A simple statistic for comparing moderation of slopes and correlations

Given a linear relationship between two continuous random variables X and Y that may be moderated by a third, Z, the extent to which the correlation ρ is (un)moderated by Z is equivalent to the extent to which the regression coefficients β _y and β _x are (un)moderated by Z iff the variance ratio σ _y ²/σ _x ² is constant over the range or states of Z. Otherwise, moderation of slopes and of correlations must diverge. Most of the literature on this issue focuses on tests for heterogeneity of variance in Y, and a test for this ratio has not been investigated. Given that regression coefficients are proportional to ρ via this ratio, accurate tests, and estimations of it would have several uses.This paper presents such a test for both a discrete and continuous moderator and evaluates its Type I error rate and power under unequal sample sizes and departures from normality. It also provides a unified approach to modeling moderated slopes and correlations with categorical moderators via structural equations models.