Two Stage Cluster Sampling Based Asymptotic Inferences in Survey Population Models for Longitudinal Count and Categorical Data

Abstract

Sutradhar (2008, Sankhya B, 18-33) has studied a GLMM (generalized linear mixed model) for count data in a finite population setup where a sample of families/clusters were chosen from a finite population using PPS (probability proportional to size) sampling scheme. The properties of the estimators were studied through a simulation study. In this paper we consider (1) an auto-regressive type dynamic (ARTD) model for longitudinal count data, and (2) a MDL (multinomial dynamic logits) model for longitudinal categorical/multinomial data; and develop a practically important two-stage cluster sampling (TSCS) design weights based estimating equations for the regression and dynamic dependence parameters involved in both ARTD and MDL models in a finite population setup. Specifically, the survey weighted generalized quasi-likelihood (WGQL) and the survey weighted maximum likelihood (WML) estimation approaches are used for the count and multinomial models, respectively. We also demonstrate in the paper that the proposed TSCS based WGQL and WML estimators specially for the regression parameters (of main interest) are consistent, whereas the existing ‘working’ longitudinal correlations based GEE (generalized estimating equations) fails to produce consistent estimates and/or consistent variance estimates. The variances of the regression estimates along with their unbiased estimates are derived for both count and multinomial models. Furthermore, the asymptotic normality of the regression estimators are developed based on a large number of independent but non-identical clusters under non-overlapping stratums.