# treatment_effect_estimation_with_datadriven_variable_decomposition__76a4766b.pdf

Treatment Effect Estimation with Data-Driven Variable Decomposition

Kun Kuang,1,2 Peng Cui,1,2 Bo Li,3 Meng Jiang,4 Shiqiang Yang,1,2 Fei Wang5

1Tsinghua National Laboratory for Information Science and Technology 2Department of Computer Science and Technology, Tsinghua University 3School of Economics and Management, Tsinghua University 4Department of Computer Science, University of Illinois Urbana-Champaign 5Department of Healthcare Policy and Research, Weill Cornell Medical School, Cornell University kk14@mails.tsinghua.edu.cn, cuip@tsinghua.edu.cn, libo@sem.tsinghua.edu.cn mjiang89@illinois.edu, yangshq@tsinghua.edu.cn, feiwang03@gmail.com

One fundamental problem in causal inference is the treatment effect estimation in observational studies when variables are confounded. Control for confounding effect is generally handled by propensity score. But it treats all observed variables as confounders and ignores the adjustment variables, which have no influence on treatment but are predictive of the outcome. Recently, it has been demonstrated that the adjustment variables are effective in reducing the variance of estimated treatment effect. However, how to automatically separate the confounders and adjustment variables in observational studies is still an open problem, especially in the scenarios of high dimensional variables, which are common in big data era. In this paper, we propose a Data-Driven Variable Decomposition (D2VD) algorithm, which can 1) automatically separate confounders and adjustment variables with a data driven approach, and 2) simultaneously estimate treatment effect in observational studies with high dimensional variables. Under standard assumptions, we show experimentally that our D2VD algorithm can automatically separate the variables precisely, and estimate treatment effect more accurately and with tighter confidence intervals than the state-of-the-art methods on both synthetic data and real online advertising dataset.

Introduction Causal inference, which refers to the process of drawing a conclusion about a causal connection based on the conditions of the occurrence of an effect (Holland 1986), is a powerful statistical modeling tool for explanatory analysis. The gold standard approaches for causal inference are randomized experiments, for example, A/B testing (Lewis and Reiley 2009), where different treatments are randomly assigned to units 1. However, the fully randomized experiments are usually extremely expensive (Kohavi and Longbotham 2011) or sometimes even infeasible (Bottou et al. 2013) in many scenarios. Hence it is highly demanding to develop automatic statistical approaches to infer treatment effect in observational studies. In literature, (Rosenbaum and Rubin 1983) proposed a statistical framework for treatment effect estimation based

Copyright c 2017, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. 1Units represent the objects of treatment. For example, in online advertising campaign, the units refer to the users in the campaign.

Confounders

Adjustment Variables

Automated Variables Decomposition

Treatment Effect Estimation

Figure 1: Our causal diagram. We separate all observed variables U into three different sets: (1) Confounders X, which are associated with the treatment T and may be causally related to the outcome Y , (2) Adjustment Variables Z, which are causally related to outcome Y , but independent with treatment T, and (3) Irrelevant Variables I (Omitted), which are independent with both treatment and outcome.

on propensity score adjustment. Such framework has been widely used in observational causal study, including matching, stratification, inverse weighting and regression on propensity score (Austin 2011; Lunceford and Davidian 2004; Kuang et al. 2016). The inverse propensity weighting is a most commonly used method and has been part of a large family of causal models known as marginal structural model (Hern an, Brumback, and Robins 2000; 2002). With combination of inverse propensity weighting and regression, (Bang and Robins 2005) proposed a doubly robust estimator. These methods have been widely used in various fields, including economics (Stuart 2010), epidemiology (Funk et al. 2011), health care (Reis et al. 2015), social science (Lechner 1999) and advertising (Sun et al. 2015). The essence of these methods is to eliminate the confounding impact of confounders so that the precision of treatment effect estimation can be significantly improved. However, most of these works treat all observed variables as confounders when estimating propensity score. Eventu-

Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence (AAAI-17)

ally, in the scenarios of high dimensional variables, some of them are not confounders but are predictive of the outcome, which are denoted by adjustment variables Z as shown in our causal diagram in Fig. 1. Ignoring the adjustment variables will make the estimated treatment effect imprecise and with inflated variance. Recently, some researchers have investigated the importance of the adjustment variables. (Brookhart et al. 2006; Vander Weele and Shpitser 2011) have advocated that the adjustment variables should be included in the causal inference model. And (Sauer et al. 2013) suggested that conditioning on such adjustment variables is unnecessary to remove bias but can reduce variance in treatment effect estimation. In randomized experiments setting, (Bloniarz et al. 2016) have proved that adjusting for the adjustment variables by lasso can reduce the variance of estimated treatment effect. All these methods in observational studies assume that the causal structure, i.e. whether a variable is the cause of the treatment or outcome, is known a priori. However, the causal structure cannot be well defined by prior knowledge in most cases, especially in the scenarios of high dimensional variables in the big data era. How to automatically separate confounders and adjustment variables in observational studies is still an open problem. To address this problem, we propose a Data-Driven Variable Decomposition (D2VD) algorithm to jointly optimize confounders separation and Average Treatment Effect (ATE) estimation. More specifically, we propose a regularized integrated regression model, where a combined orthogonality and sparsity regularizer is constructed to simultaneously 1) separate the confounders and adjustment variables with a data driven approach, 2) eliminate irrelevant variables which are neither confounders nor adjustment variables to avoid overfitting, and 3) estimate the ATE in observational studies. During estimating the ATE, the separated confounders can effectively eliminate their confounding impact on treatment, while the adjustment variables can significantly reduce the variances of estimated ATE through outcome adjustment. This enables us to estimate the true ATE more accurately and with tighter confidence intervals than baseline methods. The main contributions in this paper are as follows: We study a new problem of automatically separating confounders and adjustment variables, which is critical for the precision and confidence intervals of ATE estimation in observational studies. We propose a novel data-driven variables decomposition algorithm, where a regularized integrated regression model is presented to enable confounder separation and ATE estimation simultaneously. The advantages of D2VD algorithm are demonstrated in both synthetic and real world data. It can also be straightforwardly applied into other causal inference studies, such as social marketing, health care and public policy.

Adjusted ATE Estimator In this section, we first give the notations and assumptions for the ATE estimation in observational studies, then propose a new adjusted ATE estimator by utilizing the adjustment variables for reducing the variance of estimated ATE.

Notations and Assumptions As described in our causal diagram in Fig.1, we define a treatment as a random variable T and a potential outcome as Y (t) which corresponds to a specific treatment T = t. In this paper, we only consider binary treatment, that is t {0, 1}. We define the units which received the treatment, that is T = 1, as treated units and the others with T = 0 as control units. Then for each unit indexed by i = 1, 2, , m, we observe a treatment Ti, an outcome Y obs i and a vector of variables Ui. Our observed outcome Y obs i of unit i can be denoted by:

Y obs i = Yi(Ti) = Ti Yi(1) + (1 Ti) Yi(0), (1)

In observational studies, there are three standard assumptions (Rosenbaum and Rubin 1983) for ATE estimation. Assumption 1: Stable Unit Treatment Value. The distribution of potential outcome for one unit is assumed to be unaffected by the particular treatment assignment of another unit, when given the observed variables. Assumption 2: Unconfoundedness. The distribution of treatment is independent of potential outcome when given the observed variables. Formally, T Y (0), Y (1) |U. Assumption 3: Overlap. Every unit has a nonzero probability to receive either treatment status when given the observed variables. Formally, 0 < p(T = 1|U) < 1.

