  Microbiome data require statistical models that can simultaneously decode
microbes' reaction to the environment and interactions among microbes. While a
multiresponse linear regression model seems like a straight-forward solution,
we argue that treating it as a graphical model is flawed given that the
regression coefficient matrix does not encode the conditional dependence
structure between response and predictor nodes as it does not represent the
adjacency matrix. This observation is especially important in biological
settings when we have prior knowledge on the edges from specific experimental
interventions that can only be properly encoded under a conditional dependence
model. Here, we propose a chain graph model with two sets of nodes (predictors
and responses) whose solution yields a graph with edges that indeed represent
conditional dependence and thus, agrees with the experimenter's intuition on
the average behavior of nodes under treatment. The solution to our model is
sparse via Bayesian LASSO. In addition, we propose an adaptive extension so
that different shrinkage can be applied to different edges to incorporate
edge-specific prior knowledge. Our model is computationally inexpensive through
an efficient Gibbs sampling algorithm and can account for binary, counting and
compositional responses via appropriate hierarchical structure. We apply our
model to a human gut and a soil microbial compositional datasets and we
highlight that CG-LASSO can estimate biologically meaningful network structures
in the data. The CG-LASSO software is available as an R package at
https://github.com/YunyiShen/CAR-LASSO.
