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.
【저자키워드】 correlation, Regression, heteroscedasticity, moderator effects, interaction effects,