Minimal dispersion approximately balancing weights: asymptotic properties and practical considerations

Abstract

Weighting methods are widely used to adjust for covariates in observational studies, sample surveys, and regression settings. In this paper, we study a class of recently proposed weighting methods which find the weights of minimum dispersion that approximately balance the covariates. We call these weights minimal weights and study them under a common optimization framework. The key observation is the connection between approximate covariate balance and shrinkage estimation of the propensity score. This connection leads to both theoretical and practical developments. From a theoretical standpoint, we characterize the asymptotic properties of minimal weights and show that, under standard smoothness conditions on the propensity score function, minimal weights are consistent estimates of the true inverse probability weights. Also, we show that the resulting weighting estimator is consistent, asymptotically normal, and semiparametrically efficient. From a practical standpoint, we present a finite sample oracle inequality that bounds the loss incurred by balancing more functions of the covariates than strictly needed. This inequality shows that minimal weights implicitly bound the number of active covariate balance constraints. We finally provide a tuning algorithm for choosing the degree of approximate balance in minimal weights. We conclude the paper with 1 arXiv:1705.00998v3 [stat.ME] 25 Apr 2019 four empirical studies that suggest approximate balance is preferable to exact balance, especially when there is limited overlap in covariate distributions. In these studies, we show that the root mean squared error of the weighting estimator can be reduced by as much as a half with approximate balance.

Balancing vs modeling approaches to weighting in practice

“…We see that the above form of the dual has similar structure to the dual form in lemma 1 of Wang and Zubizarreta. 21 It is now easy to see (following Proof of Theorem 1 of the same article) that (A4) is equivalent to…”

“…In view of these desirable features, we discuss and review the modeling and balancing approaches to weighting. With regard to the balancing approach, we focus on a wide class of weighting methods discussed by Wang and Zubizarreta (2020) 21 called minimal dispersion approximately balancing weights, or minimal weights for short. These weights are minimal in that they have minimum dispersion (eg, variance) and balance approximately (not necessarily exactly) statistics of the distribution of the observed covariates.…”

“…7,53,54 The connection between minimal weights and shrinkage estimation of the propensity scores allows several attractive large sample properties to be established for the balancing weights and the corresponding weighted estimators. Wang and Zubizarreta (2020) 21 have established these properties in the framework of estimation with incomplete outcome data, but they carry over to the setup of estimating average treatment effects in causal inference. The properties rely on smoothness conditions on the propensity score function and the mean function of the response given covariates.…”

“…In particular, we have considered the minimal dispersion approximately balancing class of weights, or minimal weights. 21 We have shown that in the framework of ATT estimation, minimal weights estimate a normalized version of the inverse probability weights under a form of shrinkage estimation of the propensity score. In order to choose the tuning parameter for the optimization problem for the minimal weights, we have proposed a slight variant to the algorithm given in the article by Wang and Zubizarreta (2020).…”

Covariate Distribution Balance via Propensity Scores

“…Among the aforementioned papers, only Chan et al (2016) and Wong and Chan (2018) provide formal inference procedures for treatment effect measures. The implementation of their procedures, however, requires appropriately choosing tuning parameters, which is usually a delicate task, is tied to a given treatment effect parameter of interest (the average treatment effect), and requires modelling assumptions about the outcome data; see also Wang and Zubizarreta (2018) and Hirshberg and Wager (2018) for related unpublished work. Most recently, Han et al (2019) propose a hybrid framework that combines PS models and calibration weights and is also suitable for estimating quantile treatment effects.…”