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class="vector-toc-numb">2</span> <span>In modeling</span> </div> </a> <button aria-controls="toc-In_modeling-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle In modeling subsection</span> </button> <ul id="toc-In_modeling-sublist" class="vector-toc-list"> <li id="toc-In_ANOVA" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_ANOVA"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>In ANOVA</span> </div> </a> <ul id="toc-In_ANOVA-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Qualitative_and_quantitative_interactions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Qualitative_and_quantitative_interactions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Qualitative and quantitative interactions</span> </div> </a> <ul id="toc-Qualitative_and_quantitative_interactions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Unit_treatment_additivity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Unit_treatment_additivity"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Unit treatment additivity</span> </div> </a> <ul id="toc-Unit_treatment_additivity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Categorical_variables" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Categorical_variables"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Categorical variables</span> </div> </a> <ul id="toc-Categorical_variables-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Designed_experiments" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Designed_experiments"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Designed experiments</span> </div> </a> <ul id="toc-Designed_experiments-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Model_size" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Model_size"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>Model size</span> </div> </a> <ul id="toc-Model_size-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_regression" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_regression"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.7</span> <span>In regression</span> </div> </a> <ul id="toc-In_regression-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Interaction_plots" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Interaction_plots"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Interaction plots</span> </div> </a> <button aria-controls="toc-Interaction_plots-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Interaction plots subsection</span> </button> <ul id="toc-Interaction_plots-sublist" class="vector-toc-list"> <li id="toc-Example:_Interaction_of_species_and_air_temperature_and_their_effect_on_body_temperature" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Example:_Interaction_of_species_and_air_temperature_and_their_effect_on_body_temperature"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Example: Interaction of species and air temperature and their effect on body temperature</span> </div> </a> <ul id="toc-Example:_Interaction_of_species_and_air_temperature_and_their_effect_on_body_temperature-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example:_effect_of_stroke_severity_and_treatment_on_recovery" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Example:_effect_of_stroke_severity_and_treatment_on_recovery"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Example: effect of stroke severity and treatment on recovery</span> </div> </a> <ul id="toc-Example:_effect_of_stroke_severity_and_treatment_on_recovery-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Hypothesis_tests_for_interactions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Hypothesis_tests_for_interactions"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Hypothesis tests for interactions</span> </div> </a> <button aria-controls="toc-Hypothesis_tests_for_interactions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Hypothesis tests for interactions subsection</span> </button> <ul id="toc-Hypothesis_tests_for_interactions-sublist" class="vector-toc-list"> <li id="toc-Example:_Interaction_of_temperature_and_time_in_cookie_baking" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Example:_Interaction_of_temperature_and_time_in_cookie_baking"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Example: Interaction of temperature and time in cookie baking</span> </div> </a> <ul id="toc-Example:_Interaction_of_temperature_and_time_in_cookie_baking-sublist" class="vector-toc-list"> <li id="toc-ANOVA_model_1:_no_interaction_term;_yield_~_temperature_+_time" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#ANOVA_model_1:_no_interaction_term;_yield_~_temperature_+_time"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1.1</span> <span>ANOVA model 1: no interaction term; yield ~ temperature + time</span> </div> </a> <ul id="toc-ANOVA_model_1:_no_interaction_term;_yield_~_temperature_+_time-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-ANOVA_model_2:_include_interaction_term;_yield_~_temperature_*_time" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#ANOVA_model_2:_include_interaction_term;_yield_~_temperature_*_time"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1.2</span> <span>ANOVA model 2: include interaction term; yield ~ temperature * time</span> </div> </a> <ul id="toc-ANOVA_model_2:_include_interaction_term;_yield_~_temperature_*_time-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Examples</span> </div> </a> <ul id="toc-Examples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span 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href="https://eu.wikipedia.org/wiki/Interakzio_(estatistika)" title="Interakzio (estatistika) – Basque" lang="eu" hreflang="eu" data-title="Interakzio (estatistika)" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Interaction_(statistiques)" title="Interaction (statistiques) – French" lang="fr" hreflang="fr" data-title="Interaction (statistiques)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Interakcja_(statystyka)" title="Interakcja (statystyka) – Polish" lang="pl" hreflang="pl" data-title="Interakcja (statystyka)" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%92%D0%B7%D0%B0%D0%B8%D0%BC%D0%BE%D0%B4%D0%B5%D0%B9%D1%81%D1%82%D0%B2%D0%B8%D0%B5_(%D1%81%D1%82%D0%B0%D1%82%D0%B8%D1%81%D1%82%D0%B8%D0%BA%D0%B0)" title="Взаимодействие (статистика) – Russian" lang="ru" hreflang="ru" data-title="Взаимодействие (статистика)" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Interaction_(statistics)" title="Interaction (statistics) – Sundanese" lang="su" hreflang="su" data-title="Interaction (statistics)" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a 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href="/wiki/File:GSS_sealevel_interaction.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/GSS_sealevel_interaction.png/220px-GSS_sealevel_interaction.png" decoding="async" width="220" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/GSS_sealevel_interaction.png/330px-GSS_sealevel_interaction.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6c/GSS_sealevel_interaction.png/440px-GSS_sealevel_interaction.png 2x" data-file-width="1210" data-file-height="880" /></a><figcaption>Interaction effect of education and ideology on concern about sea level rise</figcaption></figure> <p>In <a href="/wiki/Statistics" title="Statistics">statistics</a>, an <b>interaction</b> may arise when considering the relationship among three or more <a href="/wiki/Variable_(statistics)" class="mw-redirect" title="Variable (statistics)">variables</a>, and describes a situation in which the effect of one <a href="/wiki/Causal_variable" class="mw-redirect" title="Causal variable">causal variable</a> on an outcome depends on the state of a second causal variable (that is, when effects of the two causes are not <a href="/wiki/Additive_map" title="Additive map">additive</a>).<sup id="cite_ref-Dodge_1-0" class="reference"><a href="#cite_note-Dodge-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Although commonly thought of in terms of <a href="/wiki/Causality" title="Causality">causal relationships</a>, the concept of an interaction can also describe non-causal associations (then also called <a href="/wiki/Moderation_(statistics)" title="Moderation (statistics)"><i>moderation</i></a> or <i>effect modification</i>). Interactions are often considered in the context of <a href="/wiki/Regression_analysis" title="Regression analysis">regression analyses</a> or <a href="/wiki/Factorial_experiments" class="mw-redirect" title="Factorial experiments">factorial experiments</a>. </p><p>The presence of interactions can have important implications for the interpretation of <a href="/wiki/Statistical_model" title="Statistical model">statistical models</a>. If two variables of interest interact, the relationship between each of the interacting variables and a third "dependent variable" depends on the value of the other interacting variable. In practice, this makes it more difficult to predict the consequences of changing the value of a variable, particularly if the variables it interacts with are hard to measure or difficult to control. </p><p>The notion of "interaction" is closely related to that of moderation that is common in social and health science research: the interaction between an explanatory variable and an environmental variable suggests that the effect of the explanatory variable has been moderated or modified by the environmental variable.<sup id="cite_ref-Dodge_1-1" class="reference"><a href="#cite_note-Dodge-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Introduction">Introduction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interaction_(statistics)&action=edit&section=1" title="Edit section: Introduction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An <b>interaction variable</b> or <b>interaction feature</b> is a variable constructed from an original set of variables to try to represent either all of the interaction present or some part of it. In exploratory statistical analyses it is common to use products of original variables as the basis of testing whether interaction is present with the possibility of substituting other more realistic interaction variables at a later stage. When there are more than two explanatory variables, several interaction variables are constructed, with pairwise-products representing pairwise-interactions and higher order products representing higher order interactions. </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Quantitative_interaction.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Quantitative_interaction.svg/250px-Quantitative_interaction.svg.png" decoding="async" width="250" height="169" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Quantitative_interaction.svg/375px-Quantitative_interaction.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Quantitative_interaction.svg/500px-Quantitative_interaction.svg.png 2x" data-file-width="360" data-file-height="243" /></a><figcaption>The binary factor <i>A</i> and the quantitative variable <i>X</i> interact (are non-additive) when analyzed with respect to the outcome variable <i>Y</i>.</figcaption></figure> <p>Thus, for a response <i>Y</i> and two variables <i>x</i><sub>1</sub> and <i>x</i><sub>2</sub> an <i>additive</i> model would be: </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 Y=c+ax_{1}+bx_{2}+{\text{error}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <mi>c</mi> <mo>+</mo> <mi>a</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>b</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>error</mtext> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y=c+ax_{1}+bx_{2}+{\text{error}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a4d75798f14e3d9d0cfd2f9f6054973b9a1b471" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.712ex; height:2.509ex;" alt="{\displaystyle Y=c+ax_{1}+bx_{2}+{\text{error}}\,}"></span></dd></dl> <p>In contrast to this, </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 Y=c+ax_{1}+bx_{2}+d(x_{1}\times x_{2})+{\text{error}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <mi>c</mi> <mo>+</mo> <mi>a</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>b</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>×</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>error</mtext> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y=c+ax_{1}+bx_{2}+d(x_{1}\times x_{2})+{\text{error}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97cf9964b9ac0695f551d3fed620a071433b91e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.186ex; height:2.843ex;" alt="{\displaystyle Y=c+ax_{1}+bx_{2}+d(x_{1}\times x_{2})+{\text{error}}\,}"></span></dd></dl> <p>is an example of a model with an <i>interaction</i> between variables <i>x</i><sub>1</sub> and <i>x</i><sub>2</sub> ("error" refers to the <a href="/wiki/Random_variable" title="Random variable">random variable</a> whose value is that by which <i>Y</i> differs from the <a href="/wiki/Expected_value" title="Expected value">expected value</a> of <i>Y</i>; see <a href="/wiki/Errors_and_residuals_in_statistics" class="mw-redirect" title="Errors and residuals in statistics">errors and residuals in statistics</a>). Often, models are presented without the interaction term <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 d(x_{1}\times x_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>×</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x_{1}\times x_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/717a0bb1601ffa1921978fbd45f217292d59d694" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.633ex; height:2.843ex;" alt="{\displaystyle d(x_{1}\times x_{2})}"></span>, but this confounds the main effect and interaction effect (i.e., without specifying the interaction term, it is possible that any main effect found is actually due to an interaction). </p><p>Moreover, the <a href="/wiki/Principle_of_marginality" title="Principle of marginality">hierarchical principle</a> rules that if a model includes interaction between variables, it is also necessary to include the main effects, regardless of their own statistical significance.