#### Date of Award

2014

#### Degree Type

Dissertation

#### Degree Name

Ph.D.

#### Department

Mathematics and Statistics

#### First Advisor

Paul, S. R.

#### Keywords

Pure sciences, Biological sciences, Behrens-fisher analogs, Beta-binomial model, Discrete and survival data, Negative binomial model, Over-dispersion, Score test

#### Rights

CC BY-NC-ND 4.0

#### Abstract

Discrete data often exhibit variation greater or smaller than predicted by a simple model. Negative binomial distribution and beta-binomial distribution are popular and widely used to accommodate the extra-Poisson and extra-binomial variations respectively in analyzing discrete data. Weibull distribution is one of the most popular distributions in survival data analysis. Often both discrete and survival data appear in groups and it may be of interest to compare certain characteristics of two groups of such data. The purpose of this dissertation is to deal with Behrens-Fisher analogs for data that follow negative binomial, beta-binomial and Weibull distributions. We first develop six test procedures, namely, LR, LR ( bc ), T 2 , T 2 ( bc ), T 1 and T N , for testing the equality of two negative binomial means assuming unequal dispersion parameters. A simulation study is conducted to compare the performance of the test procedures. Two sets of data are analyzed. For small to moderate sample sizes, the statistic T 1 shows best overall performance. For large sample sizes, all six statistics perform well and are found similar in terms of maintaining size and power. We, then, develop eight test procedures, namely, LR, C ml , C kmm , C qb , C qs , C eq , C rs and C ars , for testing the equality of proportions in two beta-binomial distributions where the dispersion parameters are assumed unknown and unequal. These test procedures are compared through simulation studies and data analysis. The LR test is observed to maintain the nominal level reasonably well accompanied with the best power performance. The next best is the performance of the statistic C eq in terms of nominal level and power. Last but not least, we develop four test procedures, namely, LR, C ml , C cr and C tg , for testing the equality of scale parameters of two Weibull distributions where the shape parameters are unequal and compare these statistics through simulation studies and data analysis. For small sample sizes, the statistics LR and C ml hold nominal level most effectively. The statistic C cr shows highest power although its level is also higher (liberal). For moderate and large sample sizes the overall performance of the statistic LR is found to be superior to others.

#### Recommended Citation

Alam, S.M. Khurshid, "Behrens-Fisher Analogs for Discrete and Survival Data" (2014). *Electronic Theses and Dissertations*. 5105.

http://scholar.uwindsor.ca/etd/5105