Summary: Matched sampling is a standard technique for controlling bias in observational studies due to specific covariates. Since {\it P. R. Rosenbaum} and the first author [ibid. 70, 41-55 (1983;

Zbl 0522.62091)], multivariate matching methods based on estimated propensity scores have been used with increasing frequency in medical, educational, and sociological applications.
We obtain analytic expressions for the effect of matching using linear propensity score methods with normal distributions. These expressions cover cases where the propensity score is either known, or estimated using either discriminant analysis or logistic regression, as is typically done in current practice. The results show that matching using estimated propensity scores not only reduces bias along the population propensity score, but also controls variation of components orthogonal to it. Matching on estimated rather than population propensity scores can therefore lead to relatively large variance reduction, as much as a factor of two in common matching settings where close matches are possible.
Approximations are given for the magnitude of this variance reduction, which can be computed using estimates obtained from the matching pools. Related expressions for bias reduction are also presented which suggest that, in difficult matching situations, the use of population scores leads to greater bias reduction than the use of estimated scores.