On the uncanny relationship between non-normality and moderated multiple regression

Moderated multiple regression is one of the most established, popular methods to model non-linear associations in the social sciences. A mostly unacknowledged fact is that a particular type of non-normality can make the coefficient capturing this association non-zero. To further understand this connection, a theoretical investigation was conducted. A generalization of Isserli’s 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 non-symmetric, 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 non-normality is mimicking an interaction effect.