Pair-Wise Family-Based Correlation Model for Spatial Count Data

When linear, binary or count responses are collected from a series of (spatial) locations, the responses from adjacent/neighboring locations are likely to be correlated. To model the correlation between the responses from two adjacent locations, many existing studies assume that the two locations belong to a family and their responses are correlated through the random effects of common locations shared by them. In developing a correlation model for similar spatial responses, a recent study, however, used a much broader concept that the two selected neighboring locations have their own family and the correlation between the responses from these two locations are formulated by exploiting the correlations among the random effects belonging to both families. But, this study was confined to the linear spatial responses only. In this paper, we consider spatial count responses such as the number of lip cancer cases collected from a series of (spatial) locations, and develop a correlation model for spatial counts by following the recent pair-wise family-based spatial correlation model for the linear data. The correlation models for spatial linear and count responses are generally different. As far as the estimation of the parameters of the proposed correlation model for spatial counts is concerned, we develop a second-order moments-based GQL (generalized quasi-likelihood) approach for the estimation of the regression parameters, and a fourth-order moments-based GQL approach for the estimation of both variance and correlation of the random effects. It is demonstrated through an intensive simulation study that the proposed GQL approach works quite well in estimating all parameters of the underlying correlation model developed for spatial counts. The proposed model and the estimation methodology have been illustrated through an analysis of the well-known Scottish lip cancer data.