Adjusted ATE Estimator The important goal of causal inference in observational studies is to evaluate the ATE on outcome Y . The ATE represents the mean (average) difference between the potential outcome of units under treated and control status. Formally, the ATE is defined as:

ATE = E Y (1) Y (0) , (2)

where Y (1) and Y (0) represent the potential outcome of unit with treatment status as treated T = 1 and control T = 0, respectively. E( ) refers to the expectation function. The Eq. (2) is infeasible, because for each unit, we can only observe one potential outcome corresponding to its treatment status, treated or control. This is called the counterfactual problem (Chan et al. 2010). One can address this counterfactual problem by approximating the unobserved potential outcome. The simplest approach is to directly compare the average outcome between the treated and control units. However, in observational studies, comparing two samples directly is likely to have bias if the treatment assignment is not random, as confounding impact is not taken into account (Chan et al. 2010). To unbiasedly evaluate the ATE in observational studies, one have to control the impact of confounders. Under the assumptions (1,2,3), (Rosenbaum and Rubin 1983) introduced the propensity score to summarize the information required to control the confounders. The propensity score, denoted by e(U), was defined as the probability with treated status (T = 1) of a unit given all variables U. Actually, only confounders X are associated with the treatment, therefore

e(U) = p(T = 1|U) = p(T = 1|X) = e(X). (3)

Based on the propensity score, (Rosenbaum 1987) proposed the transformed outcome Y to address the counterfactual

problem in Eq. (2) with Inverse Propensity Weighting (IPW) estimator ATEIP W , see also (Athey and Imbens 2016). The transformed outcome Y is defined as

Y = Y obs T e(U) e(U) (1 e(U)) = Y obs T e(X) e(X) (1 e(X)), (4)

and the IPW estimator is defined as ATEIP W = E(Y ) = E Y obs T e(X) e(X) (1 e(X)) . (5)

However, most previous approaches based on propensity score usually treat all observed variables as confounders when estimating the propensity score. This will make the estimated treatment effect imprecise and with inflated variance because some variables could be non-confounders and have direct impact on outcome. Therefore, based on our causal diagram as shown in Fig. 1, we propose to separate all observed variables U into three sets, the confounders X, the adjustment variables Z and irrelevant variables I (Omitted in Fig.1). And then, we propose a new adjusted estimator by incorporating adjustment variables to reduce the variance of estimated ATE under following assumption. Assumption 4: Separateness. The observed variables U can be decomposed into three sets, that is U = (X, Z, I), where X are confounders, Z are adjustment variables and I are irrelevant variables. With assumption 4, we introduce our adjusted transformed outcome Y + based on Y with the definition as

Y + = Y obs φ(Z) T e(X) e(X) (1 e(X)), (6)

where φ(Z) helps to reduce the variance among Y , which are associated with Z. Then we propose the adjusted estimator ATEadj as

ATEadj = E(Y +) = E Y obs φ(Z) T e(X) e(X) (1 e(X)) . (7)

And our adjusted estimator has following properties. Firstly, under assumptions 1-4, our adjusted estimator ATEadj is unbiased, that is

E(Y +|X) = E(Y (1) Y (0)|X). (8)

This property is obvious with the Pearl s back-door criterion (Pearl 2009). Since the conditioning set X blocks all back door paths linking treatment T and outcome Y , while not contains any descendants of T in our causal diagram. Secondly, the asymptotic variance of our adjusted estimator ATEadj is no greater than IPW estimator ATEIP W. Comparing the Eq. (5) and (7), we know that the IPW estimator only considered the confounders X when provided all variables U, while our estimator utilized the adjustment variables Z to make adjustments on outcome Y for reducing variance. The similar adjustments have been proved can reduce the variance of ATE estimation in randomized experiments by (Bloniarz et al. 2016).

Automated Variables Decomposition D2VD Algorithm With Eq. (8), we can get E(Y +) = E(Y (1) Y (0)), and obtain the estimated ATE by regressing our adjusted transformed outcome Y + against the variables U and minimizing

the following objective function

minimize Y + h(U) 2. (9)

Then we can estimate the ATE by E(h(U)). In practice, we specify φ(Z) and h(U) as linear functions with coefficient vector α and γ, that is

φ(Z) = Zα, h(U) = Uγ, (10)

and adopt linear-logistic regression to evaluate the propensity score e(X) with coefficient vector β:

e(X) = p(T = 1|X) = 1/(1 + exp( Xβ)). (11)

