An extension to the Johnson-Neyman test for generalized linear models with curvilinear effects

The Johnson-Neyman technique is widely used in observational research to identify regions of a moderator variable where the linear relationship between a predictor and an outcome variable is statistically significant. In practice, moderation is often assessed using the pick-a-point approach, which examines effects at arbitrarily chosen moderator values and can produce incomplete or misleading inferences, especially when the underlying relationships are curvilinear. Few procedures have been proposed to address the need to evaluate the effect of moderators in curvilinear relationships between outcome and predictor considering quadratic terms or more complex relationships. However, such curvilinear relationships are fundamental in psychology, medicine, economics, and other disciplines to better explain phenomena where outcomes do not change in a strictly linear manner with respect to influencing variables. This paper proposes a new approach to extend the Johnson-Neyman test to evaluate the effect of a moderator in curvilinear relationships and provides accessible tools for its implementation alongside the conventional pick-a-point approach. This extension covers both linear regression and generalized linear regression models. The curvilinear extension of the Johnson-Neyman method is validated by a simulation study and applied to a real data set to investigate the relationship between cardiac vagal tone and cooperation in school-aged children, taking into account the moderating role of age.