Semi-parametric analysis of overdispersed count and metric data with varying follow-up times: Asymptotic theory and small sample approximations.

Count data are common endpoints in clinical trials, for example magnetic resonance imaging lesion counts in multiple sclerosis. They often exhibit high levels of overdispersion, that is variances are larger than the means. Inference is regularly based on negative binomial regression along with maximum-likelihood estimators. Although this approach can account for heterogeneity it postulates a common overdispersion parameter across groups. Such parametric assumptions are usually difficult to verify, especially in small trials. Therefore, novel procedures that are based on asymptotic results for newly developed rate and variance estimators are proposed in a general framework. Moreover, in case of small samples the procedures are carried out using permutation techniques. Here, the usual assumption of exchangeability under the null hypothesis is not met due to varying follow-up times and unequal overdispersion parameters. This problem is solved by the use of studentized permutations leading to valid inference methods for situations with (i) varying follow-up times, (ii) different overdispersion parameters, and (iii) small sample sizes.