The robust estimation method for a finite mixture of Poisson mixed-effect models

When analyzing clustered count data derived from several latent subpopulations, the finite mixture of the Poisson mixed-effect model is an immediate strategy to model the underlying heterogeneity. Within the generalized linear mixed model framework, parameters in such a model are often estimated through the residual maximum likelihood estimation approach. However, the method is vulnerable to outliers. To develop robust estimators, the minimum Hellinger distance (MHD) estimation approach has been proposed by Xiang et al. (Xiang, L., Yau, K.K.W., Lee, A.H., Hui, Y.V., 2008. Minimum Hellinger distance estimation for k-component Poisson mixture with random effects. Biometrics 64, 508–518) with the random effects following a normal distribution. In some circumstances, there is little information available on the joint distributional form of the random effects. Without prescribing a parametric form for the random effects distribution, we consider embedding the non-parametric maximum likelihood (NPML) approach within the MHD estimation to extend the robust estimation method for a finite mixture of Poisson mixed-effect models. The NPML estimation not only avoids the problem of numerical integration in deriving the MHD estimating equations, but also enhances the robustness characteristic because of its resistance to possible misspecification of the parametric distribution for the random effects. The performance of the new method is evaluated and compared with that of the existing MHD estimation using simulations. Application to analyze a real data set of recurrent urinary tract infections is illustrated.