<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> <div class="mw-heading mw-heading2"><h2 id="In_modeling">In modeling</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interaction_(statistics)&action=edit&section=2" title="Edit section: In modeling"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="In_ANOVA">In ANOVA</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interaction_(statistics)&action=edit&section=3" title="Edit section: In ANOVA"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A simple setting in which interactions can arise is a <a href="/wiki/Factorial_experiment" title="Factorial experiment">two-factor experiment</a> analyzed using <a href="/wiki/Analysis_of_Variance" class="mw-redirect" title="Analysis of Variance">Analysis of Variance</a> (ANOVA). Suppose we have two binary factors <i>A</i> and <i>B</i>. For example, these factors might indicate whether either of two treatments were administered to a patient, with the treatments applied either singly, or in combination. We can then consider the average treatment response (e.g. the symptom levels following treatment) for each patient, as a function of the treatment combination that was administered. The following table shows one possible situation: </p> <table cellpadding="5" cellspacing="0" align="center"> <tbody><tr> <th> </th> <th style="background:#ffdead;border-left:1px solid black;border-top:1px solid black;"><i>B</i> = 0 </th> <th style="background:#ffdead;border-top:1px solid black;border-right:1px solid black;"><i>B</i> = 1 </th></tr> <tr> <th style="background:#ffdead;border-left:1px solid black;border-top:1px solid black;"><i>A</i> = 0 </th> <th style="border-left:1px solid black;">6 </th> <th style="border-right:1px solid black;">7 </th></tr> <tr> <th style="background:#ffdead;border-bottom:1px solid black;border-left:1px solid black;"><i>A</i> = 1 </th> <th style="border-bottom:1px solid black;border-left:1px solid black;">4 </th> <th style="border-bottom:1px solid black;border-right:1px solid black;">5 </th></tr></tbody></table> <p>In this example, there is no interaction between the two treatments — their effects are additive. The reason for this is that the difference in mean response between those subjects receiving treatment <i>A</i> and those not receiving treatment <i>A</i> is −2 regardless of whether treatment <i>B</i> is administered (−2 = 4 − 6) or not (−2 = 5 − 7). Note that it automatically follows that the difference in mean response between those subjects receiving treatment <i>B</i> and those not receiving treatment <i>B</i> is the same regardless of whether treatment <i>A</i> is administered (7 − 6 = 5 − 4). </p><p>In contrast, if the following average responses are observed </p> <table cellpadding="5" cellspacing="0" align="center"> <tbody><tr> <th> </th> <th style="background:#ffdead;border-left:1px solid black;border-top:1px solid black;"><i>B</i> = 0 </th> <th style="background:#ffdead;border-top:1px solid black;border-right:1px solid black;"><i>B</i> = 1 </th></tr> <tr> <th style="background:#ffdead;border-left:1px solid black;border-top:1px solid black;"><i>A</i> = 0 </th> <th style="border-left:1px solid black;">1 </th> <th style="border-right:1px solid black;">4 </th></tr> <tr> <th style="background:#ffdead;border-bottom:1px solid black;border-left:1px solid black;"><i>A</i> = 1 </th> <th style="border-bottom:1px solid black;border-left:1px solid black;">7 </th> <th style="border-bottom:1px solid black;border-right:1px solid black;">6 </th></tr></tbody></table> <p>then there is an interaction between the treatments — their effects are not additive. Supposing that greater numbers correspond to a better response, in this situation treatment <i>B</i> is helpful on average if the subject is not also receiving treatment <i>A</i>, but is detrimental on average if given in combination with treatment <i>A</i>. Treatment <i>A</i> is helpful on average regardless of whether treatment <i>B</i> is also administered, but it is more helpful in both absolute and relative terms if given alone, rather than in combination with treatment <i>B</i>. Similar observations are made for this particular example in the next section. </p> <div class="mw-heading mw-heading3"><h3 id="Qualitative_and_quantitative_interactions">Qualitative and quantitative interactions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interaction_(statistics)&action=edit&section=4" title="Edit section: Qualitative and quantitative interactions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In many applications it is useful to distinguish between qualitative and quantitative interactions.<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> A quantitative interaction between <i>A</i> and <i>B</i> is a situation where the magnitude of the effect of <i>B</i> depends on the value of <i>A</i>, but the direction of the effect of <i>B</i> is constant for all <i>A</i>. A qualitative interaction between <i>A</i> and <i>B</i> refers to a situation where both the magnitude and direction of each variable's effect can depend on the value of the other variable. </p><p>The table of means on the left, below, shows a quantitative interaction — treatment <i>A</i> is beneficial both when <i>B</i> is given, and when <i>B</i> is not given, but the benefit is greater when <i>B</i> is not given (i.e. when <i>A</i> is given alone). The table of means on the right shows a qualitative interaction. <i>A</i> is harmful when <i>B</i> is given, but it is beneficial when <i>B</i> is not given. Note that the same interpretation would hold if we consider the benefit of <i>B</i> based on whether <i>A</i> is given. </p> <table cellpadding="5" cellspacing="0" align="center"> <tbody><tr> <th> </th> <th style="background:#ffdead;border-left:1px solid black;border-top:1px solid black;"><i>B</i> = 0 </th> <th style="background:#ffdead;border-top:1px solid black;border-right:1px solid black;"><i>B</i> = 1 </th> <th> </th> <th> </th> <th> </th> <th> </th> <th> </th> <th> </th> <th style="background:#ffdead;border-left:1px solid black;border-top:1px solid black;"><i>B</i> = 0 </th> <th style="background:#ffdead;border-top:1px solid black;border-right:1px solid black;"><i>B</i> = 1 </th></tr> <tr> <th style="background:#ffdead;border-left:1px solid black;border-top:1px solid black;"><i>A</i> = 0 </th> <th style="border-left:1px solid black;">2 </th> <th style="border-right:1px solid black;">1 </th> <th> </th> <th> </th> <th> </th> <th> </th> <th> </th> <th style="background:#ffdead;border-left:1px solid black;border-top:1px solid black;"><i>A</i> = 0 </th> <th style="border-left:1px solid black;">2 </th> <th style="border-right:1px solid black;">6 </th></tr> <tr> <th style="background:#ffdead;border-bottom:1px solid black;border-left:1px solid black;"><i>A</i> = 1 </th> <th style="border-bottom:1px solid black;border-left:1px solid black;">5 </th> <th style="border-bottom:1px solid black;border-right:1px solid black;">3 </th> <th> </th> <th> </th> <th> </th> <th> </th> <th> </th> <th style="background:#ffdead;border-left:1px solid black;border-bottom:1px solid black;"><i>A</i> = 1 </th> <th style="border-left:1px solid black;border-bottom:1px solid black;">5 </th> <th style="border-right:1px solid black;border-bottom:1px solid black;">3 </th></tr></tbody></table> <p>The distinction between qualitative and quantitative interactions depends on the order in which the variables are considered (in contrast, the property of additivity is invariant to the order of the variables). In the following table, if we focus on the effect of treatment <i>A</i>, there is a quantitative interaction — giving treatment <i>A</i> will improve the outcome on average regardless of whether treatment <i>B</i> is or is not already being given (although the benefit is greater if treatment <i>A</i> is given alone). However, if we focus on the effect of treatment <i>B</i>, there is a qualitative interaction — giving treatment <i>B</i> to a subject who is already receiving treatment <i>A</i> will (on average) make things worse, whereas giving treatment <i>B</i> to a subject who is not receiving treatment <i>A</i> will improve the outcome on average. </p> <table cellpadding="5" cellspacing="0" align="center"> <tbody><tr> <th> </th> <th style="background:#ffdead;border-left:1px solid black;border-top:1px solid black;"><i>B</i> = 0 </th> <th style="background:#ffdead;border-top:1px solid black;border-right:1px solid black;"><i>B</i> = 1 </th></tr> <tr> <th style="background:#ffdead;border-left:1px solid black;border-top:1px solid black;"><i>A</i> = 0 </th> <th style="border-left:1px solid black;">1 </th> <th style="border-right:1px solid black;">4 </th></tr> <tr> <th style="background:#ffdead;border-bottom:1px solid black;border-left:1px solid black;"><i>A</i> = 1 </th> <th style="border-bottom:1px solid black;border-left:1px solid black;">7 </th> <th style="border-bottom:1px solid black;border-right:1px solid black;">6 </th></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Unit_treatment_additivity">Unit treatment additivity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interaction_(statistics)&action=edit&section=5" title="Edit section: Unit treatment additivity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In its simplest form, the assumption of treatment unit additivity states that the observed response <i>y</i><sub><i>ij</i></sub> from experimental unit <i>i</i> when receiving treatment <i>j</i> can be written as the sum <i>y</i><sub><i>ij</i></sub> = <i>y</i><sub><i>i</i></sub> + <i>t</i><sub><i>j</i></sub>.<sup id="cite_ref-Kempthorne_(1979)_5-0" class="reference"><a href="#cite_note-Kempthorne_(1979)-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Cox1958_2_6-0" class="reference"><a href="#cite_note-Cox1958_2-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> The assumption of unit treatment additivity implies that every treatment has exactly the same additive effect on each experimental unit. Since any given experimental unit can only undergo one of the treatments, the assumption of unit treatment additivity is a hypothesis that is not directly falsifiable, according to Cox<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (April 2010)">citation needed</span></a></i>]</sup> and Kempthorne.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (April 2010)">citation needed</span></a></i>]</sup> </p><p>However, many consequences of treatment-unit additivity can be falsified.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (April 2010)">citation needed</span></a></i>]</sup> For a randomized experiment, the assumption of treatment additivity implies that the variance is constant for all treatments. Therefore, by contraposition, a necessary condition for unit treatment additivity is that the variance is constant.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (April 2010)">citation needed</span></a></i>]</sup> </p><p>The property of unit treatment additivity is not invariant under a change of scale,<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (April 2010)">citation needed</span></a></i>]</sup> so statisticians often use transformations to achieve unit treatment additivity. If the response variable is expected to follow a parametric family of probability distributions, then the statistician may specify (in the protocol for the experiment or observational study) that the responses be transformed to stabilize the variance.<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> In many cases, a statistician may specify that logarithmic transforms be applied to the responses, which are believed to follow a multiplicative model.<sup id="cite_ref-Cox1958_2_6-1" class="reference"><a href="#cite_note-Cox1958_2-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Bailey_on_eelworms_9-0" class="reference"><a href="#cite_note-Bailey_on_eelworms-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>The assumption of unit treatment additivity was enunciated in experimental design by Kempthorne<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (April 2010)">citation needed</span></a></i>]</sup> and Cox<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (April 2010)">citation needed</span></a></i>]</sup>. Kempthorne's use of unit treatment additivity and randomization is similar to the design-based analysis of finite population survey sampling. </p><p>In recent years, it has become common<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (April 2010)">citation needed</span></a></i>]</sup> to use the terminology of Donald Rubin, which uses counterfactuals. Suppose we are comparing two groups of people with respect to some attribute <i>y</i>. For example, the first group might consist of people who are given a standard treatment for a medical condition, with the second group consisting of people who receive a new treatment with unknown effect. Taking a "counterfactual" perspective, we can consider an individual whose attribute has value <i>y</i> if that individual belongs to the first group, and whose attribute has value <i>τ</i>(<i>y</i>) if the individual belongs to the second group. The assumption of "unit treatment additivity" is that <i>τ</i>(<i>y</i>) = <i>τ</i>, that is, the "treatment effect" does not depend on <i>y</i>. Since we cannot observe both <i>y</i> and τ(<i>y</i>) for a given individual, this is not testable at the individual level. However, unit treatment additivity implies that the <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution functions</a> <i>F</i><sub>1</sub> and <i>F</i><sub>2</sub> for the two groups satisfy <i>F</i><sub>2</sub>(<i>y</i>)  = <i>F</i><sub>1</sub>(<i>y − τ</i>), as long as the assignment of individuals to groups 1 and 2 is independent of all other factors influencing <i>y</i> (i.e. there are no <a href="/wiki/Confounding_variable" class="mw-redirect" title="Confounding variable">confounders</a>). Lack of unit treatment additivity can be viewed as a form of interaction between the treatment assignment (e.g. to groups 1 or 2), and the baseline, or untreated value of <i>y</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Categorical_variables">Categorical variables</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interaction_(statistics)&action=edit&section=6" title="Edit section: Categorical variables"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sometimes the interacting variables are categorical variables rather than real numbers and the study might then be dealt with as an <a href="/wiki/Analysis_of_variance" title="Analysis of variance">analysis of variance</a> problem. For example, members of a population may be classified by religion and by occupation. If one wishes to predict a person's height based only on the person's religion and occupation, a simple <i>additive</i> model, i.e., a model without interaction, would add to an overall average height an adjustment for a particular religion and another for a particular occupation. A model with interaction, unlike an <a href="/wiki/Additive_model" title="Additive model">additive model</a>, could add a further adjustment for the "interaction" between that religion and that occupation. This example may cause one to suspect that the word <i>interaction</i> is something of a misnomer. </p><p>Statistically, the presence of an interaction between categorical variables is generally tested using a form of <a href="/wiki/Analysis_of_variance" title="Analysis of variance">analysis of variance</a> (ANOVA). If one or more of the variables is continuous in nature, however, it would typically be tested using moderated multiple regression.<sup id="cite_ref-Overton2001_10-0" class="reference"><a href="#cite_note-Overton2001-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> This is so-called because a moderator is a variable that affects the strength of a relationship between two other variables. </p> <div class="mw-heading mw-heading3"><h3 id="Designed_experiments">Designed experiments</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interaction_(statistics)&action=edit&section=7" title="Edit section: Designed experiments"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Genichi_Taguchi" title="Genichi Taguchi">Genichi Taguchi</a> contended<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> that interactions could be eliminated from a <a href="/wiki/System" title="System">system</a> by appropriate choice of response variable and transformation. However <a href="/wiki/George_Box" class="mw-redirect" title="George Box">George Box</a> and others have argued that this is not the case in general.