In statistics and regression analysis, moderation (also known as effect modification) occurs when the relationship between two variables depends on a third variable. The third variable is referred to as the moderator variable (or effect modifier) or simply the moderator (or modifier). The effect of a moderating variable is characterized statistically as an interaction; that is, a categorical (e.g., sex, ethnicity, class) or continuous (e.g., age, level of reward) variable that is associated with the direction and/or magnitude of the relation between dependent and independent variables. Specifically within a correlational analysis framework, a moderator is a third variable that affects the zero-order correlation between two other variables, or the value of the slope of the dependent variable on the independent variable. In analysis of variance (ANOVA) terms, a basic moderator effect can be represented as an interaction between a focal independent variable and a factor that specifies the appropriate conditions for its operation.
Example
Moderation analysis in the behavioral sciences involves the use of linear multiple regression analysis or causal modelling. To quantify the effect of a moderating variable in multiple regression analyses, regressing random variable Y on X, an additional term is added to the model. This term is the interaction between X and the proposed moderating variable. Thus, for a response Y and two variables: x1 and moderating variable x2,:
Y = b 0 + b 1 x 1 + b 2 x 2 + b 3 ( x 1 × x 2 ) + ε {\displaystyle Y=b_{0}+b_{1}x_{1}+b_{2}x_{2}+b_{3}(x_{1}\times x_{2})+\varepsilon \,}
In this case, the role of x2 as a moderating variable is accomplished by evaluating b3, the parameter estimate for the interaction term. See linear regression for discussion of statistical evaluation of parameter estimates in regression analyses. For binary outcomes analyzed with logistic regression, however, the coefficient of an interaction term does not necessarily measure moderation on the probability scale. Because the logistic transformation is nonlinear, an interaction can be significant on the log-odds scale but not on the probability scale, or vice versa; probability-scale moderation can instead be evaluated using differences in marginal probabilities.
Multicollinearity in moderated regression
In moderated regression analysis, a new interaction predictor ( x 1 x 2 {\displaystyle x_{1}x_{2}} ) is calculated. However, the new interaction term may be correlated with the two main effects terms used to calculate it. This is the problem of multicollinearity in moderated regression. Multicollinearity tends to cause coefficients to be estimated with higher standard errors and hence greater uncertainty. Mean-centering (subtracting raw scores from the mean) may reduce multicollinearity, resulting in more interpretable regression coefficients. However, it does not affect the overall model fit.
Post-hoc probing of interactions Like simple main effect analysis in ANOVA, in post-hoc probing of interactions in regression, we are examining the simple slope of one independent variable at the specific values of the other independent variable. Below is an example of probing two-way interactions. In what follows, the regression equation with two variables A and B and an interaction term A*B,
Y = b 0 + b 1 A + b 2 B + b 3 A ∗ B + ε {\displaystyle Y=b_{0}+b_{1}A+b_{2}B+b_{3}A*B+\varepsilon }
will be considered.
Two categorical independent variables If both of the independent variables are categorical variables, we can analyze the results of the regression for one independent variable at a specific level of the other independent variable. For example, suppose that both A and B are single dummy coded (0,1) variables, and that A represents ethnicity (0 = European Americans, 1 = East Asians) and B represents the condition in the study (0 = control, 1 = experimental). Then the interaction effect shows whether the effect of condition on the dependent variable Y is different for European Americans and East Asians and whether the effect of ethnic status is different for the two conditions. The coefficient of A shows the ethnicity effect on Y for the control condition, while the coefficient of B shows the effect of imposing the experimental condition for European American participants. To probe if there is any significant difference between European Americans and East Asians in the experimental condition, we can simply run the analysis with the condition variable reverse-coded (0 = experimental, 1 = control), so that the coefficient for ethnicity represents the ethnicity effect on Y in the experimental condition. In a similar vein, if we want to see whether the treatment has an effect for East Asian participants, we can reverse code the ethnicity variable (0 = East Asians, 1 = European Americans).
One categorical and one continuous independent variable
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