Most methods for analysing cluster-correlated biological data implicitly assume the ignorability of cluster sizes. When this assumption fails, the resulting inferences may be asymptotically invalid. Hoffman et al. (2001) proposed a simple but computationally intensive method, based on a large number of within-cluster resamples and associated separate estimating equations, that leads to asymptotically valid inferences whether the cluster sizes are ignorable or not. We study a simple method, based on a single inverse cluster size-weighted estimating equation, that avoids resampling and yet leads to asymptotically valid inferences. Simulation results are presented to assess the performance of the proposed method. We also propose Wald tests for ignorability of cluster sizes.

Balanced repeated replication, commonly used to estimate the variances of nonlinear estimators from stratified sampling designs, can run into problems in estimating a ratio of small domain totals or in the case of poststratification or unit nonresponse adjustment involving ratio weighting within cells if some cell sizes are small. This problem is mainly caused by a sharp perturbation of the weights to construct replicate estimates. To avoid extreme replicate estimates, Robert Fay proposed a modified balanced repeated replication method using gentler perturbation of weights. Theoretical properties of the modified method are studied here. The method is extended to the case of imputation for item nonresponse. Both smooth and nonsmooth estimators are studied.