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Model_size">Model size</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interaction_(statistics)&action=edit&section=8" title="Edit section: Model size"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given <i>n</i> predictors, the number of terms in a linear model that includes a constant, every predictor, and every possible interaction is <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 {\tbinom {n}{0}}+{\tbinom {n}{1}}+{\tbinom {n}{2}}+\cdots +{\tbinom {n}{n}}=2^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>0</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>1</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tbinom {n}{0}}+{\tbinom {n}{1}}+{\tbinom {n}{2}}+\cdots +{\tbinom {n}{n}}=2^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec080ad5db8b5f7ded668e1538319a07edd095fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:32.028ex; height:3.343ex;" alt="{\displaystyle {\tbinom {n}{0}}+{\tbinom {n}{1}}+{\tbinom {n}{2}}+\cdots +{\tbinom {n}{n}}=2^{n}}"></span>. Since this quantity grows exponentially, it readily becomes impractically large. One method to limit the size of the model is to limit the order of interactions. For example, if only two-way interactions are allowed, the number of terms becomes <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 {\tbinom {n}{0}}+{\tbinom {n}{1}}+{\tbinom {n}{2}}=1+{\tfrac {1}{2}}n+{\tfrac {1}{2}}n^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>0</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>1</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>n</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tbinom {n}{0}}+{\tbinom {n}{1}}+{\tbinom {n}{2}}=1+{\tfrac {1}{2}}n+{\tfrac {1}{2}}n^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf6236b8f13e64757e449d2ad62c788e13261149" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:32.13ex; height:3.509ex;" alt="{\displaystyle {\tbinom {n}{0}}+{\tbinom {n}{1}}+{\tbinom {n}{2}}=1+{\tfrac {1}{2}}n+{\tfrac {1}{2}}n^{2}}"></span>. The below table shows the number of terms for each number of predictors and maximum order of interaction. </p> <table class="wikitable" style="text-align: right;"> <caption>Number of terms </caption> <tbody><tr> <th rowspan="2">Predictors </th> <th colspan="5">Including up to <i>m</i>-way interactions </th></tr> <tr> <th>2</th> <th>3</th> <th>4</th> <th>5</th> <th>∞ </th></tr> <tr> <th scope="row">1 </th> <td>2</td> <td>2</td> <td>2</td> <td>2</td> <td>2 </td></tr> <tr> <th scope="row">2 </th> <td>4</td> <td>4</td> <td>4</td> <td>4</td> <td>4 </td></tr> <tr> <th scope="row">3 </th> <td>7</td> <td>8</td> <td>8</td> <td>8</td> <td>8 </td></tr> <tr> <th scope="row">4 </th> <td>11</td> <td>15</td> <td>16</td> <td>16</td> <td>16 </td></tr> <tr> <th scope="row">5 </th> <td>16</td> <td>26</td> <td>31</td> <td>32</td> <td>32 </td></tr> <tr> <th scope="row">6 </th> <td>22</td> <td>42</td> <td>57</td> <td>63</td> <td>64 </td></tr> <tr> <th scope="row">7 </th> <td>29</td> <td>64</td> <td>99</td> <td>120</td> <td>128 </td></tr> <tr> <th scope="row">8 </th> <td>37</td> <td>93</td> <td>163</td> <td>219</td> <td>256 </td></tr> <tr> <th scope="row">9 </th> <td>46</td> <td>130</td> <td>256</td> <td>382</td> <td>512 </td></tr> <tr> <th scope="row">10 </th> <td>56</td> <td>176</td> <td>386</td> <td>638</td> <td>1,024 </td></tr> <tr> <th scope="row">11 </th> <td>67</td> <td>232</td> <td>562</td> <td>1,024</td> <td>2,048 </td></tr> <tr> <th scope="row">12 </th> <td>79</td> <td>299</td> <td>794</td> <td>1,586</td> <td>4,096 </td></tr> <tr> <th scope="row">13 </th> <td>92</td> <td>378</td> <td>1,093</td> <td>2,380</td> <td>8,192 </td></tr> <tr> <th scope="row">14 </th> <td>106</td> <td>470</td> <td>1,471</td> <td>3,473</td> <td>16,384 </td></tr> <tr> <th scope="row">15 </th> <td>121</td> <td>576</td> <td>1,941</td> <td>4,944</td> <td>32,768 </td></tr> <tr> <th scope="row">20 </th> <td>211</td> <td>1,351</td> <td>6,196</td> <td>21,700</td> <td>1,048,576 </td></tr> <tr> <th scope="row">25 </th> <td>326</td> <td>2,626</td> <td>15,276</td> <td>68,406</td> <td>33,554,432 </td></tr> <tr> <th scope="row">50 </th> <td>1,276</td> <td>20,876</td> <td>251,176</td> <td>2,369,936</td> <td>10<sup>15</sup> </td></tr> <tr> <th scope="row">100 </th> <td>5,051</td> <td>166,751</td> <td>4,087,976</td> <td>79,375,496</td> <td>10<sup>30</sup> </td></tr> <tr> <th scope="row">1,000 </th> <td>500,501</td> <td>166,667,501</td> <td>10<sup>10</sup></td> <td>10<sup>12</sup></td> <td>10<sup>300</sup> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="In_regression">In regression</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interaction_(statistics)&action=edit&section=9" title="Edit section: In regression"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The most general approach to modeling interaction effects involves regression, starting from the elementary version given above: </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 Y=c+ax_{1}+bx_{2}+d(x_{1}\times x_{2})+{\text{error}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <mi>c</mi> <mo>+</mo> <mi>a</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>b</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>×</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>error</mtext> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y=c+ax_{1}+bx_{2}+d(x_{1}\times x_{2})+{\text{error}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97cf9964b9ac0695f551d3fed620a071433b91e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.186ex; height:2.843ex;" alt="{\displaystyle Y=c+ax_{1}+bx_{2}+d(x_{1}\times x_{2})+{\text{error}}\,}"></span></dd></dl> <p>where the interaction term <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 (x_{1}\times x_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>×</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1}\times x_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d90e16b6afe26d9dcbb8ba66e842509cb6b3667" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.418ex; height:2.843ex;" alt="{\displaystyle (x_{1}\times x_{2})}"></span> could be formed explicitly by multiplying two (or more) variables, or implicitly using factorial notation in modern statistical packages such as <a href="/wiki/Stata" title="Stata">Stata</a>. The components <i>x</i><sub>1</sub> and <i>x</i><sub>2</sub> might be measurements or {0,1} <a href="/wiki/Dummy_variable_(statistics)" title="Dummy variable (statistics)">dummy variables</a> in any combination. Interactions involving a dummy variable multiplied by a measurement variable are termed <i>slope dummy variables</i>,<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> because they estimate and test the difference in slopes between groups 0 and 1. </p><p>When measurement variables are employed in interactions, it is often desirable to work with centered versions, where the variable's mean (or some other reasonably central value) is set as zero. Centering can make the main effects in interaction models more interpretable, as it reduces the <a href="/wiki/Multicollinearity" title="Multicollinearity">multicollinearity</a> between the interaction term and the main effects.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> The coefficient <i>a</i> in the equation above, for example, represents the effect of <i>x</i><sub>1</sub> when <i>x</i><sub>2</sub> equals zero. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Tea_party_interaction.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Tea_party_interaction.png/220px-Tea_party_interaction.png" decoding="async" width="220" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Tea_party_interaction.png/330px-Tea_party_interaction.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/80/Tea_party_interaction.png/440px-Tea_party_interaction.png 2x" data-file-width="1254" data-file-height="912" /></a><figcaption>Interaction of education and political party affecting beliefs about climate change</figcaption></figure><p>Regression approaches to interaction modeling are very general because they can accommodate additional predictors, and many alternative specifications or estimation strategies beyond <a href="/wiki/Ordinary_least_squares" title="Ordinary least squares">ordinary least squares</a>. <a href="/wiki/Robust_regression" title="Robust regression">Robust</a>, <a href="/wiki/Quantile_regression" title="Quantile regression">quantile</a>, and mixed-effects (<a href="/wiki/Multilevel_model" title="Multilevel model">multilevel</a>) models are among the possibilities, as is <a href="/wiki/Generalized_linear_model" title="Generalized linear model">generalized linear modeling</a> encompassing a wide range of categorical, ordered, counted or otherwise limited dependent variables. The graph depicts an education*politics interaction, from a probability-weighted <a href="/wiki/Logit_regression" class="mw-redirect" title="Logit regression">logit regression</a> analysis of survey data.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><div class="mw-heading mw-heading2"><h2 id="Interaction_plots">Interaction plots</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interaction_(statistics)&action=edit&section=10" title="Edit section: Interaction plots"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Interaction plots, also called <a href="/wiki/Moderation_(statistics)#Two_continuous_independent_variables" title="Moderation (statistics)">simple-slope plots</a>, show possible interactions among variables. </p> <div class="mw-heading mw-heading3"><h3 id="Example:_Interaction_of_species_and_air_temperature_and_their_effect_on_body_temperature">Example: Interaction of species and air temperature and their effect on body temperature</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interaction_(statistics)&action=edit&section=11" title="Edit section: Example: Interaction of species and air temperature and their effect on body temperature"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider a study of the body temperature of different species at different air temperatures, in degrees Fahrenheit. The data are shown in the table below. </p><p><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Body_temperature_species_data_2.png" class="mw-file-description" title="Body temperature species data"><img alt="Body temperature species data" src="//upload.wikimedia.org/wikipedia/commons/6/67/Body_temperature_species_data_2.png" decoding="async" width="390" height="139" class="mw-file-element" data-file-width="390" data-file-height="139" /></a></span> </p><p>The interaction plot may use either the air temperature or the species as the x axis. The second factor is represented by lines on the interaction plot. </p><p><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Interaction_plot_body_temp.svg" class="mw-file-description" title="interaction plot body temp"><img alt="interaction plot body temp" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Interaction_plot_body_temp.svg/570px-Interaction_plot_body_temp.svg.png" decoding="async" width="570" height="334" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Interaction_plot_body_temp.svg/855px-Interaction_plot_body_temp.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Interaction_plot_body_temp.svg/1140px-Interaction_plot_body_temp.svg.png 2x" data-file-width="570" data-file-height="334" /></a></span> </p><p><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Interaction_plot_body_temp_2.svg" class="mw-file-description" title="interaction plot body temp 2"><img alt="interaction plot body temp 2" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Interaction_plot_body_temp_2.svg/796px-Interaction_plot_body_temp_2.svg.png" decoding="async" width="796" height="334" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Interaction_plot_body_temp_2.svg/1194px-Interaction_plot_body_temp_2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Interaction_plot_body_temp_2.svg/1592px-Interaction_plot_body_temp_2.svg.png 2x" data-file-width="796" data-file-height="334" /></a></span> </p><p>There is an interaction between the two factors (air temperature and species) in their effect on the response (body temperature), because the effect of the air temperature depends on the species. The interaction is indicated on the plot because the lines are not parallel. </p> <div class="mw-heading mw-heading3"><h3 id="Example:_effect_of_stroke_severity_and_treatment_on_recovery">Example: effect of stroke severity and treatment on recovery</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interaction_(statistics)&action=edit&section=12" title="Edit section: Example: effect of stroke severity and treatment on recovery"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As a second example, consider a clinical trial on the interaction between stroke severity and the efficacy of a drug on patient survival. The data are shown in the table below. </p><p><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Interaction_stroke_survival_data.png" class="mw-file-description" title="interaction stroke survival data"><img alt="interaction stroke survival data" src="//upload.wikimedia.org/wikipedia/commons/f/f3/Interaction_stroke_survival_data.png" decoding="async" width="402" height="132" class="mw-file-element" data-file-width="402" data-file-height="132" /></a></span> </p><p><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Interaction_plot_stroke_survival.svg" class="mw-file-description" title="interaction plot stroke survival"><img alt="interaction plot stroke survival" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Interaction_plot_stroke_survival.svg/624px-Interaction_plot_stroke_survival.svg.png" decoding="async" width="624" height="334" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Interaction_plot_stroke_survival.svg/936px-Interaction_plot_stroke_survival.