Modeling urge urinary incontinence data acknowledging non-Poisson dispersion of counts within individual provides a major model fit improvement.
Kevin Dykstra(1), N.H. Prins (1), A. Darekar(2), P.H. van der Graaf(2)
(1) qPharmetra LLC, Andover, MA, USA, (2) Pfizer Ltd, Sandwich, UK
Objectives: Daily number of urge urinary incontinence (UUI) episodes is a count endpoint for assessing overactive bladder disease activity and commonly modeled using Poisson (PS) regression. PS assumes equi dispersion, meaning that mean is equal to the variance. However, UUI data are over-dispersed, i.e. individual variance > individual mean. The observed UUI distribution is very skewed and contains a large number of zeroes, which result in poor model fits under the standard PS models. The generalized Poisson (GP) distribution flexibly handles over-, equi- as well as underdispersion, and recently successfully described underdispersed Likert pain scores . We wanted to evaluate if the GP distribution could also successfully describe the distribution of overdispersed count data in a model of UUI.
Methods: Placebo UUI count data from 200 patients participating in 2 studies were modeled as UUI=UUI(base)*(1-plmax*(1-exp(-k*t))) in WinBUGs assuming PS or GP count distribution. Parameter plmax was logit-transformed to ensure that predicted UUI did not go below zero and a Gamma distributed between subject variability was assumed on λ (PS) or λ1 (GP), λ2 (GP) and plmax (PS, GP). The GP equation  features a lambda (λ1) and a dispersion factor (λ2, with λ2<0 pointing to under-, λ2=0 being equi- and λ2>0 indicating over dispersion), where the mean is defined by λ1/(1- λ2) and the variance by λ1/(1- λ2)^3. In case λ2=0 the GP model collapses to a PS. Models were compared by the significance of the dispersion parameter estimate, the Deviance Information Criterion (DIC) and ability to capture mean trends and observed variability using VPC.
Results: The mean trend in the data was equally well captured by both models. The GP model estimated a λ2 of 0.37 (SE 0.02) that was highly significantly different from zero, thus confirming within-subject over dispersion of counts. Furthermore, the DIC of the GP model dropped a few hundred points indicating a highly significant model fit improvement over the PS model. The VPC showed that the PS model under predicted observed variance, while the GP model captured variance remarkably well.
Conclusions: The GP model was found to be superior to the PS model in terms of describing the variability observed in UUI count data and yielding a significantly positive dispersion parameter. The GP distribution for UUI data will as such provide more accurate inferences, such as between treatment comparisons and clinical trial simulations.
 Plan & Karlsson (2009). New models for handling correlated underdispersed Likert pain scores. PAGE 2009.
 Consul & Jain (1973). A generalization of the Poisson distribution. Technometrics 15, 791-799