#### Title

A STUDY OF THE VARIANCE ESTIMATORS OF THE MANTEL-HAENSZEL LOG-ODDS-RATIO ESTIMATE.

#### Date of Award

1988

#### Degree Type

Dissertation

#### Degree Name

Ph.D.

#### Department

Mathematics and Statistics

#### Keywords

Mathematics.

#### Rights

CC BY-NC-ND 4.0

#### Abstract

Mantel and Haenszel (1959) proposed a "summary relative risk formula" for a set of K 2 $\times$ 2 tables. This formula, $\\psi\sb{\rm MH}$, has been the most popular estimator of the common odds ratio in K 2 $\times$ 2 tables. Hauck (1979), Breslow and Liang (1982), Flanders (1985), Robins, Breslow and Greenland (1986) and others have proposed several estimators for the variance of $\\theta\sb{\rm MH}$ = log $\\psi\sb{\rm MH}$. However, the variance of $\\theta\sb{\rm MH}$ is not finite, since $\\theta\sb{\rm MH}$ can take infinite values with positive probability. This dissertation presents an approximation $\tilde\theta\sb{\rm MH}$ of $\\theta\sb{\rm MH}$ such that $\\theta\sb{\rm MH}$ = $\tilde\theta\sb{\rm MH}$ + Z where Z is a random variable that converges in probability to zero under very general conditions. In particular, this result is shown to hold for the "large stratum" case (where the number of tables remains fixed but the size of each table increases) and the "Sparse Data" case (where the sizes of the tables remain small, but the number of tables increases). Several new variance estimators are derived. $\{\rm V}\sb{\rm P}$, originally presented in Phillips and Holland (1986) is shown to be equivalent to $\{\rm V}\sb{\rm US},$ the Robins-Breslow-Greenland estimator. The derivation of $\{\rm V}\sb{\rm P}$ is different from the RBG method. Another estimator $\{\rm V}\sb{\rm PL}$ is also presented. In addition, a simpler estimator in terms of the marginal totals and $\psi$ is derived in the special case of 1:M matching. Also, empirical estimators which are functions whose only arguments are the numerator and denominator of $\\psi\sb{\rm MH}$ are proposed for quick, easy calculations. Extensive simulations and examples using real data illustrate the usefulness of the proposed estimators.Dept. of Mathematics and Statistics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis1988 .P455. Source: Dissertation Abstracts International, Volume: 48-10, Section: B, page: 2995. Thesis (Ph.D.)--University of Windsor (Canada), 1988.

#### Recommended Citation

PHILLIPS, ABRAHAM., "A STUDY OF THE VARIANCE ESTIMATORS OF THE MANTEL-HAENSZEL LOG-ODDS-RATIO ESTIMATE." (1988). *Electronic Theses and Dissertations*. 817.

https://scholar.uwindsor.ca/etd/817