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Interaction_plot_stroke_survival.svg/1248px-Interaction_plot_stroke_survival.svg.png 2x" data-file-width="624" data-file-height="334" /></a></span> </p><p>In the interaction plot, the lines for the mild and moderate stroke groups are parallel, indicating that the drug has the same effect in both groups, so there is no interaction. The line for the severe stroke group is not parallel to the other lines, indicating that there is an interaction between stroke severity and drug effect on survival. The line for the severe stroke group is flat, indicating that, among these patients, there is no difference in survival between the drug and placebo treatments. In contrast, the lines for the mild and moderate stroke groups slope down to the right, indicating that, among these patients, the placebo group has lower survival than drug-treated group. </p> <div class="mw-heading mw-heading2"><h2 id="Hypothesis_tests_for_interactions">Hypothesis tests for interactions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interaction_(statistics)&action=edit&section=13" title="Edit section: Hypothesis tests for interactions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Analysis of variance and regression analysis are used to test for significant interactions. </p> <div class="mw-heading mw-heading3"><h3 id="Example:_Interaction_of_temperature_and_time_in_cookie_baking">Example: Interaction of temperature and time in cookie baking</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interaction_(statistics)&action=edit&section=14" title="Edit section: Example: Interaction of temperature and time in cookie baking"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Is the yield of good cookies affected by the baking temperature and time in the oven? The table shows data for 8 batches of cookies. </p><p><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Interaction_cookie_yield_data.png" class="mw-file-description" title="interaction cookie yield data"><img alt="interaction cookie yield data" src="//upload.wikimedia.org/wikipedia/commons/b/b7/Interaction_cookie_yield_data.png" decoding="async" width="238" height="178" class="mw-file-element" data-file-width="238" data-file-height="178" /></a></span> </p><p><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Interaction_plot_cookie_baking.svg" class="mw-file-description" title="interaction plot cookie baking"><img alt="interaction plot cookie baking" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/Interaction_plot_cookie_baking.svg/495px-Interaction_plot_cookie_baking.svg.png" decoding="async" width="495" height="334" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/Interaction_plot_cookie_baking.svg/743px-Interaction_plot_cookie_baking.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/16/Interaction_plot_cookie_baking.svg/990px-Interaction_plot_cookie_baking.svg.png 2x" data-file-width="495" data-file-height="334" /></a></span> </p><p>The data show that the yield of good cookies is best when either (i) temperature is high and time in the oven is short, or (ii) temperature is low and time in the oven is long. If the cookies are left in the oven for a long time at a high temperature, there are burnt cookies and the yield is low. </p><p>From the graph and the data, it is clear that the lines are not parallel, indicating that there is an interaction. This can be tested using analysis of variance (ANOVA). The first ANOVA model will not include the interaction term. That is, the first ANOVA model ignores possible interaction. The second ANOVA model will include the interaction term. That is, the second ANOVA model explicitly performs a hypothesis test for interaction. </p> <div class="mw-heading mw-heading4"><h4 id="ANOVA_model_1:_no_interaction_term;_yield_~_temperature_+_time"><span id="ANOVA_model_1:_no_interaction_term.3B_yield_.7E_temperature_.2B_time"></span>ANOVA model 1: no interaction term; yield ~ temperature + time</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interaction_(statistics)&action=edit&section=15" title="Edit section: ANOVA model 1: no interaction term; yield ~ temperature + time"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Cookie_anova_model_1.png" class="mw-file-description" title="cookie anova model 1"><img alt="cookie anova model 1" src="//upload.wikimedia.org/wikipedia/commons/5/57/Cookie_anova_model_1.png" decoding="async" width="309" height="121" class="mw-file-element" data-file-width="309" data-file-height="121" /></a></span> </p><p>In the ANOVA model that ignores interaction, neither temperature nor time has a significant effect on yield (p=0.91), which is clearly the incorrect conclusion. The more appropriate ANOVA model should test for possible interaction. </p> <div class="mw-heading mw-heading4"><h4 id="ANOVA_model_2:_include_interaction_term;_yield_~_temperature_*_time"><span id="ANOVA_model_2:_include_interaction_term.3B_yield_.7E_temperature_.2A_time"></span>ANOVA model 2: include interaction term; yield ~ temperature * time</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interaction_(statistics)&action=edit&section=16" title="Edit section: ANOVA model 2: include interaction term; yield ~ temperature * time"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Cookie_anova_model_2_v_2.png" class="mw-file-description" title="Cookie anova model 2"><img alt="Cookie anova model 2" src="//upload.wikimedia.org/wikipedia/commons/6/64/Cookie_anova_model_2_v_2.png" decoding="async" width="406" height="127" class="mw-file-element" data-file-width="406" data-file-height="127" /></a></span> </p><p>The temperature:time interaction term is significant (p=0.000180). Based on the interaction test and the interaction plot, it appears that the effect of time on yield depends on temperature and vice versa. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interaction_(statistics)&action=edit&section=17" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Real-world examples of interaction include: </p> <ul><li><i>Interaction</i> between adding sugar to coffee and stirring the coffee. Neither of the two individual variables has much effect on sweetness but a combination of the two does.</li> <li><i>Interaction</i> between adding <a href="/wiki/Carbon" title="Carbon">carbon</a> to <a href="/wiki/Steel" title="Steel">steel</a> and <a href="/wiki/Quenching" title="Quenching">quenching</a>. Neither of the two individually has much effect on <a href="/wiki/Tensile_strength" class="mw-redirect" title="Tensile strength">strength</a> but a combination of the two has a dramatic effect.</li> <li><i>Interaction</i> between smoking and inhaling <a href="/wiki/Asbestos" title="Asbestos">asbestos</a> fibres: Both raise lung carcinoma risk, but exposure to asbestos <i>multiplies</i> the cancer risk in smokers and non-smokers. Here, the <i>joint effect</i> of inhaling asbestos and smoking is higher than the sum of both effects.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup></li> <li><i>Interaction</i> between genetic risk factors for <a href="/wiki/Diabetes_mellitus_type_2" class="mw-redirect" title="Diabetes mellitus type 2">type 2 diabetes</a> and diet (specifically, a "western" dietary pattern). The western dietary pattern was shown to increase diabetes risk for subjects with a high "genetic risk score", but not for other subjects.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup></li> <li><i>Interaction</i> between education and political orientation, affecting general-public perceptions about climate change. For example, US surveys often find that acceptance of the reality of <a href="/wiki/Anthropogenic_climate_change" class="mw-redirect" title="Anthropogenic climate change">anthropogenic climate change</a> rises with education among moderate or liberal survey respondents, but declines with education among the most conservative.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> Similar interactions have been observed to affect some non-climate science or environmental perceptions,<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> and to operate with science literacy or other knowledge indicators in place of education.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interaction_(statistics)&action=edit&section=18" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Analysis_of_variance" title="Analysis of variance">Analysis of variance</a></li> <li><a href="/wiki/Factorial_experiment" title="Factorial experiment">Factorial experiment</a></li> <li><a href="/wiki/Generalized_randomized_block_design" title="Generalized randomized block design">Generalized randomized block design</a></li> <li><a href="/wiki/Linear_model" title="Linear model">Linear model</a></li> <li><a href="/wiki/Main_effect" title="Main effect">Main effect</a></li> <li><a href="/wiki/Tukey%27s_test_of_additivity" title="Tukey's test of additivity">Tukey's test of additivity</a></li></ul> <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=Interaction_(statistics)&action=edit&section=19" 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 mw-references-columns"><ol class="references"> <li id="cite_note-Dodge-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Dodge_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Dodge_1-1"><sup><i><b>b</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="CITEREFDodge2003" class="citation book cs1">Dodge, Y. (2003). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/oxforddictionary0000unse"><i><span></span></i>The Oxford Dictionary of Statistical Terms<i><span></span></i></a></span>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-920613-1" title="Special:BookSources/978-0-19-920613-1"><bdi>978-0-19-920613-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Oxford+Dictionary+of+Statistical+Terms&rft.pub=Oxford+University+Press&rft.date=2003&rft.isbn=978-0-19-920613-1&rft.aulast=Dodge&rft.aufirst=Y.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Foxforddictionary0000unse&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteraction+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCox1984" class="citation journal cs1">Cox, D.R. (1984). "Interaction". <i>International Statistical Review</i>. <b>52</b> (1): <span class="nowrap">1–</span>25. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1403235">10.2307/1403235</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1403235">1403235</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Statistical+Review&rft.atitle=Interaction&rft.volume=52&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E25&rft.date=1984&rft_id=info%3Adoi%2F10.2307%2F1403235&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1403235%23id-name%3DJSTOR&rft.aulast=Cox&rft.aufirst=D.R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteraction+%28statistics%29" 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="CITEREFJamesWittenHastieTibshirani2021" class="citation book cs1">James, Gareth; Witten, Daniela; Hastie, Trevor; Tibshirani, Robert (2021). <a rel="nofollow" class="external text" href="https://link.springer.com/book/10.1007/978-1-0716-1418-1"><i>An introduction to statistical learning: with applications in R</i></a> (Second ed.). New York, NY: Springer. p. 103. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-0716-1418-1" title="Special:BookSources/978-1-0716-1418-1"><bdi>978-1-0716-1418-1</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">29 October</span> 2024</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+statistical+learning%3A+with+applications+in+R&rft.place=New+York%2C+NY&rft.pages=103&rft.edition=Second&rft.pub=Springer&rft.date=2021&rft.isbn=978-1-0716-1418-1&rft.aulast=James&rft.aufirst=Gareth&rft.au=Witten%2C+Daniela&rft.au=Hastie%2C+Trevor&rft.au=Tibshirani%2C+Robert&rft_id=https%3A%2F%2Flink.springer.com%2Fbook%2F10.1007%2F978-1-0716-1418-1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteraction+%28statistics%29" 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="CITEREFPeto1982" class="citation book cs1">Peto, D. P. (1982). "Statistical aspects of cancer trials". <i>Treatment of Cancer</i> (First ed.). London: Chapman and Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-412-21850-X" title="Special:BookSources/0-412-21850-X"><bdi>0-412-21850-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Statistical+aspects+of+cancer+trials&rft.btitle=Treatment+of+Cancer&rft.place=London&rft.edition=First&rft.pub=Chapman+and+Hall&rft.date=1982&rft.isbn=0-412-21850-X&rft.aulast=Peto&rft.aufirst=D.+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteraction+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-Kempthorne_(1979)-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Kempthorne_(1979)_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKempthorne1979" class="citation book cs1"><a href="/wiki/Oscar_Kempthorne" title="Oscar Kempthorne">Kempthorne, Oscar</a> (1979). <i>The Design and Analysis of Experiments</i> (Corrected reprint of (1952) Wiley ed.). Robert E. Krieger. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-88275-105-4" title="Special:BookSources/978-0-88275-105-4"><bdi>978-0-88275-105-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Design+and+Analysis+of+Experiments&rft.edition=Corrected+reprint+of+%281952%29+Wiley&rft.pub=Robert+E.+Krieger&rft.date=1979&rft.isbn=978-0-88275-105-4&rft.aulast=Kempthorne&rft.aufirst=Oscar&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteraction+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-Cox1958_2-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-Cox1958_2_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Cox1958_2_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCox1958" class="citation book cs1"><a href="/wiki/David_R._Cox" class="mw-redirect" title="David R. Cox">Cox, David R.</a> (1958). <i>Planning of experiments</i>. Wiley. Chapter 2. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-57429-5" title="Special:BookSources/0-471-57429-5"><bdi>0-471-57429-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Planning+of+experiments&rft.pages=Chapter+2&rft.pub=Wiley&rft.date=1958&rft.isbn=0-471-57429-5&rft.aulast=Cox&rft.aufirst=David+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteraction+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHinkelmann,_Klaus_and_Kempthorne,_Oscar2008" class="citation book cs1">Hinkelmann, Klaus and <a href="/wiki/Oscar_Kempthorne" title="Oscar Kempthorne">Kempthorne, Oscar</a> (2008). <i>Design and Analysis of Experiments, Volume I: Introduction to Experimental Design</i> (Second ed.). Wiley. Chapters 5-6. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-72756-9" title="Special:BookSources/978-0-471-72756-9"><bdi>978-0-471-72756-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Design+and+Analysis+of+Experiments%2C+Volume+I%3A+Introduction+to+Experimental+Design&rft.pages=Chapters+5-6&rft.edition=Second&rft.pub=Wiley&rft.date=2008&rft.isbn=978-0-471-72756-9&rft.au=Hinkelmann%2C+Klaus+and+Kempthorne%2C+Oscar&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteraction+%28statistics%29" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHinkelmann,_Klaus_and_Kempthorne,_Oscar2008" class="citation book cs1">Hinkelmann, Klaus and <a href="/wiki/Oscar_Kempthorne" title="Oscar Kempthorne">Kempthorne, Oscar</a> (2008). <i>Design and Analysis of Experiments, Volume I: Introduction to Experimental Design</i> (Second ed.). Wiley. Chapters 7-8. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-72756-9" title="Special:BookSources/978-0-471-72756-9"><bdi>978-0-471-72756-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Design+and+Analysis+of+Experiments%2C+Volume+I%3A+Introduction+to+Experimental+Design&rft.pages=Chapters+7-8&rft.edition=Second&rft.pub=Wiley&rft.date=2008&rft.isbn=978-0-471-72756-9&rft.au=Hinkelmann%2C+Klaus+and+Kempthorne%2C+Oscar&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteraction+%28statistics%29" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</span></span> </li> <li id="cite_note-Bailey_on_eelworms-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bailey_on_eelworms_9-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBailey2008" class="citation book cs1">Bailey, R. A. (2008). <a rel="nofollow" class="external text" href="http://www.maths.qmul.ac.uk/~rab/DOEbook/"><i>Design of Comparative Experiments</i></a>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-68357-9" title="Special:BookSources/978-0-521-68357-9"><bdi>978-0-521-68357-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Design+of+Comparative+Experiments&rft.pub=Cambridge+University+Press&rft.date=2008&rft.isbn=978-0-521-68357-9&rft.aulast=Bailey&rft.aufirst=R.