In the specifications of Eq. (10, 11), we have assumed the knowledge of the variables decomposition U = (X, Z, I). Nevertheless, we don t know the exact separation in practice. Hence we use the full set of observed variables U to replace X and Z instead, and propose a data-driven approach to automatically separate confounders and adjustment variables. We update our objective function in Eq. (9) as:

minimize (Y obs Uα) W(β) Uγ 2 2, (12)

i=1 log(1 + exp((1 2Ti) Uiβ)) < τ,

α 1 λ, β 1 δ, γ 1 η, α β 2 2 = 0.

where W(β) := T e(U) e(U) (1 e(U)) and m i=1 log(1 + exp((1 2Ti) Uiβ)) represents the loss function when estimating the propensity score. refers to Hadamard product. With the formula α β 2 2 = 0, the coefficient vector α is optimized for separating the adjustment variables Z and β is for separating confounders X form variables U. In particular, we employ an orthogonal regularizer on α and β to ensure the separation of confounders and adjustment variables. In addition, we add L1 penalties on α, β and γ to eliminate irrelevant variables I to further reduce variance and address the sparseness problem of variables. These lead to the following optimization problem, which is to minimize J (α, β, γ). J (α, β, γ) = f(α, β, γ) + g(α, β, γ), (13) f(α, β, γ) = (Y obs Uα) W(β) Uγ 2 2 + μ α β 2 2 + τ m i=1 log(1 + exp((1 2Ti) Uiβ)), g(α, β, γ) = λ α 1 + δ β 1 + η γ 1.

With the operator splitting property of proximal gradient algorithm (Parikh and Boyd 2013), we can get the optimized parameter (i.e., α(t+1)) at the tth iteration by proximal operator proxκ(t)g of function g( ) with the step size κ(t):

α(t+1) = proxκ(t)g α(t) κ(t) f( )

α refers to the gradient of function f( ) on the variable α and

proxκ(t)g(x) =

xi κ(t) λ xi κ(t) λ 0 |xi| κ(t) λ xi + κ λ xi κ(t) λ (15)

The λ in Eq. (15) is the coefficient of parameter α in function g( ). If the optimized parameter is β, then it should be δ and it should be η for optimizing parameter γ.

Algorithm 1 Data-Driven Variable Decomposition (D2VD)

Require: Initialization J (0) = J (α(0), β(0), γ(0)). Ensure: J (0) 0, J (t+1) < J (t)

1: for t = 0, 1, 2, do 2: Calculate f( )

γ 3: α(t+1) = OPTIMIZATION(α, t) 4: β(t+1) = OPTIMIZATION(β, t) 5: γ(t+1) = OPTIMIZATION(γ, t) 6: J (t+1) = J (α(t+1), β(t+1), γ(t+1)) 7: end for

With the proximal gradient algorithm, we can minimize the objective function in Eq. (13). That is, starting from some random initialization on α, β, γ, we solve each them alternatively with the other two parameters as fixed and step by step until convergence. Our Data-Driven Variable Decomposition algorithm is described in Algorithm 1. During each iteration in Algorithm 1, we update the parameters α, β and γ with OPTIMIZATION as described in Algorithm 2, where the function ˆfκ( ) is defined as:

ˆfκ(x, y) = f(y) + (x y) f( )

T + (1/(2κ)) x y 2 2. (16)

And the gradients of the function f(α, β, γ) with the respect to the variables (α, β, γ) are: f( )

α = 2(W(β) 1T U)T R + 2μα β β,

β = 2 (Y Uα) 1T W (β)

+ UT (T exp(Uβ)) + 2μα β α, f( )

where R = (Y Uα) W(β) Uγ and W (β)

(2T 1) exp (1 2T) Uβ (1 2T) 1T U.

With the optimized parameters ˆα, ˆβ and ˆγ by Algorithm 1, we can separate the confounders as X = {Ui : ˆβi = 0}, adjustment variables as Z = {Ui : ˆαi = 0} and estimate the ATE as ATED2V D = E(Uˆγ).

