Logit confidence intervals and the inverse sinh transformation

The logit Wald and Wilson score intervals for the binomial proportion are both symmetrical on the logit scale. They are closely related: the widths of the two intervals on the logit scale are shown to be related by a simple sinh function. Similarly, the Woolf logit Wald interval for the odds ratio and the analogous interval for the relative risk may be shortened by inverse sinh transformation. This modified interval can be calculated even when one of the cells contains a zero. It then takes a very simple form in which the appropriate centile of the chi-square distribution on one degree of freedom is substituted for the zero. These suggested modified intervals require evaluation of coverage probabilities.