The Generalized Linear Mixed Model for Finite Normal Mixtures with Application to Tendon Fibrilogenesis Data

Abstract

We propose the generalized linear mixed model for finite normal mixtures (GLMFM), as well as the estimation procedures for the GLMFM model, which are widely applicable to the hierarchical dataset with small number of individual units and multi-modal distributions at the lowest level of clustering. The modeling task is two-fold: (a). to model the lowest level cluster as a finite mixtures of the normal distribution; and (b). to model the properly transformed mixture proportions, means and standard deviations of the lowest-level cluster as a linear hierarchical structure. We propose the robust generalized weighted likelihood estimators and the new cubic-inverse weight for the estimation of the finite mixture model (Zhan et al., 2011). We propose two robust methods for estimating the GLMFM model, which accommodate the contaminations on all clustering levels, the standard-two-stage approach (Chervoneva et al., 2011, co-authored) and a robust joint estimation. Our research was motivated by the data obtained from the tendon fibril experiment reported in Zhang et al. (2006). Our statistical methodology is quite general and has potential application in a variety of relatively complex statistical modeling situations.