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{{Short description|Statistical term}}
[[File:GSS sealevel interaction.png|thumb|Interaction effect of education and ideology on concern about sea level rise]]In [[statistics]], an '''interaction''' may arise when considering the relationship among three or more variables, and describes a situation in which the effect of one causal variable on an outcome depends on the state of a second causal variable (that is, when effects of the two causes are not [[additive map|additive]]).<ref name=Dodge>{{cite book | last=Dodge | first=Y. | year=2003 | title=''The Oxford Dictionary of Statistical Terms'' | publisher=Oxford University Press | isbn=978-0-19-920613-
The presence of interactions can have important implications for the interpretation of statistical models.
▲[[File:GSS sealevel interaction.png|thumb|Interaction effect of education and ideology on concern about sea level rise]]In [[statistics]], an '''interaction'''<ref name=Dodge>{{cite book | last=Dodge | first=Y. | year=2003 | title=''The Oxford Dictionary of Statistical Terms'' | publisher=Oxford University Press | isbn=0-19-920613-9}}</ref><ref>{{cite journal | doi=10.2307/1403235 | last=Cox | first=D.R. | year=1984 | title=Interaction | journal=International Statistical Review | volume=52 | pages=1–25 | jstor=1403235 | issue=1 | publisher=International Statistical Review / Revue Internationale de Statistique}}</ref> may arise when considering the relationship among three or more variables, and describes a situation in which the simultaneous influence of two variables on a third is not [[additive map|additive]]. Most commonly, interactions are considered in the context of [[regression analysis|regression analyses]].
The notion of "interaction" is closely related to that of
▲The presence of interactions can have important implications for the interpretation of statistical models. If two variables of interest interact, the relationship between each of the interacting variables and a third "dependent variable" depends on the value of the other interacting variable. In practice, this makes it more difficult to predict the consequences of changing the value of a variable, particularly if the variables it interacts with are hard to measure or difficult to control.
▲The notion of "interaction" is closely related to that of "[[Moderation (statistics)|moderation]]" that is common in social and health science research: the interaction between an explanatory variable and an environmental variable suggests that the effect of the explanatory variable has been moderated or modified by the environmental variable.<ref name=Dodge />
==Introduction==
An '''interaction variable''' or '''interaction feature''' is a variable constructed from an original set of variables to try to represent either all of the interaction present or some part of it. In exploratory statistical analyses it is common to use products of original variables as the basis of testing whether interaction is present with the possibility of substituting other more realistic interaction variables at a later stage. When there are more than two explanatory variables, several interaction variables are constructed, with pairwise-products representing pairwise-interactions and higher order products representing higher order interactions.
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==In modeling==
===In ANOVA===
A simple setting in which interactions can arise is a [[factorial experiment|two-factor experiment]] analyzed using [[Analysis of Variance]] (ANOVA). Suppose we have two binary factors ''A'' and ''B''. For example, these factors might indicate whether either of two treatments were administered to a patient, with the treatments applied either singly, or in combination. We can then consider the average treatment response (e.g. the symptom levels following treatment) for each patient, as a function of the treatment combination that was administered. The following table shows one possible situation:
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===Qualitative and quantitative interactions===
In many applications it is useful to distinguish between qualitative and quantitative interactions.<ref>{{cite book | last=Peto | first=
The table of means on the left, below, shows a quantitative interaction — treatment ''A'' is beneficial both when ''B'' is given, and when ''B'' is not given, but the benefit is greater when ''B'' is not given (i.e. when ''A'' is given alone). The table of means on the right shows a qualitative interaction. ''A'' is harmful when ''B'' is given, but it is beneficial when ''B'' is not given. Note that the same interpretation would hold if we consider the benefit of ''B'' based on whether ''A'' is given.
