We propose new model-based methods for unit non-response in two-stage survey samples. A commonly used design-based adjustment weights respondents by the inverse of the estimated response rate in each cluster (method WT). This approach is consistent if the response probabilities are constant within clusters but is potentially inefficient when the estimated cluster response rates are very variable. Clusters can be collapsed to increase precision, but this may introduce bias. We consider here the model-based approach to survey inference that treats the clusters as random effects. We note that, from a model-based perspective, a missing data mechanism that assumes that the response rate varies across clusters is non-ignorable, and we propose the term cluster-specific non-ignorable (CSNI) non-response to describe this mechanism. We show that the standard random-effects model estimator RE of the population mean is biased under CSNI non-response, and we propose two modifications of RE to correct this bias. One approach includes the observed response rate as a cluster level covariate (method RERR), and the other is based on a probit model for response (method NI1). The RERR approach is simpler than NI1 but approximate, in that uncertainty in estimating the response rates is not taken into account. In addition, a simple method that corrects the bias of RE by reweighting (method RWRE) is also discussed. We show by simulations that estimators from RERR and NI1 can correct the bias of RE under CSNI non-response and have comparable or lower root-mean-squared error than WT in a variety of simulation settings, and RWRE has similar performance to WT. We also consider another non-ignorable response model estimate of the population mean (NI2) that removes the bias of WT, RWRE, RERR and NI1 under an outcome-specific non-ignorable response mechanism where non-response depends directly on the individual level survey outcomes. However, that estimate is not robust to model misspecification. The various methods are compared on a data set from the Detroit Dental Health Project. Copyright 2007 Royal Statistical Society.
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