Complexity Analysis The complexity of our D2VD algorithm is dominated by the step of calculating the gradients of function f(α, β, γ) with respect to the variables. The complexity of f( )

γ are all O(mn), where m is the sample size and n is the dimension of all observed variables. With considering that only constant time operations is involved in the for-loop and while-loop in our algorithms, therefore, the complexity of our D2VD algorithm is O(mn).

Parameters Tuning The main challenge of parameters tuning for ATE estimation in observational studies is that there is no ground truth about the true ATE in practice.

Algorithm 2 OPTIMIZATION(o, t)

1: Set κ = 1 2: while 1 do 3: Let o(t+1) = proxκg o(t) κ f( )

4: break if f(o(t+1)) ˆfκ(o(t+1), o(t)) 5: Update κ = 1

2κ 6: end while 7: return ot+1

To address this challenge, we employed the matching method to evaluate the ATE and set it as approximal ground truth like (Athey and Imbens 2016) did. Specifically, for each unit i, find its closest match among the units with opposite treatment status:

match(i) = arg minj:Tj=1 Ti Ui Uj 2 2. (17)

We drop unit i if match(i) > ϵ, that makes the matching approximate to exactly matching. We can estimate ATE with the matching estimator by comparing the average outcome between the matched treated and control units sets, and set it as approximal ground truth , denoted by ATEmatching. With the approximal ground truth , we can tune parameters of our algorithm with cross validation.

Experiments We apply our algorithm on the synthetic dataset and real online advertising dataset to estimate the ATE.

Baseline Estimators We implement the following baseline estimators to evaluate the ATE for comparison. Direct Estimator ATEdir: It evaluates the ATE by directly comparing the average outcome between the treated and control units. It ignores the confounding effect of confounders on treatment. IPW Estimator ATEIP W (Rosenbaum and Rubin 1983): It evaluates the ATE via reweighting observations with inverse of propensity score. It treats all variables as confounders and ignores the adjustment variables. Doubly Robust Estimator ATEDR (Bang and Robins 2005): It evaluates the ATE by combination of IPW and regression methods. It ignores the separation of confounders and adjustment variables. Non-Separation Estimator ATED2V D( ): It s a weakened version of our D2V D estimator. It has no variables separation step by setting coefficient μ = 0 in Eq. (13). In this paper, we implemented ATEIP W and ATEDR with lasso regression for variables selection. The difference between ATEDR and ATED2V D( ) is that the former estimates ATE sequentially but the latter does with joint optimization.

Experiments on Synthetic Data Dataset To generate the synthetic dataset, we set the sample size m = {1000, 5000} and the dimension of observed variables n = {50, 100, 200}. We first generate the variables

Table 1: Results on synthetic dataset: the true ATE is 1. The Bias refers to the absolute error between the true and estimated ATE, that is Bias = | ATE ATE|. SD, MAE and RMSE represent the standard deviations, mean absolute errors and root mean square errors of ATE after 50 times independently experiments, respectively.

