A New Family of Regression Models for Correlated Count Data with Cubic Variance Function

Poisson and negative binomial regression models are commonly used in modelling count data which are not correlated. However, due to grouping of subjects among other reasons, correlated count data arise frequently in statistical analysis. Random effect models such as random effect Poisson (REP) and random effect negative binomial (RENB) models have been proposed in the literature for modelling correlated count data. Nevertheless, REP is only suitable for modelling data with equal mean and variance whereas RENB is restricted to data where the variance is a quadratic function of the mean. On the other hand, statistical models such as generalized Poisson, inverse trinomial and strict arcsine models are commonly used in modelling count data with very long tails because of their nature whereby variances are cubic function of the mean. To the best of our knowledge, there isn’t any appropriate statistical regression model which can be used in modelling overdispersed and/or long-tailed correlated count data and with variance as a cubic function of the mean. In this study, we develop random effect models for overdispersed correlated count data using a family of discrete distributions with cubic variance functions as the baseline. We will extend the original generalized Poisson, inverse trinomial and strict arcsine models by allowing random effects in the model. Maximum likelihood estimation and a new generating function based estimation procedure will be considered for the new random effect models. Model fitting will be applied on simulated data as well as a real life medical data set consisting of frequency counts. We expect to produce three random effect models which can be used in modelling overdispersed correlated count data with
cubic variance functions of the mean. Apart from contributing new models and inference methods on big data, this study would significantly improve regression modelling in healthcare, amongst others.