A semiparametric Bayesian GLM for longitudinal data and outcome dependent sampling

Thomas M. Menino Convention & Exhibition Center 

We propose a semiparametric Bayesian GLM for longitudinal outcomes and for outcome dependent sampling (ODS). We build on recently introduced semiparametric Bayesian GLM with a Dirichlet process prior on the GLM baseline distribution (DP-GLM).
The application to ODS is motivated by a study of sepsis patients. The outcome is Acute Respiratory Distress Syndrome (ARDS) by day 7. Due to high costs one covariate, glycocalyx (gcx) degradation is only recorded in a subset of patients, with fixed sampling probabilities as a function of the outcome.
We use a conditional likelihood, using only the sampling model for patients with recorded gcx, and accounting for the ODS. We show that posterior inference can be characterized similarly to independent sampling under the basic DP-GLM, with the only modification being an additional factor in the expression for the Levy intensity for the posterior on the non-parametric baseline measure. The factor arises from the ODS.

The extension to longitudinal outcomes introduces an extension of the basic DP-GLM model to dependent repeat measurements. Representing dependence by way of a copula model allows us to retain marginally the GLM regressions structure as before, including in particular the desired interpretation of parameters and inference. Similar to the ODS extension we can show that the posterior distribution on the random baseline measure remains in a similar form. The copula function introduces an additional factor in the Levy intensity.

Posterior characterizations in both extensions allow us to proceed with similar posterior simulation strategies as under the basic DP-GLM model. We will briefly describe posterior inference and illustrate it in two case studies.

Dependent Dirichlet process

Normalized random measures

Density regression