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<title>Confounding</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Confounding</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">"Confounding factor" redirects here. For the company, see <a href="Confounding_Factor" class="mw-redirect" title="Confounding Factor">Confounding Factor</a>. For the psychological state, see <a href="Confusion" title="Confusion">Confusion</a>.</div>

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<p>In <a href="Causal_inference" title="Causal inference">causal inference</a>, a <b>confounder</b><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> is a variable that influences both the <a href="Dependent_and_independent_variables" title="Dependent and independent variables">dependent variable and independent variable</a>, causing a <a href="Spurious_relationship" title="Spurious relationship">spurious association</a>. Confounding is a <a href="Causality" title="Causality">causal</a> concept, and as such, cannot be described in terms of correlations or associations.<sup id="cite_ref-Pearl_2009_2-0" class="reference"><a href="#cite_note-Pearl_2009-2"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Greenland_Pearl_Robbins_1999_4-0" class="reference"><a href="#cite_note-Greenland_Pearl_Robbins_1999-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> The existence of confounders is an important quantitative explanation why <a href="Correlation_does_not_imply_causation" title="Correlation does not imply causation">correlation does not imply causation</a>. Some <a href="Causal_notation" title="Causal notation">notations</a> are explicitly designed to identify the existence, possible existence, or non-existence of confounders in causal relationships between elements of a system.

</p><p>Confounders are threats to <a href="Internal_validity" title="Internal validity">internal validity</a>.<sup id="cite_ref-Shadish2002_5-0" class="reference"><a href="#cite_note-Shadish2002-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Example">Example</h2></div>

<p>Let's assume that a trucking company owns a fleet of trucks made by two different manufacturers. Trucks made by one manufacturer are called "A Trucks" and trucks made by the other manufacturer are called "B Trucks." We want to find out whether A Trucks or B Trucks get better fuel economy. We measure fuel and miles driven for a month and calculate the MPG for each truck. We then run the appropriate analysis, which determines that there is a statistically significant trend that A Trucks are more fuel efficient than B Trucks. Upon further reflection, however, we also notice that A Trucks are more likely to be assigned highway routes, and B Trucks are more likely to be assigned city routes. This is a confounding variable. The confounding variable makes the results of the analysis unreliable. It is quite likely that we are just measuring the fact that highway driving results in better fuel economy than city driving.

</p><p>In statistics terms, the make of the truck is the independent variable, the fuel economy (MPG) is the dependent variable and the amount of city driving is the confounding variable. To fix this study, we have several choices. One is to randomize the truck assignments so that A trucks and B Trucks end up with equal amounts of city and highway driving. That eliminates the confounding variable. Another choice is to quantify the amount of city driving and use that as a second independent variable. A third choice is to segment the study, first comparing MPG during city driving for all trucks, and then run a separate study comparing MPG during highway driving.

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<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<p>Confounding is defined in terms of the data generating model. Let <i>X</i> be some <a href="Independent_variable" class="mw-redirect" title="Independent variable">independent variable</a>, and <i>Y</i> some <a href="Dependent_variable" class="mw-redirect" title="Dependent variable">dependent variable</a>. To estimate the effect of <i>X</i> on <i>Y</i>, the statistician must suppress the effects of <a href="Extraneous_variable" class="mw-redirect" title="Extraneous variable">extraneous variables</a> that influence both <i>X</i> and <i>Y</i>. We say that <i>X</i> and <i>Y</i> are confounded by some other variable <i>Z</i> whenever <i>Z</i> causally influences both <i>X</i> and <i>Y</i>.

</p><p>Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(y\mid {\text{do}}(x))}">

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<annotation encoding="application/x-tex">{\displaystyle P(y\mid {\text{do}}(x))}</annotation>

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</math></span><img src="./b5258d65f19ff77beaee45a4c4689da7679d1428.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.241ex; height:2.843ex;" alt="{\displaystyle P(y\mid {\text{do}}(x))}" loading="lazy"></span> be the probability of event <i>Y</i> = <i>y</i> under the hypothetical intervention <i>X</i> = <i>x</i>. <i>X</i> and <i>Y</i> are not confounded if and only if the following holds:

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</style><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(y\mid {\text{do}}(x))=P(y\mid x)}">

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<annotation encoding="application/x-tex">{\displaystyle P(y\mid {\text{do}}(x))=P(y\mid x)}</annotation>

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<p>for all values <i>X</i> = <i>x</i> and <i>Y</i> = <i>y</i>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(y\mid x)}">

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<annotation encoding="application/x-tex">{\displaystyle P(y\mid x)}</annotation>

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</math></span><img src="./2b116c9f7f5c5082d4901cdc4859d7bd46fde280.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.977ex; height:2.843ex;" alt="{\displaystyle P(y\mid x)}" loading="lazy"></span> is the <a href="Conditional_probability" title="Conditional probability">conditional probability</a> upon seeing <i>X</i> = <i>x</i>. Intuitively, this equality states that <i>X</i> and <i>Y</i> are not confounded whenever the observationally witnessed association between them is the same as the association that would be measured in a <a href="Controlled_experiment" class="mw-redirect" title="Controlled experiment">controlled experiment</a>, with <i>x</i> <a href="Randomize" class="mw-redirect" title="Randomize">randomized</a>.

</p><p>In principle, the defining equality <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(y\mid {\text{do}}(x))=P(y\mid x)}">

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(y\mid {\text{do}}(x))=P(y\mid x)}</annotation>

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</math></span><img src="./e3db7a118c5aadd999f166bc0f44713ec9868408.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.317ex; height:2.843ex;" alt="{\displaystyle P(y\mid {\text{do}}(x))=P(y\mid x)}" loading="lazy"></span> can be verified from the data generating model, assuming we have all the equations and probabilities associated with the model. This is done by simulating an intervention <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{do}}(X=x)}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{do}}(X=x)}</annotation>

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</math></span><img src="./9c8861dde2eb157d613062daee98a69e10170f72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.672ex; height:2.843ex;" alt="{\displaystyle {\text{do}}(X=x)}" loading="lazy"></span> (see <a href="Bayesian_network" title="Bayesian network">Bayesian network</a>) and checking whether the resulting probability of <i>Y</i> equals the conditional probability <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(y\mid x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(y\mid x)}</annotation>

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</math></span><img src="./2b116c9f7f5c5082d4901cdc4859d7bd46fde280.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.977ex; height:2.843ex;" alt="{\displaystyle P(y\mid x)}" loading="lazy"></span>. It turns out, however, that graph structure alone is sufficient for verifying the equality <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(y\mid {\text{do}}(x))=P(y\mid x)}">

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<annotation encoding="application/x-tex">{\displaystyle P(y\mid {\text{do}}(x))=P(y\mid x)}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Control">Control</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Controlling_for_a_variable" title="Controlling for a variable">Controlling for a variable</a></div>

<p>Consider a researcher attempting to assess the effectiveness of drug <i>X</i>, from population data in which drug usage was a patient's choice. The data shows that gender (<i>Z</i>) influences a patient's choice of drug as well as their chances of recovery (<i>Y</i>). In this scenario, gender <i>Z</i> confounds the relation between <i>X</i> and Y since <i>Z</i> is a cause of both <i>X</i> and <i>Y</i>:

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<p>We have that

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<annotation encoding="application/x-tex">{\displaystyle P(y\mid {\text{do}}(x))\neq P(y\mid x)}</annotation>