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===Unit treatment additivity===
In its simplest form, the assumption of treatment unit additivity states that the observed response ''y''<sub>''ij''</sub> from experimental unit ''i'' when receiving treatment ''j'' can be written as the sum ''y''<sub>''ij''</sub> = ''y''<sub>''i''</sub> + ''t''<sub>''j''</sub>.<ref name="Kempthorne (1979)">{{cite book |author-link=Oscar Kempthorne |last=Kempthorne |first=Oscar |year=1979 |title=The Design and Analysis of Experiments |edition=Corrected reprint of (1952) Wiley |publisher=Robert E. Krieger |isbn=978-0-88275-105-4 }}</ref><ref name=Cox1958_2>{{cite book |author-link=David R. Cox |last=Cox |first=David R. |year=1958 |title=Planning of experiments |publisher=Wiley |isbn=0-471-57429-5 |at=Chapter 2 }}</ref><ref>{{cite book
|author=Hinkelmann, Klaus and [[Oscar Kempthorne|Kempthorne, Oscar]]▼
In its simplest form, the assumption of treatment unit additivity states that the observed response ''y''<sub>''ij''</sub> from experimental unit ''i'' when receiving treatment ''j'' can be written as the sum ''y''<sub>''ij''</sub> = ''y''<sub>''i''</sub> + ''t''<sub>''j''</sub>.<ref>Kempthorne (1979)</ref><ref name=Cox1958_2>Cox (1958), Chapter 2</ref><ref>Hinkelmann & Kempthorne (2008), Chapters 5-6</ref> The assumption of unit treatment additivity implies that every treatment has exactly the same additive effect on each experimental unit. Since any given experimental unit can only undergo one of the treatments, the assumption of unit treatment additivity is a hypothesis that is not directly falsifiable, according to Cox{{Citation needed|date=April 2010}} and Kempthorne.{{Citation needed|date=April 2010}}▼
|year=2008▼
|title=Design and Analysis of Experiments, Volume I: Introduction to Experimental Design▼
|edition=Second▼
|publisher=Wiley▼
|isbn=978-0-471-72756-9▼
▲
However, many consequences of treatment-unit additivity can be falsified.{{Citation needed|date=April 2010}} For a randomized experiment, the assumption of treatment additivity implies that the variance is constant for all treatments. Therefore, by contraposition, a necessary condition for unit treatment additivity is that the variance is constant.{{Citation needed|date=April 2010}}
The property of unit treatment additivity is not invariant under a change of scale,{{Citation needed|date=April 2010}} so statisticians often use transformations to achieve unit treatment additivity. If the response variable is expected to follow a parametric family of probability distributions, then the statistician may specify (in the protocol for the experiment or observational study) that the responses be transformed to stabilize the variance.<ref>
|author=Hinkelmann, Klaus and [[Oscar Kempthorne|Kempthorne, Oscar]]▼
|edition=Second
|publisher=Wiley
|isbn=978-0-471-72756-9
The assumption of unit treatment additivity was enunciated in experimental design by Kempthorne{{Citation needed|date=April 2010}} and Cox{{Citation needed|date=April 2010}}. Kempthorne's use of unit treatment additivity and randomization is similar to the design-based analysis of finite population survey sampling.
In recent years, it has become common{{Citation needed|date=April 2010}} to use the terminology of Donald Rubin, which uses counterfactuals. Suppose we are comparing two groups of people with respect to some attribute ''y''. For example, the first group might consist of people who are given a standard treatment for a medical condition, with the second group consisting of people who receive a new treatment with unknown effect. Taking a "counterfactual" perspective, we can consider an individual whose attribute has value ''y'' if that individual belongs to the first group, and whose attribute has value ''τ''(''y'') if the individual belongs to the second group. The assumption of "unit treatment additivity" is that ''τ''(''y'') = ''τ'', that is, the "treatment effect" does not depend on ''y''. Since we cannot observe both ''y'' and τ(''y'') for a given individual, this is not testable at the individual level. However, unit treatment additivity
''F''<sub>2</sub>(''y'') = ''F''<sub>1</sub>(''y − τ''), as long as the assignment of individuals to groups 1 and 2 is independent of all other factors influencing ''y'' (i.e. there are no [[confounding variable|confounders]]). Lack of unit treatment additivity can be viewed as a form of interaction between the treatment assignment (e.g. to groups 1 or 2), and the baseline, or untreated value of ''y''.