n n = 50 n = 100 n = 200 T/m Estimator Bias SD MAE RMSE Bias SD MAE RMSE Bias SD MAE RMSE ATEdir 0.418 0.409 0.479 0.582 0.302 0.490 0.472 0.571 0.405 0.628 0.574 0.720 ATEIP W + lasso 0.078 0.310 0.252 0.317 0.097 0.356 0.295 0.366 0.073 0.328 0.267 0.320 T = Tlogit ATEDR + lasso 0.060 0.181 0.152 0.189 0.067 0.190 0.155 0.199 0.081 0.181 0.169 0.190 m = 1000 ATED2V D( ) 0.053 0.138 0.124 0.146 0.064 0.130 0.117 0.144 0.018 0.170 0.128 0.162 ATED2V D 0.045 0.108 0.091 0.116 0.019 0.114 0.093 0.115 0.067 0.144 0.130 0.152 ATEdir 0.418 0.170 0.418 0.451 0.659 0.181 0.659 0.681 0.523 0.412 0.555 0.653 ATEIP W + lasso 0.036 0.201 0.163 0.202 0.034 0.222 0.194 0.213 0.032 0.341 0.274 0.325 T = Tlogit ATEDR + lasso 0.051 0.079 0.071 0.094 0.106 0.075 0.114 0.127 0.055 0.084 0.086 0.096 m = 5000 ATED2V D( ) 0.112 0.080 0.118 0.137 0.114 0.102 0.121 0.150 0.164 0.076 0.164 0.179 ATED2V D 0.033 0.072 0.061 0.078 0.023 0.073 0.061 0.073 0.042 0.068 0.062 0.076 ATEdir 0.664 0.387 0.670 0.766 0.273 0.445 0.436 0.518 0.380 0.766 0.691 0.848 ATEIP W + lasso 0.266 0.279 0.319 0.384 0.298 0.295 0.328 0.417 0.191 0.482 0.403 0.514 T = Tmissp ATEDR + lasso 0.138 0.187 0.174 0.231 0.253 0.197 0.269 0.320 0.050 0.218 0.170 0.222 m = 1000 ATED2V D( ) 0.269 0.162 0.270 0.313 0.129 0.162 0.170 0.206 0.175 0.207 0.236 0.269 ATED2V D 0.066 0.113 0.102 0.129 0.019 0.119 0.101 0.120 0.059 0.177 0.149 0.184 ATEdir 0.446 0.180 0.446 0.480 0.587 0.323 0.587 0.662 0.778 0.246 0.778 0.812 ATEIP W + lasso 0.148 0.133 0.161 0.198 0.172 0.167 0.199 0.239 0.142 0.224 0.206 0.263 T = Tmissp ATEDR + lasso 0.119 0.073 0.123 0.139 0.100 0.067 0.107 0.120 0.127 0.079 0.127 0.148 m = 5000 ATED2V D( ) 0.112 0.070 0.119 0.132 0.058 0.067 0.069 0.086 0.068 0.055 0.073 0.086 ATED2V D 0.033 0.055 0.052 0.063 0.039 0.068 0.066 0.075 0.032 0.047 0.049 0.055

U = (X, Z, I) = (x1, , xnx, z1, , znz, i1, , ini) with independent gaussian distributions as

x1, , xnx, z1, , znz, i1, , ini iid N(0, 1),

where nx, nz and ni represent the dimension of confounders X, adjustment variables Z and irrelevant variables I, respectively. And nx = 0.2 n, nz = 0.2 n, ni = 0.6 n. To test the robustness of all estimators, we generate the binary treatment variable T from a logistic function (Tlogit) and a misspecified function (Tmissp) as

Tlogit Bernoulli(1/(1 + exp( nx i=1 xi))) and Tmissp = 1 if nx i=1 xi > 0.5, Tmissp = 0 otherwise.

The outcome Y is generated as

Y = nx j= nx

2 xj ωj + nz j=1 zk ρk + T + N(0, 2),

In this dataset, the features (x nx

2 +1, , xnx) are correlated to the treatment and outcome, simulating a confounding effect. The true treatment effect in this dataset is 1.

ATE Estimation To evaluate the performance of our proposed method, we carry out the experiments 50 times independently. Based on our estimated ATE, we calculate the Bias, SD, MAE and RMSE, and report the results in Tab.1, where the smaller Bias, SD, MAE and RMSE are better. From Tab.1, we have following observations. First, the direct estimator is failed (with large Bias) under different settings because it did not consider the confounding effect. Second, the IPW estimator can unbiasedly (with small Bias) estimate the ATE when T = Tlogit, but with a big Bias when propensity score model is misspecified by setting T = Tmissp. With combination of IPW and

regression models, DR estimator can get better performance than IPW estimator, especially when T = Tmissp. Third, our D2V D( ) estimator, which has no variables separation step, can get the similar results with DR estimator. But with considering the separation between confounders and adjustment variables, our D2V D estimator can improve the accuracy (smaller Bias) and reduce the variance (smaller SD) for ATE estimation from D2V D( ), DR and other baseline estimators under different settings.