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</math></span><img src="./444c4f8aedecade7891f10b5464a21274534ea90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.317ex; height:2.843ex;" alt="{\displaystyle P(y\mid {\text{do}}(x))\neq P(y\mid x)}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_2" class="reference nourlexpansion" style="font-weight:bold;">2</span></td></tr></tbody></table>

<p>because the observational quantity contains information about the correlation between <i>X</i> and <i>Z</i>, and the interventional quantity does not (since <i>X</i> is not correlated with <i>Z</i> in a randomized experiment). It can be shown<sup id="cite_ref-Pearl_1993_6-0" class="reference"><a href="#cite_note-Pearl_1993-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> that, in cases where only observational data is available, an unbiased estimate of the desired quantity <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(y\mid {\text{do}}(x))}">

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<annotation encoding="application/x-tex">{\displaystyle P(y\mid {\text{do}}(x))}</annotation>

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be obtained by "adjusting" for all confounding factors, namely, conditioning on their various values and averaging the result. In the case of a single confounder <i>Z</i>, this leads to the "adjustment formula":

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<annotation encoding="application/x-tex">{\displaystyle P(y\mid {\text{do}}(x))=\sum _{z}P(y\mid x,z)P(z)}</annotation>

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</math></span><img src="./924e54a5557adcda16381b310c40737d51877bc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.824ex; height:5.509ex;" alt="{\displaystyle P(y\mid {\text{do}}(x))=\sum _{z}P(y\mid x,z)P(z)}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_3" class="reference nourlexpansion" style="font-weight:bold;">3</span></td></tr></tbody></table>

<p>which gives an unbiased estimate for the causal effect of <i>X</i> on <i>Y</i>. The same adjustment formula works when there are multiple confounders except, in this case, the choice of a set <i>Z</i> of variables that would guarantee unbiased estimates must be done with caution. The criterion for a proper choice of variables is called the Back-Door<sup id="cite_ref-Pearl_1993_6-1" class="reference"><a href="#cite_note-Pearl_1993-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-pearl09-causal-diagrams_7-0" class="reference"><a href="#cite_note-pearl09-causal-diagrams-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> and requires that the chosen set <i>Z</i> "blocks" (or intercepts) every path between <i>X</i> and <i>Y</i> that contains an arrow into X. Such sets are called "Back-Door admissible" and may include variables which are not common causes of <i>X</i> and <i>Y</i>, but merely proxies thereof.

</p><p>Returning to the drug use example, since <i>Z</i> complies with the Back-Door requirement (i.e., it intercepts the one Back-Door path <span class="mwe-math-element mwe-math-element-inline"><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\leftarrow Z\rightarrow Y}">

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<annotation encoding="application/x-tex">{\displaystyle X\leftarrow Z\rightarrow Y}</annotation>

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</math></span><img src="./fe5d8215d19000272946102bc3e7c5d313aa7124.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.662ex; height:2.176ex;" alt="{\displaystyle X\leftarrow Z\rightarrow Y}" loading="lazy"></span>), the Back-Door adjustment formula is valid:

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<mtext>recovered</mtext>

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<mtext>recovered</mtext>

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<mtr>

<mtd>

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<mtext>recovered</mtext>

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<mtext>female</mtext>

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<mtext>female</mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}P(Y={\text{recovered}}\mid {\text{do}}(x={\text{give drug}}))={}&P(Y={\text{recovered}}\mid X={\text{give drug}},Z={\text{male}})P(Z={\text{male}})\\&{}+P(Y={\text{recovered}}\mid X={\text{give drug}},Z={\text{female}})P(Z={\text{female}})\end{aligned}}}</annotation>

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</math></span><img src="./fc3f2b95caf225ab70f572420525bd734ea1b80b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:106.267ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}P(Y={\text{recovered}}\mid {\text{do}}(x={\text{give drug}}))={}&P(Y={\text{recovered}}\mid X={\text{give drug}},Z={\text{male}})P(Z={\text{male}})\\&{}+P(Y={\text{recovered}}\mid X={\text{give drug}},Z={\text{female}})P(Z={\text{female}})\end{aligned}}}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_4" class="reference nourlexpansion" style="font-weight:bold;">4</span></td></tr></tbody></table>

<p>In this way the physician can predict the likely effect of administering the drug from observational studies in which the conditional probabilities appearing on the right-hand side of the equation can be estimated by regression.

</p><p>Contrary to common beliefs, adding covariates to the adjustment set <i>Z</i> can introduce bias.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> A typical counterexample occurs when <i>Z</i> is a common effect of <i>X</i> and <i>Y</i>,<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> a case in which <i>Z</i> is not a confounder (i.e., the null set is Back-door admissible) and adjusting for <i>Z</i> would create bias known as "<a href="Collider_(epidemiology)" class="mw-redirect" title="Collider (epidemiology)">collider</a> bias" or "<a href="Berkson's_paradox" title="Berkson's paradox">Berkson's paradox</a>." Controls that are not good confounders are sometimes called <a href="Bad_control" title="Bad control">bad controls</a>.

</p><p>In general, confounding can be controlled by adjustment if and only if there is a set of observed covariates that satisfies the Back-Door condition. Moreover, if <i>Z</i> is such a set, then the adjustment formula of Eq. (3) is valid.<sup id="cite_ref-Pearl_1993_6-2" class="reference"><a href="#cite_note-Pearl_1993-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-pearl09-causal-diagrams_7-1" class="reference"><a href="#cite_note-pearl09-causal-diagrams-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Pearl's do-calculus provides all possible conditions under which <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(y\mid {\text{do}}(x))}">

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<annotation encoding="application/x-tex">{\displaystyle P(y\mid {\text{do}}(x))}</annotation>

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</math></span><img src="./b5258d65f19ff77beaee45a4c4689da7679d1428.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.241ex; height:2.843ex;" alt="{\displaystyle P(y\mid {\text{do}}(x))}" loading="lazy"></span> can be estimated, not necessarily by adjustment.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

<p>According to Morabia (2011),<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> the word <i>confounding</i> derives from the <a href="Medieval_Latin" title="Medieval Latin">Medieval Latin</a> verb "confundere", which meant "mixing", and was probably chosen to represent the confusion (from Latin: con=with + fusus=mix or fuse together) between the cause one wishes to assess and other causes that may affect the outcome and thus confuse, or stand in the way of the desired assessment. Greenland, Robins and Pearl<sup id="cite_ref-Greenland_Robins_Pearl_1999_12-0" class="reference"><a href="#cite_note-Greenland_Robins_Pearl_1999-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> note an early use of the term "confounding" in causal inference by John Stuart Mill in 1843.

</p><p><a href="Ronald_Fisher" title="Ronald Fisher">Fisher</a> introduced the word "confounding" in his 1935 book "The Design of Experiments"<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> to refer specifically to a consequence of <a href="Blocking_(statistics)" title="Blocking (statistics)">blocking</a> (i.e., <a href="Partition_of_a_set" title="Partition of a set">partitioning</a>) the set of treatment combinations in a <a href="Factorial_experiment" title="Factorial experiment">factorial experiment</a>, whereby certain interactions may be "confounded with blocks". This popularized the notion of confounding in statistics, although Fisher was concerned with the control of heterogeneity in experimental units, not with causal inference.

