  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.