+A.&rft_id=http%3A%2F%2Fwww.maths.qmul.ac.uk%2F~rab%2FDOEbook%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteraction+%28statistics%29" class="Z3988"></span> Pre-publication chapters are available on-line.</span> </li> <li id="cite_note-Overton2001-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-Overton2001_10-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOverton,_R._C.2001" class="citation journal cs1">Overton, R. C. (2001). "Moderated multiple regression for interactions involving categorical variables: a statistical control for heterogeneous variance across two groups". <i>Psychol Methods</i>. <b>6</b> (3): <span class="nowrap">218–</span>33. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1037%2F1082-989X.6.3.218">10.1037/1082-989X.6.3.218</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/11570229">11570229</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Psychol+Methods&rft.atitle=Moderated+multiple+regression+for+interactions+involving+categorical+variables%3A+a+statistical+control+for+heterogeneous+variance+across+two+groups&rft.volume=6&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E218-%3C%2Fspan%3E33&rft.date=2001&rft_id=info%3Adoi%2F10.1037%2F1082-989X.6.3.218&rft_id=info%3Apmid%2F11570229&rft.au=Overton%2C+R.+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteraction+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.qualitytrainingportal.com/resources/doe/taguchi_concepts.htm">"Design of Experiments - Taguchi Experiments"</a>. <i>www.qualitytrainingportal.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2015-11-27</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.qualitytrainingportal.com&rft.atitle=Design+of+Experiments+-+Taguchi+Experiments&rft_id=http%3A%2F%2Fwww.qualitytrainingportal.com%2Fresources%2Fdoe%2Ftaguchi_concepts.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteraction+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGeorge_E._P._Box1990" class="citation journal cs1"><a href="/wiki/George_E._P._Box" title="George E. P. Box">George E. P. Box</a> (1990). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100610131759/http://cqpi.engr.wisc.edu/system/files/r046.pdf">"Do interactions matter?"</a> <span class="cs1-format">(PDF)</span>. <i>Quality Engineering</i>. <b>2</b>: <span class="nowrap">365–</span>369. <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%2F08982119008962728">10.1080/08982119008962728</a>. Archived from <a rel="nofollow" class="external text" href="http://cqpi.engr.wisc.edu/system/files/r046.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2010-06-10<span class="reference-accessdate">. Retrieved <span class="nowrap">2009-07-28</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Quality+Engineering&rft.atitle=Do+interactions+matter%3F&rft.volume=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E365-%3C%2Fspan%3E369&rft.date=1990&rft_id=info%3Adoi%2F10.1080%2F08982119008962728&rft.au=George+E.+P.+Box&rft_id=http%3A%2F%2Fcqpi.engr.wisc.edu%2Fsystem%2Ffiles%2Fr046.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteraction+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">Hamilton, L.C. 1992. <i>Regression with Graphics: A Second Course in Applied Statistics</i>. Pacific Grove, CA: Brooks/Cole. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0534159009" title="Special:BookSources/978-0534159009">978-0534159009</a></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIacobucciSchneiderPopovichBakamitsos2016" class="citation journal cs1">Iacobucci, Dawn; Schneider, Matthew J.; Popovich, Deidre L.; Bakamitsos, Georgios A. (2016). <a rel="nofollow" class="external text" href="https://doi.org/10.3758%2Fs13428-015-0624-x">"Mean centering helps alleviate "micro" but not "macro" multicollinearity"</a>. <i>Behavior Research Methods</i>. <b>48</b> (4): <span class="nowrap">1308–</span>1317. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.3758%2Fs13428-015-0624-x">10.3758/s13428-015-0624-x</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1554-3528">1554-3528</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/26148824">26148824</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Behavior+Research+Methods&rft.atitle=Mean+centering+helps+alleviate+%22micro%22+but+not+%22macro%22+multicollinearity&rft.volume=48&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1308-%3C%2Fspan%3E1317&rft.date=2016&rft.issn=1554-3528&rft_id=info%3Apmid%2F26148824&rft_id=info%3Adoi%2F10.3758%2Fs13428-015-0624-x&rft.aulast=Iacobucci&rft.aufirst=Dawn&rft.au=Schneider%2C+Matthew+J.&rft.au=Popovich%2C+Deidre+L.&rft.au=Bakamitsos%2C+Georgios+A.&rft_id=https%3A%2F%2Fdoi.org%2F10.3758%252Fs13428-015-0624-x&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteraction+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHamiltonSaito2015" class="citation journal cs1">Hamilton, L.C.; Saito, K. (2015). "A four-party view of U.S. environmental concern". <i>Environmental Politics</i>. <b>24</b> (2): <span class="nowrap">212–</span>227. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2015EnvPo..24..212H">2015EnvPo..24..212H</a>. <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%2F09644016.2014.976485">10.1080/09644016.2014.976485</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:154762226">154762226</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Environmental+Politics&rft.atitle=A+four-party+view+of+U.S.+environmental+concern&rft.volume=24&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E212-%3C%2Fspan%3E227&rft.date=2015&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A154762226%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1080%2F09644016.2014.976485&rft_id=info%3Abibcode%2F2015EnvPo..24..212H&rft.aulast=Hamilton&rft.aufirst=L.C.&rft.au=Saito%2C+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteraction+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLee,_P._N.2001" class="citation journal cs1">Lee, P. N. (2001). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1740104">"Relation between exposure to asbestos and smoking jointly and the risk of lung cancer"</a>. <i>Occupational and Environmental Medicine</i>. <b>58</b> (3): <span class="nowrap">145–</span>53. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1136%2Foem.58.3.145">10.1136/oem.58.3.145</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1740104">1740104</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/11171926">11171926</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Occupational+and+Environmental+Medicine&rft.atitle=Relation+between+exposure+to+asbestos+and+smoking+jointly+and+the+risk+of+lung+cancer&rft.volume=58&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E145-%3C%2Fspan%3E53&rft.date=2001&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC1740104%23id-name%3DPMC&rft_id=info%3Apmid%2F11171926&rft_id=info%3Adoi%2F10.1136%2Foem.58.3.145&rft.au=Lee%2C+P.+N.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC1740104&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteraction+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLu,_Q.2009" class="citation journal cs1">Lu, Q.; et al. (2009). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2676999">"Genetic predisposition, Western dietary pattern, and the risk of type 2 diabetes in men"</a>. <i>Am J Clin Nutr</i>. <b>89</b> (5): <span class="nowrap">1453–</span>1458. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.3945%2Fajcn.2008.27249">10.3945/ajcn.2008.27249</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2676999">2676999</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/19279076">19279076</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Am+J+Clin+Nutr&rft.atitle=Genetic+predisposition%2C+Western+dietary+pattern%2C+and+the+risk+of+type+2+diabetes+in+men&rft.volume=89&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1453-%3C%2Fspan%3E1458&rft.date=2009&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC2676999%23id-name%3DPMC&rft_id=info%3Apmid%2F19279076&rft_id=info%3Adoi%2F10.3945%2Fajcn.2008.27249&rft.au=Lu%2C+Q.&rft.au=%3CPlease+add+first+missing+authors+to+populate+metadata.%3E&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC2676999&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteraction+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHamilton2011" class="citation journal cs1">Hamilton, L.C. (2011). <a rel="nofollow" class="external text" href="https://scholars.unh.edu/cgi/viewcontent.cgi?article=1388&context=soc_facpub">"Education, politics and opinions about climate change: Evidence for interaction effects"</a>. <i><a href="/wiki/Climatic_Change_(journal)" title="Climatic Change (journal)">Climatic Change</a></i>. <b>104</b> (2): <span class="nowrap">231–</span>242. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2011ClCh..104..231H">2011ClCh..104..231H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10584-010-9957-8">10.1007/s10584-010-9957-8</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:16481640">16481640</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Climatic+Change&rft.atitle=Education%2C+politics+and+opinions+about+climate+change%3A+Evidence+for+interaction+effects&rft.volume=104&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E231-%3C%2Fspan%3E242&rft.date=2011&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A16481640%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs10584-010-9957-8&rft_id=info%3Abibcode%2F2011ClCh..104..231H&rft.aulast=Hamilton&rft.aufirst=L.C.&rft_id=https%3A%2F%2Fscholars.unh.edu%2Fcgi%2Fviewcontent.cgi%3Farticle%3D1388%26context%3Dsoc_facpub&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteraction+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcCright2011" class="citation journal cs1">McCright, A. M. (2011). "Political orientation moderates Americans' beliefs and concern about climate change". <i><a href="/wiki/Climatic_Change_(journal)" title="Climatic Change (journal)">Climatic Change</a></i>. <b>104</b> (2): <span class="nowrap">243–</span>253. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2011ClCh..104..243M">2011ClCh..104..243M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10584-010-9946-y">10.1007/s10584-010-9946-y</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:152795205">152795205</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Climatic+Change&rft.atitle=Political+orientation+moderates+Americans%27+beliefs+and+concern+about+climate+change&rft.volume=104&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E243-%3C%2Fspan%3E253&rft.date=2011&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A152795205%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs10584-010-9946-y&rft_id=info%3Abibcode%2F2011ClCh..104..243M&rft.aulast=McCright&rft.aufirst=A.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteraction+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHamiltonSaito2015" class="citation journal cs1">Hamilton, Lawrence C.; Saito, Kei (2015). "A four-party view of US environmental concern". <i>Environmental Politics</i>. <b>24</b> (2): <span class="nowrap">212–</span>227. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2015EnvPo..24..212H">2015EnvPo..24..212H</a>. <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%2F09644016.2014.976485">10.1080/09644016.2014.976485</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:154762226">154762226</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Environmental+Politics&rft.atitle=A+four-party+view+of+US+environmental+concern&rft.volume=24&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E212-%3C%2Fspan%3E227&rft.date=2015&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A154762226%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1080%2F09644016.2014.976485&rft_id=info%3Abibcode%2F2015EnvPo..24..212H&rft.aulast=Hamilton&rft.aufirst=Lawrence+C.&rft.au=Saito%2C+Kei&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteraction+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKahanJenkins-SmithBraman2011" class="citation journal cs1">Kahan, D.M.; Jenkins-Smith, H.; Braman, D. (2011). <a rel="nofollow" class="external text" href="https://scholarship.law.gwu.edu/cgi/viewcontent.cgi?article=1269&context=faculty_publications">"Cultural cognition of scientific consensus"</a>. <i>Journal of Risk Research</i>. <b>14</b> (2): <span class="nowrap">147–</span>174. <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%2F13669877.2010.511246">10.1080/13669877.2010.511246</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/10.1080%2F13669877.2010.511246">10.1080/13669877.2010.511246</a></span>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:216092368">216092368</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+Risk+Research&rft.atitle=Cultural+cognition+of+scientific+consensus&rft.volume=14&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E147-%3C%2Fspan%3E174&rft.date=2011&rft_id=info%3Ahdl%2F10.1080%2F13669877.2010.511246&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A216092368%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1080%2F13669877.2010.511246&rft.aulast=Kahan&rft.aufirst=D.M.&rft.au=Jenkins-Smith%2C+H.&rft.au=Braman%2C+D.&rft_id=https%3A%2F%2Fscholarship.law.gwu.edu%2Fcgi%2Fviewcontent.cgi%3Farticle%3D1269%26context%3Dfaculty_publications&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteraction+%28statistics%29" class="Z3988"></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHamiltonCutlerSchaefer2012" class="citation journal cs1">Hamilton, L.C.; Cutler, M.J.; Schaefer, A. (2012). "Public knowledge and concern about polar-region warming". <i><a href="/wiki/Polar_Geography" title="Polar Geography">Polar Geography</a></i>. <b>35</b> (2): <span class="nowrap">155–</span>168. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2012PolGe..35..155H">2012PolGe..35..155H</a>. <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%2F1088937X.2012.684155">10.1080/1088937X.2012.684155</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:12437794">12437794</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Polar+Geography&rft.atitle=Public+knowledge+and+concern+about+polar-region+warming&rft.volume=35&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E155-%3C%2Fspan%3E168&rft.date=2012&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A12437794%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1080%2F1088937X.2012.684155&rft_id=info%3Abibcode%2F2012PolGe..35..155H&rft.aulast=Hamilton&rft.aufirst=L.C.&rft.au=Cutler%2C+M.J.&rft.au=Schaefer%2C+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteraction+%28statistics%29" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interaction_(statistics)&action=edit&section=20" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/David_R._Cox" class="mw-redirect" title="David R. Cox">Cox, David R.</a> and Reid, Nancy M. (2000) <i>The theory of design of experiments</i>, Chapman & Hall/CRC. