Multi-level Modelling Under Informative Sampling

We consider a model dependent approach for multi-level modelling that accounts for informative probability sampling, and compare it with the use of probability weighting as proposed by Pfeffermann et al. (1998a). The new modelling approach consists of first extracting the hierarchical model holding for the sample data as a function of the corresponding population model and the first and higher level sample selection probabilities, and then fitting the resulting sample model using Bayesian methods. An important implication of the use of this approach is that the sample selection probabilities feature in the analysis as additional outcome values that strengthen the estimators. A simulation experiment is carried out in order to study and compare the performance of the two approaches. The simulation study indicates that both approaches perform generally equally well in terms of point estimation, but the model dependent approach yields confidence (credibility) intervals with better coverage properties. A robustness simulation study is performed, which allows to assess the impact of misspecification of the models assumed for the sample selection probabilities under informative sampling schemes.