Efficiency of case-control studies with multiple controls per case: continuous or dichotomous data

The statistical efficiency of a study design which matches each case with k controls is compared with that of the standard matched-pairs setup. For the situation in which the distribution of measurements on the factor under investigation is continuous, the comparison criterion used is the reciprocal of the variance ratio of the corresponding case-control difference estimators; for dichotomous observations, the Pitman efficiency of Miettinen's test [1969] or of Pike and Morrow's test [1970] relative to McNemar's test is used. When the variances for continuously distributed case and control measurements are equal to the same result, 2k/(k + 1), is obtained for both situations and, more generally, the efficiency of k1 controls relative to k2 controls is given by k1(k2 + 1)/k2(k1 + 1). The Pitman efficiency is compared with the "practical efficiency" for large-sample dichotomous situations.