Confidence intervals for the common odds ratio based on the inverse sinh transformation

This paper proposes two new approximate confidence limit methods for the common odds ratio from multiple 2 × 2 tables. The two new procedures, based on the asymptotic distribution of Woolf estimator and Mantel-Haenszel estimator, associate with inverse sinh transformation. We employ three pseudo-frequency methods to calculate confidence intervals in order to avoid the interval failure caused by the presence of zero cells in multiple 2 × 2 tables. We develop the modified inverse sinh intervals for the common odds ratio which add one pseudo-frequency (c₁) to all the cells before computing the point estimate of common odds ratio and another pseudo-frequency (c₂) to all the cells before computing the standard error estimate. The simulation is to evaluate the 22 confidence intervals, including Woolf, Mantel-Haenszel, their inverse sinh intervals, and their pseudo-frequency modified inverse sinh intervals, in terms of their coverage probabilities and average log lengths. Simulation results demonstrate that the adjusted inverse sinh intervals by two different pseudo-frequencies perform quite well when c₂ is slightly greater than c₁ since the coverage probabilities of them are closer to confidence level of 95%. Larger values of c₂ lead to narrow intervals and low coverage probabilities. We also find that inverse sinh intervals are shorter than untransformed intervals based on Woolf estimator and Mantel-Haenszel estimator, respectively. These procedures were illustrated with two clinical trials.