===Categorical variables===
Sometimes the interacting variables are categorical variables rather than real numbers and the study might then be dealt with as an [[analysis of variance]] problem. For example, members of a population may be classified by religion and by occupation. If one wishes to predict a person's height based only on the person's religion and occupation, a simple ''additive'' model, i.e., a model without interaction, would add to an overall average height an adjustment for a particular religion and another for a particular occupation. A model with interaction, unlike an [[additive model]], could add a further adjustment for the "interaction" between that religion and that occupation. This example may cause one to suspect that the word ''interaction'' is something of a misnomer.
Statistically, the presence of an interaction between categorical variables is generally tested using a form of [[analysis of variance]] (ANOVA). If one or more of the variables is continuous in nature, however, it would typically be tested using moderated multiple regression.<ref name=Overton2001>{{Cite journal
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| doi = 10.1037/1082-989X.6.3.218
| pmid = 11570229
}}</ref> This is so-called because a moderator is a variable that affects the strength of a relationship between two other variables.▼
▲}}</ref> This is so-called because a moderator is a variable that affects the strength of a relationship between two other variables.
===Designed experiments===
[[Genichi Taguchi]] contended<ref>{{Cite web|title = Design of Experiments - Taguchi Experiments|url = http://www.qualitytrainingportal.com/resources/doe/taguchi_concepts.htm|website = www.qualitytrainingportal.com|
| author =
| author-link = George E. P. Box
| year = 1990
| title = Do interactions matter?
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| pages = 365–369
| url = http://cqpi.engr.wisc.edu/system/files/r046.pdf
|
| archive-url = https://web.archive.org/web/20100610131759/http://cqpi.engr.wisc.edu/system/files/r046.pdf
}}</ref>▼
| archive-date = 2010-06-10
| url-status = dead
| doi = 10.1080/08982119008962728
▲ }}</ref>
===Model size===
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where the interaction term <math>(x_1\times x_2)</math> could be formed explicitly by multiplying two (or more) variables, or implicitly using factorial notation in modern statistical packages such as [[Stata]]. The components ''x''<sub>1</sub> and ''x''<sub>2</sub> might be measurements or {0,1} [[dummy variable (statistics)|dummy variable]]s in any combination. Interactions involving a dummy variable multiplied by a measurement variable are termed ''slope dummy variables'',<ref>Hamilton, L.C. 1992. ''Regression with Graphics: A Second Course in Applied Statistics''. Pacific Grove, CA: Brooks/Cole. {{ISBN|978-0534159009}}</ref> because they estimate and test the difference in slopes between groups 0 and 1.
When measurement variables are employed in interactions, it is often desirable to work with centered versions, where the variable's mean (or some other reasonably central value) is set as zero. Centering
[[File:Tea party interaction.png|thumb|Interaction of education and political party affecting beliefs about climate change]]Regression approaches to interaction modeling are very general because they can accommodate additional predictors, and many alternative specifications or estimation strategies beyond [[ordinary least squares]]. [[Robust regression|Robust]], [[Quantile regression|quantile]], and mixed-effects ([[Multilevel model|multilevel]]) models are among the possibilities, as is [[generalized linear model]]ing encompassing a wide range of categorical, ordered, counted or otherwise limited dependent variables. The graph depicts an education*politics interaction, from a probability-weighted [[logit regression]] analysis of survey data.<ref>{{cite journal | last1 = Hamilton
==Interaction plots==
Interaction plots, also called [[Moderation (statistics)#Two continuous independent variables|simple-slope plots]], show possible interactions among variables.
===Example: Interaction of species and air temperature and their effect on body temperature===
Consider a study of the body temperature of different species at different air temperatures, in degrees Fahrenheit. The data are shown in the table below.
