Incorporating nonlinearity into mediation analyses

Abstract

Background: Mediation is an important issue considered in the behavioral, medical, and social sciences. It addresses situations where the effect of a predictor variable X on an outcome variable Y is explained to some extent by an intervening, mediator variable M. Methods for addressing mediation have been available for some time. While these methods continue to undergo refinement, the relationships underlying mediation are commonly treated as linear in the outcome Y, the predictor X, and the mediator M. These relationships, however, can be nonlinear. Methods are needed for assessing when mediation relationships can be treated as linear and for estimating them when they are nonlinear.

Methods: Existing adaptive regression methods based on fractional polynomials are extended here to address nonlinearity in mediation relationships, but assuming those relationships are monotonic as would be consistent with theories about directionality of such relationships.

Results: Example monotonic mediation analyses are provided assessing linear and monotonic mediation of the effect of family functioning (X) on a child's adaptation (Y) to a chronic condition by the difficulty (M) for the family in managing the child's condition. Example moderated monotonic mediation and simulation analyses are also presented.

Conclusions: Adaptive methods provide an effective way to incorporate possibly nonlinear monotonicity into mediation relationships.

Keywords: Adaptive regression; Childhood chronic conditions; Fractional polynomials; Mediation; Moderated mediation; Nonlinearity.

Fig. 1

Linear mediation relationships

Fig. 2

Monotonic mediation relationships (assuming $q\prime \ne 0$ and $p\prime \ne 0$).

Fig. 3

Normal plot for the linear mediation model for child adaptation as a function of family functioning as mediated by difficulty with independent omitted factors or errors

Fig. 4

Normal plot for the monotonic mediation model for child adaptation as a function of family functioning as mediated by difficulty controlling for having Crohn’s disease or a bowel disorder with independent omitted factors or errors

Fig. 5

Standardized residual plot for the monotonic mediation model for child adaptation as a function of family functioning as mediated by difficulty controlling for having Crohn’s disease or a bowel disorder with independent omitted factors or errors