Covariate balance is crucial for unconfounded descriptive or causal comparisons. However, lack of balance is common in observational studies. This article considers weighting strategies for balancing covariates. We define a general class of weights---the balancing weights---that balance the weighted distributions of the covariates between treatment groups. These weights incorporate the propensity score to weight each group to an analyst-selected target population. This class unifies existing weighting methods, including commonly used weights such as inverse-probability weights as special cases. General large-sample results on nonparametric estimation based on these weights are derived. We further propose a new weighting scheme, the overlap weights, in which each unit's weight is proportional to the probability of that unit being assigned to the opposite group. The overlap weights are bounded, and minimize the asymptotic variance of the weighted average treatment effect among the class of balancing weights. The overlap weights also possess a desirable small-sample exact balance property, based on which we propose a new method that achieves exact balance for means of any selected set of covariates. Two applications illustrate these methods and compare them with other approaches.Comment: 33 pages, 5 figures, 5 table

Randomized experiments are the "gold standard" for estimating causal effects, yet often in practice, chance imbalances exist in covariate distributions between treatment groups. If covariate data are available before units are exposed to treatments, these chance imbalances can be mitigated by first checking covariate balance before the physical experiment takes place. Provided a precise definition of imbalance has been specified in advance, unbalanced randomizations can be discarded, followed by a rerandomization, and this process can continue until a randomization yielding balance according to the definition is achieved. By improving covariate balance, rerandomization provides more precise and trustworthy estimates of treatment effects.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1008 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Objective: To examine the persistence of the original treatment effects 10 years after the Diabetes Control and Complications Trial (DCCT) in the follow-up Epidemiology of Diabetes Interventions and Complications (EDIC) study. In the DCCT, intensive therapy aimed at nearnormal glycemia reduced the risk of microvascular complications of type 1 diabetes mellitus compared with conventional therapy.Methods: Retinopathy was evaluated by fundus photography in 1211 subjects at EDIC year 10. Further 3-step progression on the Early Treatment Diabetic Retinopathy Study scale from DCCT closeout was the primary outcome.Results: After 10 years of EDIC follow-up, there was no significant difference in mean glycated hemoglobin levels (8.07% vs 7.98%) between the original treatment groups. Nevertheless, compared with the former conven-tional treatment group, the former intensive group had significantly lower incidences from DCCT close of further retinopathy progression and proliferative retinopathy or worse (hazard reductions, 53%-56%; PϽ.001). The risk (hazard) reductions at 10 years of EDIC were attenuated compared with the 70% to 71% over the first 4 years of EDIC (PϽ.001). The persistent beneficial effects of former intensive therapy were largely explained by the difference in glycated hemoglobin levels during DCCT. Conclusion:The persistent difference in diabetic retinopathy between former intensive and conventional therapy ("metabolic memory") continues for at least 10 years but may be waning.

The seminal work of Morgan & Rubin (2012) considers rerandomization for all the units at one time.In practice, however, experimenters may have to rerandomize units sequentially. For example, a clinician studying a rare disease may be unable to wait to perform an experiment until all the experimental units are recruited. Our work offers a mathematical framework for sequential rerandomization designs, where the experimental units are enrolled in groups. We formulate an adaptive rerandomization procedure for balancing treatment/control assignments over some continuous or binary covariates, using Mahalanobis distance as the imbalance measure. We prove in our key result that given the same number of rerandomizations, in expected value, under certain mild assumptions, sequential rerandomization achieves better covariate balance than rerandomization at one time.

When conducting a randomized experiment, if an allocation yields treatment groups that differ meaningfully with respect to relevant covariates, groups should be rerandomized. The process involves specifying an explicit criterion for whether an allocation is acceptable, based on a measure of covariate balance, and rerandomizing units until an acceptable allocation is obtained. Here we illustrate how rerandomization could have improved the design of an already conducted randomized experiment on vocabulary and mathematics training programs, then provide a rerandomization procedure for covariates that vary in importance, and finally offer other extensions for rerandomization, including methods addressing computational efficiency. When covariates vary in a priori importance, better balance should be required for more important covariates. Rerandomization based on Mahalanobis distance preserves the joint distribution of covariates, but balances all covariates equally. Here we propose rerandomizing based on Mahalanobis distance within tiers of covariate importance. Because balancing covariates in one tier will in general also partially balance covariates in other tiers, for each subsequent tier we explicitly balance only the components orthogonal to covariates in more important tiers.