</p><p>According to Vandenbroucke (2004)<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> it was <a href="Leslie_Kish" title="Leslie Kish">Kish</a><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> who used the word "confounding" in the sense of "incomparability" of two or more groups (e.g., exposed and unexposed) in an observational study. Formal conditions defining what makes certain groups "comparable" and others "incomparable" were later developed in <a href="Epidemiology" title="Epidemiology">epidemiology</a> by Greenland and Robins (1986)<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> using the counterfactual language of <a href="Jerzy_Neyman" title="Jerzy Neyman">Neyman</a> (1935)<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> and <a href="Donald_Rubin" title="Donald Rubin">Rubin</a> (1974).<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> These were later supplemented by graphical criteria such as the Back-Door condition (<a href="Judea_Pearl" title="Judea Pearl">Pearl</a> 1993; Greenland, Robins and Pearl 1999).<sup id="cite_ref-Greenland_Robins_Pearl_1999_12-1" class="reference"><a href="#cite_note-Greenland_Robins_Pearl_1999-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Pearl_1993_6-3" class="reference"><a href="#cite_note-Pearl_1993-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

</p><p>Graphical criteria were shown to be formally equivalent to the counterfactual definition<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> but more transparent to researchers relying on process models.

</p>

<div class="mw-heading mw-heading2"><h2 id="Types">Types</h2></div>

<p>In the case of <a href="Risk_assessment" title="Risk assessment">risk assessments</a> evaluating the magnitude and nature of risk to <a href="Human" title="Human">human</a> <a href="Health" title="Health">health</a>, it is important to control for confounding to isolate the effect of a particular hazard such as a food additive, <a href="Pesticide" title="Pesticide">pesticide</a>, or new drug. For prospective studies, it is difficult to recruit and screen for volunteers with the same background (age, diet, education, geography, etc.), and in historical studies, there can be similar variability. Due to the inability to control for variability of volunteers and human studies, confounding is a particular challenge. For these reasons, <a href="Experiment" title="Experiment">experiments</a> offer a way to avoid most forms of confounding.

</p><p>In some disciplines, confounding is categorized into different types. In <a href="Epidemiology" title="Epidemiology">epidemiology</a>, one type is "confounding by indication",<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> which relates to confounding from <a href="Observational_studies" class="mw-redirect" title="Observational studies">observational studies</a>. Because prognostic factors may influence treatment decisions (and bias estimates of treatment effects), controlling for known prognostic factors may reduce this problem, but it is always possible that a forgotten or unknown factor was not included or that factors interact complexly. Confounding by indication has been described as the most important limitation of observational studies. Randomized trials are not affected by confounding by indication due to <a href="Random_assignment" title="Random assignment">random assignment</a>.

</p><p>Confounding variables may also be categorised according to their source. The choice of measurement instrument (operational confound), situational characteristics (procedural confound), or inter-individual differences (person confound).

</p>

<ul><li>An <b>operational confounding</b> can occur in both <a href="Experiment" title="Experiment">experimental</a> and non-experimental research designs. This type of confounding occurs when a measure designed to assess a particular construct inadvertently measures something else as well.<sup id="cite_ref-Pelham_21-0" class="reference"><a href="#cite_note-Pelham-21"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup></li>

<li>A <b>procedural confounding</b> can occur in a laboratory experiment or a <a href="Quasi-experiment" title="Quasi-experiment">quasi-experiment</a>. This type of confound occurs when the researcher mistakenly allows another variable to change along with the manipulated independent variable.<sup id="cite_ref-Pelham_21-1" class="reference"><a href="#cite_note-Pelham-21"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup></li>

<li>A <b>person confounding</b> occurs when two or more groups of units are analyzed together (e.g., workers from different occupations), despite varying according to one or more other (observed or unobserved) characteristics (e.g., gender).<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<p>Say one is studying the relation between birth order (1st child, 2nd child, etc.) and the presence of <a href="Down_Syndrome" class="mw-redirect" title="Down Syndrome">Down Syndrome</a> in the child. In this scenario, maternal age would be a confounding variable:

</p>

<ol><li>Higher maternal age is directly associated with Down Syndrome in the child</li>

<li>Higher maternal age is directly associated with Down Syndrome, regardless of birth order (a mother having her 1st vs 3rd child at age 50 confers the same risk)</li>

<li>Maternal age is directly associated with birth order (the 2nd child, except in the case of twins, is born when the mother is older than she was for the birth of the 1st child)</li>

<li>Maternal age is not a consequence of birth order (having a 2nd child does not change the mother's age)</li></ol>

<p>In <a href="Risk_assessment" title="Risk assessment">risk assessments</a>, factors such as age, gender, and educational levels often affect health status and so should be controlled. Beyond these factors, researchers may not consider or have access to data on other causal factors. An example is on the study of smoking tobacco on human health. Smoking, drinking alcohol, and diet are lifestyle activities that are related. A risk assessment that looks at the effects of smoking but does not control for alcohol consumption or diet may overestimate the risk of smoking.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> Smoking and confounding are reviewed in occupational risk assessments such as the safety of coal mining.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> When there is not a large sample population of non-smokers or non-drinkers in a particular occupation, the risk assessment may be biased towards finding a negative effect on health.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Decreasing_the_potential_for_confounding">Decreasing the potential for confounding</h2></div>

<p>A reduction in the potential for the occurrence and effect of confounding factors can be obtained by increasing the types and numbers of comparisons performed in an analysis. If measures or manipulations of core constructs are confounded (i.e. operational or procedural confounds exist), subgroup analysis may not reveal problems in the analysis. Additionally, increasing the number of comparisons can create other problems (see <a href="Multiple_comparisons" class="mw-redirect" title="Multiple comparisons">multiple comparisons</a>).

</p><p><a href="Peer_review" title="Peer review">Peer review</a> is a process that can assist in reducing instances of confounding, either before study implementation or after analysis has occurred. Peer review relies on collective expertise within a discipline to identify potential weaknesses in study design and analysis, including ways in which results may depend on confounding. Similarly, <a href="Replication_(statistics)" title="Replication (statistics)">replication</a> can test for the robustness of findings from one study under alternative study conditions or alternative analyses (e.g., controlling for potential confounds not identified in the initial study).

</p><p>Confounding effects may be less likely to occur and act similarly at multiple times and locations. In selecting study sites, the environment can be characterized in detail at the study sites to ensure sites are ecologically similar and therefore less likely to have confounding variables. Lastly, the relationship between the environmental variables that possibly confound the analysis and the measured parameters can be studied. The information pertaining to environmental variables can then be used in site-specific models to identify residual variance that may be due to real effects.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup>

</p><p>Depending on the type of study design in place, there are various ways to modify that design to actively exclude or control confounding variables:<sup id="cite_ref-Hennekens1987_27-0" class="reference"><a href="#cite_note-Hennekens1987-27"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup>

</p>

<ul><li><a href="Case-control_study" class="mw-redirect" title="Case-control study">Case-control studies</a> assign confounders to both groups, cases and controls, equally. For example, if somebody wanted to study the cause of myocardial infarct and thinks that the age is a probable confounding variable, each 67-year-old infarct patient will be matched with a healthy 67-year-old "control" person. In case-control studies, matched variables most often are the age and sex. Drawback: Case-control studies are feasible only when it is easy to find controls, <i>i.e.</i> persons whose status vis-à-vis all known potential confounding factors is the same as that of the case's patient: Suppose a case-control study attempts to find the cause of a given disease in a person who is 1) 45 years old, 2) African-American, 3) from <a href="Alaska#Demographics" title="Alaska">Alaska</a>, 4) an avid football player, 5) vegetarian, and 6) working in education. A theoretically perfect control would be a person who, in addition to not having the disease being investigated, matches all these characteristics and has no diseases that the patient does not also have—but finding such a control would be an enormous task.</li>

