SUMMARY Suppose the k x 1 vector x has a multivariate normal distribution with mean vector p and variance matrix E. Suppose the probability of an event of interest given x is a probit plane with regression vector 8 and suppose that incidence of the event of interest is p (O <p < 1). Based on independent samples of equal size from the event and nonevent groups, for large samples, the distribution of the two-sample Hotelling's T2 when * t 0 is noncentral chi- squared with k degrees of freedom. It depends only on p and f'E8. Thus, for p given and specified levels of a relative risk parameter, one can determine sample sizes giving power e for testing 8 is zero, i.e. relative risk is unity, at level a. The relation to work of Yang (1978) is discussed. A number of recent papers have considered sample size determination in case-control studies, i.e. studies in which one samples cases and controls separately and wishes to test from possibly multivariate measurements made on the two groups whether there is an asso- ciation between 'caseness', e.g. occurrence of a particular disease, and the levels of the measurements. Yang (1978) appears to be the only worker who has considered determination of sample size in the context of a comparison of multivariate continuous responses between cases and noncases taking into account that it may be appropriate to consider that differences in distribution for cases and noncases may be more profound than differences between means. Yang assumes that in a total population, i.e. cases plus noncases, the characteristics which may be associated with caseness have a nonsingular multivariate normal distribution of dimensionality k, with mean vector p and variance-covariance matrix E. Yang also assumes that p(x), the probability that an individual with characteristics x will be a case is given by a probit plane, that is p(x) = P{fo + P'(x - p)}, where ?P(z) is the cumulative distribution function of a normal variate with mean zero and variance unity, ,' is a row vector of k con- stants and P is also a constant. Yang points out that if one has a sample of cases of size n, and a sample of noncases of size n., the hypothesis 8 = 0, that is p(x) = p, independent of x, is appropriately tested by a two-sample Hotelling T2 test, where E is estimated using pooled sums of squares of deviations and cross-products of deviations from the sample mean vector of each sample. The parameters relevant to Yang's sample size computation are p, the expected proportionate incidence over the total population of cases and noncases, a quantity Po = yp, 0 < y < 1, and 0 = pr {p(x) , pO}, where x is a random k-vector from the total popu- lation. It is then argued that for 0 01 < ( 0' 10 and y near 2, and moderate to large samples,