Interaction effect

In statistics, an interaction 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. Most commonly, interactions are considered in the context of regression analyses.

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" 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.

An "interaction variable" 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.

Thus, for a response Y and two variables x₁ and x₂ an additive model would be:

In contrast to this,

is an example of a model with an interaction between variables x₁ and x₂ ("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 ${\displaystyle d(x_{1}\times x_{2})}$, 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).

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