<li><a href="Cohort_study" title="Cohort study">Cohort studies</a>: A degree of matching is also possible and it is often done by only admitting certain age groups or a certain sex into the study population, creating a cohort of people who share similar characteristics and thus all cohorts are comparable in regard to the possible confounding variable. For example, if age and sex are thought to be confounders, only 40 to 50 years old males would be involved in a cohort study that would assess the myocardial infarct risk in cohorts that either are physically active or inactive. Drawback: In cohort studies, the overexclusion of input data may lead researchers to define too narrowly the set of similarly situated persons for whom they claim the study to be useful, such that other persons to whom the causal relationship does in fact apply may lose the opportunity to benefit from the study's recommendations. Similarly, "over-stratification" of input data within a study may reduce the sample size in a given stratum to the point where generalizations drawn by observing the members of that stratum alone are not <a href="Statistical_significance" title="Statistical significance">statistically significant</a>.</li>

<li><a href="Double_blinding" class="mw-redirect" title="Double blinding">Double blinding</a>: conceals from the trial population and the observers the experiment group membership of the participants. By preventing the participants from knowing if they are receiving treatment or not, the <a href="Placebo_effect" class="mw-redirect" title="Placebo effect">placebo effect</a> should be the same for the control and treatment groups. By preventing the observers from knowing of their membership, there should be no bias from researchers treating the groups differently or from interpreting the outcomes differently.</li>

<li><a href="Randomized_controlled_trial" title="Randomized controlled trial">Randomized controlled trial</a>: A method where the study population is divided randomly in order to mitigate the chances of self-selection by participants or bias by the study designers. Before the experiment begins, the testers will assign the members of the participant pool to their groups (control, intervention, parallel), using a randomization process such as the use of a random number generator. For example, in a study on the effects of exercise, the conclusions would be less valid if participants were given a choice if they wanted to belong to the control group which would not exercise or the intervention group which would be willing to take part in an exercise program. The study would then capture other variables besides exercise, such as pre-experiment health levels and motivation to adopt healthy activities. From the observer's side, the experimenter may choose candidates who are more likely to show the results the study wants to see or may interpret subjective results (more energetic, positive attitude) in a way favorable to their desires.</li>

<li><a href="Stratification_(statistics)" class="mw-redirect" title="Stratification (statistics)">Stratification</a>: As in the example above, physical activity is thought to be a behaviour that protects from myocardial infarct; and age is assumed to be a possible confounder. The data sampled is then stratified by age group – this means that the association between activity and infarct would be analyzed per each age group. If the different age groups (or age strata) yield much different <a href="Risk_ratio" class="mw-redirect" title="Risk ratio">risk ratios</a>, age must be viewed as a confounding variable. There exist statistical tools, among them Mantel–Haenszel methods, that account for stratification of data sets.</li>

<li>Controlling for confounding by measuring the known confounders and including them as <a href="Covariate" class="mw-redirect" title="Covariate">covariates</a> is <a href="Multivariate_statistics" title="Multivariate statistics">multivariable analysis</a> such as <a href="Regression_analysis" title="Regression analysis">regression analysis</a>. Multivariate analyses reveal much less information about the <i>strength</i> or <i>polarity</i> of the confounding variable than do stratification methods. For example, if multivariate analysis controls for <a href="Antidepressant" title="Antidepressant">antidepressant</a>, and it does not stratify antidepressants for <a href="Tricyclic_antidepressant" title="Tricyclic antidepressant">TCA</a> and <a href="SSRI" class="mw-redirect" title="SSRI">SSRI</a>, then it will ignore that these two classes of antidepressant have <i>opposite</i> effects on myocardial infarction, and one is much <i>stronger</i> than the other.</li></ul>

<p>All these methods have their drawbacks:

</p>

<ol><li>The best available defense against the possibility of spurious results due to confounding is often to dispense with efforts at stratification and instead conduct a <a href="Randomization" title="Randomization">randomized study</a> of a <a href="Law_of_large_numbers" title="Law of large numbers">sufficiently large</a> sample taken as a whole, such that all potential confounding variables (known and unknown) will be distributed by chance across all study groups and hence will be uncorrelated with the binary variable for inclusion/exclusion in any group.</li>

<li>Ethical considerations: In double-blind and randomized controlled trials, participants are not aware that they are recipients of <a href="Sham_treatment" class="mw-redirect" title="Sham treatment">sham treatments</a> and may be denied effective treatments.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> There is a possibility that patients only agree to invasive surgery (which carry real medical risks) under the understanding that they are receiving treatment. Although this is an ethical concern, it is not a complete account of the situation. For surgeries that are currently being performed regularly, but for which there is no concrete evidence of a genuine effect, there may be ethical issues to continue such surgeries. In such circumstances, many of people are exposed to the real risks of surgery yet these treatments may possibly offer no discernible benefit. Sham-surgery control is a method that may allow medical science to determine whether a surgical procedure is efficacious or not. Given that there are known risks associated with medical operations, it is questionably ethical to allow unverified surgeries to be conducted ad infinitum into the future.</li></ol>

<div class="mw-heading mw-heading2"><h2 id="Artifacts">Artifacts</h2></div>

<p>Artifacts are variables that should have been systematically varied, either within or across studies, but that were accidentally held constant. Artifacts are thus threats to <a href="External_validity" title="External validity">external validity</a>. Artifacts are factors that covary with the treatment and the outcome. Campbell and Stanley<sup id="cite_ref-C&S2006_29-0" class="reference"><a href="#cite_note-C&S2006-29"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> identify several artifacts. The major threats to internal validity are history, maturation, testing, instrumentation, <a href="Regression_analysis" title="Regression analysis">statistical regression</a>, selection, experimental mortality, and selection-history interactions.

</p><p>One way to minimize the influence of artifacts is to use a pretest-posttest <a href="Experimental_control" class="mw-redirect" title="Experimental control">control group</a> design. Within this design, "groups of people who are initially equivalent (at the pretest phase) are randomly assigned to receive the experimental treatment or a control condition and then assessed again after this differential experience (posttest phase)".<sup id="cite_ref-C&B2002_30-0" class="reference"><a href="#cite_note-C&B2002-30"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> Thus, any effects of artifacts are (ideally) equally distributed in participants in both the treatment and control conditions.

