="toc-background" class="nav-link active" data-scroll-target="#background"><span class="header-section-number">1</span> Background</a></li>
<li><a href="#important-substantive-cases" id="toc-important-substantive-cases" class="nav-link" data-scroll-target="#important-substantive-cases"><span class="header-section-number">2</span> Important Substantive Cases</a></li>
<li><a href="#references" id="toc-references" class="nav-link" data-scroll-target="#references"><span class="header-section-number">3</span> References</a></li>
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<h1 class="title">The Value of Accepting the Null Hypothesis</h1>
</div>
<div class="quarto-title-meta">
<div>
<div class="quarto-title-meta-heading">Author</div>
<div class="quarto-title-meta-contents">
<p>Andy Grogan-Kaylor </p>
</div>
</div>
<div>
<div class="quarto-title-meta-heading">Published</div>
<div class="quarto-title-meta-contents">
<p class="date">November 29, 2023</p>
</div>
</div>
</div>
</header>
<div class="cell">
<style type="text/css">
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<section id="background" class="level1" data-number="1">
<h1 data-number="1"><span class="header-section-number">1</span> Background</h1>
<p>In standard frequentist models, we cannot formally accept the Null Hypothesis <span class="math inline">\(H_0\)</span>, but can only reject, or fail to reject, <span class="math inline">\(H_0\)</span>.</p>
<p>Bayesian models allow one to both accept and reject <span class="math inline">\(H_0\)</span> <span class="citation" data-cites="Kruschke2018">(<a href="#ref-Kruschke2018" role="doc-biblioref">Kruschke and Liddell 2018</a>)</span>.</p>
<p>Accepting <span class="math inline">\(H_0\)</span> may be very scientifically valuable, and may have consequences for affirming similarity, universality, or treatment invariance <span class="citation" data-cites="Gallistel2009 Morey2018">(<a href="#ref-Gallistel2009" role="doc-biblioref">Gallistel 2009</a>; <a href="#ref-Morey2018" role="doc-biblioref">Morey, Homer, and Proulx 2018</a>)</span>. The ability to accept <span class="math inline">\(H_0\)</span> may also lead to a lower likelihood of the publication bias that results from frequentist methods predicated upon the rejection of <span class="math inline">\(H_0\)</span> <span class="citation" data-cites="Kruschke2018">(<a href="#ref-Kruschke2018" role="doc-biblioref">Kruschke and Liddell 2018</a>)</span>.</p>
<blockquote class="blockquote">
<p>This handout is written from a <em>Bayesian</em> perspective. However, even from a traditional <em>frequentist</em> statistical perspective, it may be helpful to think about the <em>value</em> of results that are <em>not statistically significant</em>.</p>
</blockquote>
<blockquote class="blockquote">
<p>A finding of a <em>null result</em> is dependent on having enough statistical power that one might plausibly detect an effect were an effect to exist.</p>
</blockquote>
<div style="page-break-after: always;"></div>
</section>
<section id="important-substantive-cases" class="level1" data-number="2">
<h1 data-number="2"><span class="header-section-number">2</span> Important Substantive Cases</h1>
<p>The Value of Accepting the Null Hypothesis <span class="math inline">\(H_0\)</span></p>
<div class="cell">
<div class="cell-output-display">
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<tr class="header">
<th style="text-align: left;" data-quarto-table-cell-role="th">case</th>
<th style="text-align: left;" data-quarto-table-cell-role="th">description</th>
<th style="text-align: left;" data-quarto-table-cell-role="th">H_0</th>
<th style="text-align: left;" data-quarto-table-cell-role="th">example</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: left;">Equivalence Testing</td>
<td style="text-align: left;">Equivalence Of 2 Treatments Or Interventions</td>
<td style="text-align: left;">$$\beta_1 = \beta_2$$</td>
<td style="text-align: left;">The effect of Treatment 1 is indistinguishable from the effect of Treatment 2 (especially important if one treatment is much more expensive, or time consuming than another).</td>
</tr>
<tr class="even">
<td style="text-align: left;">Equivalence Testing</td>
<td style="text-align: left;">Equivalence Of 2 Groups On An Outcome</td>
<td style="text-align: left;">$$\bar{y_1} = \bar{y_2}$$ or in multilevel modeling $$u_0 = 0$$</td>
<td style="text-align: left;">People identifying as men and people identifying as women are more similar than different with regard to psychological processes (Hyde2005).</td>
</tr>
<tr class="odd">
<td style="text-align: left;">Retiring Interventions</td>
<td style="text-align: left;">There Is No Evidence That Intervention X Is Effective</td>
<td style="text-align: left;">$$\beta_{intervention} = 0$$</td>
<td style="text-align: left;">Evidence consistently suggests that a particular treatment has near zero effect.</td>
</tr>
<tr class="even">
<td style="text-align: left;">Contextual Equivalence</td>
<td style="text-align: left;">Equivalence of a Predictor Across Contexts (Moderation)</td>
<td style="text-align: left;">$$\beta_{interaction} = 0$$ or in multilevel modeling $$u_k = 0$$</td>
<td style="text-align: left;">Warm and supportive parenting is equally beneficial across different contexts or countries.</td>
