Modelling Overdispersion and Markovian Features in Count Data
I˝aki F. Trocˇniz (1), Raymond Miller (2), Mats O. Karlsson (3)
1) Department of Pharmacy and Pharmaceutical Technology, School of Pharmacy; University of Navarra; Pamplona; Spain; (2) Pfizer Global Research and Development, Ann Arbor, MI 48108, USA; (3) Division of Pharmacokinetics and Drug Therapy, Department of Pharmaceutical Biosciences, Uppsala University, Uppsala, Uppsala, Sweden
Background: 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 (lambda) is equal to the variance of the data, and the number of counts occurring in non-overlapping intervals of time are assumed independent. However, many counting outcomes show greater variability than that predicted by the PS model, a phenomenon called overdispersion. Moreover we are currently realizing that an increasing number of pharmacodynamic variables show a certain degree of interdependency between neighbouring measurements, a feature that has been modelled incorporating Markovian elements.
Objectives: To implement and explore, in the population context, different distribution models accounting for the overdispersion and Markovian patterns in the analysis of count data.
Methods: Daily seizure count data obtained from 551 subjects during the 12 weeks 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 to account for overdispersion: (1) the Zero Inflated Poisson (ZIP), (2) the Inverse Binomial (INB), (3) the Zero Inflated Inverse Binomial, (ZINB), and (4) mixture models. The Markovian patterns were introduced estimating different lambdass and overdispersion parameters depending on whether the previous day was a seizure or non-seizure day. All analyses were performed with NONMEN VI.
Results: All were successfully implemented in NONMEM and all overdispersed models improved the fit in respect to the PS model. The INB model resulted in the best model fit to the data providing a minimum value of the objective function 7275 points lower than the PS model for six extra parameters. Including Markovian patterns in l and in the overdispersion parameter improved the fit significantly (P<0.0001). The typical population estimates of lambda if the previous day was a seizure or a non-seizure day were 0.53 and 0.32, respectively. For the case of the overdispersion parameter the values were 0.15, and 0.58, respectively.
Conclusions: The ZIP, INB, ZINB and mixture models were all capable of dealing with the overdispersion in count data and allowed the flexible incorporation of Markovian elements. They provide, in addition to the mean number of counts, additional possibilities to test placebo/drug/disease progression effects such as the degree of overdispersion, and transition probabilities.
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