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href="https://es.wikipedia.org/wiki/Correcci%C3%B3n_de_Bonferroni" title="Corrección de Bonferroni – Spanish" lang="es" hreflang="es" data-title="Corrección de Bonferroni" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%B5%D8%AD%DB%8C%D8%AD_%D8%A8%D9%86%D9%81%D8%B1%D9%88%D9%86%DB%8C" title="تصحیح بنفرونی – Persian" lang="fa" hreflang="fa" data-title="تصحیح بنفرونی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Correction_de_Bonferroni" title="Correction de Bonferroni – French" lang="fr" hreflang="fr" data-title="Correction de Bonferroni" data-language-autonym="Français" data-language-local-name="French" 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title="Statistics">statistics</a>, the <b>Bonferroni correction</b> is a method to counteract the <a href="/wiki/Multiple_comparisons_problem" title="Multiple comparisons problem">multiple comparisons problem</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Background">Background</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bonferroni_correction&action=edit&section=1" title="Edit section: Background"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The method is named for its use of the <a href="/wiki/Bonferroni_inequalities" class="mw-redirect" title="Bonferroni inequalities">Bonferroni inequalities</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Application of the method to <a href="/wiki/Confidence_intervals" class="mw-redirect" title="Confidence intervals">confidence intervals</a> was described by <a href="/wiki/Olive_Jean_Dunn" title="Olive Jean Dunn">Olive Jean Dunn</a>.<sup id="cite_ref-Dunn1961_2-0" class="reference"><a href="#cite_note-Dunn1961-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Statistical_hypothesis_testing" class="mw-redirect" title="Statistical hypothesis testing">Statistical hypothesis testing</a> is based on rejecting the <a href="/wiki/Null_hypothesis" title="Null hypothesis">null hypothesis</a> when the likelihood of the observed data would be low if the null hypothesis were true. If multiple hypotheses are tested, the probability of observing a rare event increases, and therefore, the likelihood of incorrectly rejecting a null hypothesis (i.e., making a <a href="/wiki/Type_I_error" class="mw-redirect" title="Type I error">Type I error</a>) increases.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>The Bonferroni correction compensates for that increase by testing each individual hypothesis at a significance level of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha /m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha /m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4c4e3720141261887fab9693eaf32d6a65d3837" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.69ex; height:2.843ex;" alt="{\displaystyle \alpha /m}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> is the desired overall alpha level and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> is the number of hypotheses.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> For example, if a trial is testing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=20}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>20</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=20}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/003f58fd782a8ea62aa011defdf402a9567f4f42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.464ex; height:2.176ex;" alt="{\displaystyle m=20}"></span> hypotheses with a desired overall <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =0.05}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mn>0.05</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =0.05}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fdb2e1d36bb136f99c1a76e11e8bc05906354b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.72ex; height:2.176ex;" alt="{\displaystyle \alpha =0.05}"></span>, then the Bonferroni correction would test each individual hypothesis at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =0.05/20=0.0025}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mn>0.05</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>20</mn> <mo>=</mo> <mn>0.0025</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =0.05/20=0.0025}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd4304b0e97c05b0e6490635c0e01d9f42c71d27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.765ex; height:2.843ex;" alt="{\displaystyle \alpha =0.05/20=0.0025}"></span>. </p><p>The Bonferroni correction can also be applied as a p-value adjustment: Using that approach, instead of adjusting the alpha level, each p-value is multiplied by the number of tests (with adjusted p-values that exceed 1 then being reduced to 1), and the alpha level is left unchanged. The significance decisions using this approach will be the same as when using the alpha-level adjustment approach. </p> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bonferroni_correction&action=edit&section=2" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{1},\ldots ,H_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{1},\ldots ,H_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64680698cbdb7c49a1eebd6e210f68d677c468f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.77ex; height:2.509ex;" alt="{\displaystyle H_{1},\ldots ,H_{m}}"></span> be a family of null hypotheses and let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{1},\ldots ,p_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{1},\ldots ,p_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92e43e4cbe8115165ca3559393f90caee2e86a75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.336ex; height:2.009ex;" alt="{\displaystyle p_{1},\ldots ,p_{m}}"></span> be their corresponding <a href="/wiki/P-value" title="P-value">p-values</a>. Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> be the total number of null hypotheses, and let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a6ff51ee949104fe6fae553cfbdfba29d5fac1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.095ex; height:2.009ex;" alt="{\displaystyle m_{0}}"></span> be the number of true null hypotheses (which is presumably unknown to the researcher). The <a href="/wiki/Family-wise_error_rate" title="Family-wise error rate">family-wise error rate</a> (FWER) is the probability of rejecting at least one true <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bd0312f590cc5a400008938f3cc304d42ad3986" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.731ex; height:2.509ex;" alt="{\displaystyle H_{i}}"></span>, that is, of making at least one <a href="/wiki/Type_I_error" class="mw-redirect" title="Type I error">type I error</a>. The Bonferroni correction rejects the null hypothesis for each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{i}\leq {\frac {\alpha }{m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mi>m</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{i}\leq {\frac {\alpha }{m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16358edfbf546100fa79428a8264d2a9e4788554" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.089ex; width:8.034ex; height:4.676ex;" alt="{\displaystyle p_{i}\leq {\frac {\alpha }{m}}}"></span>, thereby controlling the <a href="/wiki/Family-wise_error_rate" title="Family-wise error rate">FWER</a> at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≤</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leq \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/413e538ade3a7e83a2febbbc4f27178a056843b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.941ex; height:2.176ex;" alt="{\displaystyle \leq \alpha }"></span>. Proof of this control follows from <a href="/wiki/Boole%27s_inequality" title="Boole's inequality">Boole's inequality</a>, as follows: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{FWER}}=P\left\{\bigcup _{i=1}^{m_{0}}\left(p_{i}\leq {\frac {\alpha }{m}}\right)\right\}\leq \sum _{i=1}^{m_{0}}\left\{P\left(p_{i}\leq {\frac {\alpha }{m}}\right)\right\}\leq m_{0}{\frac {\alpha }{m}}\leq \alpha .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>FWER</mtext> </mrow> <mo>=</mo> <mi>P</mi> <mrow> <mo>{</mo> <mrow> <munderover> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mi>m</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>}</mo> </mrow> <mo>≤</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </munderover> <mrow> <mo>{</mo> <mrow> <mi>P</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mi>m</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>}</mo> </mrow> <mo>≤</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mi>m</mi> </mfrac> </mrow> <mo>≤</mo> <mi>α</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{FWER}}=P\left\{\bigcup _{i=1}^{m_{0}}\left(p_{i}\leq {\frac {\alpha }{m}}\right)\right\}\leq \sum _{i=1}^{m_{0}}\left\{P\left(p_{i}\leq {\frac {\alpha }{m}}\right)\right\}\leq m_{0}{\frac {\alpha }{m}}\leq \alpha .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/433c4c39b77aad427ecf7493c98ce43c0cdcbd1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:67.048ex; height:7.509ex;" alt="{\displaystyle {\text{FWER}}=P\left\{\bigcup _{i=1}^{m_{0}}\left(p_{i}\leq {\frac {\alpha }{m}}\right)\right\}\leq \sum _{i=1}^{m_{0}}\left\{P\left(p_{i}\leq {\frac {\alpha }{m}}\right)\right\}\leq m_{0}{\frac {\alpha }{m}}\leq \alpha .}"></span></dd></dl> <p>This control does not require any assumptions about dependence among the p-values or about how many of the null hypotheses are true.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Extensions">Extensions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bonferroni_correction&action=edit&section=3" title="Edit section: Extensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Generalization">Generalization</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bonferroni_correction&action=edit&section=4" title="Edit section: Generalization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Rather than testing each hypothesis at the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha /m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha /m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4c4e3720141261887fab9693eaf32d6a65d3837" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.69ex; height:2.843ex;" alt="{\displaystyle \alpha /m}"></span> level, the hypotheses may be tested at any other combination of levels that add up to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>, provided that the level of each test is decided before looking at the data.<sup id="cite_ref-pmid8014990_6-0" class="reference"><a href="#cite_note-pmid8014990-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> For example, for two hypothesis tests, an overall <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> of 0.05 could be maintained by conducting one test at 0.04 and the other at 0.01. </p> <div class="mw-heading mw-heading3"><h3 id="Confidence_intervals">Confidence intervals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bonferroni_correction&action=edit&section=5" title="Edit section: Confidence intervals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The procedure proposed by Dunn<sup id="cite_ref-Dunn1961_2-1" class="reference"><a href="#cite_note-Dunn1961-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> can be used to adjust <a href="/wiki/Confidence_intervals" class="mw-redirect" title="Confidence intervals">confidence intervals</a>. If one establishes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> confidence intervals, and wishes to have an overall confidence level of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9afa7876fb8b4fb8c4d8039ebed6cd1cbc4781cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.49ex; height:2.343ex;" alt="{\displaystyle 1-\alpha }"></span>, each individual confidence interval can be adjusted to the level of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-{\frac {\alpha }{m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mi>m</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-{\frac {\alpha }{m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71329d1b0a91c858b82ce1899cde06155842035b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.879ex; height:4.676ex;" alt="{\displaystyle 1-{\frac {\alpha }{m}}}"></span>.