When Do Interaction/Moderation Effects Stabilize in Linear Regression?

Two-way interaction effects in linear regression occur when the relation between two variables changes depending on the level of a third. Despite their frequent use, interactions are notoriously difficult to estimate accurately and test for statistical significance because of small effect sizes and low reliability. In this study, we used Monte Carlo simulations to establish stability thresholds for two-way interactions between continuous variables across combinations of reliability (0.7-1.0), main effect size (0.1-0.5), collinearity (0.1-0.5), and interaction effect size (0.05-0.2). Stability was defined as the consistency of estimated effect sizes across repeated samples of the same size from the same population and operationalized using modified definitions of the corridor of stability and point of stability from Schönbrodt and Perugini. Results show that the stability of interaction estimates is primarily determined by sample size and predictor reliability. The case representing a realistic psychology field study, in which researchers have limited control over variables, stabilized at $n=\mathrm{3,800}$ , requiring 72% statistical power. At $n\le 100$ , 11% to 45% of the estimates were incorrectly signed (i.e., negative when the true effect was positive). Most psychology studies enroll far fewer than 500 participants, and our results indicate many published interactions may be unstable. Analyses involving highly reliable predictors, such as group assignment in experimental designs, may stabilize at lower sample sizes because they attenuate the expected effect size less than variables with more measurement error. Researchers are encouraged to avoid routine tests of two-way interactions unless sample size and reliability are adequate and hypotheses are specified a priori.