Assessment of Interactions in Multiple Regression

With theories growing ever more complex, analytic tools for their analysis and testing need to be developed. In exploring moderator variables that are found in theory tests in both experimental and nonexperimental research, we need to be careful to assess the interactions between the moderator variable and the predictor [Page 48]variable(s) in an appropriate way. Prior to the early 1990s, many nonexperimentalists using a correlational paradigm often used it inappropriately by (a) correctly creating a product term for the moderator and independent variable and then (b) inappropriately correlating it with the dependent variable. This paradigm is inappropriate for both theoretical and empirical reasons.

Theoretically, as Jacob Cohen has argued, while the interaction is carried by the product term, it is not the product term. The product term alone also carries variance due to the main effects of the independent variable and the moderator variable. The appropriate analysis is to partial out the main effects in a multiple regression analysis, as pointed out by Saunders as long ago as 1956.

Empirically, as Schmidt has shown, the correlation between a product term and the dependent variable is sensitive to the scale numbers used in the analysis. Changing from a scale of 1 to 5 to a scale of −2 to +2 will change the correlation dramatically. The proper analysis, as Arnold and Evans have shown, results in the incremental R2 between an equation containing just main effects and one containing the main effects plus the product term being invariant under linear transformations of the data (unlike the simple correlation between the product term and the dependent variable, which changes dramatically). This invariance translates to a proper test of theory only if the measurement scales and the underlying psychological constructs are linearly related. More recent developments involving structural equation modeling do not have this limitation.