Inference with Survey Weights Roderick J.A. Little Abstract: This article considers the analysis of disproportionate stratified samples from a modelbased (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 stratumeffects models , yields as special cases standard probabilityweighted inferences favored by survey statisticians. The second class, which I call random stratumeffects models, yields estimators that behave like fixed stratumeffects estimators when the stratum sample sizes are large. In moderate samples they are compromises between estimators from fixed stratumeffects 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 meansquared error. For inference about a finite population mean, a fixed stratumeffects model leads to posterior probability intervals identical to standard randomization inference based on the stratified mean; random stratumeffects 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 nonnormal populations. For inference about a population slope, it is shown that classical designbased inference using the sample weights approximates Bayesian inference under a fixed stratumeffects model. Thus the need to model stratum effects leads to the probabilityweighted methods usually associated with designbased inference. Keywords: Bayesian methods; design consistency; JamesStein shrinkage; random effects; regression; stratified sampling; superpopulation models.
