Inference for Causal Path-Specific Effects in Longitudinal Studies with Repeated Treatments and Multiple Mediators

Abstract

In longitudinal studies, repeated treatments may be followed by different sequences of mediators and time-varying covariates. We develop semiparametric inference for path-specific effects in this setting, allowing the number, type, and ordering of intermediate variables to vary across treatment occasions. Under consistency, positivity, edge consistency, and the cross-world restrictions of a nonparametric structural equation model with independent errors, path-specific means are identified by the edge g-formula. We derive two parameterizations of the efficient influence function of the identifying functional. The first follows the observed-data factorization and uses conditional intermediate-variable densities. The second uses sequential regressions to replace the corresponding integrations and Bayes' rule to express density ratios through treatment-history regressions, thereby avoiding direct estimation of intermediate-variable densities. Each yields a one-step estimator that permits flexible nuisance estimation. We establish product-rate conditions for asymptotic linearity and semiparametric efficiency. The density-based estimator has blockwise multiple robustness, generalizing triply robust estimation, while the regression-based estimator has sequential multiple robustness. We further develop a sensitivity analysis for selected violations of cross-world independence assumptions used to identify these effects. Simulations examine finite-sample performance and robustness to model misspecifications. An application to the Framingham Heart Study examines sustained-smoking effects on systolic blood pressure along pathways through repeated body mass index measurements. The proposed methods are implemented in the R package flexPaths on Gihub at xxou/flexPaths.