WEIGHTING FOR UNEQUAL SELECTION PROBABILITIES IN MULTILEVEL MODELS

When multilevel models are estimated from survey data derived using mu ltistage sampling, unequal selection probabilities at any stage of sam pling may induce bias in standard estimators, unless the sources of th e unequal probabilities are fully controlled for in the covariates. Th is paper proposes alternative ways of weighting the estimation of a tw o-level model by using the reciprocals of the selection probabilities at each stage of sampling. Consistent estimators are obtained when bot h the sample number of level 2 units and the sample number of level 1 units within sampled level 2 units increase. Scaling of the weights is proposed to improve the properties of the estimators and to simplify computation. Variance estimators are also proposed. In a limited simul ation study the scaled weighted estimators are found to perform well, although non-negligible bias starts to arise for informative designs w hen the sample number of level 1 units becomes small. The variance est imators perform extremely well. The procedures are illustrated using d ata from the survey of psychiatric morbidity.