Debiased Causal Tree: Heterogeneous Treatment Effects Estimation with Unmeasured Confounding
Abstract
Unmeasured confounding poses a signiﬁcant threat to the validity of causal inference.
Despite that various ad hoc methods are developed to remove confoundingeffects, they are subject to certain fairly strong assumptions.
In this work, weconsider the estimation of conditional causal effects using observational data andhistorical controls, in the presence of unmeasured confounding.
Under an interpretable transportability condition, we prove the partial identiﬁability of conditionalaverage treatment effect on the treated group (CATT).
For tree-based models, a newnotion, confounding entropy , is proposed to measure the discrepancy introducedby unobserved confounders between the conditional outcome distribution of thetreated and control groups.
The confounding entropy generalizes conventionalconfounding bias, and can be estimated effectively using historical controls.
Wedevelop a new method, debiased causal tree, whose splitting rule is to minimizethe empirical risk regularized by the confounding entropy.
Notably, our methodintegrates current observational data (for empirical risk) and their historical controls(for confounding entropy) harmoniously.
We highlight that, debiased causal treecan not only estimate CATT well in the presence of unmeasured confounding, butalso is a robust estimator of conditional average treatment effect (CATE) againstthe imbalance of the treated and control populations when all confounders areobserved.
An extension of combining multiple debiased causal trees to furtherreduce biases by gradient boosting is considered.
The computational feasibilityand statistical power of our method are evidenced by simulations and a study of acredit card balance dataset.
Introduction
Conditional average treatment effect (CATE), also known as heterogeneous treatment effect, cancharacterize the causal effect of certain treatments on the outcome over the whole population as afunction of measured covariates.
Another closely related concept, conditional average treatmenteffect on the treated group (CATT), is more relevant when we focus on the effect of treatments forthe sub-population which is actually exposed to certain speciﬁc treatments; for instance, we are moreinterested in the effect on children of taking anti-depressants during pregnancy for the people whoactually took the drug [7].
The interpretable and informative nature has made it a great importance toestimate CATE and CATT from observational or experimental data in a wide spectrum of applications,including medical decision-making and personalized medicine [19, 34], evaluating advertising andsocioeconomic policies [10, 2, 21] and large-scale randomized experiments for recommendations[29, 18], etc.
In the absence of unmeasured confounding, there exist lots of methods developed to estimate CATE(CATT), including tree-based methods [3, 47, 4], meta-learners [27], R-learners [33], doubly robust learners [25], and so on.
Basically, such an unconfoundedness condition hinges on an investigator’sability to accurately measure all related covariates, which is questionable in practice, especially whenthere exist covariate-dependent measurement errors [46, 42], biased treatment allocations [23, 43]and batch effects [17, 20].
In the presence of unmeasured confounding, however, the identiﬁability of
CATE or CATT cannot be ensured in general, leading to failures of the aforementioned estimationprocedures.
To control the effect introduced by confounders, various methods have been developed, e.g., sensitivityanalyses [36], instrumental variable (IV) approaches [1, 31], and proximal causal learning [28, 32, 39].
Nonetheless, several limitations of these methods hinder their direct application in CATE/CATTestimation.
For example, we cannot derive point estimation from sensitivity analyses, and even valid
IVs can only partially identify CATE in principle [41].
Although ATE and ATT can be identiﬁed inproximal causal learning [15], the assumption that the measured covariates are correctly classiﬁedinto three types of proxies requires prior knowledge.
Recently, a line of work proposed to combine randomized trials [14] and observational studies toadjust confounders, which typically requires transportability [35] of a randomized controlled trialto external observational studies [37].
For example, [50] proposed to use the confounding functionwhich summarizes the impact of confounders to estimate heterogeneous treatment effects, wherethe confounding function is assumed to be transportable.
For a comprehensive review, see [13].
Problematically, many observational studies do not have related randomized trials due to practicaland methodological issues such as cost and restrictive inclusion/exclusion criteria.
In this work, we move further in the direction of data fusion, without requiring experimental data.
Instead, we consider the case where covariates and the outcome are collected at multiple timestamps,before and after the treatment.
We take two timestamps t ∈ { t 1 , t 2 } ( t 1 < t 2 ) , as an example, wherethe treatment is imposed at the time t 2 .
Following the potential outcome framework [38], denoteby Y t = DY (1) t + (1 − D ) Y (0)
t the observed outcomes at the time t , where Y ( d ) t is the potentialoutcome for the treatment D = d ∈ { 0 , 1 } .
We remark that Y t 1 = Y (0) t 1 since the treatment does nottake place at t 1 .
Let X
and U be time-invariant covariates and unmeasured confounders, respectively.
