Inferences and power analysis concerning two negative binomial distributions with an application to MRI lesion counts data

In comparing the mean count of two independent samples, some practitioners would use the t-test or the Wilcoxon rank sum test while others may use methods based on a Poisson model. It is not uncommon to encounter count data that exhibit overdispersion where the Poisson model is no longer appropriate. This paper deals with methods for overdispersed data using the negative binomial distribution resulting from a Poisson-Gamma mixture. We investigate the small sample properties of the likelihood-based tests and compare their performances to those of the t-test and of the Wilcoxon test. We also illustrate how these procedures may be used to compute power and sample sizes to design studies with response variables that are overdispersed count data. Although methods are based on inferences about two independent samples, sample size calculations may also be applied to problems comparing more than two independent samples. It will be shown that there is gain in efficiency when using the likelihood-based methods compared to the t-test and the Wilcoxon test. In studies where each observation is very costly, the ability to derive smaller sample size estimates with the appropriate tests is not only statistically, but also financially, appealing.