</p>

<ul><li><a href="Observational_interpretation_fallacy" title="Observational interpretation fallacy">Observational interpretation fallacy</a></li>

<li><a href="Anecdotal_evidence" title="Anecdotal evidence">Anecdotal evidence</a> – Evidence relying on personal testimony</li>

<li><a href="Causal_inference" title="Causal inference">Causal inference</a> – Branch of statistics concerned with inferring causal relationships between variables</li>

<li><a href="Epidemiological_method" title="Epidemiological method">Epidemiological method</a> – Scientific method in the specific field</li>

<li><a href="Simpson's_paradox" title="Simpson's paradox">Simpson's paradox</a> – Error in statistical reasoning with groups</li>

<li><a href="Omitted-variable_bias" title="Omitted-variable bias">Omitted-variable bias</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Also known as a <b>confounding variable</b>, <b>confounding factor</b>, <b>extraneous determinant</b>, or <b>lurking variable</b>.</span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<li id="cite_note-Pearl_2009-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Pearl_2009_2-0">^</a></b></span> <span class="reference-text">Pearl, J., (2009). <a href="Simpson's_Paradox" class="mw-redirect" title="Simpson's Paradox">Simpson's Paradox</a>, Confounding, and Collapsibility In <i>Causality: Models, Reasoning and Inference</i> (2nd ed.). New York : Cambridge University Press.</span>

</li>

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</style><cite id="CITEREFVanderWeeleShpitser2013" class="citation journal cs1">VanderWeele, T.J.; Shpitser, I. (2013). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4276366">"On the definition of a confounder"</a>. <i>Annals of Statistics</i>. <b>41</b> (1): <span class="nowrap">196–</span>220. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1304.0564">1304.0564</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1214%2F12-aos1058">10.1214/12-aos1058</a>. <a href="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/PMC4276366">4276366</a></span>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/25544784">25544784</a>.</cite></span>

</li>

<li id="cite_note-Greenland_Pearl_Robbins_1999-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Greenland_Pearl_Robbins_1999_4-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFGreenlandRobinsPearl1999" class="citation journal cs1">Greenland, S.; Robins, J. M.; Pearl, J. (1999). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Fss%2F1009211805">"Confounding and Collapsibility in Causal Inference"</a>. <i>Statistical Science</i>. <b>14</b> (1): <span class="nowrap">29–</span>46. <a href="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/10.1214%2Fss%2F1009211805">10.1214/ss/1009211805</a></span>.</cite></span>

</li>

<li id="cite_note-Shadish2002-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Shadish2002_5-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFShadishCookCampbell2002" class="citation book cs1">Shadish, W. R.; Cook, T. D.; Campbell, D. T. (2002). <i>Experimental and quasi-experimental designs for generalized causal inference</i>. Boston, MA: <a href="Houghton_Mifflin" class="mw-redirect" title="Houghton Mifflin">Houghton-Mifflin</a>.</cite></span>

</li>

<li id="cite_note-Pearl_1993-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-Pearl_1993_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Pearl_1993_6-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Pearl_1993_6-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Pearl_1993_6-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text">Pearl, J., (1993). "Aspects of Graphical Models Connected With Causality", <i>In Proceedings of the 49th Session of the International Statistical Science Institute,</i> pp. 391–401.</span>

</li>

<li id="cite_note-pearl09-causal-diagrams-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-pearl09-causal-diagrams_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-pearl09-causal-diagrams_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Pearl, J. (2009). Causal Diagrams and the Identification of Causal Effects In <i>Causality: Models, Reasoning and Inference</i> (2nd ed.). New York, NY, US: Cambridge University Press.</span>

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<li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">Fisher, R. A. (1935). The design of experiments (pp. 114–145).</span>

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<li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text">Pearl, J., (2009). <i>Causality: Models, Reasoning and Inference</i> (2nd ed.). New York, NY, US: Cambridge University Press.</span>

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<li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><cite id="CITEREFJohnston2001" class="citation journal cs1">Johnston, S. C. (2001). <a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Faje%2F154.3.276">"Identifying Confounding by Indication through Blinded Prospective Review"</a>. <i><a href="American_Journal_of_Epidemiology" title="American Journal of Epidemiology">American Journal of Epidemiology</a></i>. <b>154</b> (3): <span class="nowrap">276–</span>284. <a href="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/10.1093%2Faje%2F154.3.276">10.1093/aje/154.3.276</a></span>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/11479193">11479193</a>.</cite></span>

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<li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><cite id="CITEREFAxelson1989" class="citation journal cs1">Axelson, O. (1989). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1009818">"Confounding from smoking in occupational epidemiology"</a>. <i>British Journal of Industrial Medicine</i>. <b>46</b> (8): <span class="nowrap">505–</span>07. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1136%2Foem.46.8.505">10.1136/oem.46.8.505</a>. <a href="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/PMC1009818">1009818</a></span>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/2673334">2673334</a>.</cite></span>

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<li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text">Calow, Peter P. (2009) <i>Handbook of Environmental Risk Assessment and Management</i>, Wiley</span>

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</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFPearl1998" class="citation journal cs1">Pearl, J. (January 1998). <a rel="nofollow" class="external text" href="http://ftp.cs.ucla.edu/pub/stat_ser/R256.pdf">"Why there is no statistical test for confounding, why many think there is, and why they are almost right"</a> <span class="cs1-format">(PDF)</span>. <i>UCLA Computer Science Department, Technical Report R-256</i>.</cite></li>

<li><cite id="CITEREFMontgomery2001" class="citation book cs1">Montgomery, D. C. (2001). "Blocking and Confounding in the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{k}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2^{k}}</annotation>

</semantics>

</math></span><img src="./2d82641ae2702b0db07dd11830af27b9ee0cd196.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.251ex; height:2.676ex;" alt="{\displaystyle 2^{k}}" loading="lazy"></span> Factorial Design". <i>Design and Analysis of Experiments</i> (5th ed.). Wiley. pp. <span class="nowrap">287–</span>302. This textbook has an overview of confounding factors and how to account for them in design of experiments.</cite><span class="cs1-maint citation-comment"><code class="cs1-code">{{cite book}}</code>: CS1 maint: postscript (link)</span></li>

<li><cite id="CITEREFBrewer2000" class="citation book cs1">Brewer, M. B. (2000). "Research design and issues of validity". In Reis, H. T.; Judd, C. M. (eds.). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/handbookofresear00reis"><i>Handbook of Research</i></a></span>. New York: <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/handbookofresear00reis/page/3">3–16</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780521551281</bdi>.</cite></li>

<li><cite id="CITEREFSmith2000" class="citation book cs1">Smith, E. R. (2000). "Research design". In Reis, H. T.; Judd, C. M. (eds.). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/handbookofresear00reis"><i>Handbook of research methods in social and personality psychology</i></a></span>. New York: Cambridge University Press. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/handbookofresear00reis/page/17">17–39</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780521551281</bdi>.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://sphweb.bumc.bu.edu/otlt/MPH-Modules/BS/BS704-EP713_Confounding-EM/BS704-EP713_Confounding-EM_print.html">Tutorial: Confounding and Effect Measure Modification (Boston University School of Public Health)</a></li>

<li><a rel="nofollow" class="external text" href="http://www.stat.yale.edu/Courses/1997-98/101/linreg.htm">Linear Regression (Yale University)</a></li>

<li><a rel="nofollow" class="external text" href="http://www.une.edu.au/WebStat/unit_materials/c1_behavioural_science_research/confounds.html">Tutorial by University of New England</a></li></ul>

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</style><div id="Statistics654" style="font-size:114%;margin:0 4em"><a href="Statistics" title="Statistics">Statistics</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><a href="Outline_of_statistics" title="Outline of statistics">Outline</a></li>

<li><a href="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_statistics654" style="font-size:114%;margin:0 4em"><a href="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="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="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="Arithmetic_mean" title="Arithmetic mean">Arithmetic</a></li>

<li><a href="Arithmetic%E2%80%93geometric_mean" title="Arithmetic–geometric mean">Arithmetic-Geometric</a></li>

<li><a href="Contraharmonic_mean" title="Contraharmonic mean">Contraharmonic</a></li>

<li><a href="Cubic_mean" title="Cubic mean">Cubic</a></li>

<li><a href="Generalized_mean" title="Generalized mean">Generalized/power</a></li>

