This article considers the analysis of disproportionate stratified samples from a model-based (Bayesian) perspective. It is argued that a key element of models for such samples is that they explicitly account for differences between strata, even when the target quantity is aggregated over strata. Two general classes of models with this property are proposed. The first class, which I call fixed stratum-effects models , yields as special cases standard probability-weighted inferences favored by survey statisticians. The second class, which I call random stratum-effects models, yields estimators that behave like fixed stratum-effects estimators when the stratum sample sizes are large. In moderate samples they are compromises between estimators from fixed stratum-effects models and estimators from models that ignore stratum effects. In simple settings these are weighted estimators where the weights have been smoothed towards one, yielding in certain cases a reduction in mean-squared error. For inference about a finite population mean, a fixed stratum-effects model leads to posterior probability intervals identical to standard randomization inference based on the stratified mean; random stratum-effects models yield estimators with smoothed weights. Repeated sampling properties of these estimators and associated probability intervals are illustrated by a simulation study on normal and non-normal populations.
For inference about a population slope, it is shown that classical design-based inference using the sample weights approximates Bayesian inference under a fixed stratum-effects model. Thus the need to model stratum effects leads to the probability-weighted methods usually associated with design-based inference.