[[File:Body temperature species data 2.png|Body temperature species data]]
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==Hypothesis tests for interactions==
Analysis of variance and regression analysis are used to test for significant interactions.
===Example: Interaction of temperature and time in cookie baking===
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| pmid = 11171926
| pmc = 1740104
}}</ref>▼
*''Interaction'' between genetic risk factors for [[Diabetes mellitus type 2|type 2 diabetes]] and diet (specifically, a "western" dietary pattern). The western dietary pattern was shown to increase diabetes risk for subjects with a high "genetic risk score", but not for other subjects.<ref>{{Cite journal | author = Lu, Q. | year = 2009 | title = Genetic predisposition, Western dietary pattern, and the risk of type 2 diabetes in men | journal = Am J Clin Nutr | volume = 89 | pages = 1453–1458 | doi = 10.3945/ajcn.2008.27249 | pmid = 19279076 | display-authors = 1 | issue = 5 | author2 = <Please add first missing authors to populate metadata.>| pmc = 2676999 }}</ref>▼
▲}}</ref>
*''Interaction'' between education and political orientation, affecting general-public perceptions about climate change. For example, US surveys often find that acceptance of the reality of [[anthropogenic climate change]] rises with education among moderate or liberal survey respondents, but declines with education among the most conservative.<ref>{{cite journal | last1 = Hamilton
▲*''Interaction'' between genetic risk factors for [[Diabetes mellitus type 2|type 2 diabetes]] and diet (specifically, a "western" dietary pattern). The western dietary pattern was shown to increase diabetes risk for subjects with a high "genetic risk score", but not for other subjects.<ref>{{Cite journal | author = Lu, Q. | year = 2009 | title = Genetic predisposition, Western dietary pattern, and the risk of type 2 diabetes in men | journal = Am J Clin Nutr | volume = 89 | pages = 1453–1458 | doi = 10.3945/ajcn.2008.27249 | display-authors = 1 | issue = 5 | author2 = <Please add first missing authors to populate metadata.>| pmc = 2676999 }}</ref>
▲*''Interaction'' between education and political orientation, affecting general-public perceptions about climate change. For example, US surveys often find that acceptance of the reality of [[anthropogenic climate change]] rises with education among moderate or liberal survey respondents, but declines with education among the most conservative.<ref>Hamilton, L.C. 2011. "Education, politics and opinions about climate change: Evidence for interaction effects." ''[[Climatic Change (journal)|Climatic Change]]'' 104:231–242. {{DOI|10.1007/s10584-010-9957-8}}</ref><ref>McCright, A.M., 2011: Political orientation moderates Americans’ beliefs and concern about climate change. ''[[Climatic Change (journal)|Climatic Change]]'' DOI: 10.1007/s10584-010-9946-y</ref> Similar interactions have been observed to affect some non-climate science or environmental perceptions,<ref>Hamilton, L.C. & K. Saito. 2015. "[http://www.tandfonline.com/doi/abs/10.1080/09644016.2014.976485 A four-party view of U.S. environmental concern]." Environmental Politics 24(2):212–227. {{DOI| 10.1080/09644016.2014.976485}}</ref> and to operate with science literacy or other knowledge indicators in place of education.<ref>Kahan, D.M., H. Jenkins-Smith and D. Braman. 2011. "Cultural cognition of scientific consensus." Journal of Risk Research 14(2):147–174. doi: 10.1080/13669877.2010.511246</ref><ref>Hamilton, L.C., M.J. Cutler & A. Schaefer. 2012. "Public knowledge and concern about polar-region warming." ''[[Polar Geography]]'' 35(2):155–168. {{DOI| 10.1080/1088937X.2012.684155}}</ref>
== See also ==
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* [[Linear model]]
* [[Main effect]]
* [[Tukey's test of additivity]]
==
{{Reflist}}
==
▲* {{cite book |author=Bailey, R. A.|title=Design of Comparative Experiments|url=http://www.maths.qmul.ac.uk/~rab/DOEbook/|publisher=[http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=9780521683579 Cambridge University Press]|year=2008 |isbn=978-0-521-68357-9}} Pre-publication chapters are available on-line.
