Simultaneous inference for the comparison of overdispersed multinomial data

Abstract

Multinomial data is present when the outcome of an experiment is a discrete choice of more than two mutually exclusive alternatives and a multinomial distribution is assumed as the underlying distribution. Usually, a multinomial regression model is considered for the analysis of multinomial count data. Frequently, such data exhibits overdispersion especially if the data is acquired in clusters. The collection of data in cell cultures, litters, members of a family or classroom will lead to observations that are more similar within clusters than observations from di erent clusters. Therefore, it has to be expected that some sources of variation may di er between clusters and overdispersion is present. In addition, several treatments are often of interest for such data, causing a multiple testing problem. While for normally distributed data multiple comparison procedures proposed by Tukey and Dunnett are standard since decades, multiple treatment comparisons between several overdispersed multinomial samples have been rarely investigated. The primary objective of this thesis is to develop a method to obtain multiple hypothesis tests and simultaneous confidence intervals for the comparison of multiple polytomous vectors that lack independence among experimental units due to a collection of data in clusters. Building on previous work, overdispersion is considered and an asymptotic procedure is proposed for simultaneous inference of odds ratios between multiple multinomial samples by including an estimated dispersion parameter. To assess validity, a simulation approach utilizing the Dirichlet-multinomial distribution is applied to determine the simultaneous coverage probability of confidence intervals for different magnitudes of overdispersion. As part of this thesis, the proposed test procedure is implemented in the statistical software environment R in a user-friendly way. The application of the novel method and corresponding R-functions is described comprehensively on two real data sets from toxicological research and one data set from a social study. Especially the first example is examined in detail and an alternative approach using multiple marginal models is presented. Possible problems are discussed and suggestions for future work are outlined.