<li><a href="Geometric_mean" title="Geometric mean">Geometric</a></li>

<li><a href="Harmonic_mean" title="Harmonic mean">Harmonic</a></li>

<li><a href="Heronian_mean" title="Heronian mean">Heronian</a></li>

<li><a href="Heinz_mean" title="Heinz mean">Heinz</a></li>

<li><a href="Lehmer_mean" title="Lehmer mean">Lehmer</a></li></ul></li>

<li><a href="Median" title="Median">Median</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="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="Average_absolute_deviation" title="Average absolute deviation">Average absolute deviation</a></li>

<li><a href="Coefficient_of_variation" title="Coefficient of variation">Coefficient of variation</a></li>

<li><a href="Interquartile_range" title="Interquartile range">Interquartile range</a></li>

<li><a href="Percentile" title="Percentile">Percentile</a></li>

<li><a href="Range_(statistics)" title="Range (statistics)">Range</a></li>

<li><a href="Standard_deviation" title="Standard deviation">Standard deviation</a></li>

<li><a href="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="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="Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></li>

<li><a href="Moment_(mathematics)" title="Moment (mathematics)">Moments</a>

<ul><li><a href="Kurtosis" title="Kurtosis">Kurtosis</a></li>

<li><a href="L-moment" title="L-moment">L-moments</a></li>

<li><a href="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="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="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="Contingency_table" title="Contingency table">Contingency table</a></li>

<li><a href="Frequency_distribution" class="mw-redirect" title="Frequency distribution">Frequency distribution</a></li>

<li><a href="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="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="Partial_correlation" title="Partial correlation">Partial correlation</a></li>

<li><a href="Pearson_correlation_coefficient" title="Pearson correlation coefficient">Pearson product-moment correlation</a></li>

<li><a href="Rank_correlation" title="Rank correlation">Rank correlation</a>

<ul><li><a href="Kendall_rank_correlation_coefficient" title="Kendall rank correlation coefficient">Kendall's τ</a></li>

<li><a href="Spearman's_rank_correlation_coefficient" title="Spearman's rank correlation coefficient">Spearman's ρ</a></li></ul></li>

<li><a href="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="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="Bar_chart" title="Bar chart">Bar chart</a></li>

<li><a href="Biplot" title="Biplot">Biplot</a></li>

<li><a href="Control_chart" title="Control chart">Control chart</a></li>

<li><a href="Correlogram" title="Correlogram">Correlogram</a></li>

<li><a href="Fan_chart_(statistics)" title="Fan chart (statistics)">Fan chart</a></li>

<li><a href="Forest_plot" title="Forest plot">Forest plot</a></li>

<li><a href="Histogram" title="Histogram">Histogram</a></li>

<li><a href="Pie_chart" title="Pie chart">Pie chart</a></li>

<li><a href="Radar_chart" title="Radar chart">Radar chart</a></li>

<li><a href="Run_chart" title="Run chart">Run chart</a></li>

<li><a href="Scatter_plot" title="Scatter plot">Scatter plot</a></li>

<li><a href="Stem-and-leaf_display" title="Stem-and-leaf display">Stem-and-leaf display</a></li>

<li><a href="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_collection654" style="font-size:114%;margin:0 4em"><a href="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="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="Effect_size" title="Effect size">Effect size</a></li>

<li><a href="Missing_data" title="Missing data">Missing data</a></li>

<li><a href="Optimal_design" class="mw-redirect" title="Optimal design">Optimal design</a></li>

<li><a href="Statistical_population" title="Statistical population">Population</a></li>

<li><a href="Replication_(statistics)" title="Replication (statistics)">Replication</a></li>

<li><a href="Sample_size_determination" title="Sample size determination">Sample size determination</a></li>

<li><a href="Statistic" title="Statistic">Statistic</a></li>

<li><a href="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="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="Sampling_(statistics)" title="Sampling (statistics)">Sampling</a>

<ul><li><a href="Cluster_sampling" title="Cluster sampling">Cluster</a></li>

<li><a href="Stratified_sampling" title="Stratified sampling">Stratified</a></li></ul></li>

<li><a href="Opinion_poll" title="Opinion poll">Opinion poll</a></li>

<li><a href="Questionnaire" title="Questionnaire">Questionnaire</a></li>

<li><a href="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="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="Blocking_(statistics)" title="Blocking (statistics)">Blocking</a></li>

<li><a href="Factorial_experiment" title="Factorial experiment">Factorial experiment</a></li>

<li><a href="Interaction_(statistics)" title="Interaction (statistics)">Interaction</a></li>

<li><a href="Random_assignment" title="Random assignment">Random assignment</a></li>

<li><a href="Randomized_controlled_trial" title="Randomized controlled trial">Randomized controlled trial</a></li>

<li><a href="Randomized_experiment" title="Randomized experiment">Randomized experiment</a></li>

<li><a href="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="Adaptive_clinical_trial" class="mw-redirect" title="Adaptive clinical trial">Adaptive clinical trial</a></li>

<li><a href="Stochastic_approximation" title="Stochastic approximation">Stochastic approximation</a></li>

<li><a href="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="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="Cohort_study" title="Cohort study">Cohort study</a></li>

<li><a href="Cross-sectional_study" title="Cross-sectional study">Cross-sectional study</a></li>

<li><a href="Natural_experiment" title="Natural experiment">Natural experiment</a></li>

<li><a href="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_inference654" style="font-size:114%;margin:0 4em"><a href="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="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="Population_(statistics)" class="mw-redirect" title="Population (statistics)">Population</a></li>

<li><a href="Statistic" title="Statistic">Statistic</a></li>

<li><a href="Probability_distribution" title="Probability distribution">Probability distribution</a></li>

<li><a href="Sampling_distribution" title="Sampling distribution">Sampling distribution</a>

<ul><li><a href="Order_statistic" title="Order statistic">Order statistic</a></li></ul></li>

<li><a href="Empirical_distribution_function" title="Empirical distribution function">Empirical distribution</a>

<ul><li><a href="Density_estimation" title="Density estimation">Density estimation</a></li></ul></li>

<li><a href="Statistical_model" title="Statistical model">Statistical model</a>

<ul><li><a href="Model_specification" class="mw-redirect" title="Model specification">Model specification</a></li>

<li><a href="Lp_space" title="Lp space">L<sup><i>p</i></sup> space</a></li></ul></li>

<li><a href="Statistical_parameter" title="Statistical parameter">Parameter</a>

<ul><li><a href="Location_parameter" title="Location parameter">location</a></li>

<li><a href="Scale_parameter" title="Scale parameter">scale</a></li>

<li><a href="Shape_parameter" title="Shape parameter">shape</a></li></ul></li>

<li><a href="Parametric_statistics" title="Parametric statistics">Parametric family</a>

<ul><li><a href="Likelihood_function" title="Likelihood function">Likelihood</a> <a href="Monotone_likelihood_ratio" title="Monotone likelihood ratio"><span style="font-size: 85%;">(monotone)</span></a></li>

<li><a href="Location%E2%80%93scale_family" title="Location–scale family">Location–scale family</a></li>

<li><a href="Exponential_family" title="Exponential family">Exponential family</a></li></ul></li>

<li><a href="Completeness_(statistics)" title="Completeness (statistics)">Completeness</a></li>

<li><a href="Sufficient_statistic" title="Sufficient statistic">Sufficiency</a></li>

<li><a href="Plug-in_principle" class="mw-redirect" title="Plug-in principle">Statistical functional</a>

