Multinomial Logistic Mixed Models for Clustered Categorical Data in a Complex Survey Sampling Setup

Abstract

In a finite/survey population setup, where categorical/multinomial responses are collected from individuals belonging to a cluster, in a recent study Skinner (International Statistical Review, 87, S64-S78 2019) has modeled the means of the clustered categorical responses as a function of regression parameters, and suggested a ‘working’ correlations based GEE (generalized estimating equations) approach for the estimation of the regression parameters. However, this mean model involving only regression parameters is not justified for clustered multinomial responses because of the fact that these responses share a common cluster effect which compels the clustered correlation parameter to enter into the mean function on top of the regression parameters. Consequently, the so-called GEE approach, which requires the means to be free of correlations, is not applicable for regression analysis in the clustered multinomial setup. As a remedy, in this paper we consider a multinomial mixed model which accommodates the clustered correlation parameter in the mean functions. For inferences in the present finite population setup, as the GQL (generalized quasi-likelihood) approach is known to produce consistent and more efficient estimate than the MM (method of moments) approach in an infinite population setup, we estimate the regression parameters of primary interest by using the first order response based survey weighted GQL (WGQL) approach. For the estimation of the random effects variance (also known as clustered correlation) parameter, as it is of secondary interest, we use the second order response based survey weighted MM (WMM) approach, which is simpler than the corresponding WGQL estimation approach. The estimation steps are presented clearly for the benefit to the practitioners. Also because, in practice, survey practitioners such as statistical agencies frequently deal with a large health or socio-economic data set at national or state levels, for example, we make sure for their benefit that our proposed WGQL and WMM estimators are consistent. Thus, the asymptotic properties such as asymptotic unbiasedness and consistency for both regression and clustered correlation parameters are studied in details. The asymptotic normality property, for the benefit of constructing confidence interval for the main regression parameters, is also studied.