Zero-Truncated Models applied to the Nigerian National Health Insurance Data

Abstract

In this paper we have applied zero-truncated univariate and bivariate distributions to the NHIS data with two outcomes variables that are strongly over dispersed. Because of strong correlation between the outcome variables, we have fitted bivariate Poisson model that encompass covariance between the two variables. In the univariate
case, we found that the Type II zero truncated generalized Poisson will be most suitable to each of the outcome variables. Other models also equally perform well, but the type II zero-truncated generalized Poisson (ZTGP2) is most preferable because of the ease of its implementation. For the bivariate models, we recognize that the zero-truncated Marshall & Olkin (1985) bivariate NB model does not perform well. Our preferred model would be the version of zero-truncated bivariate Poisson model proposed in Holgate(1964) and recently represented in AlMuhayfith et al. (2016). Our results indicate that this model is most suitable. Further it captures the covariance between the two outcome variables. All the models are implemented in SAS PROC NLMIXED. For each distribution considered, MLE estimation based on the log-likelihood functions are obtained using the Adaptive Gaussian Quadrature (usually with 32 q-points) and then optimized by using the Newton-Raphson algorithm. Starting values are obtained from those obtained from employing the Poisson or Negative binomial models.