Matching when covariables are normally distributed

Abstract

SUMMARY The use of matched pairs to reduce effects of concomitant variation in observational studies is considered when covariables have multivariate normal distributions. A case is matched by the nearest control, provided the control lies within a specified distance; otherwise the case is unmatched. Simulation studies indicate that if the number of cases to be matched is less than half the number of controls to choose from, cases rarely compete for the same control, so that matched pairs are effectively independent. The probability, P1, of being able to construct a matched pair, together with moments and other parameters of the distributions of matched cases and controls, are expressed in terms of single or double integrals. These have been evaluated. The relationship of P1 to the number of covariables, the number of controls to choose from and other parameters is outlined. The optimal radius of search depends on the method of analysis. Matching can double the precision of a study. Some key word8: Analysis of covariance; Bias; Matched controls; Noncentral chi squared; Observational study; Precision; Probability of a match. In biomedical studies involving the comparison of cases with normal controls, matched pairs are often used to reduce unwanted effects of concomitant variation. The method is especialy useful in observational studies. When cases and controls are exactly matched the design is fully efficient (Cochran, 1965), and Ury (1975) has tabulated the gain from using multiple exactly matched controls. In practice, however, difficulties arise in forming closely matched pairs when several concomitant variables are involved. McKinlay (1974) has discussed the situation when covariables are qualitative or categorized. The purpose of this paper is to examine these problems analytically when concomitant variables are continuous and to present some practical conclusions.