Our work is motivated by the following identity
[Equation]
which holds regardless of U
In general, the confounding bias [6, 50] b t does not vanish due to U .
However, the smaller b t is, the closer CATT could be to the difference in conditional expectations.
We then propose a tree-based method, which, at high level, aims to (i) ﬁnd an optimal partition of thefeature space such that the confounding bias is smallest within each small region, and (ii) estimate theleft hand of (1).
Despite that the confounding bias is unavailable in general, we can represent it withtree-based models using data collected at t
= t 1 since Y t 1 = Y (0) t
and it is known which individualwould receive the treatment.
If boundaries of the optimal partition that leads to the smallest historicalconfounding bias change “mildly” over time, i.e., it is transportable, the partition derived at time t 1 isalso nearly optimal at t 2 .
The two steps can proceed simultaneously, and we calibrate and formalizethe basic idea in Section 2.2.
Our contribution is summarized as follows: (i) we introduce a new notion, confounding entropy, tocharacterize the discrepancy between the conditional distributions of the treated and control groupsthat generalizes the confounding bias, and propose a tractable and scalable method, debiased causaltree, to estimate CATT from observational data before and after the treatment by developing a newcriterion for node splitting, which minimizes both empirical risk and confounding entropy at the sametime; (ii) to further reduce the bias, we integrate multiple causal trees by gradient boosting, which werefer to as Gradient Boosting Causal Tree ( GBCT ); (iii) under the unconfoundedness condition, weshow that GBCT is robust to imbalance of the treated and control populations and outperforms othercompared methods, which is obvious in severely imbalanced cases.
Methodology
We ﬁrst introduce the problem setup, related notation, and basic ideas behind our proposed method.
2.1 Problem Setup
To begin with, we consider the case of binary treatment for simplicity. Our method can be easilyextended to multiple treatments.
Let Y ( d ) t ∈ R be the potential outcome at the time t, t ∈ { t 1 , . . .
, t m }for the treatment D = d ∈ { 0 , 1 } .
We impose the treatment at the time t = t m and let X ∈ [0 , 1] pbe a time-invariant p -dimensional covariate.
Before the treatment occurs, the historical outcomesare denoted by Y t k
= Y (0) t k , 1 ≤ k ≤ m − 1 , t 1 < · · · < t m − 1 < t m .
Given n triplets of i.i.d.samples { ( X i , { Y i,t k } mk =1 , D i ) } ni =1 with Y i,t m = D i Y (1) i,t m + (1 − D i ) Y (0) i,t m , our goal is to estimateη ∗ ( x ) ≡ E ( Y (1) t m − Y (0) t m | D = 1 , X = x ) .
In general, η ∗ is not identiﬁable due to unmeasuredconfounding; however we can prove that it is partially identiﬁable under certain transportabilitycondition.
In the following, we denote the indicator function of event A by I( A ) .
Equation (1) implies that, η ∗ ( x ) ≈ δ t m ( x ) ≡ E ( Y (1) t m | D
= 1 , X = x ) − E ( Y (0) t | D = 0 ,
X = x )if the confounding bias b t m ( x ) ≡ E ( Y (0) t m | D = 1 , X = x ) − E ( Y (0) t m | D = 0 , X = x ) ≈ 0 .
Ingeneral, (cid:12)(cid:12) E X [ b t m ( X )] (cid:12)(cid:12) ≥ c 0 > 0 due to unmeasured confounding.
Proper stratiﬁcation [26], notably,can help mitigate confounding effects by introducing multiple subgroups within which confoundersremain almost constant.
Therefore, we propose to ﬁnd the optimal partition of the feature space, e.g.,{ Q j } qj =1 with ∪ qj =1 Q j = [0 , 1] p , such that [Equation]
for each j = 1 , . . . , q . Alternatively, we can obtain such a partition by the following optimization
[Equation]
Once ( Q ∗ 1 , . . .
, Q ∗ q ) being derived, by (1), we can choose
[Equation]
) to approximate η ∗ ( · ) , where δ t m ( Q ∗ j )
=
E ( Y (1) t m | D
= 1 , X ∈ Q ∗ j ) − E ( Y (0) t | D = 0 , X ∈ Q ∗ j )is a constant function that approximates δ t m ( x ) for x ∈ Q ∗ j .
In (3), choosing a constant functionto approximate δ t m ( x ) within Q ∗ j is only for simplicity, and local polynomials of any degree areapplicable.
As expected, η ( x ) is close to η ∗ ( x ) provided that (2) implies max 1 ≤ j ≤ q b 2 t m ( Q ∗ j )
≈ 0 .
A major difﬁculty of solving (2) is the unavailability of b t m ( Q j ) , 1 ≤ j ≤ q , since Y (0) t m is acounterfactual for D = 1 .
