A Class of Directed Acyclic Graphs with Mixed Data Types in Mediation Analysis

We propose a unified class of generalized structural equation models (GSEMs) with data of mixed types in mediation analysis, including continuous, categorical, and count variables. Such models extend substantially the classical linear structural equation model to accommodate many data types arising from the application of mediation analysis. Invoking the hierarchical modeling approach, we specify GSEMs by a copula joint distribution of outcome variable, mediator and exposure variable, in which marginal distributions are built upon generalized linear models (GLMs) with confounding factors. We discuss the identifiability conditions for the causal mediation effects in the counterfactual paradigm as well as the issue of mediation leakage, and develop an asymptotically efficient profile maximum likelihood estimation and inference for two key mediation estimands, natural direct effect and natural indirect effect, in different scenarios of mixed data types. The proposed new methodology is illustrated by a motivating epidemiological study that aims to investigate whether the tempo of reaching infancy BMI peak (delay or on time), an important early life growth milestone, may mediate the association between prenatal exposure to phthalates and pubertal health outcomes.