We determine factors, both hospital-specific and market area-specific, associated with hospitals' propensities for discharging Medicare stroke patients to skilled nursing facilities (SNF's) in California and Florida. 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. It is found that 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. We develop fully Bayesian inference that takes into account uncertainty about the hyperparameters, and we find that this also allows us to test for overdispersion in a natural way. The number of observed zeros (i.e., hospitals that sent no stroke patients to a SNF) is excessive compared to the number expected from a standard logistic regression model and is fit better by the hierarchical betabinomial model. The factors associated with discharge to SNF differ between California and Florida. In California the case-mix index and percent Medicaid admissions of the hospital, as well as the per capita income for the area and whether there is a rehabilitation facility in the area, are associated with discharge rates to SNF's. In Florida, whether there is a rehabilitation facility in the area is the only factor that exhibits association with discharge rates to SNF's.