Probing Interactions in Fixed and Multilevel Regression: Inferential and Graphical Techniques

TLDR

Conditional relations where one predictor’s effect depends on another are commonly examined as multiplicative interactions in fixed‑ and random‑effects regression, yet the widely used Johnson‑Neyman technique is limited to categorical‑by‑continuous interactions in fixed‑effects models. The article aims to extend the Johnson‑Neyman method so it can test a broader range of interactions in both fixed‑ and random‑effects regression. The authors review existing probing methods, derive analytic expressions to generalize these tests, and illustrate the advantages of the extended Johnson‑Neyman technique over simple slopes with three empirical examples.

Abstract

Many important research hypotheses concern conditional relations in which the effect of one predictor varies with the value of another. Such relations are commonly evaluated as multiplicative interactions and can be tested in both fixed- and random-effects regression. Often, these interactive effects must be further probed to fully explicate the nature of the conditional relation. The most common method for probing interactions is to test simple slopes at specific levels of the predictors. A more general method is the Johnson-Neyman (J-N) technique. This technique is not widely used, however, because it is currently limited to categorical by continuous interactions in fixed-effects regression and has yet to be extended to the broader class of random-effects regression models. The goal of our article is to generalize the J-N technique to allow for tests of a variety of interactions that arise in both fixed- and random-effects regression. We review existing methods for probing interactions, explicate the analytic expressions needed to expand these tests to a wider set of conditions, and demonstrate the advantages of the J-N technique relative to simple slopes with three empirical examples.