# Estimation of Direct and Indirect Causal Effects in Longitudinal Studies

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## Abstract

The causal effect of a treatment on an outcome is generally mediated by several intermediate variables. Estimation of the component of the causal effect of a treatment that is mediated by a given intermediate variable (the indirect effect of the treatment), and the component that is not mediated by that intermediate variable (the direct effect of the treatment) is often relevant to mechanistic understanding and to the design of clinical and public health interventions. Under the assumption of no-unmeasured confounders, Robins & Greenland (1992) and Pearl (2000), develop two identifiability results for direct and indirect causal effects. They define an individual direct effect as the counterfactual effect of a treatment on an outcome when the intermediate variable is set at the value it would have had if the individual had not been treated, and the population direct effect as the mean of these individual counterfactual direct effects. The identifiability result developed by Robins & Greenland (1992) relies on an additional ``No-Interaction Assumption'', while the identifiability result developed by Pearl (2000) relies on a particular assumption about conditional independence in the population being sampled. Both assumptions are considered very restrictive. As a result, estimation of direct and indirect effects has been considered infeasible in many settings. We show that the identifiability result of Pearl (2000), also holds under a new conditional independence assumption which states that, within strata of baseline covariates, the individual direct effect at a fixed level of the intermediate variable is independent of the no-treatment counterfactual intermediate variable. We argue that our assumption is typically less restrictive than both the assumption of Pearl (2000), and the ``No-interaction Assumption'' of Robins & Greenland (1992). We also generalize the current definition of the direct (and indirect) effect of a treatment as the population mean of individual counterfactual direct (and indirect) effects to 1) a general parameter of the population distribution of individual counterfactual direct (and indirect) effects, and 2) change of a general parameter of the population distribution of the appropriate counterfactual treatment-specific outcome. Subsequently, we generalize our identifiability result for the mean to identifiability results for these generally defined direct effects. We also discuss methods for modelling, testing, and estimation, and we illustrate our results throughout using an example drawn from the treatment of HIV infection.

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