Editing Interaction (statistics) (section)

==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.

[[Image:Quantitative interaction.svg|right|thumb|250px|The binary factor ''A'' and the quantitative variable ''X'' interact (are non-additive) when analyzed with respect to the outcome variable ''Y''.]]

Thus, for a response ''Y'' and two variables ''x''<sub>1</sub> and ''x''<sub>2</sub> an ''additive'' model would be:

:<math>Y = c + ax_1 + bx_2 + \text{error}\,</math>

In contrast to this,

:<math>Y = c + ax_1 + bx_2 + d(x_1\times x_2) + \text{error} \,</math>

is an example of a model with an ''interaction'' between variables ''x''<sub>1</sub> and ''x''<sub>2</sub> ("error" refers to the [[random variable]] whose value is that by which ''Y'' differs from the [[expected value]] of ''Y''; see [[errors and residuals in statistics]]). Often, models are presented without the interaction term <math>d(x_1\times x_2)</math>, but this confounds the main effect and interaction effect (i.e., without specifying the interaction term, it is possible that any main effect found is actually due to an interaction). 

Moreover, the [[Principle of marginality|hierarchical principle]] rules that if a model includes interaction between variables, it is also necessary to include the main effects, regardless of their own statistical significance.<ref>{{cite book |last1=James |first1=Gareth |last2=Witten |first2=Daniela |last3=Hastie |first3=Trevor |last4=Tibshirani |first4=Robert |title=An introduction to statistical learning: with applications in R |date=2021 |publisher=Springer |location=New York, NY |isbn=978-1-0716-1418-1 |page=103 |edition=Second |url=https://link.springer.com/book/10.1007/978-1-0716-1418-1 |access-date=29 October 2024}}</ref>