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Alternative estimators of the common odds ratio for 2 X 2 tables.

Hauck (1984) reported on the comparative efficiencies and biases, as evident from simulation studies, of three estimators of a common odds ratio or common log-odds ratio. These three estimators included both the conditional and the unconditional maximum likelihood (ML) estimators and the estimator based on the Mantel-Haenszel (MH, 1959) procedure. But particular care needs to be taken in the interpretation of seeming biases in the estimation of the common odds ratio. Reasonably, we might look for unbiasedness in the log-odds ratio rather than in the odds ratio itself-odds ratio estimates are likely to be highly right-skewed around the true parameter value, while the log-odds ratio estimate would tend to show more symmetric behavior. If an estimator shows unbiasedness relative to the log-odds ratio, it will necessarily show positive bias relative to the odds ratio itself. This is because, in the presence of some variation, the arithmetic mean will exceed the geometric mean; if the geometric mean is the true odds ratio, the arithmetic mean will exceed the true odds ratio. Another consequence is that if two estimators are unbiased relative to the log-odds ratio, then the one showing the greater variability in estimation of the log-odds ratio will ordinarily show greater bias in the estimation of the odds ratio. [For log-normally distributed estimates, which is not exactly the present case, the bias in the arithmetic mean due to variability in the logarithmic estimates is known to be the factor exp(r 2/2).] This kind of thing can be recognized from Hauck's results, where one can see that the bias in estimates of the odds ratio reflects both the bias and the variance in estimates of the log-odds ratio.