A simple statistic for comparing moderation of slopes and correlations

Area of Science:

• Statistics
• Psychometrics
• Quantitative Psychology

Background:

• Linear relationships between continuous variables (X, Y) can be moderated by a third variable (Z).
• Equivalence between moderated correlation (ρ) and moderated regression coefficients (β) depends on a constant variance ratio.
• Existing literature often overlooks the variance ratio, focusing primarily on heterogeneity of variance in Y.

Purpose of the Study:

• To investigate the conditions under which moderated correlations and regression slopes diverge.
• To develop and evaluate a novel statistical test for the variance ratio, crucial for assessing moderation equivalence.
• To provide a unified framework for modeling moderated relationships using structural equation models.

Main Methods:

• Development of a statistical test for the variance ratio applicable to both discrete and continuous moderators.
• Evaluation of the test's Type I error rate and statistical power under conditions of unequal sample sizes and non-normal data.
• Application of structural equation models for a unified approach to modeling moderated slopes and correlations with categorical moderators.

Main Results:

• The proposed test for the variance ratio is effective in assessing the equivalence of moderation effects on correlations and regression coefficients.
• The study provides performance metrics (Type I error, power) for the new test across various conditions.
• Structural equation modeling offers a flexible approach for analyzing complex moderated relationships.

Conclusions:

• The variance ratio is a critical factor in determining the equivalence of moderation effects on correlations and regression slopes.
• The developed test offers a valuable tool for researchers analyzing moderated relationships, particularly when the variance ratio is not constant.
• This work enhances the understanding and modeling of moderation in statistical analyses.