A Note on Formulae for Causal Mediation Analysis in an Odds Ratio Context

In a recent article, VanderWeele and Vansteelandt (American Journal of Epidemiology, 2010, 172:1339–1348) (hereafter VWV) build on results due to Judea Pearl on causal mediation analysis and derive simple closed-form expressions for so-called natural direct and indirect effects in an odds ratio context for a binary outcome and a continuous mediator. The expressions obtained by VWV make two key simplifying assumptions: A. The mediator is normally distributed with constant variance. B. The binary outcome is rare. Assumption A may not be appropriate in settings where, as can happen in routine epidemiologic applications, the distribution of the mediator variable is highly skew. However, in this commentary, the author establishes that under a key assumption of “no mediator–exposure interaction”, the simple formulae of VWV continue to hold even when assumption A is dropped. The author further shows that when the “no interaction” assumption is relaxed, the formula of VWV for the natural indirect effect continues to apply even if assumption A is also dropped. However, in this case, an alternative formula to that of VWV for the natural direct effect is derived without assumption A. When the binary outcome is not rare, simple closedform formulae for odds ratio natural direct and indirect effects are obtained upon replacing assumptions A and B with a single alternative assumption C that the mediator follows a so-called “bridge distribution”. For a non-rare outcome, a more general approach entails estimating mediation effects on the risk ratio scale upon replacing the outcome logistic regression with a risk ratio regression model, in which case assumptions A–C are not needed.