</tr>
<tr class="odd">
<td style="text-align: left;">Family Member Equivalence</td>
<td style="text-align: left;">Equivalence of a Predictor Across Family Members</td>
<td style="text-align: left;">$$\beta_{parent1} = \beta_{parent2}$$</td>
<td style="text-align: left;">Parenting from one parent is equivalent to parenting from another parent</td>
</tr>
<tr class="even">
<td style="text-align: left;">Full Mediation</td>
<td style="text-align: left;">Association of x and y Is Completely Mediated; No Direct Effect</td>
<td style="text-align: left;">$$\beta_{xmy} \neq 0$$ $$\beta_{xy} = 0$$</td>
<td style="text-align: left;">The relationship of the treatment and the outcome is completely mediated by mechanism m.</td>
</tr>
<tr class="odd">
<td style="text-align: left;">No Mediation</td>
<td style="text-align: left;">No Indirect Effect; Association of x and y Is Not Mediated by m</td>
<td style="text-align: left;">$$\beta_{xmy} = 0$$ $$\beta_{xy} \neq 0$$</td>
<td style="text-align: left;">The relationship of the treatment and the outcome is not mediated at all by mechanism m.</td>
</tr>
<tr class="even">
<td style="text-align: left;">Theory Simplification</td>
<td style="text-align: left;">Removing An Association From A Theory</td>
<td style="text-align: left;">$$\beta_x = 0$$</td>
<td style="text-align: left;">There is no evidence that x is associated with y.</td>
</tr>
<tr class="odd">
<td style="text-align: left;">Theory Rejection</td>
<td style="text-align: left;">Rejecting A Theory</td>
<td style="text-align: left;">$$\beta_{theory} = 0$$</td>
<td style="text-align: left;">There is strong evidence (contra Theory X) that x is not associated with y.</td>
</tr>
</tbody>
</table>
</div>
</div>
</section>
<section id="references" class="level1 unnumbered" data-number="3">
</section>
<div id="quarto-appendix" class="default"><section class="quarto-appendix-contents" role="doc-bibliography"><h2 class="anchored quarto-appendix-heading">3 References</h2><div id="refs" class="references csl-bib-body hanging-indent" role="list">
<div id="ref-Gallistel2009" class="csl-entry" role="listitem">
Gallistel, C R. 2009. <span>“<span class="nocase">The importance of proving the null</span>.”</span> <em>Psychological Review</em> 116 (2): 439–53. <a href="https://doi.org/10.1037/a0015251">https://doi.org/10.1037/a0015251</a>.
</div>
<div id="ref-Hyde2005" class="csl-entry" role="listitem">
Hyde, Janet Shibley. 2005. <span>“The Gender Similarities Hypothesis.”</span> <em>American Psychologist</em> 60 (6): 581–92. <a href="https://doi.org/10.1037/0003-066X.60.6.581">https://doi.org/10.1037/0003-066X.60.6.581</a>.
</div>
<div id="ref-Kruschke2018" class="csl-entry" role="listitem">
Kruschke, John K, and Torrin M Liddell. 2018. <span>“The Bayesian New Statistics: Hypothesis Testing, Estimation, Meta-Analysis, and Power Analysis from a Bayesian Perspective.”</span> <em>Psychonomic Bulletin <span>&</span> Review</em> 25 (1): 178–206. <a href="https://doi.org/10.3758/s13423-016-1221-4">https://doi.org/10.3758/s13423-016-1221-4</a>.
</div>
<div id="ref-Morey2018" class="csl-entry" role="listitem">
Morey, Richard D., Saskia Homer, and Travis Proulx. 2018. <span>“Beyond Statistics: Accepting the Null Hypothesis in Mature Sciences.”</span> <em>Advances in Methods and Practices in Psychological Science</em>. <a href="https://doi.org/10.1177/2515245918776023">https://doi.org/10.1177/2515245918776023</a>.
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},
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interactiveBorder: 10,
theme: 'quarto',
placement: 'bottom-start'
};
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try { href = new URL(href).hash; } catch {}
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selectedAnnoteEl = annoteEl;
}
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elementsIds.forEach((elId) => {
const div = window.document.getElementById(elId);
if (div) {
div.remove();
}
});
selectedAnnoteEl = undefined;
};
// Attach click handler to the DT
const annoteDls = window.document.querySelectorAll('dt[data-target-cell]');
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annoteDlNode.addEventListener('click', (event) => {
const clickedEl = event.target;
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unselectCodeLines();
const activeEl = window.document.querySelector('dt[data-target-cell].code-annotation-active');
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activeEl.classList.remove('code-annotation-active');
}
selectCodeLines(clickedEl);
clickedEl.classList.add('code-annotation-active');
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unselectCodeLines();
clickedEl.classList.remove('code-annotation-active');
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el,
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citeDiv.classList.add('csl-entry');
var biblioDiv = window.document.getElementById('ref-' + cite);
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citeDiv.innerHTML = biblioDiv.innerHTML;
}
popup.appendChild(citeDiv);
});
return popup.innerHTML;
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});
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</div> <!-- /content -->
</body></html>