Probing curvilinear-by-linear interactions when the predictors are randomly sampled.

Curvilinear effects and curvilinear-by-linear interactions are hypothesized, tested, and probed in various fields of the social and behavioral sciences. When such effects are found, researchers traditionally have tested the simple slope, and more recently have utilized the Johnson-Neyman technique, often at the values of the sample statistics of the predictor(s). Both approaches implicitly assume that values on the predictors have been sampled according to a fixed sampling plan. More typically in social and behavioral research, however, values on the predictors can be more appropriately considered as randomly sampled from a multivariate population distribution. I show analytically and through simulation that for regression models containing a curvilinear-by-linear interaction, fixed and random regression models produce the same estimates but different standard errors of the simple slope at values of the sample statistics of the predictors. When values on the predictors are randomly sampled, treating them as fixed when testing the simple slope or generating conditional Johnson-Neyman confidence bands at values of the sample statistics of the predictors can lead to inflated Type I error rates and inaccurate coverage rates. Recommendations for researchers are provided, and directions for future research are discussed.