Fortunately, since D i is known for each i and all units stay untreatedbefore t m , b t ( Q j ) for all t ∈ { t 1 , . . .
, t m − 1 } are identiﬁable.
Instead of (2), consider the followingoptimization
[Equation]
The partition Q ∗∗ = { Q ∗∗ j } qj =1 of the feature space yields minimal historical confounding bias.
Ifconfounding effects are independent of time, Q ∗∗ j = Q ∗ j holds for each j = 1 , . . .
, q .
We imposethe following condition to ensure transportability from the historical optimal partition to its currentcounterpart.
Assumption 1.
For the partition { Q ∗∗ j } qj =1 deﬁned in (4) , B ( t ) ≡ log (cid:8) (cid:80) qj
=1 b 2 t ( Q ∗∗ j ) (cid:9) : t (cid:55)→
B ( t ) ∈ R is a Lipschitz continuous function with parameter L ; that is,(cid:12)(cid:12) B ( τ [Equation]
( τ 2 ) (cid:12)(cid:12) ≤ L | τ 1 − τ 2 | , for any τ 1 , τ 2 ≥ 0 .
Assumption 1 is testable empirically for t ∈ [0 , t m ) , provided that historical controls are collectedat sufﬁciently many timestamps.
Under Assumption 1, we prove in the Appendix that Q ∗ j ≈ Q ∗∗ jin effect ; that is, the solution to (4) is transferrable or transportable to the current data source.
Thepartial identiﬁability of η ∗ ( x ) is justiﬁed by the following theorem.
Theorem 1.
Under Assumption 1, for the partition { Q ∗∗ j } qj =1 of [0 , 1] p deﬁned in (4) , it holds that
[Equation]
where η ∗ ( Q ∗∗ j ) =
E (cid:0) Y (1) t m − Y
(0) t m | D = 1 , X ∈ Q ∗∗ j (cid:1) and C L depends only on L and t m − t 1 .
The ﬁrst term in (5) denotes the confounding bias, and hinges on the transportability assumed in
Assumption 1, while the second term measures the error of using a piecewise constant functionto approximate η ∗ ( · ) .
Theorem 1 shows that η ∗ ( · ) is identiﬁable up to a band, which is similarin spirit to CATE estimation by IV [41].
Interestingly, the width of band in our case can be closeenough to zero under stronger assumptions.
For example, the confounding bias can be small ifthe stratiﬁcation { Q ∗∗ j } qj =1 can effectively adjust for confounding, and under certain smoothnesscondition, the approximation error decays to zero rapidly as the maximal volume of Q ∗∗ j tends tozero.
Our goal is to develop a computational feasible method to solve (4) and estimate η ∗ ( x ) as in(3) substituting Q ∗ j with Q ∗∗ j , for each j .
2.2 Debiased Causal Tree
When it comes to stratifying or segmenting the feature space, the perhaps mostly used algorithms aretree-based methods [22], including classiﬁcation and regression tree (CART) [9], random forest [8],and gradient boosting trees [16].
A regression tree T ( x ; Q , µ ) consists of two components: a set ofleafs that partition the feature space, and the corresponding parameter µ = µ ( Q ) =
( µ ( Q j )) , where
Q j is the region partitioned by the j -th leaf, 1 ≤ j ≤ q , and q is the total number of leafs.
Typically,given a partition Q and n pairs of i.i.d samples { ( X i , y i ) } ni =1 , the optimal data driven estimator is
T ( x ; Q , (cid:98) µ ) = (cid:80) qj
[-qj =
-]=1 I { x ∈ Q j } (cid:98) µ ( Q j ) , where
[Equation]
We deﬁne the auxiliary trees T ( x ; Q , θ ( d ) t ) to represent confounding biases, where θ ( d ) t =
( θ ( d ) t,j ) suchthat θ ( d ) t,j = θ ( d ) t ( Q j ) ﬁts the conditional mean E ( Y (0) t | D = d, X ∈ Q j ) for d = 0 , 1 , t < t m , anddeﬁne the main tree T ( x ; Q , µ ( d ) ) to learn CATT, where µ ( d ) =
( µ ( d ) j ) such that µ ( d ) j = µ ( d ) ( Q j )ﬁts the conditional mean E ( Y ( d ) t m | D
= d, X ∈ Q j ) for d = 0 , 1 .
Auxiliary and main trees share thesame spplitting rules.
Confounding entropy.
Treating the partition { Q j } qj =1 as a parameter to be optimized, by (6), weestimate b t ( Q j ) for each t ∈ { t 1 , . . .
