Bootstrap Confidence Intervals for Proportions of Unequal Sized Groups Adjusted for Overdispersion

Group testing is a method of pooling a number of units together and performing a single test on the resulting group. It is an appealing option when few individual units are thought to be infected leading to reduced costs of testing as compared to individually testing the units. Group testing aims to identify the positive groups in all the groups tested or to estimate the proportion of positives (p) in a population. Interval estimation methods of the proportions in group testing for unequal group sizes adjusted for overdispersion have been examined. Lately improvement in statistical methods allows the construction of highly accurate confidence intervals (CIs). The aim here is to apply group testing for estimation and generate highly accurate Bootstrap confidence intervals (CIs) for the proportion of defective or positive units in particular. This study provided a comparison of several proven methods of constructing CIs for a binomial proportion after adjusting for overdispersion in group testing with groups of unequal sizes. Bootstrap resampling was applied on data simulated from binomial distribution, and confidence intervals with high coverage probabilities were produced. This data was assumed to be overdispersed and independent between groups but correlated within these groups. Interval estimation methods based on the Wald, the Logit and Complementary log-log (CLL) functions were considered. The criterion used in the comparisons is mainly the coverage probabilities attained by nominal 95% CIs, though interval width is also regarded. Bootstrapping produced CIs with high coverage probabilities for each of the three interval methods.

Estimating Equations for Estimation of Mcdonald Generalized Beta— Binomial Parameters

There has been a considerable recent attention in modeling over dispersed binomial data occurring in toxicology, biology, clinical medicine, epidemiology and other similar fields using a class of Binomial mixture distribution such as Beta Binomial distribution (BB) and Kumaraswamy-Binomial distribution (KB). A new three-parameter binomial mixture distribution namely, McDonald Generalized Beta Binomial (McGBB) distribution has been developed which is superior to KB and BB since studies have shown that it gives a better fit than the KB and BB distribution on both real life data set and on the extended simulation study in handling over dispersed binomial data. The dispersion parameter will be treated as nuisance in the analysis of proportions since our interest is in the parameters of McGBB distribution. In this paper, we consider estimation of parameters of this MCGBB model using Quasi-likelihood (QL) and Quadratic estimating functions (QEEs) with dispersion. By varying the coefficients of the QEE’s we obtain four sets of estimating equations which in turn yield four sets of estimates. We compare small sample relative efficiencies of the estimates based on QEEs and quasi-likelihood with the maximum likelihood estimates. The comparison is performed using real life data sets arising from alcohol consumption practices and simulated data. These comparisons show that estimates based on optimal QEEs and QL are highly efficient and are the best among all estimates investigated.