Non-linear Approximations of Shape and Location Parameters in the Poisson Inverse Gaussian Model in Analysis of Infectious Count Data

Abstract

Statistical models create a basis for analysis of infectious disesase count. These data sets exhibit unique characteristics such as low counts, delayed reporting, underreporting amoung others. The tendency to model these counts using linear models with their simplicity is common with most research. Further, the assumption of a fixed dispersion in modeling infectious disease counts is quite high. Prediction relating to infectious disease counts have been based on the Poisson model framework. The extension of the Poisson models such NB and PIG distributions have gained popularity over the recent past in modeling count responses showing over dispersion relative to the Poisson distribution. In this study we propose non-linear models for these data sets, modeling the mean and dispersion parameters as additive terms. Negative Binomial (NB) and Poisson Inverse Gaussian (PIG) glm models with a fixed and a varying dispersion parameter and compare them with NB GAM and PIG GAM with both mean and dispersion modeled as additive terms. The model are fitted to over dispersed infectious counts, Salmonella Hadar data set. Residual plots are constructed to explore the quality of fits and analysis goodness of fit is carried out to access the best fitting model. The study results reveal better performance of the PIG models on both the linear and non linear model platforms. Further, modelling both the mean and dispersion proved better as compared to models assuming the dispersion as a constant.