Kahn-Raftery (1996) Discharge Rates of Medicare Stroke Patients to Skilled Nursing Facilities: Bayesian Logistic Regression With Unobserved Heterogeneity

Summary

This paper generalizes logistic regression to a beta-binomial hierarchical model to handle unobserved heterogeneity and overdispersion in grouped binary data, applied to hospitals' rates of discharging Medicare stroke patients to skilled nursing facilities (SNFs) in California and Florida. The data show far more hospitals with a zero SNF-discharge rate than a standard logistic model predicts; rather than attribute this to distinct "policies," the authors model it as extra-binomial variation via a second-stage beta distribution whose parameters carry the covariate information. The resulting posterior mean of each hospital's rate is (approximately) a shrinkage estimator — a weighted average of the logistic-regression ensemble estimate and the hospital's own observed rate — and the fully Bayesian treatment of the hyperparameters yields a natural test for overdispersion.

Key Claims

• Beta-binomial hierarchical model. The number of SNF discharges is binomial given a hospital-specific rate $p_i$, and $p_i$ is drawn from a beta second-stage distribution whose (hyper)parameters depend on covariates through a regression — a random-effects / overdispersion generalization of logistic regression that admits extra-binomial variation.
• Excess zeros = overdispersion, not policy. The large number of hospitals with zero SNF discharges (even among those with many stroke patients) exceeds what a logistic model implies; the beta-binomial model fits it as unobserved heterogeneity in the true rates rather than as a separate zero-generating mechanism.
• Posterior mean is a shrinkage estimator. The estimate for each hospital is approximately a weighted average of the covariate-driven ensemble (logistic-regression) estimate and the observed rate, with weight shifting toward the ensemble when the hospital has few discharges (little own-information) — the empirical-Bayes borrowing-strength logic.
• Fully Bayesian, with overdispersion testing. Propagating uncertainty about the hyperparameters (via Laplace approximation and locally uniform priors, in the spirit of Raftery's approximate Bayes factors for GLMs) gives honest inference and a natural way to test for overdispersion.
• Substantive findings. Determinants of SNF discharge differ by state: in California the case-mix index, percent-Medicaid admissions, area per-capita income, and presence of an area rehabilitation facility matter; in Florida only the presence of an area rehabilitation facility is associated with discharge rates.

Concepts Introduced or Extended

• Bayesian Logistic Regression
• Count-Data Regression
• Bayesian Hierarchical Model

Entities Mentioned

• Michael J. Kahn
• Adrian E. Raftery

Quotes

"Logistic regression is generalized to the case of a betabinomial, hierarchical model, in which covariate information is included in the hyperparameters of the second-stage beta distribution."

"The posterior mean of the proportion discharged to SNF is approximately a weighted average (i.e., shrinkage estimator) of the logistic regression estimator and the observed rate."

My Take

The paper is a clean applied demonstration that "too many zeros" in a binary-rate dataset is usually a symptom of overdispersion, and that a beta-binomial hierarchy is the principled fix — it lets each unit have its own latent rate while borrowing strength through covariates, which is exactly the shrinkage/empirical-Bayes idea instantiated for proportions. Methodologically it is a nice period piece: it gets full Bayesian inference for a hierarchical GLM through Laplace approximations rather than MCMC (Raftery's approximate-Bayes-factor machinery), the pragmatic route just before Gibbs sampling took over this class of models. For the wiki it links the Bayesian logistic and count/overdispersion material to the hierarchical-shrinkage thread, and its overdispersion-testing angle is the constructive counterpart to simply noting that a binomial model underfits.