Sample size determination in retrospective case-control studies

Abstract

Suppose the k × 1 vector x has a multivariate normal distribution with mean vector μ and variance matrix σ. Suppose the probability of an event of interest given x is a probit plane with regression vector β and suppose that incidence of the event of interest is p (0 < 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 T 2 when β ┼ 0 is noncentral chisquared with k degrees of freedom. It depends only on p and β'σβ. Thus, for p given and specified levels of a relative risk parameter, one can determine sample sizes giving power ε for testing β is zero, i.e. relative risk is unity, at level α. The relation to work of Yang (1978) is discussed.