Direct effect and indirect effect on an outcome under nonlinear modeling

Exact formulae relating parameters in conditional and reduced generalized linear models are introduced where the reduced model omits a continuous mediator from the conditional model. For certain link functions including logit, the natural direct effect and the natural indirect effect of the counterfactual method are smaller in magnitude than, respectively, the direct effect used by the difference method and the indirect effect by the product method. Contrary to what is implicitly assumed in Jiang and VanderWeele [11] for logit link, the total effect of the counterfactual method and the total effect used for the difference method are generally not the same. They are equal to each other only under special situations. For accelerated failure time models the difference method and the product method are equivalent regardless of censoring or not, a result stated in VanderWeele [6] in the absence of censorship but proved in a misleading manner. For proportional hazards models, maximum likelihood analysis indicates that these two methods can be equivalent in the absence of censorship. In the case of logit link, one can focus on the treatment effect on the marginalized odds instead of the odds of the marginalized event so that the product method would be equivalent to the difference method. Similarly, for the proportional hazards model, one can focus on the treatment effect on the marginalized hazards instead of the hazards for the reduced model.