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-58488-195-X" title="Special:BookSources/1-58488-195-X">1-58488-195-X</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSouthwood1978" class="citation journal cs1">Southwood, K.E. (1978). "Substantive Theory and Statistical Interaction: Five Models". <i><a href="/wiki/The_American_Journal_of_Sociology" class="mw-redirect" title="The American Journal of Sociology">The American Journal of Sociology</a></i>. <b>83</b> (5): <span class="nowrap">1154–</span>1203. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F226678">10.1086/226678</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:143521842">143521842</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Journal+of+Sociology&rft.atitle=Substantive+Theory+and+Statistical+Interaction%3A+Five+Models&rft.volume=83&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1154-%3C%2Fspan%3E1203&rft.date=1978&rft_id=info%3Adoi%2F10.1086%2F226678&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A143521842%23id-name%3DS2CID&rft.aulast=Southwood&rft.aufirst=K.E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteraction+%28statistics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBramborClark2006" class="citation journal cs1">Brambor, T.; Clark, W. R. (2006). "Understanding Interaction Models: Improving Empirical Analyses". <i>Political Analysis</i>. <b>14</b> (1): <span class="nowrap">63–</span>82. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fpan%2Fmpi014">10.1093/pan/mpi014</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Political+Analysis&rft.atitle=Understanding+Interaction+Models%3A+Improving+Empirical+Analyses&rft.volume=14&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E63-%3C%2Fspan%3E82&rft.date=2006&rft_id=info%3Adoi%2F10.1093%2Fpan%2Fmpi014&rft.aulast=Brambor&rft.aufirst=T.&rft.au=Clark%2C+W.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteraction+%28statistics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHayesMatthes2009" class="citation journal cs1">Hayes, A. F.; Matthes, J. (2009). <a rel="nofollow" class="external text" href="https://doi.org/10.3758%2FBRM.41.3.924">"Computational procedures for probing interactions in OLS and logistic regression: SPSS and SAS implementations"</a>. <i>Behavior Research Methods</i>. <b>41</b> (3): <span class="nowrap">924–</span>936. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.3758%2FBRM.41.3.924">10.3758/BRM.41.3.924</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/19587209">19587209</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Behavior+Research+Methods&rft.atitle=Computational+procedures+for+probing+interactions+in+OLS+and+logistic+regression%3A+SPSS+and+SAS+implementations&rft.volume=41&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E924-%3C%2Fspan%3E936&rft.date=2009&rft_id=info%3Adoi%2F10.3758%2FBRM.41.3.924&rft_id=info%3Apmid%2F19587209&rft.aulast=Hayes&rft.aufirst=A.+F.&rft.au=Matthes%2C+J.&rft_id=https%3A%2F%2Fdoi.org%2F10.3758%252FBRM.41.3.924&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteraction+%28statistics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBalliSørensen2012" class="citation journal cs1">Balli, H. O.; Sørensen, B. E. (2012). "Interaction effects in econometrics". <i>Empirical Economics</i>. <b>43</b> (x): <span class="nowrap">1–</span>21. <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=10.1.1.691.4349">10.1.1.691.4349</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/10.1007%2Fs00181-012-0604-2">10.1007/s00181-012-0604-2</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:53504187">53504187</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Empirical+Economics&rft.atitle=Interaction+effects+in+econometrics&rft.volume=43&rft.issue=x&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E21&rft.date=2012&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.691.4349%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A53504187%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs00181-012-0604-2&rft.aulast=Balli&rft.aufirst=H.+O.&rft.au=S%C3%B8rensen%2C+B.+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteraction+%28statistics%29" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interaction_(statistics)&action=edit&section=21" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20160303165623/http://pages.towson.edu/mchamber/chapter7eco306.pdf">"Using Indicator and Interaction Variables"</a> <span class="cs1-format">(PDF)</span>. Archived from <a rel="nofollow" class="external text" href="http://pages.towson.edu/mchamber/chapter7eco306.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2016-03-03<span class="reference-accessdate">. Retrieved <span class="nowrap">2010-02-03</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Using+Indicator+and+Interaction+Variables&rft_id=http%3A%2F%2Fpages.towson.edu%2Fmchamber%2Fchapter7eco306.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteraction+%28statistics%29" class="Z3988"></span> <span style="font-size:85%;">(158 <a href="/wiki/Kibibyte" class="mw-redirect" title="Kibibyte">KiB</a>)</span></li> <li><a rel="nofollow" class="external text" href="http://davis.foulger.info/papers/statisticalInteraction1979.htm">Credibility and the Statistical Interaction Variable: Speaking Up for Multiplication as a Source of Understanding</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20100212121505/http://www.ruf.rice.edu/~branton/interaction/faqfund.htm">Fundamentals of Statistical Interactions: What is the difference between "main effects" and "interaction effects"? 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colspan="2"><div> <ul><li><a href="/wiki/Outline_of_statistics" title="Outline of statistics">Outline</a></li> <li><a href="/wiki/List_of_statistics_articles" title="List of statistics articles">Index</a></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Descriptive_statistics636" style="font-size:114%;margin:0 4em"><a href="/wiki/Descriptive_statistics" title="Descriptive statistics">Descriptive statistics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Continuous_probability_distribution" class="mw-redirect" title="Continuous probability distribution">Continuous data</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Central_tendency" title="Central tendency">Center</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mean" title="Mean">Mean</a> <ul><li><a href="/wiki/Arithmetic_mean" title="Arithmetic mean">Arithmetic</a></li> <li><a href="/wiki/Arithmetic%E2%80%93geometric_mean" title="Arithmetic–geometric mean">Arithmetic-Geometric</a></li> <li><a href="/wiki/Contraharmonic_mean" title="Contraharmonic mean">Contraharmonic</a></li> <li><a href="/wiki/Cubic_mean" title="Cubic mean">Cubic</a></li> <li><a href="/wiki/Generalized_mean" title="Generalized mean">Generalized/power</a></li> <li><a href="/wiki/Geometric_mean" title="Geometric mean">Geometric</a></li> <li><a href="/wiki/Harmonic_mean" title="Harmonic mean">Harmonic</a></li> <li><a href="/wiki/Heronian_mean" title="Heronian mean">Heronian</a></li> <li><a href="/wiki/Heinz_mean" title="Heinz mean">Heinz</a></li> <li><a href="/wiki/Lehmer_mean" title="Lehmer mean">Lehmer</a></li></ul></li> <li><a href="/wiki/Median" title="Median">Median</a></li> <li><a href="/wiki/Mode_(statistics)" title="Mode (statistics)">Mode</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Statistical_dispersion" title="Statistical dispersion">Dispersion</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Average_absolute_deviation" title="Average absolute deviation">Average absolute deviation</a></li> <li><a href="/wiki/Coefficient_of_variation" title="Coefficient of variation">Coefficient of variation</a></li> <li><a href="/wiki/Interquartile_range" title="Interquartile range">Interquartile range</a></li> <li><a href="/wiki/Percentile" title="Percentile">Percentile</a></li> <li><a href="/wiki/Range_(statistics)" title="Range (statistics)">Range</a></li> <li><a href="/wiki/Standard_deviation" title="Standard deviation">Standard deviation</a></li> <li><a href="/wiki/Variance#Sample_variance" title="Variance">Variance</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Shape_of_the_distribution" class="mw-redirect" title="Shape of the distribution">Shape</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></li> <li><a href="/wiki/Moment_(mathematics)" title="Moment (mathematics)">Moments</a> <ul><li><a href="/wiki/Kurtosis" title="Kurtosis">Kurtosis</a></li> <li><a href="/wiki/L-moment" title="L-moment">L-moments</a></li> <li><a href="/wiki/Skewness" title="Skewness">Skewness</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Count_data" title="Count data">Count data</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Index_of_dispersion" title="Index of dispersion">Index of dispersion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Summary tables</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Contingency_table" title="Contingency table">Contingency table</a></li> <li><a href="/wiki/Frequency_distribution" class="mw-redirect" title="Frequency distribution">Frequency distribution</a></li> <li><a href="/wiki/Grouped_data" title="Grouped data">Grouped data</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Dependence</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Partial_correlation" title="Partial correlation">Partial correlation</a></li> <li><a href="/wiki/Pearson_correlation_coefficient" title="Pearson correlation coefficient">Pearson product-moment correlation</a></li> <li><a href="/wiki/Rank_correlation" title="Rank correlation">Rank correlation</a> <ul><li><a href="/wiki/Kendall_rank_correlation_coefficient" title="Kendall rank correlation coefficient">Kendall's τ</a></li> <li><a href="/wiki/Spearman%27s_rank_correlation_coefficient" title="Spearman's rank correlation coefficient">Spearman's ρ</a></li></ul></li> <li><a href="/wiki/Scatter_plot" title="Scatter plot">Scatter plot</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Statistical_graphics" title="Statistical graphics">Graphics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bar_chart" title="Bar chart">Bar chart</a></li> <li><a href="/wiki/Biplot" title="Biplot">Biplot</a></li> <li><a href="/wiki/Box_plot" title="Box plot">Box plot</a></li> <li><a href="/wiki/Control_chart" title="Control chart">Control chart</a></li> <li><a href="/wiki/Correlogram" title="Correlogram">Correlogram</a></li> <li><a href="/wiki/Fan_chart_(statistics)" title="Fan chart (statistics)">Fan chart</a></li> <li><a href="/wiki/Forest_plot" title="Forest plot">Forest plot</a></li> <li><a href="/wiki/Histogram" title="Histogram">Histogram</a></li> <li><a href="/wiki/Pie_chart" title="Pie chart">Pie chart</a></li> <li><a href="/wiki/Q%E2%80%93Q_plot" title="Q–Q plot">Q–Q plot</a></li> <li><a href="/wiki/Radar_chart" title="Radar chart">Radar chart</a></li> <li><a href="/wiki/Run_chart" title="Run chart">Run chart</a></li> <li><a href="/wiki/Scatter_plot" title="Scatter plot">Scatter plot</a></li> <li><a href="/wiki/Stem-and-leaf_display" title="Stem-and-leaf display">Stem-and-leaf display</a></li> <li><a href="/wiki/Violin_plot" title="Violin plot">Violin plot</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Data_collection636" style="font-size:114%;margin:0 4em"><a href="/wiki/Data_collection" title="Data collection">Data collection</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Design_of_experiments" title="Design of experiments">Study design</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Effect_size" title="Effect size">Effect size</a></li> <li><a href="/wiki/Missing_data" title="Missing data">Missing data</a></li> <li><a href="/wiki/Optimal_design" class="mw-redirect" title="Optimal design">Optimal design</a></li> <li><a href="/wiki/Statistical_population" title="Statistical population">Population</a></li> <li><a href="/wiki/Replication_(statistics)" title="Replication (statistics)">Replication</a></li> <li><a href="/wiki/Sample_size_determination" title="Sample size determination">Sample size determination</a></li> <li><a href="/wiki/Statistic" title="Statistic">Statistic</a></li> <li><a href="/wiki/Statistical_power" class="mw-redirect" title="Statistical power">Statistical power</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Survey_methodology" title="Survey methodology">Survey methodology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Sampling_(statistics)" title="Sampling (statistics)">Sampling</a> <ul><li><a href="/wiki/Cluster_sampling" title="Cluster sampling">Cluster</a></li> <li><a href="/wiki/Stratified_sampling" title="Stratified sampling">Stratified</a></li></ul></li> <li><a href="/wiki/Opinion_poll" title="Opinion poll">Opinion poll</a></li> <li><a href="/wiki/Questionnaire" title="Questionnaire">Questionnaire</a></li> <li><a href="/wiki/Standard_error" title="Standard error">Standard error</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Experiment" title="Experiment">Controlled experiments</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Blocking_(statistics)" title="Blocking (statistics)">Blocking</a></li> <li><a href="/wiki/Factorial_experiment" title="Factorial experiment">Factorial experiment</a></li> <li><a class="mw-selflink selflink">Interaction</a></li> <li><a href="/wiki/Random_assignment" title="Random assignment">Random assignment</a></li> <li><a href="/wiki/Randomized_controlled_trial" title="Randomized controlled trial">Randomized controlled trial</a></li> <li><a href="/wiki/Randomized_experiment" title="Randomized experiment">Randomized experiment</a></li> <li><a href="/wiki/Scientific_control" title="Scientific control">Scientific control</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Adaptive designs</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adaptive_clinical_trial" class="mw-redirect" title="Adaptive clinical trial">Adaptive clinical trial</a></li> <li><a href="/wiki/Stochastic_approximation" title="Stochastic approximation">Stochastic approximation</a></li> <li><a href="/wiki/Up-and-Down_Designs" class="mw-redirect" title="Up-and-Down Designs">Up-and-down designs</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Observational_study" title="Observational study">Observational studies</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cohort_study" title="Cohort study">Cohort study</a></li> <li><a href="/wiki/Cross-sectional_study" title="Cross-sectional study">Cross-sectional study</a></li> <li><a href="/wiki/Natural_experiment" title="Natural experiment">Natural experiment</a></li> <li><a href="/wiki/Quasi-experiment" title="Quasi-experiment">Quasi-experiment</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Statistical_inference636" style="font-size:114%;margin:0 4em"><a href="/wiki/Statistical_inference" title="Statistical inference">Statistical inference</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Statistical_theory" title="Statistical theory">Statistical theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Population_(statistics)" class="mw-redirect" title="Population (statistics)">Population</a></li> <li><a href="/wiki/Statistic" title="Statistic">Statistic</a></li> <li><a href="/wiki/Probability_distribution" title="Probability distribution">Probability distribution</a></li> <li><a href="/wiki/Sampling_distribution" title="Sampling distribution">Sampling distribution</a> <ul><li><a href="/wiki/Order_statistic" title="Order