Approximate tests and interval estimation of the common relative risk in the combination of 2×2 tables

Consider I pairs of mutually independent binomial variates xoi and x1i, with corresponding parameters poi and p1i and sample sizes, noi and n1,, where i = 1, ..., I. We assume that the relative risk is constant, that is, b = Pt /izpoi (i = 1, ..., I). Further let qji = 1-pji, and we assume throughout that 0 < pji, qji < 1 for i = 1, ..., I and j = 0,1. We consider asymptotic results as noi's and n1j's become large, but I is fixed. Extending the results of Cochran (1954) for the common odds ratio, Radhakrishna (1965) derived an asymptotically efficient test of the hypothesis b = 1. Recently, for I = 1, Koopman (1984) heuristically derived a useful approximate method for finding confidence limits for 0. In the present paper we use the methods of score statistics, particularly as given by Bartlett (1953a, 1953b, 1955) to derive, for the general case, the asymptotically efficient tests of b = 1, and of its homogeneity over the I tables, as well as approximate confidence limits for the common 0. The test of 0 = 1 is identical with that of Radhakrishna, and Koopman's limits are a special case of these more general results. Thus this rather simple theory gives a coherent structured methodology for analysing ratios of binomial probabilities parallel to that usually employed for common odds ratios; see, for example, Gart (1970, 1971).