A Survey on Causal Inference <s> Confounder balancing. <s> This paper addresses strategies for selecting variables for adjustment in non-experimental comparative effectiveness research and uses causal graphs to illustrate the causal network that relates treatment to outcome. Variables in the causal network take on multiple structural forms. Adjustment for a common cause pathway between treatment and outcome can remove confounding, whereas adjustment for other structural types may increase bias. For this reason, variable selection would ideally be based on an understanding of the causal network; however, the true causal network is rarely known. Therefore, we describe more practical variable selection approaches based on background knowledge when the causal structure is only partially known. These approaches include adjustment for all observed pretreatment variables thought to have some connection to the outcome, all known risk factors for the outcome, and all direct causes of the treatment or the outcome. Empirical approaches, such as forward and backward selection and automatic high-dimensional proxy adjustment, are also discussed. As there is a continuum between knowing and not knowing the causal, structural relations of variables, we recommend addressing variable selection in a practical way that involves a combination of background knowledge and empirical selection and that uses high-dimensional approaches. This empirical approach can be used to select from a set of a priori variables based on the researcher's knowledge to be included in the final analysis or to identify additional variables for consideration. This more limited use of empirically derived variables may reduce confounding while simultaneously reducing the risk of including variables that may increase bias. Copyright (c) 2013 John Wiley & Sons, Ltd. <s> BIB001 </s> A Survey on Causal Inference <s> Confounder balancing. <s> We provide a principled way for investigators to analyze randomized experiments when the number of covariates is large. Investigators often use linear multivariate regression to analyze randomized experiments instead of simply reporting the difference of means between treatment and control groups. Their aim is to reduce the variance of the estimated treatment effect by adjusting for covariates. If there are a large number of covariates relative to the number of observations, regression may perform poorly because of overfitting. In such cases, the least absolute shrinkage and selection operator (Lasso) may be helpful. We study the resulting Lasso-based treatment effect estimator under the Neyman–Rubin model of randomized experiments. We present theoretical conditions that guarantee that the estimator is more efficient than the simple difference-of-means estimator, and we provide a conservative estimator of the asymptotic variance, which can yield tighter confidence intervals than the difference-of-means estimator. Simulation and data examples show that Lasso-based adjustment can be advantageous even when the number of covariates is less than the number of observations. Specifically, a variant using Lasso for selection and ordinary least squares (OLS) for estimation performs particularly well, and it chooses a smoothing parameter based on combined performance of Lasso and OLS. <s> BIB002 </s> A Survey on Causal Inference <s> Confounder balancing. <s> Estimating treatment effect plays an important role on decision making in many fields, such as social marketing, healthcare, and public policy. The key challenge on estimating treatment effect in the wild observational studies is to handle confounding bias induced by imbalance of the confounder distributions between treated and control units. Traditional methods remove confounding bias by re-weighting units with supposedly accurate propensity score estimation under the unconfoundedness assumption. Controlling high-dimensional variables may make the unconfoundedness assumption more plausible, but poses new challenge on accurate propensity score estimation. One strand of recent literature seeks to directly optimize weights to balance confounder distributions, bypassing propensity score estimation. But existing balancing methods fail to do selection and differentiation among the pool of a large number of potential confounders, leading to possible underperformance in many high dimensional settings. In this paper, we propose a data-driven Differentiated Confounder Balancing (DCB) algorithm to jointly select confounders, differentiate weights of confounders and balance confounder distributions for treatment effect estimation in the wild high dimensional settings. The synergistic learning algorithm we proposed is more capable of reducing the confounding bias in many observational studies. To validate the effectiveness of our DCB algorithm, we conduct extensive experiments on both synthetic and real datasets. The experimental results clearly demonstrate that our DCB algorithm outperforms the state-of-the-art methods. We further show that the top features ranked by our algorithm generate accurate prediction of online advertising effect. <s> BIB003
The aforementioned sample re-weighting methods could achieve balance in the sense that the observed variables are considered equally as confounders. However, in real cases, not all the observed variables are confounders. Some of the variables, named as adjustment variables, are only predictive to the outcome, and some others might be irrelevant variables. Adjusting on the adjustment variables by Lasso, although it cannot reduce the bias, helps decrease the variance BIB002 BIB001 . While including the irrelevant variables would cause overfitting. Based on the separateness assumption that the observed variables can be decomposed into confounders, adjusted variables and the irrelevant variables, in  , the Data-Driven Variable Decomposition (D 2 VD) algorithm is proposed to distinguish the confounders and adjustment variables, and meanwhile, eliminate the irrelevant variables. In detail, the adjusted outcome is written as: where z is the adjustment variables. Therefore, the ATE estimator of D 2 VD is: . To get ATE D 2 VD , the Y * D 2 VD is regressed on all observed variables. The objective function is l 2 loss between Y * D 2 VD and the linear regression function on all observed variables, along with sparse regularization to distinguish the confounder, adjusted variables, and irrelevant variables. However, little prior knowledge about the interactions among observed variables is provided in practice, and the data are usually high-dimensional and noisy. To solve this, Differentiated Confounder Balancing (DCB) algorithm BIB003 is proposed to select and differentiate confounders to balance the distributions. Overall, DCB balances the distributions by re-weighting both the samples and confounders.