statistic">Order statistic</a></li></ul></li> <li><a href="/wiki/Empirical_distribution_function" title="Empirical distribution function">Empirical distribution</a> <ul><li><a href="/wiki/Density_estimation" title="Density estimation">Density estimation</a></li></ul></li> <li><a href="/wiki/Statistical_model" title="Statistical model">Statistical model</a> <ul><li><a href="/wiki/Model_specification" class="mw-redirect" title="Model specification">Model specification</a></li> <li><a href="/wiki/Lp_space" title="Lp space">L<sup><i>p</i></sup> space</a></li></ul></li> <li><a href="/wiki/Statistical_parameter" title="Statistical parameter">Parameter</a> <ul><li><a href="/wiki/Location_parameter" title="Location parameter">location</a></li> <li><a href="/wiki/Scale_parameter" title="Scale parameter">scale</a></li> <li><a href="/wiki/Shape_parameter" title="Shape parameter">shape</a></li></ul></li> <li><a href="/wiki/Parametric_statistics" title="Parametric statistics">Parametric family</a> <ul><li><a href="/wiki/Likelihood_function" title="Likelihood function">Likelihood</a> <a href="/wiki/Monotone_likelihood_ratio" title="Monotone likelihood ratio"><span style="font-size:85%;">(monotone)</span></a></li> <li><a href="/wiki/Location%E2%80%93scale_family" title="Location–scale family">Location–scale family</a></li> <li><a href="/wiki/Exponential_family" title="Exponential family">Exponential family</a></li></ul></li> <li><a href="/wiki/Completeness_(statistics)" title="Completeness (statistics)">Completeness</a></li> <li><a href="/wiki/Sufficient_statistic" title="Sufficient statistic">Sufficiency</a></li> <li><a href="/wiki/Plug-in_principle" class="mw-redirect" title="Plug-in principle">Statistical functional</a> <ul><li><a href="/wiki/Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li> <li><a href="/wiki/U-statistic" title="U-statistic">U</a></li> <li><a href="/wiki/V-statistic" title="V-statistic">V</a></li></ul></li> <li><a href="/wiki/Optimal_decision" title="Optimal decision">Optimal decision</a> <ul><li><a href="/wiki/Loss_function" title="Loss function">loss function</a></li></ul></li> <li><a href="/wiki/Efficiency_(statistics)" title="Efficiency (statistics)">Efficiency</a></li> <li><a href="/wiki/Statistical_distance" title="Statistical distance">Statistical distance</a> <ul><li><a href="/wiki/Divergence_(statistics)" title="Divergence (statistics)">divergence</a></li></ul></li> <li><a href="/wiki/Asymptotic_theory_(statistics)" title="Asymptotic theory (statistics)">Asymptotics</a></li> <li><a href="/wiki/Robust_statistics" title="Robust statistics">Robustness</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Frequentist_inference" title="Frequentist inference">Frequentist inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Point_estimation" title="Point estimation">Point estimation</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Estimating_equations" title="Estimating equations">Estimating equations</a> <ul><li><a href="/wiki/Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">Maximum likelihood</a></li> <li><a href="/wiki/Method_of_moments_(statistics)" title="Method of moments (statistics)">Method of moments</a></li> <li><a href="/wiki/M-estimator" title="M-estimator">M-estimator</a></li> <li><a href="/wiki/Minimum_distance_estimation" class="mw-redirect" title="Minimum distance estimation">Minimum distance</a></li></ul></li> <li><a href="/wiki/Bias_of_an_estimator" title="Bias of an estimator">Unbiased estimators</a> <ul><li><a href="/wiki/Minimum-variance_unbiased_estimator" title="Minimum-variance unbiased estimator">Mean-unbiased minimum-variance</a> <ul><li><a href="/wiki/Rao%E2%80%93Blackwell_theorem" title="Rao–Blackwell theorem">Rao–Blackwellization</a></li> <li><a href="/wiki/Lehmann%E2%80%93Scheff%C3%A9_theorem" title="Lehmann–Scheffé theorem">Lehmann–Scheffé theorem</a></li></ul></li> <li><a href="/wiki/Median-unbiased_estimator" class="mw-redirect" title="Median-unbiased estimator">Median unbiased</a></li></ul></li> <li><a href="/wiki/Plug-in_principle" class="mw-redirect" title="Plug-in principle">Plug-in</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Interval_estimation" title="Interval estimation">Interval estimation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Confidence_interval" title="Confidence interval">Confidence interval</a></li> <li><a href="/wiki/Pivotal_quantity" title="Pivotal quantity">Pivot</a></li> <li><a href="/wiki/Likelihood_interval" class="mw-redirect" title="Likelihood interval">Likelihood interval</a></li> <li><a href="/wiki/Prediction_interval" title="Prediction interval">Prediction interval</a></li> <li><a href="/wiki/Tolerance_interval" title="Tolerance interval">Tolerance interval</a></li> <li><a href="/wiki/Resampling_(statistics)" title="Resampling (statistics)">Resampling</a> <ul><li><a href="/wiki/Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li> <li><a href="/wiki/Jackknife_resampling" title="Jackknife resampling">Jackknife</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Statistical_hypothesis_testing" class="mw-redirect" title="Statistical hypothesis testing">Testing hypotheses</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/One-_and_two-tailed_tests" title="One- and two-tailed tests">1- & 2-tails</a></li> <li><a href="/wiki/Power_(statistics)" title="Power (statistics)">Power</a> <ul><li><a href="/wiki/Uniformly_most_powerful_test" title="Uniformly most powerful test">Uniformly most powerful test</a></li></ul></li> <li><a href="/wiki/Permutation_test" title="Permutation test">Permutation test</a> <ul><li><a href="/wiki/Randomization_test" class="mw-redirect" title="Randomization test">Randomization test</a></li></ul></li> <li><a href="/wiki/Multiple_comparisons" class="mw-redirect" title="Multiple comparisons">Multiple comparisons</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Parametric_statistics" title="Parametric statistics">Parametric tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio</a></li> <li><a href="/wiki/Score_test" title="Score test">Score/Lagrange multiplier</a></li> <li><a href="/wiki/Wald_test" title="Wald test">Wald</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/List_of_statistical_tests" title="List of statistical tests">Specific tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Z-test" title="Z-test"><i>Z</i>-test <span style="font-size:85%;">(normal)</span></a></li> <li><a href="/wiki/Student%27s_t-test" title="Student's t-test">Student's <i>t</i>-test</a></li> <li><a href="/wiki/F-test" title="F-test"><i>F</i>-test</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Goodness_of_fit" title="Goodness of fit">Goodness of fit</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chi-squared_test" title="Chi-squared test">Chi-squared</a></li> <li><a href="/wiki/G-test" title="G-test"><i>G</i>-test</a></li> <li><a href="/wiki/Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov–Smirnov</a></li> <li><a href="/wiki/Anderson%E2%80%93Darling_test" title="Anderson–Darling test">Anderson–Darling</a></li> <li><a href="/wiki/Lilliefors_test" title="Lilliefors test">Lilliefors</a></li> <li><a href="/wiki/Jarque%E2%80%93Bera_test" title="Jarque–Bera test">Jarque–Bera</a></li> <li><a href="/wiki/Shapiro%E2%80%93Wilk_test" title="Shapiro–Wilk test">Normality <span style="font-size:85%;">(Shapiro–Wilk)</span></a></li> <li><a href="/wiki/Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio test</a></li> <li><a href="/wiki/Model_selection" title="Model selection">Model selection</a> <ul><li><a href="/wiki/Cross-validation_(statistics)" title="Cross-validation (statistics)">Cross validation</a></li> <li><a href="/wiki/Akaike_information_criterion" title="Akaike information criterion">AIC</a></li> <li><a href="/wiki/Bayesian_information_criterion" title="Bayesian information criterion">BIC</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Rank_statistics" class="mw-redirect" title="Rank statistics">Rank statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Sign_test" title="Sign test">Sign</a> <ul><li><a href="/wiki/Sample_median" class="mw-redirect" title="Sample median">Sample median</a></li></ul></li> <li><a href="/wiki/Wilcoxon_signed-rank_test" title="Wilcoxon signed-rank test">Signed rank <span style="font-size:85%;">(Wilcoxon)</span></a> <ul><li><a href="/wiki/Hodges%E2%80%93Lehmann_estimator" title="Hodges–Lehmann estimator">Hodges–Lehmann estimator</a></li></ul></li> <li><a href="/wiki/Mann%E2%80%93Whitney_U_test" title="Mann–Whitney U test">Rank sum <span style="font-size:85%;">(Mann–Whitney)</span></a></li> <li><a href="/wiki/Nonparametric_statistics" title="Nonparametric statistics">Nonparametric</a> <a href="/wiki/Analysis_of_variance" title="Analysis of variance">anova</a> <ul><li><a href="/wiki/Kruskal%E2%80%93Wallis_test" title="Kruskal–Wallis test">1-way <span style="font-size:85%;">(Kruskal–Wallis)</span></a></li> <li><a href="/wiki/Friedman_test" title="Friedman test">2-way <span style="font-size:85%;">(Friedman)</span></a></li> <li><a href="/wiki/Jonckheere%27s_trend_test" title="Jonckheere's trend test">Ordered alternative <span style="font-size:85%;">(Jonckheere–Terpstra)</span></a></li></ul></li> <li><a href="/wiki/Van_der_Waerden_test" title="Van der Waerden test">Van der Waerden test</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Bayesian_inference" title="Bayesian inference">Bayesian inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bayesian_probability" title="Bayesian probability">Bayesian probability</a> <ul><li><a href="/wiki/Prior_probability" title="Prior probability">prior</a></li> <li><a href="/wiki/Posterior_probability" title="Posterior probability">posterior</a></li></ul></li> <li><a href="/wiki/Credible_interval" title="Credible interval">Credible interval</a></li> <li><a href="/wiki/Bayes_factor" title="Bayes factor">Bayes factor</a></li> <li><a href="/wiki/Bayes_estimator" title="Bayes estimator">Bayesian estimator</a> <ul><li><a href="/wiki/Maximum_a_posteriori_estimation" title="Maximum a posteriori estimation">Maximum posterior estimator</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="CorrelationRegression_analysis636" style="font-size:114%;margin:0 4em"><div class="hlist"><ul><li><a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></li><li><a href="/wiki/Regression_analysis" title="Regression analysis">Regression analysis</a></li></ul></div></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pearson_product-moment_correlation_coefficient" class="mw-redirect" title="Pearson product-moment correlation coefficient">Pearson product-moment</a></li> <li><a href="/wiki/Partial_correlation" title="Partial correlation">Partial correlation</a></li> <li><a href="/wiki/Confounding" title="Confounding">Confounding variable</a></li> <li><a href="/wiki/Coefficient_of_determination" title="Coefficient of determination">Coefficient of determination</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Regression_analysis" title="Regression analysis">Regression analysis</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Errors_and_residuals" title="Errors and residuals">Errors and residuals</a></li> <li><a href="/wiki/Regression_validation" title="Regression validation">Regression validation</a></li> <li><a href="/wiki/Mixed_model" title="Mixed model">Mixed effects models</a></li> <li><a href="/wiki/Simultaneous_equations_model" title="Simultaneous equations model">Simultaneous equations models</a></li> <li><a href="/wiki/Multivariate_adaptive_regression_splines" class="mw-redirect" title="Multivariate adaptive regression splines">Multivariate adaptive regression splines (MARS)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Linear_regression" title="Linear regression">Linear regression</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Simple_linear_regression" title="Simple linear regression">Simple linear regression</a></li> <li><a href="/wiki/Ordinary_least_squares" title="Ordinary least squares">Ordinary least squares</a></li> <li><a href="/wiki/General_linear_model" title="General linear model">General linear model</a></li> <li><a href="/wiki/Bayesian_linear_regression" title="Bayesian linear regression">Bayesian regression</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Non-standard predictors</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nonlinear_regression" title="Nonlinear regression">Nonlinear regression</a></li> <li><a href="/wiki/Nonparametric_regression" title="Nonparametric regression">Nonparametric</a></li> <li><a href="/wiki/Semiparametric_regression" title="Semiparametric regression">Semiparametric</a></li> <li><a href="/wiki/Isotonic_regression" title="Isotonic regression">Isotonic</a></li> <li><a href="/wiki/Robust_regression" title="Robust regression">Robust</a></li> <li><a href="/wiki/Homoscedasticity_and_heteroscedasticity" title="Homoscedasticity and heteroscedasticity">Homoscedasticity and Heteroscedasticity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Generalized_linear_model" title="Generalized linear model">Generalized linear model</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Exponential_family" title="Exponential family">Exponential families</a></li> <li><a href="/wiki/Logistic_regression" title="Logistic regression">Logistic <span style="font-size:85%;">(Bernoulli)</span></a> / <a href="/wiki/Binomial_regression" title="Binomial regression">Binomial</a> / <a href="/wiki/Poisson_regression" title="Poisson regression">Poisson regressions</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Partition_of_sums_of_squares" title="Partition of sums of squares">Partition of variance</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Analysis_of_variance" title="Analysis of variance">Analysis of variance (ANOVA, anova)</a></li> <li><a href="/wiki/Analysis_of_covariance" title="Analysis of covariance">Analysis of covariance</a></li> <li><a href="/wiki/Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Multivariate ANOVA</a></li> <li><a href="/wiki/Degrees_of_freedom_(statistics)" title="Degrees of freedom (statistics)">Degrees of freedom</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Categorical_/_Multivariate_/_Time-series_/_Survival_analysis636" style="font-size:114%;margin:0 4em"><a href="/wiki/Categorical_variable" title="Categorical variable">Categorical</a> / <a href="/wiki/Multivariate_statistics" title="Multivariate statistics">Multivariate</a> / <a href="/wiki/Time_series" title="Time series">Time-series</a> / <a href="/wiki/Survival_analysis" title="Survival analysis">Survival analysis</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Categorical_variable" title="Categorical variable">Categorical</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cohen%27s_kappa" title="Cohen's kappa">Cohen's kappa</a></li> <li><a href="/wiki/Contingency_table" title="Contingency table">Contingency table</a></li> <li><a href="/wiki/Graphical_model" title="Graphical model">Graphical model</a></li> <li><a href="/wiki/Poisson_regression" title="Poisson regression">Log-linear model</a></li> <li><a href="/wiki/McNemar%27s_test" title="McNemar's test">McNemar's test</a></li> <li><a href="/wiki/Cochran%E2%80%93Mantel%E2%80%93Haenszel_statistics" title="Cochran–Mantel–Haenszel