A Survey on Causal Inference <s> Re-weighting Methods <s> Abstract : The results of observational studies are often disputed because of nonrandom treatment assignment. For example, patients at greater risk may be overrepresented in some treatment group. This paper discusses the central role of propensity scores and balancing scores in the analysis of observational studies. The propensity score is the (estimated) conditional probability of assignment to a particular treatment given a vector of observed covariates. Both large and small sample theory show that adjustment for the scalar propensity score is sufficient to remove bias due to all observed covariates. Applications include: matched sampling on the univariate propensity score which is equal percent bias reducing under more general conditions than required for discriminant matching, multivariate adjustment by subclassification on balancing scores where the same subclasses are used to estimate treatment effects for all outcome variables and in all subpopulations, and visual representation of multivariate adjustment by a two-dimensional plot. (Author) <s> BIB001 </s> A Survey on Causal Inference <s> Re-weighting Methods <s> Abstract Direct adjustment or standardization applies population weights to subclass means in an effort to estimate population quantities from a sample that is not representative of the population. Direct adjustment has several attractive features, but when there are many subclasses it can attach large weights to small quantities of data, often in a fairly erratic manner. In the extreme, direct adjustment can attach infinite weight to nonexistent data, a noticeable inconvenience in practice. This article proposes a method of model-based direct adjustment that preserves the attractive features of conventional direct adjustment while stabilizing the weights attached to small subclasses. The sample mean and conventional direct adjustment are both special cases of model-based direct adjustment under two different extreme models for the subclass-specific selection probabilities. The discussion of this method provides some insights into the behavior of true and estimated propensity scores: the estimated scores ... <s> BIB002 </s> A Survey on Causal Inference <s> Re-weighting Methods <s> Abstract In applied problems it is common to specify a model for the conditional mean of a response given a set of regressors. A subset of the regressors may be missing for some study subjects either by design or happenstance. In this article we propose a new class of semiparametric estimators, based on inverse probability weighted estimating equations, that are consistent for parameter vector α0 of the conditional mean model when the data are missing at random in the sense of Rubin and the missingness probabilities are either known or can be parametrically modeled. We show that the asymptotic variance of the optimal estimator in our class attains the semiparametric variance bound for the model by first showing that our estimation problem is a special case of the general problem of parameter estimation in an arbitrary semiparametric model in which the data are missing at random and the probability of observing complete data is bounded away from 0, and then deriving a representation for the efficient score... <s> BIB003 </s> A Survey on Causal Inference <s> Re-weighting Methods <s> In observational studies, investigators have no control over the treatment assignment. The treated and non-treated (that is, control) groups may have large differences on their observed covariates, and these differences can lead to biased estimates of treatment effects. Even traditional covariance analysis adjustments may be inadequate to eliminate this bias. The propensity score, defined as the conditional probability of being treated given the covariates, can be used to balance the covariates in the two groups, and therefore reduce this bias. In order to estimate the propensity score, one must model the distribution of the treatment indicator variable given the observed covariates. Once estimated the propensity score can be used to reduce bias through matching, stratification (subclassification), regression adjustment, or some combination of all three. In this tutorial we discuss the uses of propensity score methods for bias reduction, give references to the literature and illustrate the uses through applied examples. © 1998 John Wiley & Sons, Ltd. <s> BIB004 </s> A Survey on Causal Inference <s> Re-weighting Methods <s> Summary The goal of this article is to construct doubly robust (DR) estimators in ignorable missing data and causal inference models. In a missing data model, an estimator is DR if it remains consistent when either (but not necessarily both) a model for the missingness mechanism or a model for the distribution of the complete data is correctly specified. Because with observational data one can never be sure that either a missingness model or a complete data model is correct, perhaps the best that can be hoped for is to find a DR estimator. DR