Modelling overdispersion and Markovian features in count data

The number of counts (events) per unit of time is a discrete response variable that is generally analyzed with the Poisson distribution (PS) model. The PS model makes two assumptions: the mean number of counts (λ) is assumed equal to the variance, and counts occurring in non-overlapping intervals are assumed independent. However, many counting outcomes show greater variability than predicted by the PS model, a phenomenon called overdispersion. The purpose of this study was to implement and explore, in the population context, different distribution models accounting for overdispersion and Markov patterns in the analysis of count data. Daily seizures count data obtained from 551 subjects during the 12-week screening phase of a double-blind, placebo-controlled, parallel-group multicenter study performed in epileptic patients with medically refractory partial seizures, were used in the current investigation. The following distribution models were fitted to the data: PS, Zero-Inflated PS (ZIP), Negative Binomial (NB), and Zero-Inflated Negative Binomial (ZINB) models. Markovian features were introduced estimating different λs and overdispersion parameters depending on whether the previous day was a seizure or a non-seizure day. All analyses were performed with NONMEM VI. All models were successfully implemented and all overdispersed models improved the fit with respect to the PS model. The NB model resulted in the best description of the data. The inclusion of Markovian features in λ and in the overdispersion parameter improved the fit significantly (P < 0.001). The plot of the variance versus mean daily seizure count profiles, and the number of transitions, are suggested as model performance tools reflecting the capability to handle overdispersion and Markovian features, respectively.