Quantifying and Testing Indirect Effects in Simple Mediation Models When the Constituent Paths Are Nonlinear

Abstract

Most treatments of indirect effects and mediation in the statistical methods literature and the corresponding methods used by behavioral scientists have assumed linear relationships between variables in the causal system. Here we describe and extend a method first introduced by Stolzenberg (1980) for estimating indirect effects in models of mediators and outcomes that are nonlinear functions but linear in their parameters. We introduce the concept of the instantaneous indirect effect of X on Y through M and illustrate its computation and describe a bootstrapping procedure for inference. Mplus code as well as SPSS and SAS macros are provided to facilitate the adoption of this approach and ease the computational burden on the researcher.

Notes

¹We offer our appreciation, indebtedness, and thanks to Carsten K. W. De Dreu and Daniel R. Ames, who generously donated their data for use as examples in this article.

²Examples of functions that are not differentiable at every point, yet are commonly encountered in the behavioral sciences, are step functions, splines, and reciprocal functions like 1/X. If functions that are not differentiable at every point within the range of X and M in one's data, care must be taken to avoid evaluating the indirect effect at those values of X at which the instantaneous indirect effect is actually undefined.

³Note that indirect effects operate on Y in the context of direct effects as well. So although a change in X from point x might increase Y through M, it is possible that the change in X could yield a net decrease in Y if the direct effect of X is in the opposite direction and large enough to compensate for the indirect effect.

⁴ Ames and Flynn (2007) report their analysis using standardized variables. Our analysis is based on the unstandardized variables, and thus all model coefficients we report are unstandardized coefficients, which is the customary metric in causal modeling.

⁵The coefficient for the square of assertiveness was statistically significant in the model of instrumental outcomes as was the coefficient for instrumental outcomes in the model of leadership ability.

⁶Although we defer to Carsten De Dreu's expertise and judgment as to what is the most sensible and parsimonious functional form of this relationship, we did find in the data we were provided that an exponential function fit as well if not better than a linear function. We make this modification to De Dreu's model merely to illustrate the application of the method to a model with a nonlinear M ⇉ Y association.

⁷The Mplus code we provide requires the user to enter the square of X or M as data at input when estimating a quadratic model. If centering is going to be undertaken, this should be done prior to squaring. Our SPSS and SAS macros automatically generate the square of X or M when estimating a quadratic model, so the user needs to provide only the centered values of X or M at input.