Total Effect Decomposition in Mediation Analysis with a Binary Outcome

Introduction: Mediation analysis is an important tool for prevention research. The goal of mediation analysis is to decompose the total effect of an intervention on an outcome into a direct and indirect effect (MacKinnon, 2008). The indirect effect is the part of the total effect that can be explained by a mediator variable and the part of the total effect that is not explained by the mediator variable is the direct effect. The indirect effect is traditionally calculated as either the product of the a and b coefficients, ab, or as the difference between the c and c’ coefficients, c-c’. These estimators have been shown to be mathematically equivalent when the mediator and outcome are continuous and the model equations are estimated with linear regression. However, when the outcome is binary and logisitc regression is used to estimate the model equations, ab and c-c’ no longer coincide (MacKinnon & Dwyer, 1993). The difference between the estimators is caused by the non-collapsibility of the odds ratio in logistic regression (VanderWeele, 2015). An important recent advancement in mediation analysis is the application of the potential outcomes framework. The potential outcomes framework provides a non-parametric decomposition of the total effect into the direct and indirect effect (Pearl, 2001). The total effect in the potential outcomes framework is defined as the sum of the direct and indirect effect, that is ab+c’. This total effect is referred to the marginal total effect, as this total effect is marginalized over all values of the mediator. Recently, authors outlined conditions for which a type of third variable, a confounder, affects the non-collapsibilty of odds ratio in logistic regression (Pang, Kaufman, & Platt, 2016). The goal of this paper is to extend that work to conditions for which a mediator affects the non-collapsibiliby of odds ratios in logisitc regression.

Method: A Monte Carlo simulation study was used to investigate the statistical performance of logistic regression when estimating total, direct, and indirect effects for a single mediator model. Simulation conditions included zero, small, medium, and large effect sizes for the a, b, and c’ paths in the single mediator model and sample sizes of 50, 100, 200, and 1000.

Results: Simulation results suggest the total effect differs from the marginal total effect as a function of the strength of the mediator-outcome relation, affecting the magnitude of the non-collapsibility effect.

Conclusions: Potential outcomes methods provide insight into the problem of non-collapsibility in odds ratios in logistic regression by providing a non-parametric decomposition of the total effect.