Causal inference

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Mediation analysis aims at assessing the effect of some exposures on an outcome both through and around some mediators. While multi-mediator mediation analysis has been addressed in recent literature, the case with multiple exposures received little attention. With the presence of multiple exposure, we consider regularizations that allow variable selection and effects estimations simultaneously, hence stabilize model fit and account for model uncertainty. In the framework of linear structural equation models, we analytically show that a two-stage approach that regularizes on regression coefficients does not guarantee unimodal posterior distribution, and a product-in-coefficient approach that regularizes on direct and indirect effects tend to penalize excessively. We instead propose a regularized difference-in-coefficient approach that is free of these pitfalls. Using connections between frequentist regularizations and Bayesian hierarchical models with Laplace priors, we develop an efficient Markov chain Monte Carlo algorithm for posterior inference. Through simulations, we show that the proposed approach has good empirical performance in estimation. The methodology is illustrated using two data from reproductive epidemiology.