Weighted estimation in multilevel ordinal and binary models in the presence of informative sampling designs

Multilevel models are often fitted to survey data gathered with a complex multistage sampling design. However, if such a design is informative, in the sense that the inclusion probabilities depend on the response variable even after conditioning on the covariates, the standard maximum likelihood estimators are biased. In this paper, following the Pseudo Maximum Likelihood approach of Skinner (1989), we propose a probability-weighted estimation procedure for multilevel ordinal and binary models which eliminates the bias generated by the informativeness of the design. The reciprocals of the inclusion probabilities at each sampling stage are used toweight the log-likelihood function and the weighted estimators obtained in this way are tested by means of a simulation study for the simple case of a binary random intercept model with and without covariates. The variance estimators are obtained by a boostrap procedure. The maximization of the weighted log-likelihood of the model is domne by the NLMIXED procedure of the SA, which is based on adaptive Gaussian quadrature. Also the bootstrap estimation of variances is implemented in the SAS environment