Allocation of subjects to test null relative risks smaller than one

Allocating a proportion k'=1/(1+ radicalr(0)) of subjects to an intervention is a practical approach to approximately maximize power for testing whether an intervention reduces relative risk of disease below a null ratio r(0)<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> or =r(0)>0.33,2/3 for 0.33> or =r(0)>0.17 and 3/4 for 0.17> or =r(0)> or =0.10. Both k' and k'(s) are easily calculated and invariant to alterations in disease rate estimates under null and alternative hypotheses, when r(0) 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 size and sample size), for likelihood score tests. Compared to equal allocation, k' and k'(s) reduced sample sizes by amounts ranging from approximately 5.5 per cent for r(0)=0.50 to approximately 24 per cent for r(0)=0.10. These sample size savings may be particularly important for large studies of prophylactic interventions such as vaccines. While k' was derived from variance minimization for an arcsine transformation, we do not recommend the arcsine test, since its true size exceeded the nominal value. In contrast, the true size for the uncorrected score test was less than the nominal size. A skewness correction made the size of the score test very close to the nominal 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(0)<1.