Weighting for unequal selection probabilities in multilevel models

D. Pfeffermann, C. J. Skinner*, D. J. Holmes, H. Goldstein, J. Rasbash

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

313 Scopus citations

Abstract

When multilevel models are estimated from survey data derived using multistage sampling, unequal selection probabilities at any stage of sampling may induce bias in standard estimators, unless the sources of the unequal probabilities are fully controlled for in the covariates. This paper proposes alternative ways of weighting the estimation of a two-level model by using the reciprocals of the selection probabilities at each stage of sampling. Consistent estimators are obtained when both 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 simulation study the scaled weighted estimators are found to perform well, although non-negligible bias starts to arise for informative designs when the sample number of level 1 units becomes small. The variance estimators perform extremely well. The procedures are illustrated using data from the survey of psychiatric morbidity.

Original languageEnglish
Pages (from-to)23-40
Number of pages18
JournalJournal of the Royal Statistical Society. Series B: Statistical Methodology
Volume60
Issue number1
DOIs
StatePublished - 1998

Keywords

  • Hierarchical linear model
  • Iterative generalized least squares
  • Multistage sampling
  • Pseudolikelihood
  • Scaled weights
  • Variance components

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