<ul><li><a href="Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li>

<li><a href="Optimal_decision" title="Optimal decision">Optimal decision</a>

<ul><li><a href="Loss_function" title="Loss function">loss function</a></li></ul></li>

<li><a href="Efficiency_(statistics)" title="Efficiency (statistics)">Efficiency</a></li>

<li><a href="Statistical_distance" title="Statistical distance">Statistical distance</a>

<ul><li><a href="Divergence_(statistics)" title="Divergence (statistics)">divergence</a></li></ul></li>

<li><a href="Asymptotic_theory_(statistics)" title="Asymptotic theory (statistics)">Asymptotics</a></li>

<li><a href="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="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="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="Estimating_equations" title="Estimating equations">Estimating equations</a>

<ul><li><a href="Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">Maximum likelihood</a></li>

<li><a href="Method_of_moments_(statistics)" title="Method of moments (statistics)">Method of moments</a></li>

<li><a href="M-estimator" title="M-estimator">M-estimator</a></li>

<li><a href="Minimum_distance_estimation" class="mw-redirect" title="Minimum distance estimation">Minimum distance</a></li></ul></li>

<li><a href="Bias_of_an_estimator" title="Bias of an estimator">Unbiased estimators</a>

<ul><li><a href="Minimum-variance_unbiased_estimator" title="Minimum-variance unbiased estimator">Mean-unbiased minimum-variance</a>

<ul><li><a href="Rao%E2%80%93Blackwell_theorem" title="Rao–Blackwell theorem">Rao–Blackwellization</a></li>

<li><a href="Lehmann%E2%80%93Scheff%C3%A9_theorem" title="Lehmann–Scheffé theorem">Lehmann–Scheffé theorem</a></li></ul></li>

<li><a href="Median-unbiased_estimator" class="mw-redirect" title="Median-unbiased estimator">Median unbiased</a></li></ul></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="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="Confidence_interval" title="Confidence interval">Confidence interval</a></li>

<li><a href="Pivotal_quantity" title="Pivotal quantity">Pivot</a></li>

<li><a href="Likelihood_interval" class="mw-redirect" title="Likelihood interval">Likelihood interval</a></li>

<li><a href="Prediction_interval" title="Prediction interval">Prediction interval</a></li>

<li><a href="Tolerance_interval" title="Tolerance interval">Tolerance interval</a></li>

<li><a href="Resampling_(statistics)" title="Resampling (statistics)">Resampling</a>

<ul><li><a href="Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li>

<li><a href="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="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="One-_and_two-tailed_tests" title="One- and two-tailed tests">1- & 2-tails</a></li>

<li><a href="Power_(statistics)" title="Power (statistics)">Power</a>

<ul><li><a href="Uniformly_most_powerful_test" title="Uniformly most powerful test">Uniformly most powerful test</a></li></ul></li>

<li><a href="Permutation_test" title="Permutation test">Permutation test</a>

<ul><li><a href="Randomization_test" class="mw-redirect" title="Randomization test">Randomization test</a></li></ul></li>

<li><a href="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="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="Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio</a></li>

<li><a href="Score_test" title="Score test">Score/Lagrange multiplier</a></li>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="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="Z-test" title="Z-test"><i>Z</i>-test <span style="font-size: 85%;">(normal)</span></a></li>

<li><a href="Student's_t-test" title="Student's t-test">Student's <i>t</i>-test</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="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="Chi-squared_test" title="Chi-squared test">Chi-squared</a></li>

<li><a href="Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov–Smirnov</a></li>

<li><a href="Anderson%E2%80%93Darling_test" title="Anderson–Darling test">Anderson–Darling</a></li>

<li><a href="Lilliefors_test" title="Lilliefors test">Lilliefors</a></li>

<li><a href="Jarque%E2%80%93Bera_test" title="Jarque–Bera test">Jarque–Bera</a></li>

<li><a href="Shapiro%E2%80%93Wilk_test" title="Shapiro–Wilk test">Normality <span style="font-size: 85%;">(Shapiro–Wilk)</span></a></li>

<li><a href="Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio test</a></li>

<li><a href="Model_selection" title="Model selection">Model selection</a>

<ul><li><a href="Cross-validation_(statistics)" title="Cross-validation (statistics)">Cross validation</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="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="Sample_median" class="mw-redirect" title="Sample median">Sample median</a></li></ul></li>

<li><a href="Wilcoxon_signed-rank_test" title="Wilcoxon signed-rank test">Signed rank <span style="font-size: 85%;">(Wilcoxon)</span></a>

<ul><li><a href="Hodges%E2%80%93Lehmann_estimator" title="Hodges–Lehmann estimator">Hodges–Lehmann estimator</a></li></ul></li>

<li><a href="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="Nonparametric_statistics" title="Nonparametric statistics">Nonparametric</a> <a href="Analysis_of_variance" title="Analysis of variance">anova</a>

<ul><li><a href="Kruskal%E2%80%93Wallis_test" title="Kruskal–Wallis test">1-way <span style="font-size: 85%;">(Kruskal–Wallis)</span></a></li>

<li><a href="Friedman_test" title="Friedman test">2-way <span style="font-size: 85%;">(Friedman)</span></a></li>

<li><a href="Jonckheere's_trend_test" title="Jonckheere's trend test">Ordered alternative <span style="font-size: 85%;">(Jonckheere–Terpstra)</span></a></li></ul></li>

<li><a href="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="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="Bayesian_probability" title="Bayesian probability">Bayesian probability</a>

<ul><li><a href="Prior_probability" title="Prior probability">prior</a></li>

<li><a href="Posterior_probability" title="Posterior probability">posterior</a></li></ul></li>

<li><a href="Credible_interval" title="Credible interval">Credible interval</a></li>

<li><a href="Bayes_factor" title="Bayes factor">Bayes factor</a></li>

<li><a href="Bayes_estimator" title="Bayes estimator">Bayesian estimator</a>

<ul><li><a href="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_analysis654" style="font-size:114%;margin:0 4em"><div class="hlist"><ul><li><a href="Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></li><li><a href="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="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="Pearson_product-moment_correlation_coefficient" class="mw-redirect" title="Pearson product-moment correlation coefficient">Pearson product-moment</a></li>

<li><a href="Partial_correlation" title="Partial correlation">Partial correlation</a></li>

<li><a href="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="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="Errors_and_residuals" title="Errors and residuals">Errors and residuals</a></li>

<li><a href="Regression_validation" title="Regression validation">Regression validation</a></li>

<li><a href="Mixed_model" title="Mixed model">Mixed effects models</a></li>

<li><a href="Simultaneous_equations_model" title="Simultaneous equations model">Simultaneous equations models</a></li>

<li><a href="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="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="Simple_linear_regression" title="Simple linear regression">Simple linear regression</a></li>

<li><a href="Ordinary_least_squares" title="Ordinary least squares">Ordinary least squares</a></li>

<li><a href="General_linear_model" title="General linear model">General linear model</a></li>

<li><a href="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="Nonlinear_regression" title="Nonlinear regression">Nonlinear regression</a></li>

<li><a href="Nonparametric_regression" title="Nonparametric regression">Nonparametric</a></li>

<li><a href="Semiparametric_regression" title="Semiparametric regression">Semiparametric</a></li>

<li><a href="Isotonic_regression" title="Isotonic regression">Isotonic</a></li>

<li><a href="Robust_regression" title="Robust regression">Robust</a></li>

<li><a href="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="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="Exponential_family" title="Exponential family">Exponential families</a></li>

