  We consider modeling a binary response variable together with a set of
covariates for two groups under observational data. The grouping variable can
be the confounding variable (the common cause of treatment and outcome),
gender, case/control, ethnicity, etc. Given the covariates and a binary latent
variable, the goal is to construct two directed acyclic graphs (DAGs), while
sharing some common parameters. The set of nodes, which represent the
variables, are the same for both groups but the directed edges between nodes,
which represent the causal relationships between the variables, can be
potentially different. For each group, we also estimate the effect size for
each node. We assume that each group follows a Gaussian distribution under its
DAG. Given the parent nodes, the joint distribution of DAG is conditionally
independent due to the Markov property of DAGs. We introduce the concept of
Gaussian DAG-probit model under two groups and hence doubly Gaussian DAG-probit
model. To estimate the skeleton of the DAGs and the model parameters, we took
samples from the posterior distribution of doubly Gaussian DAG-probit model via
MCMC method. We validated the proposed method using a comprehensive simulation
experiment and applied it on two real datasets. Furthermore, we validated the
results of the real data analysis using well-known experimental studies to show
the value of the proposed grouping variable in the causality domain.
