Biostatistics (2004), 5, 2, pp. 263-275
Biostatistics Vol. 5 No. 2 © Oxford University Press 2004; all rights reserved.
Optimal multivariate matching before randomization
Department of Statistics, The Wharton School, University of Pennsylvania, 400 Jon M. Huntsman Hall, 3730 Walnut Street, Philadelphia, PA 19104-6340, USA
Center for Statistical Sciences, Department of Community Health, Brown University School of Medicine, Providence, RI 02912, USA
Department of Pediatrics, University of Pennsylvania, School of Medicine, 3535 Market Street, STE 1029, Philadelphia, PA 19104-3309,USA
Department of Statistics, The Wharton School, University of Pennsylvania, 473 Jon M. Huntsman Hall, 3730 Walnut Street, Philadelphia, PA 19104-6340, USA
rosenbaum{at}stat.wharton.upenn.edu
* To whom correspondence should be addressed.
Although blocking or pairing before randomization is a basic principle of experimental design, the principle is almost invariably applied to at most one or two blocking variables. Here, we discuss the use of optimal multivariate matching prior to randomization to improve covariate balance for many variables at the same time, presenting an algorithm and a case-study of its performance. The method is useful when all subjects, or large groups of subjects, are randomized at the same time. Optimal matching divides a single group of 
subjects into 
pairs to minimize covariate differences within pairs—the so-called nonbipartite matching problem—then one subject in each pair is picked at random for treatment, the other being assigned to control. Using the baseline covariate data for 132 patients from an actual, unmatched, randomized experiment, we construct 66 pairs matching for 14 covariates. We then create 10\,000 unmatched and 10\,000 matched randomized experiments by repeatedly randomizing the 132 patients, and compare the covariate balance with and without matching. By every measure, every one of the 14 covariates was substantially better balanced when randomization was performed within matched pairs. Even after covariance adjustment for chance imbalances in the 14 covariates, matched randomizations provided more accurate estimates than unmatched randomizations, the increase in accuracy being equivalent to, on average, a 7% increase in sample size. In randomization tests of no treatment effect, matched randomizations using the signed rank test had substantially higher power than unmatched randomizations using the rank sum test, even when only 2 of 14 covariates were relevant to a simulated response. Unmatched randomizations experienced rare disasters which were consistently avoided by matched randomizations.
Keywords: Experimental design; Matched experiment; Network optimization; Randomized experiment