<sup id="cite_ref-Dunn1961_2-2" class="reference"><a href="#cite_note-Dunn1961-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Continuous_problems">Continuous problems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bonferroni_correction&action=edit&section=6" title="Edit section: Continuous problems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When searching for a signal in a continuous parameter space there can also be a problem of multiple comparisons, or look-elsewhere effect. For example, a physicist might be looking to discover a particle of unknown mass by considering a large range of masses; this was the case during the Nobel Prize winning detection of the <a href="/wiki/Higgs_boson" title="Higgs boson">Higgs boson</a>. In such cases, one can apply a continuous generalization of the Bonferroni correction by employing <a href="/wiki/Bayesian_statistics" title="Bayesian statistics">Bayesian</a> logic to relate the effective number of trials, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span>, to the prior-to-posterior volume ratio.<sup id="cite_ref-Bayer2020_7-0" class="reference"><a href="#cite_note-Bayer2020-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Alternatives">Alternatives</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bonferroni_correction&action=edit&section=7" title="Edit section: Alternatives"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Family-wise_error_rate#Controlling_procedures" title="Family-wise error rate">Family-wise error rate § Controlling procedures</a></div> <p>There are alternative ways to control the <a href="/wiki/Family-wise_error_rate" title="Family-wise error rate">family-wise error rate</a>. For example, the <a href="/wiki/Holm%E2%80%93Bonferroni_method" title="Holm–Bonferroni method">Holm–Bonferroni method</a> and the <a href="/wiki/%C5%A0id%C3%A1k_correction" title="Šidák correction">Šidák correction</a> are universally more powerful procedures than the Bonferroni correction, meaning that they are always at least as powerful. But unlike the Bonferroni procedure, these methods do not control the <a href="/wiki/Expected_number" class="mw-redirect" title="Expected number">expected number</a> of Type I errors per family (the per-family Type I error rate).<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Criticism">Criticism</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bonferroni_correction&action=edit&section=8" title="Edit section: Criticism"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>With respect to <a href="/wiki/Family-wise_error_rate" title="Family-wise error rate">FWER</a> control, the Bonferroni correction can be conservative if there are a large number of tests and/or the test statistics are positively correlated.<sup id="cite_ref-Moran2003_9-0" class="reference"><a href="#cite_note-Moran2003-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>Multiple-testing corrections, including the Bonferroni procedure, increase the probability of <a href="/wiki/Type_I_and_type_II_errors#Type_II_error" title="Type I and type II errors">Type II errors</a> when null hypotheses are false, i.e., they reduce <a href="/wiki/Statistical_power" class="mw-redirect" title="Statistical power">statistical power</a>.<sup id="cite_ref-Nakagawa2004_10-0" class="reference"><a href="#cite_note-Nakagawa2004-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Moran2003_9-1" class="reference"><a href="#cite_note-Moran2003-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bonferroni_correction&action=edit&section=9" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Bonferroni, C. E., Teoria statistica delle classi e calcolo delle probabilità, Pubblicazioni del R Istituto Superiore di Scienze Economiche e Commerciali di Firenze 1936</span> </li> <li id="cite_note-Dunn1961-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Dunn1961_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Dunn1961_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Dunn1961_2-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFDunn1961" class="citation journal cs1">Dunn, Olive Jean (1961). <a rel="nofollow" class="external text" href="http://sci2s.ugr.es/keel/pdf/algorithm/articulo/1961-Bonferroni_Dunn-JASA.pdf">"Multiple Comparisons Among Means"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Journal_of_the_American_Statistical_Association" title="Journal of the American Statistical Association">Journal of the American Statistical Association</a></i>. <b>56</b> (293): 52–64. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.309.1277">10.1.1.309.1277</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F01621459.1961.10482090">10.1080/01621459.1961.10482090</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+American+Statistical+Association&rft.atitle=Multiple+Comparisons+Among+Means&rft.volume=56&rft.issue=293&rft.pages=52-64&rft.date=1961&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.309.1277%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1080%2F01621459.1961.10482090&rft.aulast=Dunn&rft.aufirst=Olive+Jean&rft_id=http%3A%2F%2Fsci2s.ugr.es%2Fkeel%2Fpdf%2Falgorithm%2Farticulo%2F1961-Bonferroni_Dunn-JASA.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABonferroni+correction" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMittelhammerJudgeMiller2000" class="citation book cs1"><a href="/wiki/Ron_C._Mittelhammer" title="Ron C. 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Cambridge University Press. pp. 73–74. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-62394-0" title="Special:BookSources/978-0-521-62394-0"><bdi>978-0-521-62394-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Econometric+Foundations&rft.pages=73-74&rft.pub=Cambridge+University+Press&rft.date=2000&rft.isbn=978-0-521-62394-0&rft.aulast=Mittelhammer&rft.aufirst=Ron+C.&rft.au=Judge%2C+George+G.&rft.au=Miller%2C+Douglas+J.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DfycmsfkK6RQC%26pg%3DPA73&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABonferroni+correction" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMiller1966" class="citation book cs1">Miller, Rupert G. 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