On the uncanny relationship between nonnormality and moderated multiple regression

Moderated multiple regression is one of the most established, popular methods to model nonlinear associations in social sciences. A mostly unacknowledged fact is that a particular type of nonnormality can make the coefficient capturing this association nonzero. To further understand this connection, a theoretical investigation was conducted. A generalization of Isserlis' theorem from multivariate normal densities to all elliptical densities is presented. Through this generalization, it was found that the family of elliptical densities (which includes the multivariate normal) cannot generate a product-interaction term. Moreover, asymmetry in lower and/or higher dimensions can induce a product-interaction term. Special case studies are presented where the variables are unidimensional symmetric, but jointly nonsymmetric, resulting in a moderated multiple regression model. A call is made for researchers to think carefully and decide when they have a true interaction term, theorized a priori, and when nonnormality is mimicking an interaction effect. (PsycInfo Database Record (c) 2026 APA, all rights reserved).