estimators, in contrast to standard likelihood-based or (nonaugmented) inverse probability-weighted estimators, give the analyst two chances, instead of only one, to make a valid inference. In a causal inference model, an estimator is DR if it remains consistent when either a model for the treatment assignment mechanism or a model for the distribution of the counterfactual data is correctly specified. Because with observational data one can never be sure that a model for the treatment assignment mechanism or a model for the counterfactual data is correct, inference based on DR estimators should improve upon previous approaches. Indeed, we present the results of simulation studies which demonstrate that the finite sample performance of DR estimators is as impressive as theory would predict. The proposed method is applied to a cardiovascular clinical trial. <s> BIB005 </s> A Survey on Causal Inference <s> Re-weighting Methods <s> Comment on ``Performance of Double-Robust Estimators When ``Inverse Probability'' Weights Are Highly Variable'' [arXiv:0804.2958] <s> BIB006 </s> A Survey on Causal Inference <s> Re-weighting Methods <s> 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. <s> BIB007 </s> A Survey on Causal Inference <s> Re-weighting Methods <s> Propensity score matching and weighting are popular methods when estimating causal eects in observational studies. Beyond the assumption of unconfoundedness, however, these methods also require the model for the propensity score to be correctly specied. The recently proposed covariate balancing propensity score (CBPS) methodology increases the robustness to model misspecication by directly optimizing sample covariate balance between the treatment and control groups. In this paper, we extend the CBPS to a continuous treatment. We propose the covariate balancing generalized propensity score (CBGPS) methodology, which minimizes the association between covariates and the treatment. We develop both parametric and nonparametric approaches and show their superior performance over the standard maximum likelihood estimation in a simulation study. The CBGPS methodology is applied to an observational study, whose goal is to estimate the causal eects of political advertisements on campaign contributions. We also provide open-source software that implements the proposed methods. <s> BIB008 </s> A Survey on Causal Inference <s> Re-weighting Methods <s> AbstractInverse probability weighting (IPW) is widely used in empirical work in economics and other disciplines. As Gaussian approximations perform poorly in the presence of “small denominators,” t... <s> BIB009
Due to the existence of confounders, the covariate distributions of the treated group and control group are different, which leads to the selection bias problem as described in Section 2.4. In other words, the treatment assignment is correlated with covariates in the observational data. Sample re-weighting is an effective approach to overcome the selection bias. By assigning appropriate weight to each unit in the observational data, a pseudo-population can be created on which the distributions of the treated group and control group are similar. In sample re-weighting methods, a key concept is balancing score. Balancing score b(x) is a general weighting score, which is the function of x satisfying: W ⊥ ⊥ x |b(x)  , where W is the treatment assignment and x is the background variables. There are various designs of the balancing score, and apparently, the most trivial balancing score is b(x) = x. Besides, propensity score is also a special case of balancing score. Definition 9. Propensity score: The propensity score is defined as the conditional probability of treatment given background variables BIB001 : In detail, a propensity score indicates the probability of a unit being assigned to a particular treatment given a set of observed covariates. Balancing scores that incorporate propensity score are the most common approach. A summarization of the algorithms mentioned in this section is shown in Fig 2. The propensity score based sample reweighting will be introduced in the next section, followed by methods that weigh both samples and the covariates. 3.1.1 Propensity score based sample re-weighting. Propensity scores can be used to reduce selection bias by equating groups based on these covariates. Inverse propensity weighting (IPW) BIB002 BIB001 , also named as inverse probability of treatment weighting (IPTW), assigns a weight r to each sample: where W is the treatment assignment (W = 1 denotes being treated group; W = 0 denotes the control group), and e(x) is the propensity score defined in Eqn.  . After re-weighting, the IPW estimator of average treatment effect (ATE) is:  Both large and small sample theory show that adjustment for the scalar propensity score is enough to remove bias due to all observed covariates BIB001 . The propensity score can be used to balance the covariates in the treatment and control groups and therefore reduce the bias through matching, stratification (subclassification), regression adjustment, or some combination of all three. BIB004 discusses the use of propensity score to reduce the bias, which also provides examples and detailed discussions. However, in practice, the correctness of the IPW estimator highly relies on the correctness of the propensity score estimation, and slightly misspecification of propensity scores would cause ATE estimation error dramatically  . To handle this dilemma, Doubly Robust estimator (DR) BIB003 , also named as Augmented IPW (AIPW), is proposed. DR estimator combines the propensity score weighting with the outcome regression, so that the estimator is robust even when one of the propensity score or outcome regression is incorrect (but not both). In detail, the DR estimator is formalized as: wherem(1, x i ) andm(0, x i ) are the regression model estimations of treated and control outcomes. The DR estimator is consistent and therefore asymptotically unbiased, if either the propensity score is correct or the model correctly reflects the true relationship among exposure and confounders with the outcome  . In reality, one definitely cannot guarantee whether one model can accurately explain the relationship among variables. The combination of outcome regression with weighting by propensity score ensures that the estimators are robust to misspecification of one of these models BIB005 BIB006 BIB003  . The DR estimator consults outcomes to make the IPW estimator robust when propensity score estimation is not correct. An alternative way is to improve the estimation of propensity scores. In the IPW estimator, propensity score serves as both the probability of being treated and the covariate balancing score, covariate balancing propensity score (CBPS)  is proposed to exploit such dual characteristics. In particular, CBPS estimates propensity scores by solving the following problem: wherex i = f (x i ) is a predefined vector-valued measurable function of x i . By solving the above problem, CBPS directly constructs the covariate balancing score from the estimated parametric propensity score, which increase the robustness to the misspecification of the propensity score model. An extension of CBPS is the covariate balancing generalized propensity score (CBGPS) BIB008 , which enables to handle the treatment with continuous value. Due to the continuous valued treatment, it's hard to directly minimized the covariates distribution distance between the control and treated group. CBGPS solves this problem by mitigating the definition of the balancing score. Based on the definition, the treatment assignment is conditionally independent of the background variables, CBGPS directly minimize the correlation between the treatment assignment and the covariates after weighting. In specific, the objective of CBGPS is to learn a propensity score based weight so that the weighted correlation between the treatment assignment and the covariates are minimized: where p(t * |x * ) is the propensity score, and p(t * |x * ) is the balancing weight, t * and x * is the treatment assignment and the background variables after centering and orthogonalizing (i.e., normalization). In a nutshell, both CBPS and CBGPS learns the propensity score based sample weight directly towards the covariate balancing goal, which can alleviates negative effect brought by model misspecification of propensity score. Another drawback of the IPW estimator is that it might be unstable if the estimated propensity scores are small. If the probability of either treatment assignment is small, the logistic regression model can become unstable around the tails, causing the IPW to also be less stable. To overcome this issue, trimming is routinely employed as a regularization strategy, which eliminates the samples whose propensity scores are less than a pre-defined threshold  . However, this approach is highly sensitive to the amount of trimming BIB009 . Also, theoretical results in BIB009 show that the small probability of propensity scores and the trimming procedure may result in different non-Gaussian asymptotic distribution of IPW estimator. Based on this observation, a two-way robustness IPW estimation algorithm is proposed in BIB009 . This method combines subsampling with a local polynomial regression based trimming bias corrector, so that it is robust to both small propensity score and the large scale of trimming threshold. An alternative approach to overcome the instability of IPW under small propensity scores is to redesign the sample weight so that the weight is bounded. In BIB007 , the overlap weight is proposed, in which each unit's weight is proportional to the probability of that unit being assigned to the opposite group. In detail, the overlap weight h(x) is defined as h(x) ∝ 1 − e(x), where e(x) is the propensity score. The overlap weight is bounded within the interval [0, 0.5], and thus it is less sensitive to extreme vale of propensity score. Recent theoretical results show that the overlap weight has the minimum asymptotic variance among all balancing weights BIB007 .