In a placebo-controlled vaccine efficacy trial or a trial of equivalence of vaccines, one may wish to show that relative risk of disease is less than a specified value R0, not equal to one. This paper compares three methods for estimating relative risk in the binomial setting, based on a logarithmic transformation, likelihood scores, and a Poisson approximation. Exact power and size of test are calculated by enumeration of possible binomial outcomes, and power is approximated from asymptotic formulations. Although the score method is generally preferable, for most studies of practical interest the log and score methods are comparable, and the Poisson method is also appropriate for small risks, up to about 0.05. When true and null relative risks are less than one, unequal allocation of study individuals can increase power, and the asymptotic formula for the log method may substantially underestimate power; in such a study the power approximation for the score method is more reliable, even if the log method is used in analysis. Exact power calculations are helpful in planning studies. The log and Poisson methods, but not the score method, apply readily in the case of unequal follow-up.