Distribution of Odds Ratio in 2 × 2 Contingency Table: Adjustment for Correlation

Abstract

The Log-odds ratio for 2 × 2 contingency tables is often approximated by a normal distribution with an approximated variance. Hwang and Biswas (2008) illustrated that the standard expression for the variance should be modified in the presence of correlation. They also provided an adjustment to this variance expression when a single 2 × 2 table is available with matched-pair data. In this present paper, we first provide the required adjustment for multiple 2 × 2 tables, theoretically and also with reference to some data examples. We illustrate that this variance-adjusted normal approximation is a better approximation for such data. We provide two examples, one of which came from a late-phase clinical trial. As the theoretical development of this research depends on the existence of a bivariate binomial distribution, a multivariate (and hence bivariate) binomial distribution is motivated and derived. We then provide a suitably correlation adjusted Mantel–Haenszel test procedure.

Key Words:

• Bivariate binomial distribution
• Log-odds ratio
• Mantel–Haenszel estimator
• Normal approximation

ACKNOWLEDGMENTS

The authors thank two anonymous referees for there valuable suggestions, which led some improvement over an earlier version of the paper. Part of the research was done when the first author was visiting the Institute of Statistical Science, Academia Sinica at Taiwan. The first author thanks the institute for its hospitality. The first author also thanks Dr. Suman Bhattacharya for his interest and some discussions during the preparation of the paper.

Notes

Note. Adjusted cutoff point is the same as the unadjusted cutoff when ρ = 0. The bivariate distributions are generated using the methodology of section 5. Corresponding (p ₀, α) values are also given.