AUTHOR=Smithson Michael 

TITLE=A Simple Statistic for Comparing Moderation of Slopes and Correlations

JOURNAL=Frontiers in Psychology

VOLUME=3

YEAR=2012

URL=https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2012.00231

DOI=10.3389/fpsyg.2012.00231

ISSN=1664-1078

ABSTRACT=<p>Given a linear relationship between two continuous random variables <italic>X</italic> and <italic>Y</italic> that may be moderated by a third, <italic>Z</italic>, the extent to which the correlation ρ is (un)moderated by <italic>Z</italic> is equivalent to the extent to which the regression coefficients <italic>β<sub>y</sub></italic> and <italic>β<sub>x</sub></italic> are (un)moderated by <italic>Z</italic> iff the variance ratio <inline-formula><mml:math id="M1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo class="MathClass-bin">∕</mml:mo><mml:msubsup><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:math></inline-formula> is constant over the range or states of <italic>Z</italic>. Otherwise, moderation of slopes and of correlations must diverge. Most of the literature on this issue focuses on tests for heterogeneity of variance in <italic>Y</italic>, 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.</p>