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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]]
|year=2008
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When measurement variables are employed in interactions, it is often desirable to work with centered versions, where the variable's mean (or some other reasonably central value) is set as zero. Centering can make the main effects in interaction models more interpretable, as it reduces the [[multicollinearity]] between the interaction term and the main effects.<ref>{{Cite journal|last1=Iacobucci|first1=Dawn|last2=Schneider|first2=Matthew J.|last3=Popovich|first3=Deidre L.|last4=Bakamitsos|first4=Georgios A.|date=2016|title=Mean centering helps alleviate "micro" but not "macro" multicollinearity|journal=Behavior Research Methods|language=en|volume=48|issue=4|pages=1308–1317|doi=10.3758/s13428-015-0624-x|pmid=26148824 |issn=1554-3528|doi-access=free}}</ref> The coefficient ''a'' in the equation above, for example, represents the effect of ''x''<sub>1</sub> when ''x''<sub>2</sub> equals zero.
[[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 | first1 = L.C. | last2 = Saito | first2 = K. | year = 2015 | title = A four-party view of U.S. environmental concern | journal = Environmental Politics | volume = 24 | issue = 2| pages = 212–227 | doi = 10.1080/09644016.2014.976485 | bibcode = 2015EnvPo..24..212H | s2cid = 154762226 }}</ref>
==Interaction plots==
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*''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>
*''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 | first1 = L.C. | year = 2011 | title = Education, politics and opinions about climate change: Evidence for interaction effects | url = https://scholars.unh.edu/cgi/viewcontent.cgi?article=1388&context=soc_facpub| journal = [[Climatic Change (journal)|Climatic Change]] | volume = 104 | issue = 2| pages = 231–242 | doi = 10.1007/s10584-010-9957-8 | bibcode = 2011ClCh..104..231H | s2cid = 16481640 }}</ref><ref>{{cite journal |last=McCright |first=A. M. |year=2011 |title=Political orientation moderates Americans' beliefs and concern about climate change |journal=[[Climatic Change (journal)|Climatic Change]] |doi=10.1007/s10584-010-9946-y |volume=104 |issue=2 |pages=243–253 |bibcode=2011ClCh..104..243M |s2cid=152795205 }}</ref> Similar interactions have been observed to affect some non-climate science or environmental perceptions,<ref>{{Cite journal | doi=10.1080/09644016.2014.976485|title = A four-party view of US environmental concern| journal=Environmental Politics| volume=24| issue=2| pages=212–227|year = 2015|last1 = Hamilton|first1 = Lawrence C.| last2=Saito| first2=Kei| bibcode=2015EnvPo..24..212H |s2cid = 154762226}}</ref> and to operate with science literacy or other knowledge indicators in place of education.<ref>{{cite journal | last1 = Kahan | first1 = D.M. | last2 = Jenkins-Smith | first2 = H. | last3 = Braman | first3 = D. | year = 2011 | title = Cultural cognition of scientific consensus | url = https://scholarship.law.gwu.edu/cgi/viewcontent.cgi?article=1269&context=faculty_publications| journal = Journal of Risk Research | volume = 14 | issue = 2| pages = 147–174 | doi = 10.1080/13669877.2010.511246 | hdl = 10.1080/13669877.2010.511246 | s2cid = 216092368 | hdl-access = free }}</ref><ref>{{cite journal | last1 = Hamilton | first1 = L.C. | last2 = Cutler | first2 = M.J. | last3 = Schaefer | first3 = A. | year = 2012 | title = Public knowledge and concern about polar-region warming | journal = [[Polar Geography]] | volume = 35 | issue = 2| pages = 155–168 | doi = 10.1080/1088937X.2012.684155 | bibcode = 2012PolGe..35..155H | s2cid = 12437794 }}</ref>
== See also ==
* [[Analysis of variance]]
* [[Factorial experiment]]
* [[Generalized randomized block design]]
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