Variables Decomposition As we described before, with the optimized ˆα and ˆβ, our algorithm can separate the confounders as X = {Ui : ˆβi = 0} and adjustment variables as Z = {Ui : ˆαi = 0}. To demonstrate the performance of automated variables decomposition of our algorithm, we carry out the experiments 50 times independently and record the true positive rate (TPR) and true negative rate (TNR) in Tab. 3. The formulations of TPR and TNR for separated confounders X are defined as

TPR = #{ ˆβi =0,βi =0}

#{βi =0} , TNR = #{ ˆβi=0,βi=0}

#{βi=0} . (18)

In the same way, we calculate the TPR and TNR for separated adjustment variables Z via Eq. (18) by using α. Tab. 3 shows that our D2VD algorithm can separate the confounders X more precisely when T = Tlogit, comparing with T = Tmissp. This is because of the logistic assumption of treatment assignment in our algorithm is correct. Even if setting T = Tmissp, our algorithm can still precisely separate the confounders and adjustment variables. This enables us to estimate the ATE more accurately and with tighter confidence intervals than the state-of-the-art methods.

Table 2: The top ranked features by their absolute ATE estimated with our D2VD estimator ATED2V D, comparing with the baseline estimator ATEIP W and ATEDR. The ATEmatching is the approximal ground truth by matching method, n/a means that we cannot obtain the ATE from matching method since the number of matching samples are not sufficient.

No. Features ATED2V D (SD) ATEIP W (SD) ATEDR (SD) ATEmatching 1 No. friends (> 166) 0.295 (0.018) 0.240 (0.026) 0.297(0.021) 0.276 2 Age (> 33) -0.284 (0.014) -0.235 (0.029) -0.302(0.068) -0.263 3 Share Album to Strangers 0.229 (0.030) 0.236 (0.030) -0.034(0.021) n/a 4 With Online Payment 0.226 (0.019) 0.260 (0.029) 0.244(0.028) n/a 5 With High-Definition Head Portrait 0.218 (0.028) 0.203 (0.032) 0.237(0.046) n/a 6 With We Chat Album 0.191 (0.014) 0.237 (0.021) 0.097(0.050) n/a 7 With Delicacy Plugin 0.124 (0.038) -0.253 (0.037) 0.067(0.051) 0.099 8 Device (i OS) 0.100 (0.024) 0.206 (0.012) 0.060(0.021) 0.085 9 Add friends by Drift Bottle -0.098 (0.012) 0.016 (0.019) -0.115(0.015) -0.032 10 Gender (Male) -0.073 (0.017) -0.240 (0.029) 0.065(0.055) -0.097

Table 3: Separation results of confounders X and adjustment variables Z. The closer to 1 for TPR and TNR is better.

T = Tlogit n = 50 n = 100 n = 200 m TPR TNR TPR TNR TPR TNR

m = 1000 X 1.000 0.917 0.977 0.948 0.966 0.906 Z 1.000 0.973 1.000 0.983 1.000 0.984

m = 5000 X 1.000 0.923 1.000 0.887 0.994 0.989 Z 1.000 0.975 1.000 0.987 1.000 0.994 T = Tmissp

m = 1000 X 1.000 0.844 0.997 0.866 0.867 0.977 Z 1.000 0.982 1.000 0.987 1.000 0.983

m = 5000 X 1.000 0.843 1.000 0.837 0.998 0.965 Z 1.000 0.986 1.000 0.990 1.000 0.994

Experiments on Real World data

Dataset The real online advertising dataset we used is collected during Sep. 2015 from Tencent We Chat App2. In We Chat, each user can share posts to his/her friends and receive posts from friends as like in the Twitter and Facebook. The advertisers can push advertisements to users, by merging them into list of the user s wallposts. There are two types of feedback on the advertisements: Like and Dislike . The online advertising campaign is about LONGCHAMP handbags for young ladies3. This campaign contains 14,891 user feedbacks with Like and 93,108 Dislikes. For each user, we have 56 features including (1) demographic attributes, such as age, gender, (2) number of friends, (3) device (i OS or Android), and (4) the user settings on We Chat, for example, whether allowing strangers to see his/her album ( Share Album to Strangers ) and whether installing the online payment service ( With Online Payment ).

