## Theses and Dissertations

#### DOI

https://doi.org/10.25772/DY3A-VY55

1996

Dissertation

#### Degree Name

Doctor of Philosophy

Biostatistics

#### First Advisor

W. Hans Carter, Jr.

#### Abstract

In certain experimental situations, the data observed are pseudo-proportional. By this, we observe the numerator, the number of responders, but the denominator, the total number, is random and, possibly, unobserved. In such situations, the data often exhibit "extra" variability due to the randomness of the denominator. Analysis of these data should account for this overdispersion. Several authors have proposed parametric approaches to this problem. Finney [Biometrika (1949) 36, 239-256] proposed the use of a mixture of the binomial and Poisson distributions. Anscombe [Annals of Applied Biology (1949) 36, 203-205] discussed the use of a mixture of the binomial and negative binomial distributions. Margolin et al. [Proceedings of the National Academy of Sciences (1981) 78, 3779-3783] suggested the use of a gamma mixture of Poisson distributions. While it is true that these approaches provide a means of handling overdispersion, the choice of the distributions used often is based on mathematical convenience. To avoid making full distributional assumptions, Kim [Unpublished Ph.D. dissertation (1991), UCLA] applied quasi-likelihood methodology to overdispersed binomial data. In his method, he assumed that the conditional mean and variance of the numerator was that of a binomial distribution, and that the mean of the distribution of the denominator was known, but that its variance was unknown. Using conditional arguments, he arrived at the unconditional mean and variance of the numerator. With the form of the first two moments of the numerator, Kim used quasi-likelihood method of moments estimation to get estimates of the unknown parameters. In this dissertation, Kim's results are generalized to allow the denominator to have an unknown mean and variance, and estimation is achieved using a generalized quasilikelihood method of moments technique. In addition, methodology is developed that allows for overdispersion in both the numerator and denominator. Here, extended quasilikelihood techniques are used for estimation of any unknown parameters. Properties of the estimators were studied via a simulation study which suggests the extended quasilikelihood estimates are not asymptotically normally distributed. Also, goodness-of-link testing is described for use in maximum likelihood, as well as, quasi-likelihood estimation. The methods developed here are illustrated by the analysis of data from a colony formation assay involving serial dilutions.

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#### Is Part Of

VCU University Archives

#### Is Part Of

VCU Theses and Dissertations

10-4-2016

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