A Survey on Causal Inference <s> Stratification 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> Stratification Methods <s> Propensity score methods are used to estimate a treatment effect with observational data. This paper considers the formation of propensity score subclasses by investigating different methods for determining subclass boundaries and the number of subclasses used. We compare several methods: balancing a summary of the observed information matrix and equal-frequency subclasses. Subclasses that balance the inverse variance of the treatment effect reduce the mean squared error of the estimates and maximize the number of usable subclasses. <s> BIB002 </s> A Survey on Causal Inference <s> Stratification Methods <s> Many scientific problems require that treatment comparisons be adjusted for posttreatment variables, but the estimands underlying standard methods are not causal effects. To address this deficiency, we propose a general framework for comparing treatments adjusting for posttreatment variables that yields principal effects based on principal stratification. Principal stratification with respect to a posttreatment variable is a cross-classification of subjects defined by the joint potential values of that posttreatment variable tinder each of the treatments being compared. Principal effects are causal effects within a principal stratum. The key property of principal strata is that they are not affected by treatment assignment and therefore can be used just as any pretreatment covariate. such as age category. As a result, the central property of our principal effects is that they are always causal effects and do not suffer from the complications of standard posttreatment-adjusted estimands. We discuss briefly that such principal causal effects are the link between three recent applications with adjustment for posttreatment variables: (i) treatment noncompliance, (ii) missing outcomes (dropout) following treatment noncompliance. and (iii) censoring by death. We then attack the problem of surrogate or biomarker endpoints, where we show, using principal causal effects, that all current definitions of surrogacy, even when perfectly true, do not generally have the desired interpretation as causal effects of treatment on outcome. We go on to forrmulate estimands based on principal stratification and principal causal effects and show their superiority. <s> BIB003 </s> A Survey on Causal Inference <s> Stratification Methods <s> Choosing optimal (or at least better) policies is an important problem in domains from medicine to education to finance and many others. One approach to this problem is through controlled experiments/trials - but controlled experiments are expensive. Hence it is important to choose the best policies on the basis of observational data. This presents two difficult challenges: (i) missing counterfactuals, and (ii) selection bias. This paper presents theoretical bounds on estimation errors of counterfactuals from observational data by making connections to domain adaptation theory. It also presents a principled way of choosing optimal policies using domain adversarial neural networks. We illustrate the effectiveness of domain adversarial training together with various features of our algorithm on a semi-synthetic breast cancer dataset and a supervised UCI dataset (Statlog). <s> BIB004
Stratification, also named as subclassification or blocking  , is a representative method to adjust the confounders. The idea of stratification is to adjust the bias that stems from the difference between the treated group and the control group by splitting the entire group into homogeneous subgroups (blocks). Ideally, in each subgroup, the treated group and the control group are similar under certain measurements over the covariates, therefore, the units in the same subgroup can be viewed as sampled from the data under randomized controlled trials. Based on the homogeneity of each subgroup, the treatment effect within each subgroup (i.e., CATE) can be calculated through the method developed on RCTs data. After having the CATE of each subgroup, the treatment effect over the interested group can be obtained by combining the CATEs of subgroups belonging to that group, as shown in BIB004 . In the following, we adopt the calculation of ATE as an example. In detail, if we separate the whole dataset into J blocks, the ATE is estimated as: whereȲ t (j) andȲ c (j) are the average of the treated outcome and control outcome in the j-th block, respectively. q(j) = N (j) N is the portion of the units in the j-th block to the whole units. Stratification effectively decreases the bias of ATE estimation compared with the difference-estimator where ATE is estimated as: The bias of the difference-estimator is: While, the bias of the stratification estimator is the weighted average of the within-block bias: Compared with the difference estimator, the stratification estimator reduces the bias per covariate by the factor: whereX t,k (j) (X c,k (j)) is the average of k-th covariate of treated (control) group in j-th block, andX t,k (X c,k ) is the average of k-th covariate in the whole treated (control) group. The key component of stratification methods is how to create the blocks and how to combine the created blocks. The equal frequency BIB001 is a common strategy to create blocks. Equal-frequency approach split the block by the appearance probability, such as the propensity score, so that the covariates have the same appearance probability (i.e., the propensity score) in each subgroup (block). The ATE is estimated by weighted average of each block's CATE, with the weight as the fraction of the units in this block. However, this approach suffers from high variance due to the insufficient overlap between treated and control groups in the blocks whose propensity score is very high or low. To reduce the variance, in BIB002 , the blocks, which divided according to the propensity score, are re-weighted by the inverse variance of the block-specific treatment effect. Although this method reduces the variance of equal-frequency method, it unavoidably increases the estimation bias. The stratification methods described above are all splitting the blocks according to the pre-treatment variables. However, in some real-world applications, it is required to compare the outcome conditioned on some posttreatment variables, denoted as S. For example, the "surrogate" markers of disease progression (i.e., intermediate outcome) like CD4 count and measures of viral load in AIDS are the post-treatment variables BIB003 . In the studies comparing drugs for AIDS patients, the researchers are interested in the effect of AIDS drugs on group with CD4 count lower than 200 cell/mm 3 . However, directly comparing the observed outcomes on the group with S obs < 200 is not the true effect because the compared two subgroups: {i : W i = 1, S obs < 200} and {j : W j = 0, S obs < 20}, where S obs is the observed post-treatment values, have great discrepency if the treatment has effect on the intermediate results. To solve this, principle stratification BIB003 constructs the subgroup based on the potential values of the pre-treatment variables. Analogous to the potential outcome defined in 2.1, potential pre-treatment variables value, denoted as S(W = w), is the potential value of S under treatment with value w. With the nature assumption that potential value of S is independent of the treatment assignment, the treatment effect of subgroup can be obtained by comparing the outcomes of two sets: {Y obs i : where v 1 and v 2 are two post-treatment values. The comparison based on the potential values of post-treatment variables ensures that the compared two set are similar, so that the obtained treatment effect is the true effect.