Memorial Session for Paul Rathouz

In this presentation I will discuss collaborative work with Paul Rathouz. I will review semiparametric estimation procedures for case-control study designs with longitudinal follow-up data as well as analogous estimation procedures for time-dependent outcome dependent sampling designs for binary data. I will then move on to describe ours and others work on outcome dependent sampling study designs and analysis procedures for longitudinal binary and quantitative response data. Finally, I will describe analysis procedures based on outcome dependent sampling designs for a semiparametric extension of the generalized linear model class. This semiparametric class of GLMs was introduced and has been developed by Paul.  

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. 

Power and sample size calculations for generalized linear models (GLMs) are, somewhat surprisingly, still largely ad hoc. Beyond linear regression, applied statisticians typically have to use methods tailored to a specific GLM, predictor, or distribution — what Paul Rathouz called "bespoke" solutions. Paul recognized an opportunity to define effect sizes that are general enough for routine study planning yet simple enough to be practical for applied investigators. He proposed two such measures, which he affectionately named 2SLiP and P2R2. In joint work with Paul and Shijie Yuan, we present these measures as a general framework for power and sample size calculations across GLMs. The framework accommodates arbitrary predictors and adjusters while requiring only limited information at the planning stage. We justify the framework theoretically and show its practical value through simulations and applied examples. This work grew out of Paul's gift for finding general solutions to the practical problems that emerge through collaboration.