Odds Ratio - Statistical Inference

Statistical Inference

Several approaches to statistical inference for odds ratios have been developed.

One approach to inference uses large sample approximations to the sampling distribution of the log odds ratio (the natural logarithm of the odds ratio). If we use the joint probability notation defined above, the population log odds ratio is

If we observe data in the form of a contingency table

then the probabilities in the joint distribution can be estimated as

where p̂ = n_ij / n, with n = n₁₁ + n₁₀ + n₀₁ + n₀₀ being the sum of all four cell counts. The sample log odds ratio is

.

The distribution of the log odds ratio is approximately normal with:

The standard error for the log odds ratio is approximately

.

This is an asymptotic approximation, and will not give a meaningful result if any of the cell counts are very small. If L is the sample log odds ratio, an approximate 95% confidence interval for the population log odds ratio is L ± 1.96SE. This can be mapped to exp(L − 1.96SE), exp(L + 1.96SE) to obtain a 95% confidence interval for the odds ratio. If we wish to test the hypothesis that the population odds ratio equals one, the two-sided p-value is 2P(Z< −|L|/SE), where P denotes a probability, and Z denotes a standard normal random variable.

An alternative approach to inference for odds ratios looks at the distribution of the data conditionally on the marginal frequencies of X and Y. An advantage of this approach is that the sampling distribution of the odds ratio can be expressed exactly.