Klaas Prins
qPharmetra LLC, Andover, MA, USA
Objectives: Traditionally, count data are modeled using the Poisson distribution that assumes mean and variance are the same. However, there are examples where variance is less (under dispersion) or more (over dispersion) than the mean count. A variety of distributions, including the Generalized Poisson (GP, [2]) are used to model the central tendency and data dispersion. Generally, the model fits improve substantially by estimating dispersion [1,3], but it has not been reported yet if this also leads to increased power to detect a certain effect. In the current simulation study a case of under dispersion this will be investigated.
Methods: The true model was based on under dispersed daily micturition frequency. The placebo model included a monoexponential decrease in base count (11 voids/d) reaching steady state effect after 8 weeks. The drug effect was modeled directly proportional to placebo effect via an Emax model (ED50 defined at 50 mg). Using PsN’s [4] stochastic simulation-estimation (sse), an underdispersed data set (lambda=22, dispersion=-1) was simulated using a datasets of varying treatment arm size (n=10, 30, 50, 75, 100) including placebo, 25, 50, 100 and 200mg as treatments. The model was re-estimated with a placebo and a placebo+drug effect model and the proportion ofplacebo+drug effect models being statistically superior to their corresponding placebo model was used to define the power to detect a drug effect (alpha=5%). This procedure was performed under the GP (estimating dispersion) and the Poisson distribution (by fixing dispersion to zero, making the GP collapse to a Poisson) and the success rates were compared between the two distributions.
Results: The model estimating the dispersion alongside the mean count (GP) was able to detect dose response at much lower treatment arm size than the Poisson. At n=10 and n=30, the 33% and 37% of the GP models detected a drug effect versus 1% and 3% for Poisson. At n=100 Poisson detected a significant drug effect in 47% of the cases, vs 51% for GP. Around 75-100 patients per arm the success rate was similar between the distributions.
Conclusions: Acknowledging dispersion in models for under dispersed count data using the GP distribution improves the power to estimate central tendencies in the data at relatively low treatment arm size. With increasing treatment arm size, this power advantage gradually fades out and becomes independent of the estimation of the degree of dispersion.
References:
[1] Plan (2014) Modeling and Simulation of Count Data. CPT-PSP
[2] Consul & Jain (1973). A Generalization of the Poisson distribution. Technometrics 15, 791-799
[3] Prins et al. (2011). Use of a generalized Poisson model to describe micturition frequency in patients with overactive bladder disease. PAGE-meeting 2011 Athens
[4] Lindbom et al. (2005) PsN-Toolkit — A collection of computer intensive statistical methods for non-linear mixed effect modeling. Comp Meth Prog Biomed.
Reference: PAGE 24 () Abstr 3621 [www.page-meeting.org/?abstract=3621]
Poster: Methodology - Estimation Methods