Testing for homogeneity of proportions using new Mcdonald generalized beta-binomial distribution

Abstract

Testing for homogeneity of proportions in handling over-dispersion is employed in toxicology, teratology, consumers purchasing behavior, alcohol drinking behavior, in studies of dental caries in children and other similar fields. An important inference problem of interest is to compare proportions of certain characteristic in several groups. However, these proportions often exhibit variation greater than predicted by a simple binomial model. In real world applications, the binomial outcome data are widely encountered and the binomial distribution often fails to test homogeneity of proportions due to over-dispersion. The binomial proportion is assigned a continuous distribution defined on the standard unit interval as one way of handling overdispersion in the test for homogeneity of proportions. The new McDonald Generalized Beta Binomial distribution (McGBB) with three shape parameters has been shown to give better fit to binomial outcome data than the Kumaraswamy-Binomial (KB) distribution and Beta-Binomial (BB) distribution based on both simulated data and real data sets and hence considered in this work. This thesis considered derivation of the C ( ) tests based on Quasi-likelihood (QL) and Extended Quasi-likelihood (EQL) estimating functions using the new McGBB distribution which have not been done in testing homogeneity of the proportions. Simulation was done by using R package and also real data was used to calculate p-values for both C ( ) tests and LR test. The size and power of a test was compared for the simulated data and showed that C ( ) tests maintained nominal level well and had higher power than LR test. The comparison of p-values for real data showed that C ( ) tests had smaller p-values than LR test hence C ( ) tests were preffered since they require estimates only under the null hypothesis. Thus, this thesis has provided a better tests ( C ( ) tests) based on Quasi-likelihood and Extended Quasi-likelihood estimating functions for testing homogeneity of proportions in presence of overdispersion using the new McGBB distribution.