An Improved Approximate Formula for Calculating Sample Sizes for Comparing Two Binomial Distributions

If x and y are each binomially distributed with index n and parameters pl and p2 respectively, then the comparison of these two binomial distributions is usually displayed as a 2 X 2 table and Fisher's (one-sided) test may be used to test the null hypothesisp1 = p2, against the alternative hypothesis that pl > p2. The exact test is based on arguing conditionally on the observed number ofsuccesses, i.e., x + y, see Yates (1934) and Fisher (1935). The distribution of x with x + y = m fixed depends on pl and P2 only through the odds ratio 0 = (p1q2)/(q,p2), where qi 1-Pi rhe conditional distribution is Pr (x 1 0) C (n,x) C (n,y) AX/zzC(n,i)C(n,m-i)0i, where i takes the values L = max(O, m-n) to U = min(n, m). Let xc be the critical value of x for the exact test of P1 > P2 (i.e., 0 > 1 ) against the null hypothesis 0 = 1 with type I error oe, so that