To the Editor:
Causal mediation analysis is a topic of intense research.1 In this note, we derive an analytic expression for the proportion explained by mediation on the survival function scale, marginally over measured confounders. This expression has a simple form that does not depend on time, on the confounder distribution, or on the assumed models for the confounder effects on the mediator and on the outcome.
Let , , and be the exposure, mediator and time-to-event outcome of interest, respectively. In standard potential outcome/counterfactual notation2 we let be the outcome for a given subject, had the exposure been set to . Similarly, we let be the outcome for a given subject, had the exposure been set to , and had the mediator been set to whatever value it would have had, if the exposure would have been set to . We define the counterfactual survival functions and .
On the survival function scale, the total exposure effect comparing levels and is defined as
and the natural indirect and direct effects are usually defined as
respectively.3 These natural indirect and direct effects add up to the total effect. By symmetry, we may also define the natural indirect and direct effects as
respectively, which also add up to the total effect. The proportion explained by mediation is defined as the indirect effect divided with the total effect:
where is either equal to or , which correspond to the definitions of indirect effect in (1) and (2), respectively.
We make the following assumptions:
- Assumption 1 Conditional on measured confounders , there is no confounding for the exposure, mediator and outcome, and no mediator-outcome confounders that are affected by the exposure.
- Assumption 2 The mediator follows the linear regression model where is an arbitrary function of , specified by the analyst.
- Assumption 3 The outcome follows the Cox PH model , where is the unspecified baseline hazard, and is an arbitrary function of , specified by the analyst.
- Assumption 4 The outcome is rare, so that the cumulative hazard function is close to 0 for the specific time under consideration, and for all levels of .
Under the assumptions 1–4, it can be shown (see the eAppendix; http://links.lww.com/EDE/B607) that
The regression coefficients and measure the effect of the exposure on the mediator, and the effect of the mediator on the outcome, respectively. If either or , then the expression in (3) is equal to 0. The regression coefficient measures the direct effect of the exposure on the outcome. If , then the expression in (3) is equal to 1.
Although generally depends on the time , the expression in (3) does not. Thus, under assumptions 1–4 the proportion explained by mediation is approximately constant across time. Furthermore, the expression in (3) does not depend on the confounder distribution or on the effect of the confounders on the mediator and outcome. The latter is particularly useful, since it allows the analyst to use realistic and elaborate (e.g., spline) functions for and , thus reducing the risk for bias due to model misspecification without making the target parameter more complicated.
The expression in (3) can easily be estimated by fitting the mediator and outcome models, and plugging the obtained estimates of () into (3). Once has been estimated, one would typically want to compute a confidence interval around the estimate. It can be shown (see the eAppendix; http://links.lww.com/EDE/B607) that the estimate of has an asymptotic normal distribution, with a standard error that can be obtained with the delta method. Thus, we may use the standard Wald 95% confidence interval estimate 1.96 standard error.
In the eAppendix; http://links.lww.com/EDE/B607, we provide an R function that implements the proposed estimator, and illustrate the finite samples properties of the proposed estimator by a simulation study.
Department of Medical Epidemiology and Biostatistics
1. VanderWeele T. Explanation in causal inference. 2015.New York, NY: Oxford University Press;
2. Pearl J. Causality: Models, Reasoning and Inference. 2009.2nd ed. Cambridge, UK: Cambridge University Press;
3. VanderWeele TJ. Causal mediation analysis with survival data. Epidemiology. 2011;22:582–585.