A Simple Statistic for Comparing Moderation of Slopes and Correlations - Smithson, Michael (Frontiers in Psychology, 2012)
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 $\rho$ is (un)moderated by $Z$ is equivalent to the extent to which the regression coefficients $\beta_y$ and $\beta_x$ are (un)moderated by $Z$ iff the variance ratio $\sigma_y^2/\sigma_x^2$ 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 $\rho$ 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.