Stable Weights that Balance Covariates for Estimation With Incomplete Outcome Data

“…in practical methodologies that emphasize balance when paired with AIPW methodology (Athey et al., 2016; Chan et al., 2015; Graham et al., 2012, 2016; Hainmueller, 2012; Hirano et al., 2001, 2003; Imai and Ratkovic, 2014; Kallus, 2016; Wang and Zubizarreta, 2017; Zhao, 2016; Zubizarreta, 2015).”

“…et al., 1994; Rotnitzky et al., 1998), first introduced in the area of observational study, has been known as an effective method to deal with such causal missingness and still remains popular (Bang & Robins, 2005; Kang et al., 2007; Rotnitzky et al., 2012; Han & Wang, 2013; Zubizarreta, 2015).”

“Covariate balanced estimators have been traditionally used to reduce bias in observational studies and missing data models (e.g., Hainmueller, 2011; Imai and Ratkovic, 2014; Zubizarreta, 2015), but covariate selection for balancing remains an open problem.”

“The key idea is to construct sample weights by either employing propensity scores (Rosenbaum and Rubin, 1983, Kuang et al., 2017b, Austin, 2011) or optimizing weights directly (Athey et al., 2018, Zubizarreta, 2015, Hainmueller, 2012, Kuang et al., 2017a).”

“mization have been very popular recently (e.g., Chan et al., 2016; Hainmueller, 2012; Imai and Ratkovic, 2014; Li et al., 2018; Qin and Zhang, 2007; Wong and Chan, 2018; Yiu and Su, 2018; Zhao, 2019; Zubizarreta, 2015). Following the same idea, we seek functional propensity score estimates which can minimize the corre…”

“Zubizarreta [22] proposed stable balancing weights by forcing wi ≥ 0 but relaxing the constraints in (13) to | ∑n i=1 wiXi − X̄0| < d to allow for slight imbalance, controlled by a vector d to make the weights more stable.”

“…paper gives a brief review of the covariate balancing propensity score method by Imai and Ratkovic (2014), the stable balance weighting procedure by Zubizarreta (2015), the calibration balance weighting approach by Chan, et al. (2016), and the integrated propensity score technique by Sant’Anna, et…”

“The method proposed by Zubizarreta (2015), Stable Balancing Weights, is similarly designed to minimize the variance of the weights subject to the reduction of the average weight (loss function) for a set of balancing constraints.”

“Hainmueller [12], Zubizarreta [13] and Chan et al. [14] directly optimized the sample weights to attain a set of predefined balancing conditions.”

“Inverse probability weighting (e.g., Chen et al. [9]; Creemers et al. [10]; Zubizarreta [11]; Hughes et al. [12]) heavily relies on the information from complete cases, which can be problematic when the fraction of completely observed subject is small.”

“The synthetic control method (SCM), first proposed by Abadie et al. (2010, 2015), is a widely used approach in the social sciences for evaluating the unit-specific effects of large-scale, infrequent interventions on one or few treated unit(s), with the presence of longitudinal evolution of…”

“Instead, this paper applied stable balancing weights (SBWs) (Chattopadhyay et al., 2020; de los Angeles Resa & Zubizarreta, 2020; Zubizarreta, 2015), which achieved covariate balancing up to a user-specified level.”

“Most of the current research is based on univariate treatment (binary, multivalued or continuous treatment) (Imai and Ratkovic (2014); Zhu et al. (2015); Fong et al. (2018); Zubizarreta (2015a); Chan et al. (2016); Xiong et al. (2017); Yiu and Su (2018); Dong et al. (2021); Hsu et al. (2022)).”

“A recently developed optimization-based weighting method, stable balancing weights, overcomes the challenge of partial IPD by estimating weights of minimum variance that re-weight IPD to mimic researcher-specified target populations (estimands) [19].”

“Although regularization techniques can help mitigate these issues, extreme propensity scores and the imperfections of the imputation model cannot always be completely avoided (Kang and Schafer 2007; Zubizarreta 2015).”

“Imai, Ratkovic [13] and Zubizarreta [14] try to seek a better way to estimate the weight of the inverse probability to improve performance.”

“Accordingly, a different approach to weighting consists in using moment method based on appropriate balancing with respect to covariates, or better in controlling the maximal imbalance; cfr. [18], [19], [20].”

“…average treatment effect (ATE), which can be viewed as a missing data problem with only the response variable missing; see, for example, Zubizarreta [35], Chan et al. [4] and Wong and Chan [31], and Zhao [34] introduced tailored loss functions to show the connection between IPW and balancing…”

“This approach extends the direct confounder balancing method by [16, 43], commonly used in causal inference to eliminate confounding effects, to cases with multiple levels.”

“However, their meth-ods require exact fulfillment of covariate balancing conditions, implying that an equational constraint on a moment condition is necessary.”

“…case–control analysis (details in Supplementary Material A, A.2 and Figure S1) we (i) used covariate balancing by optimization-based weighting [42,43] to achieve balance between cases and controls on as many characteristics as possible (implemented in package WeightIt [44] in R version 4.5.0…”

“Different approaches to balancing weights make different choices about these functions — e.g., re-placing the 2-norm of imbalance with a stronger norm — though they all follow the same principle of minimizing a combination of imbalance and weight dispersion/complexity.”

“It is worth mentioning that variable balancing methods have also played an important role [12], [13].”

“30 34 However, unlike these approaches, our article does not aim to achieve covariate balance directly.”

“…weights can improve performance by increasing stability and covariate balance while reducing weight dispersion, thereby yielding more accurate treatment effect estimates (Chattopadhyay et al., 2020; Deshpande & Kuleshov, 2023; Graham et al., 2012; Imai & Ratkovic, 2014; Zubizarreta, 2015).”

“For ω j = θ s = 1, and λ nn ∞ , we recover the MC estimator [ABD + 21b].”

“…stable-balancing weight methods have been shown to provide ample flexibility with respect to pre-specification of covariate balance goals, and to result in bet - ter post-weighting covariate balance with larger effective sample sizes compared to propensity score weighting [15, 16 ].”