Sample sizes for estimating differences in proportions--can we keep things simple?

Sample sizes for studies that aim to estimate differences in proportions are often calculated using a confidence interval approach. In particular, the methods advocated by Day ( 1988 ) and Bristol ( 1989 ), based on asymptotic normal approximations, are commonly used. We compare these to the Wilson score approach discussed by Newcombe ( 1998 ) and show, except for extreme values, that the Day method gives results close to those of Wilson score method, while that of Bristol is consistently higher. We argue that the asymptotic normal approximation approach of Day serves as a better guide for quick calculations of sample sizes.