statistics">Cochran–Mantel–Haenszel statistics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Multivariate_statistics" title="Multivariate statistics">Multivariate</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/General_linear_model" title="General linear model">Regression</a></li> <li><a href="/wiki/Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Manova</a></li> <li><a href="/wiki/Principal_component_analysis" title="Principal component analysis">Principal components</a></li> <li><a href="/wiki/Canonical_correlation" title="Canonical correlation">Canonical correlation</a></li> <li><a href="/wiki/Linear_discriminant_analysis" title="Linear discriminant analysis">Discriminant analysis</a></li> <li><a href="/wiki/Cluster_analysis" title="Cluster analysis">Cluster analysis</a></li> <li><a href="/wiki/Statistical_classification" title="Statistical classification">Classification</a></li> <li><a href="/wiki/Structural_equation_modeling" title="Structural equation modeling">Structural equation model</a> <ul><li><a href="/wiki/Factor_analysis" title="Factor analysis">Factor analysis</a></li></ul></li> <li><a href="/wiki/Multivariate_distribution" class="mw-redirect" title="Multivariate distribution">Multivariate distributions</a> <ul><li><a href="/wiki/Elliptical_distribution" title="Elliptical distribution">Elliptical distributions</a> <ul><li><a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">Normal</a></li></ul></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Time_series" title="Time series">Time-series</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Decomposition_of_time_series" title="Decomposition of time series">Decomposition</a></li> <li><a href="/wiki/Trend_estimation" class="mw-redirect" title="Trend estimation">Trend</a></li> <li><a href="/wiki/Stationary_process" title="Stationary process">Stationarity</a></li> <li><a href="/wiki/Seasonal_adjustment" title="Seasonal adjustment">Seasonal adjustment</a></li> <li><a href="/wiki/Exponential_smoothing" title="Exponential smoothing">Exponential smoothing</a></li> <li><a href="/wiki/Cointegration" title="Cointegration">Cointegration</a></li> <li><a href="/wiki/Structural_break" title="Structural break">Structural break</a></li> <li><a href="/wiki/Granger_causality" title="Granger causality">Granger causality</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Specific tests</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dickey%E2%80%93Fuller_test" title="Dickey–Fuller test">Dickey–Fuller</a></li> <li><a href="/wiki/Johansen_test" title="Johansen test">Johansen</a></li> <li><a href="/wiki/Ljung%E2%80%93Box_test" title="Ljung–Box test">Q-statistic <span style="font-size:85%;">(Ljung–Box)</span></a></li> <li><a href="/wiki/Durbin%E2%80%93Watson_statistic" title="Durbin–Watson statistic">Durbin–Watson</a></li> <li><a href="/wiki/Breusch%E2%80%93Godfrey_test" title="Breusch–Godfrey test">Breusch–Godfrey</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Time_domain" title="Time domain">Time domain</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Autocorrelation" title="Autocorrelation">Autocorrelation (ACF)</a> <ul><li><a href="/wiki/Partial_autocorrelation_function" title="Partial autocorrelation function">partial (PACF)</a></li></ul></li> <li><a href="/wiki/Cross-correlation" title="Cross-correlation">Cross-correlation (XCF)</a></li> <li><a href="/wiki/Autoregressive%E2%80%93moving-average_model" class="mw-redirect" title="Autoregressive–moving-average model">ARMA model</a></li> <li><a href="/wiki/Box%E2%80%93Jenkins_method" title="Box–Jenkins method">ARIMA model <span style="font-size:85%;">(Box–Jenkins)</span></a></li> <li><a href="/wiki/Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Autoregressive conditional heteroskedasticity (ARCH)</a></li> <li><a href="/wiki/Vector_autoregression" title="Vector autoregression">Vector autoregression (VAR)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Frequency_domain" title="Frequency domain">Frequency domain</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Spectral_density_estimation" title="Spectral density estimation">Spectral density estimation</a></li> <li><a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li> <li><a href="/wiki/Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a></li> <li><a href="/wiki/Wavelet" title="Wavelet">Wavelet</a></li> <li><a href="/wiki/Whittle_likelihood" title="Whittle likelihood">Whittle likelihood</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Survival_analysis" title="Survival analysis">Survival</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Survival_function" title="Survival function">Survival function</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Kaplan%E2%80%93Meier_estimator" title="Kaplan–Meier estimator">Kaplan–Meier estimator (product limit)</a></li> <li><a href="/wiki/Proportional_hazards_model" title="Proportional hazards model">Proportional hazards models</a></li> <li><a href="/wiki/Accelerated_failure_time_model" title="Accelerated failure time model">Accelerated failure time (AFT) model</a></li> <li><a href="/wiki/First-hitting-time_model" title="First-hitting-time model">First hitting time</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Failure_rate" title="Failure rate">Hazard function</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nelson%E2%80%93Aalen_estimator" title="Nelson–Aalen estimator">Nelson–Aalen estimator</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Test</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Log-rank_test" class="mw-redirect" title="Log-rank test">Log-rank test</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Applications636" style="font-size:114%;margin:0 4em"><a href="/wiki/List_of_fields_of_application_of_statistics" title="List of fields of application of statistics">Applications</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Biostatistics" title="Biostatistics">Biostatistics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bioinformatics" title="Bioinformatics">Bioinformatics</a></li> <li><a href="/wiki/Clinical_trial" title="Clinical trial">Clinical trials</a> / <a href="/wiki/Clinical_study_design" title="Clinical study design">studies</a></li> <li><a href="/wiki/Epidemiology" title="Epidemiology">Epidemiology</a></li> <li><a href="/wiki/Medical_statistics" title="Medical statistics">Medical statistics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Engineering_statistics" title="Engineering statistics">Engineering statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chemometrics" title="Chemometrics">Chemometrics</a></li> <li><a href="/wiki/Methods_engineering" title="Methods engineering">Methods engineering</a></li> <li><a href="/wiki/Probabilistic_design" title="Probabilistic design">Probabilistic design</a></li> <li><a href="/wiki/Statistical_process_control" title="Statistical process control">Process</a> / <a href="/wiki/Quality_control" title="Quality control">quality control</a></li> <li><a href="/wiki/Reliability_engineering" title="Reliability engineering">Reliability</a></li> <li><a href="/wiki/System_identification" title="System identification">System identification</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Social_statistics" title="Social statistics">Social statistics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Actuarial_science" title="Actuarial science">Actuarial science</a></li> <li><a href="/wiki/Census" title="Census">Census</a></li> <li><a href="/wiki/Crime_statistics" title="Crime statistics">Crime statistics</a></li> <li><a href="/wiki/Demographic_statistics" title="Demographic statistics">Demography</a></li> <li><a href="/wiki/Econometrics" title="Econometrics">Econometrics</a></li> <li><a href="/wiki/Jurimetrics" title="Jurimetrics">Jurimetrics</a></li> <li><a href="/wiki/National_accounts" title="National accounts">National accounts</a></li> <li><a href="/wiki/Official_statistics" title="Official statistics">Official statistics</a></li> <li><a href="/wiki/Population_statistics" class="mw-redirect" title="Population statistics">Population statistics</a></li> <li><a href="/wiki/Psychometrics" title="Psychometrics">Psychometrics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Spatial_analysis" title="Spatial analysis">Spatial 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style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Experimental_design" title="Template:Experimental design"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Experimental_design" title="Template talk:Experimental design"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Experimental_design" title="Special:EditPage/Template:Experimental design"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Design_of_experiments690" style="font-size:114%;margin:0 4em"><a href="/wiki/Design_of_experiments" title="Design of experiments">Design of experiments</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Scientific_method" title="Scientific method">Scientific<br />method</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Experiment" title="Experiment">Scientific experiment</a></li> <li><a href="/wiki/Design_of_experiments" title="Design of experiments">Statistical design</a></li> <li><a href="/wiki/Scientific_control" title="Scientific control">Control</a></li> <li><a href="/wiki/Internal_validity" title="Internal validity">Internal</a> and <a href="/wiki/External_validity" title="External validity">external</a> <a href="/wiki/Validity_(statistics)" title="Validity (statistics)">validity</a></li> <li><a href="/wiki/Experimental_unit" class="mw-redirect" title="Experimental unit">Experimental unit</a></li> <li><a href="/wiki/Blind_experiment" class="mw-redirect" title="Blind experiment">Blinding</a></li></ul> <ul><li><b><a href="/wiki/Optimal_design" class="mw-redirect" title="Optimal design">Optimal design</a></b>: <a href="/wiki/Bayesian_experimental_design" title="Bayesian experimental design">Bayesian</a></li> <li><a href="/wiki/Random_assignment" title="Random assignment">Random assignment</a></li> <li><a href="/wiki/Randomization" title="Randomization">Randomization</a></li> <li><a href="/wiki/Restricted_randomization" title="Restricted randomization">Restricted randomization</a></li> <li><a href="/wiki/Replication_(statistics)" title="Replication (statistics)">Replication versus subsampling</a></li> <li><a href="/wiki/Sample_size" class="mw-redirect" title="Sample size">Sample size</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Glossary_of_experimental_design" title="Glossary of experimental design">Treatment</a><br /> and <a href="/wiki/Blocking_(statistics)" title="Blocking (statistics)">blocking</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><b><a href="/wiki/Glossary_of_experimental_design" title="Glossary of experimental design">Treatment</a></b></li> <li><a href="/wiki/Effect_size" title="Effect size">Effect size</a></li> <li><a href="/wiki/Contrast_(statistics)" title="Contrast (statistics)">Contrast</a></li> <li><a class="mw-selflink selflink">Interaction</a></li> <li><a href="/wiki/Confounding" title="Confounding">Confounding</a></li> <li><a href="/wiki/Orthogonality#Statistics,_econometrics,_and_economics" title="Orthogonality">Orthogonality</a></li> <li><b><a href="/wiki/Blocking_(statistics)" title="Blocking (statistics)">Blocking</a></b></li> <li><a href="/wiki/Covariate" class="mw-redirect" title="Covariate">Covariate</a></li> <li><a href="/wiki/Nuisance_variable" title="Nuisance variable">Nuisance variable</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Statistical_model" title="Statistical model">Models</a> <br /> and <a href="/wiki/Statistical_inference" title="Statistical inference">inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Linear_regression" title="Linear regression">Linear regression</a></li> <li><a href="/wiki/Ordinary_least_squares" title="Ordinary least squares">Ordinary least squares</a></li> <li><a href="/wiki/Bayesian_linear_regression" title="Bayesian linear regression">Bayesian</a></li></ul> <ul><li><b><a href="/wiki/Random_effect" class="mw-redirect" title="Random effect">Random effect</a></b></li> <li><a href="/wiki/Mixed_model" title="Mixed model">Mixed model</a></li> <li><a href="/wiki/Hierarchical_linear_modeling" class="mw-redirect" title="Hierarchical linear modeling">Hierarchical model:</a> <a href="/wiki/Hierarchical_Bayes_model" class="mw-redirect" title="Hierarchical Bayes model">Bayesian</a></li></ul> <ul><li><b><a href="/wiki/Analysis_of_variance" title="Analysis of variance">Analysis of variance (Anova)</a></b></li> <li><a href="/wiki/Cochran%27s_theorem" title="Cochran's theorem">Cochran's theorem</a></li> <li><a href="/wiki/Multivariate_analysis_of_variance" title="Multivariate analysis of variance"><b>Manova</b> (<i>multivariate</i>)</a></li> <li><a href="/wiki/Analysis_of_covariance" title="Analysis of covariance"><b>Ancova</b> (<i>covariance</i>)</a></li></ul> <ul><li><a href="/wiki/Comparing_means" class="mw-redirect" title="Comparing means">Compare means</a></li> <li><a href="/wiki/Multiple_comparison" class="mw-redirect" title="Multiple comparison">Multiple comparison</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Design_of_experiments" title="Design of experiments">Designs</a> <br /> <br /><a href="/wiki/Completely_randomized_design" title="Completely randomized design">Completely<br />randomized</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><b><a href="/wiki/Factorial_experiment" title="Factorial experiment">Factorial</a></b></li> <li><a href="/wiki/Fractional_factorial_design" title="Fractional factorial design">Fractional factorial</a></li> <li><a href="/wiki/Plackett-Burman_design" class="mw-redirect" title="Plackett-Burman design">Plackett–Burman</a></li> <li><a href="/wiki/Taguchi_methods" title="Taguchi methods">Taguchi</a></li></ul> <ul><li><b><a href="/wiki/Response_surface_methodology" title="Response surface methodology">Response surface methodology</a></b></li> <li><a href="/wiki/Polynomial_and_rational_function_modeling" title="Polynomial and rational function modeling">Polynomial and rational modeling</a></li> <li><a href="/wiki/Box%E2%80%93Behnken_design" title="Box–Behnken design">Box–Behnken</a></li> <li><a href="/wiki/Central_composite_design" title="Central composite design">Central composite</a></li></ul> <ul><li><b><a href="/wiki/Randomized_block_design" class="mw-redirect" title="Randomized block design">Block</a></b></li> <li><a href="/wiki/Generalized_randomized_block_design" title="Generalized randomized block design">Generalized randomized block design</a> (GRBD)</li> <li><a href="/wiki/Latin_square" title="Latin square">Latin square</a></li> <li><a href="/wiki/Graeco-Latin_square" class="mw-redirect" title="Graeco-Latin square">Graeco-Latin square</a></li> <li><a href="/wiki/Orthogonal_array" title="Orthogonal array">Orthogonal array</a></li> <li><a href="/wiki/Latin_hypercube_sampling" title="Latin hypercube sampling">Latin hypercube</a> <br /> <b><a href="/wiki/Repeated_measures_design" title="Repeated measures design">Repeated measures design</a></b></li> <li><a href="/wiki/Crossover_study" title="Crossover study">Crossover study</a></li></ul> <ul><li><b><a 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