Allocation of subjects to test null relative risks smaller than one

Allocating a proportion k' = 1/(1 + rootr(o)) of subjects to an interventio n is a practical approach to approximately maximize power for testing wheth er an intervention reduces relative risk of disease below a null ratio r(o) < 1. Furthermore, allocating k(s)', a convenient fraction close to k', to intervention performs nearly as well; for example, allocating k(s)'=3/5 for 0.5<greater than or equal to>r(o)>0.33,2/3 for 0.33 greater than or equal tor(o)>0.17 and 3/4 for 0.17 greater than or equal to r(o) greater than or equal to 0.10. Both k' and k(s)' are easily calculated and invariant to alt erations in disease rate estimates under null and alternative hypotheses, w hen ro remains constant. In examples that we studied, allocating k' (or k(s )') subjects to intervention achieved close to the minimum possible sample size, given test size and power (equivalently, maximum power, given test si ze and sample size), for likelihood score tests. Compared to equal allocati on, k' and k(s)' reduced sample sizes by amounts ranging from similar to 5. 5 per cent for r(o) = 0.50 to similar to 24 per cent for r(o) = 0.10. These sample size savings may be particularly important for large studies of pro phylactic interventions such as vaccines. While k' was derived from varianc e minimization for an arcsine transformation, we do not recommend the arcsi ne test, since its true size exceeded the nominal value. In contrast, the t rue size for the uncorrected score test was less than the nominal size. A s kewness correction made the size of the score test very close to the nomina l level and slightly increased power. We recommend using the score test, or the skewness-corrected score test, for planing studies designed to show a ratio of proportions is less than a prespecified null ratio r(o) < 1. Copyr ight (C) 2001 John Wiley & Sons, Ltd.