Survey-weighted Generalized Linear Mixed Models

Regression analysis aims at the revelation of interdependencies and causalities between variables observed in the population. That is, a structure between regressors and regressants that causes the realization of the finite population is assumed, the so-called data generating process or a superpopulation model. When data points occur in an inherent clustering, mixed models are a natural modelling approach. Given the finite population realization, a consistent estimation of the superpopulation parameters is possible. However, regression analysis seldomly takes place at the level of the finite population. Rather, a survey is conducted on the population and the analyst has to use the sample for regression modeling. Under a correct regression setup, derived estimators are consistent given the sample is non-informative. Though, these conditions are hard to verify, especially when the survey design is complex, employing clustering and unequal selection probabilities. The use of sampling weights may reduce a consequent estimation bias as they could contain additional information about the sampling process conditional on which the data generating process of the sampled units becomes closer to the one of the whole population. Common estimation procedures that allow for survey weights in generalized linear mixed models require one unique survey-weight per sampling stage which are consequently nested and correspond to the random effects analyzed in the regression. However, the data inherent clustering (e.g. students in classes in schools) possibly does not correspond to the sampling stages (e.g. blocks of houses where the students’ families live). Or the analyst has no access to the detailed sample design due to dis- closure risk or the selection of units follows an unequal sampling probability scheme. Or the survey weights vary within clusters due to calibration. Therefore, we propose an estimation procedure that allows for unit-specific survey weights: The Monte-Carlo EM (MCEM) algorithm whose complete-data log-likelihood leads to a single-level modeling problem that allows a unit-specific weighting. In the E-step, the random effects are considered to be missing data. The expected (weighted) log-likelihood is approximated via Monte-Carlo integration and maximized with respect to the regression parameters. The method’s performance is evaluated in a model-based simulation study with finite populations.