*[[David R. Cox|Cox, David R.]] and Reid, Nancy M. (2000) ''The theory of design of experiments'', Chapman & Hall/CRC. {{ISBN|1-58488-195-X}}
*{{Cite journal | doi = 10.1086/226678 | last1 = Southwood | first1 = K.E. | year = 1978 | title = Substantive Theory and Statistical Interaction: Five Models
*{{Cite journal | doi = 10.1093/pan/mpi014 | last1 = Brambor| first1 = T. | last2 = Clark | first2 = W. R. | year = 2006 | title = Understanding Interaction Models: Improving Empirical Analyses
▲|author=Hinkelmann, Klaus and [[Oscar Kempthorne|Kempthorne, Oscar]]
*{{Cite journal | doi = 10.3758/BRM.41.3.924 | last1 = Hayes | first1 = A. F. | last2 = Matthes | first2 = J. | year = 2009 | title = Computational procedures for probing interactions in OLS and logistic regression: SPSS and SAS implementations
▲|year=2008
*{{Cite journal | doi = 10.1007/s00181-012-0604-2 | last1 = Balli| first1 = H. O. | last2 = Sørensen| first2 = B. E. | year = 2012 | title = Interaction effects in econometrics
▲|title=Design and Analysis of Experiments, Volume I: Introduction to Experimental Design
▲|edition=Second
▲|publisher=Wiley
▲|isbn=978-0-471-72756-9
▲|author=[[Oscar Kempthorne|Kempthorne, Oscar]]
▲|year=1979
▲|title=The Design and Analysis of Experiments
▲*{{Cite journal | doi = 10.1086/226678 | last1 = Southwood | first1 = K.E. | year = 1978 | title = Substantive Theory and Statistical Interaction: Five Models | url = | journal = [[The American Journal of Sociology]] | volume = 83 | issue = 5| pages = 1154–1203 }}
▲*{{Cite journal | doi = 10.1093/pan/mpi014 | last1 = Brambor| first1 = T. | last2 = Clark | first2 = W. R. | year = 2006 | title = Understanding Interaction Models: Improving Empirical Analyses | url = http://pan.oxfordjournals.org/content/14/1/63.short | journal = Political Analysis | volume = 14 | issue = 1 | pages = 63–82 }}
▲*{{Cite journal | doi = 10.3758/BRM.41.3.924 | last1 = Hayes | first1 = A. F. | last2 = Matthes | first2 = J. | year = 2009 | title = Computational procedures for probing interactions in OLS and logistic regression: SPSS and SAS implementations | url = | journal = Behavior Research Methods | volume = 41 | issue = 3| pages = 924–936 | pmid = 19587209 }}
▲*{{Cite journal | doi = 10.1007/s00181-012-0604-2 | last1 = Balli| first1 = H. O. | last2 = Sørensen| first2 = B. E. | year = 2012 | title = Interaction effects in econometrics | url = | journal = Empirical Economics | volume = 43 | issue = x | pages = 1–21 }}
==External links==
*{{cite web |url= http://pages.towson.edu/mchamber/chapter7eco306.pdf |title= Using Indicator and Interaction Variables |access-date= 2010-02-03 |archive-url= https://web.archive.org/web/20160303165623/http://pages.towson.edu/mchamber/chapter7eco306.pdf |archive-date= 2016-03-03 |url-status= dead }} {{small|(158 [[Kibibyte|KiB]])}}
*[http://davis.foulger.info/papers/statisticalInteraction1979.htm Credibility and the Statistical Interaction Variable: Speaking Up for Multiplication as a Source of Understanding]
*[https://web.archive.org/web/20100212121505/http://www.ruf.rice.edu/~branton/interaction/faqfund.htm Fundamentals of Statistical Interactions: What is the difference between "main effects" and "interaction effects"? ]
{{Statistics}}
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