Experimental Settings In our experiments, we set the feedback of users about the advertisement as outcome Y . Specifically, we set the outcome Yi = 1 when the user i likes the advertisement, and Yi = 0 if user i dislikes it. And we alternatively set one of the features as the treatment T and all other features as the variables U. So that we can evaluate the ATE of each feature. During the parameters tuning, we set the matching thresh-

2http://www.wechat.com/en/ 3http://en.longchamp.com/en/womens-bags

old ϵ = 5, which make the matching estimator is close to the exactly matching. The hyper-parameters of λ, δ, τ, η and μ set as 30, 50, 90, 70 and 30 by using grid search.

ATE Estimation For each user feature, we employ our D2VD algorithm to estimate its ATE on the outcome. Tab. 2 shows the top ranked features by their absolute ATE estimated with our D2VD estimator, comparing with baseline estimators and the approximal ground truth ATEmatching. Note that the ATEmatching has very rigorous requirements on the sample size with exactly matching. For some user features, we do not have a sufficient number of samples thus we cannot derive their ATEmatching. From Tab. 2, we have following observations. O1. Our D2VD estimator evaluate the ATE more accurately than baseline estimators. With separated confounders, the ATE estimated by our D2VD estimator is closer to the approximate ground truth ATEmatching. While the IPW and DR estimators, which treat all variables as confounders, generate huge error in estimating ATE for some features, even make wrong estimation of the ATE polarity (positive of negative), such as feature With Delicacy Plugin for IPW estimator and feature Gender for DR estimator. O2. Our D2VD estimator can reduce the variance of estimated ATE from baseline estimators. With regression on separated adjustment variables, our estimator obtain smaller SD than IPW and DR estimators, where IPW estimator ignores the adjustment variables and DR estimator makes regression on all variables, ignoring the variables separation. O3. Younger ladies are with higher probability to like the advertisement about LONGCHAMP handbags. The ATE of Age(> 33) is 0.284 and Gender(Male) is 0.073, which indicate that the younger ladies have higher probability to like the advertisement. This is consistent with our intuition since the LONGCHAMP advertisement is mainly designed for young ladies as their potential customers.

Variables Decomposition Tab. 4 shows the separation results between confounders and adjusted variables when we set feature Add friends by Shake as the treatment. Shake4 is a two way function where both people using this function at the same time can see each other and make friends

4https://rumorscity.com/2014/07/25/how-to-add-friends-onwechat-7-ways/

Table 4: Confounders and adjusted variables when we set feature Add friends by Shake as treatment.

Confounders Adjustment Variables Add friends by Drift Bottle No. friends Add friends by People Nearby Age Add friends by QQ Contacts With We Chat Album Without Friends Confirmation Plugin Device

on We Chat. In Tab. 4, the confounders are many other ways for adding friends on We Chat, indicating the separated confounders have significant causal association with treatment. While the adjustment variables, for example, the No. friends and Age , are not associated with treatment but have significant effect on outcome, as shown in Tab. 2, they are the top ranked features. The results demonstrate that our proposed D2V D algorithm can precisely separate the confounders and adjustment variables in practical. With the separated confounders, our estimator can obtain an accurate ATE, and reduce the variance of estimated ATE by the adjustment variables.

In this paper, we focus on how to evaluate the average treatment effect in a more precisely way with tighter confidence intervals in observational studies. We argued that most previous causal methods based on propensity score is deficient because they usually treat all variables as confounders. Based on our causal diagram, we proposed to separate the confounders and adjustment variables from all observed variables. And we proposed a Data-Driven Variable Decomposition (D2VD) algorithm to jointly optimize the variables decomposition and ATE estimation. Experimental results on synthetic data and real world data verify the practical usefulness of our model and the effectiveness of our D2VD algorithm for ATE estimation in observational study.

Acknowledgement

This work was supported by National Program on Key Basic Research Project, No. 2015CB352300; National Natural Science Foundation of China, No. 61370022, No. 61210008, No. 71490723 and No. 71432004. Thanks for the research fund of Tsinghua-Tencent Joint Laboratory for Internet Innovation Technology. Thanks for the support of National Science Foundation IIS-1650723.

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