#### Title

On the estimation of dispersion parameters for data in the form of counts and proportions.

#### Date of Award

2004

#### Degree Type

Dissertation

#### Degree Name

Ph.D.

#### Department

Mathematics and Statistics

#### First Advisor

Paul, Sadhir R.,

#### Keywords

Statistics.

#### Rights

CC BY-NC-ND 4.0

#### Abstract

Data in the form of counts or proportions arise in biology, biomedical, toxicology, epidemiology and other similar fields. These data often exhibit variation greater or smaller than predicted by a simple model, such as, a Poisson or a binomial model. Thus, in practice, a model having an over or under dispersion parameter might be necessary. This is an important parameter to estimate for making appropriate inferences regarding the mean or the regression parameters in analyzing count data or binary data. Also, in some instances the dispersion parameter may be of interest in its own right. Several parametric or semi-parametric methods have been developed to estimate the dispersion parameter in the analysis of binary data. In this thesis, I first conduct an extensive empirical investigation, by simulations, to study the properties, in terms of mean bias, standard deviation and mean square error, of twenty six different estimators of the dispersion parameter that exist in the literature. Of these twenty six estimators I recommend an estimator Q2, based on optimal quadratic estimating equations for the mean and dispersion parameters. Further, I derive an expression for a bias-corrected maximum likelihood estimator (BCML) of this parameter using a beta-binomial model. A bias corrected maximum likelihood estimator of the dispersion parameter is also obtained for a regression situation. I then compare, by simulations, the BCML estimator, in terms of bias and efficiency, with the maximum likelihood (ML) estimator, a double extended quasi-likelihood (DEQL) estimator proposed by Lee (2004) and the estimator Q 2. A toxicological data set from Paul (1982) and a medical data set from Otake and Prentice (1984) are analyzed. The simulation results and examples indicate that the BCML estimator performs well in terms of bias and efficiency in most instances. For count data we derive a first order BCML estimator using a two-parameter negative binomial model and a (DEQL) estimator for the dispersion parameter for data involving only the mean and the dispersion parameters. The bias corrected maximum likelihood estimator of the dispersion parameter is also derived for a negative binomial regression model. The BCML estimator, in case of a two-parameter model, is compared, in terms of bias and efficiency, with the ML estimator investigated by Piegorsch (1990), the moment and the maximum extended quasi-likelihood estimators investigated by Clark and Perry (1989), and the DEQL estimator. The BCML estimator works best in term of bias and efficiency properties in most instances. The European red mites data with no covariate (Bliss and Fisher, 1953) and the Ames Salmonella assay data with one covariate (Margoline et al., 1981) are analyzed.Dept. of Mathematics and Statistics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis2004 .S24. Source: Dissertation Abstracts International, Volume: 65-10, Section: B, page: 5221. Adviser: Sadhir R. Paul. Thesis (Ph.D.)--University of Windsor (Canada), 2004.

#### Recommended Citation

Saha, Krishna K., "On the estimation of dispersion parameters for data in the form of counts and proportions." (2004). *Electronic Theses and Dissertations*. 2925.

http://scholar.uwindsor.ca/etd/2925