<li><a href="Logistic_regression" title="Logistic regression">Logistic <span style="font-size: 85%;">(Bernoulli)</span></a> / <a href="Binomial_regression" title="Binomial regression">Binomial</a> / <a href="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="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="Analysis_of_variance" title="Analysis of variance">Analysis of variance (ANOVA, anova)</a></li>

<li><a href="Analysis_of_covariance" title="Analysis of covariance">Analysis of covariance</a></li>

<li><a href="Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Multivariate ANOVA</a></li>

<li><a href="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_analysis654" style="font-size:114%;margin:0 4em"><a href="Categorical_variable" title="Categorical variable">Categorical</a> / <a href="Multivariate_statistics" title="Multivariate statistics">multivariate</a> / <a href="Time_series" title="Time series">time-series</a> / <a href="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="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="Cohen's_kappa" title="Cohen's kappa">Cohen's kappa</a></li>

<li><a href="Contingency_table" title="Contingency table">Contingency table</a></li>

<li><a href="Graphical_model" title="Graphical model">Graphical model</a></li>

<li><a href="Poisson_regression" title="Poisson regression">Log-linear model</a></li>

<li><a href="McNemar's_test" title="McNemar's test">McNemar's test</a></li>

<li><a href="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="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="General_linear_model" title="General linear model">Regression</a></li>

<li><a href="Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Manova</a></li>

<li><a href="Principal_component_analysis" title="Principal component analysis">Principal components</a></li>

<li><a href="Canonical_correlation" title="Canonical correlation">Canonical correlation</a></li>

<li><a href="Linear_discriminant_analysis" title="Linear discriminant analysis">Discriminant analysis</a></li>

<li><a href="Cluster_analysis" title="Cluster analysis">Cluster analysis</a></li>

<li><a href="Statistical_classification" title="Statistical classification">Classification</a></li>

<li><a href="Structural_equation_modeling" title="Structural equation modeling">Structural equation model</a>

<ul><li><a href="Factor_analysis" title="Factor analysis">Factor analysis</a></li></ul></li>

<li><a href="Multivariate_distribution" class="mw-redirect" title="Multivariate distribution">Multivariate distributions</a>

<ul><li><a href="Elliptical_distribution" title="Elliptical distribution">Elliptical distributions</a>

<ul><li><a href="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="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="Decomposition_of_time_series" title="Decomposition of time series">Decomposition</a></li>

<li><a href="Trend_estimation" class="mw-redirect" title="Trend estimation">Trend</a></li>

<li><a href="Stationary_process" title="Stationary process">Stationarity</a></li>

<li><a href="Seasonal_adjustment" title="Seasonal adjustment">Seasonal adjustment</a></li>

<li><a href="Exponential_smoothing" title="Exponential smoothing">Exponential smoothing</a></li>

<li><a href="Cointegration" title="Cointegration">Cointegration</a></li>

<li><a href="Structural_break" title="Structural break">Structural break</a></li>

<li><a href="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="Dickey%E2%80%93Fuller_test" title="Dickey–Fuller test">Dickey–Fuller</a></li>

<li><a href="Johansen_test" title="Johansen test">Johansen</a></li>

<li><a href="Ljung%E2%80%93Box_test" title="Ljung–Box test">Q-statistic <span style="font-size: 85%;">(Ljung–Box)</span></a></li>

<li><a href="Durbin%E2%80%93Watson_statistic" title="Durbin–Watson statistic">Durbin–Watson</a></li>

<li><a href="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="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="Autocorrelation" title="Autocorrelation">Autocorrelation (ACF)</a>

<ul><li><a href="Partial_autocorrelation_function" title="Partial autocorrelation function">partial (PACF)</a></li></ul></li>

<li><a href="Cross-correlation" title="Cross-correlation">Cross-correlation (XCF)</a></li>

<li><a href="Autoregressive%E2%80%93moving-average_model" class="mw-redirect" title="Autoregressive–moving-average model">ARMA model</a></li>

<li><a href="Box%E2%80%93Jenkins_method" title="Box–Jenkins method">ARIMA model <span style="font-size: 85%;">(Box–Jenkins)</span></a></li>

<li><a href="Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Autoregressive conditional heteroskedasticity (ARCH)</a></li>

<li><a href="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="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="Spectral_density_estimation" title="Spectral density estimation">Spectral density estimation</a></li>

<li><a href="Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li>

<li><a href="Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a></li>

<li><a href="Wavelet" title="Wavelet">Wavelet</a></li>

<li><a href="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="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="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="Kaplan%E2%80%93Meier_estimator" title="Kaplan–Meier estimator">Kaplan–Meier estimator (product limit)</a></li>

<li><a href="Proportional_hazards_model" title="Proportional hazards model">Proportional hazards models</a></li>

<li><a href="Accelerated_failure_time_model" title="Accelerated failure time model">Accelerated failure time (AFT) model</a></li>

<li><a href="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="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="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">

</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="Applications654" style="font-size:114%;margin:0 4em"><a href="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="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="Bioinformatics" title="Bioinformatics">Bioinformatics</a></li>

<li><a href="Clinical_trial" title="Clinical trial">Clinical trials</a> / <a href="Clinical_study_design" title="Clinical study design">studies</a></li>

<li><a href="Epidemiology" title="Epidemiology">Epidemiology</a></li>

<li><a href="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="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="Chemometrics" title="Chemometrics">Chemometrics</a></li>

<li><a href="Methods_engineering" title="Methods engineering">Methods engineering</a></li>

<li><a href="Probabilistic_design" title="Probabilistic design">Probabilistic design</a></li>

<li><a href="Statistical_process_control" title="Statistical process control">Process</a> / <a href="Quality_control" title="Quality control">quality control</a></li>

<li><a href="Reliability_engineering" title="Reliability engineering">Reliability</a></li>

<li><a href="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="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="Actuarial_science" title="Actuarial science">Actuarial science</a></li>

<li><a href="Census" title="Census">Census</a></li>

<li><a href="Crime_statistics" title="Crime statistics">Crime statistics</a></li>

<li><a href="Demographic_statistics" title="Demographic statistics">Demography</a></li>

<li><a href="Econometrics" title="Econometrics">Econometrics</a></li>

<li><a href="Jurimetrics" title="Jurimetrics">Jurimetrics</a></li>

<li><a href="National_accounts" title="National accounts">National accounts</a></li>

<li><a href="Official_statistics" title="Official statistics">Official statistics</a></li>

<li><a href="Population_statistics" class="mw-redirect" title="Population statistics">Population statistics</a></li>

<li><a href="Psychometrics" title="Psychometrics">Psychometrics</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Spatial_analysis" title="Spatial analysis">Spatial 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="Cartography" title="Cartography">Cartography</a></li>

<li><a href="Environmental_statistics" title="Environmental statistics">Environmental statistics</a></li>

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</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Design_of_experiments" title="Design of experiments">Designs</a> <br> <br><a href="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">

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<li><a href="Central_composite_design" title="Central composite design">Central composite</a></li></ul>

<ul><li><b><a href="Randomized_block_design" class="mw-redirect" title="Randomized block design">Block</a></b></li>

<li><a href="Generalized_randomized_block_design" title="Generalized randomized block design">Generalized randomized block design</a> (GRBD)</li>

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<li><a href="Crossover_study" title="Crossover study">Crossover study</a></li></ul>

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This article is issued from <a class="external text" title="Last edited on 2025-03-12" href="https://en.wikipedia.org/wiki/?title=Confounding&oldid=1280065491">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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