, t m − 1 } by(cid:98) b t ( Q j ) = (cid:98) [Equation] 1) t [Equation]
Q
[Equation]
) − (cid:98)
[Equation]
, where [Equation] For simplicity, we write (cid:98) θ ( d ) t ( Q j ) as (cid:98) θ ( d ) t,j .
Plugging (cid:98) b t ( Q j ) in (7) into (4), we obtain the estimatorof the optimal partitions.
Nonetheless, (cid:98) b t ( Q j ) as well as its population counterpart b t ( Q j ) onlysummarizes information of the ﬁrst-order moment, and thus can give rise to partitions that yieldestimators with large variances.
This motivates us to model the discrepancy between P t, 1 ,j =
P ( Y (0) t | D = 1 , X ∈ Q j ) and P t, 0 ,j = P ( Y (0) t
| D = 0 , X ∈ Q j ) using symmetric cross entropy[49], deﬁned as H( P t, 1 ,j , P t, 0 ,j ) =
− E P t, 1 [log( P t, 0 ,j )] − E P t, 0 ,j [log( P t, 1 ,j )] , which we refer to as 153 the confounding entropy for time t and region Q j .
For each t < t m , parametrize the working densitiesof P t, 1 ,j and P t, 0 ,j by p ( y t ; θ ∗ (1) t,j ) and p ( y t ; θ ∗ (0) t,j ) , respectively.
In our case, the parameter is theconditional mean θ ∗ ( d ) t,j = E ( Y (0) t | D = d, X ∈ Q j ) for both d = 0 , 1 .
The estimator of symmetric cross entropy between P t, 1 ,j and P t, 0 ,j can be obtained by cross-ﬁtting the negative log-likelihood;that is,
[Equation]
In particular, for Gaussian distributions, log( p ( y t
[Equation]
where, the ﬁrst term is exactly (cid:98) b 2 t ( Q j ) deﬁned in (7), and the second term is the sum of samplevariances of (cid:98) θ (0) t,j and (cid:98) θ (1) t,j .
Therefore, both confounding bias and variance can be emphasized.
Inthe end, assuming historical data collected at distinct timestamps are independent, the confoundingentropy of all historical controls
E (Π t<t m Π qj
=1 P t, 1 ,j , Π t<t m Π qj
=1 P t, 0 ,j ) is estimated by
[Equation]
Q
[Equation]
Q
[Equation]
) is deﬁned in (9) . We remark that (10) can be interpreted as a marginally independent composite likelihood estimator[44], which can be readily generalized to model potential temporal correlation among { Y t k ,i } ( m − 1) ,n k,i = .
However, for simplicity, we presume no temporal dependencies.
We also remark that, under thiscomposite likelihood framework, (cid:98) b t ( Q j ) is exactly the empirical cross entropy with working densitieslog( p ( y t ; θ t,j ( d )))
= − ( y t − µ t,j ( d ))
+ const for both d = 0 , 1 (i.e., exponential distribution).
Asillustrated, such a speciﬁcation of density functions, under which high-order moments equal to theﬁrst moment up to constants, is prone to high-order variations.
Estimating CATT.
On the basis of (6), we then can estimate η ∗ ( x ) with
[Equation]
where by (6), (cid:98) µ ( d [Equation] ( Q j
[Equation]
t m I { D i = d } ) / |{ i : X i ∈
Q j , D i =
d }| , d = 0 , 1 forany given Q , and (cid:98) Q h = { (cid:98) Q hj } qj =1 denotes the minimizer of (cid:98) H( Q ) in (10).
Here, the superscript“h” denotes the fact that only historical data are used to learn Q ∗∗ .
This two-stage estimator (cid:98) η h ( x )may suffer from potential heterogeneity between historical and current data sources, since splittingmechanism of (cid:98) η h ( x ) is independent of current data set { ( X i , Y i,t m , D i ) } ni =1 .
It relies crucially onthe transportability of historical optimal partitions { Q ∗∗ j } qj =1 .
To leverage the information of currentdata, we aim to ﬁnd a partition in which E ( Y ( d ) t m | D = d, X = x ) can be accurately learned bycorresponding trees.
Let (cid:98) m ( x, D ; Q ) =
DT ( x ; Q , (cid:98) µ (1) ( Q ))
+ (1 − D ) T ( x ; Q , (cid:98) µ (0) ( Q )) .
We aimto ﬁnd a Q such that (cid:98) m
( x, D ; Q ) ≈ DY (1) t m ( x ) + (1 − D ) Y (0) t m ( x ) meanwhile (cid:98) H( Q ) being small.
Regularized Regression Tree Estimator.
For any problem-speciﬁc loss function (cid:96) ( · , · ) , e.g.,squared loss for regression and cross-entropy for classiﬁcation, we propose the following regularized empirical risk minimization: [Equation]
[Equation]
where λ denotes the tuning parameter, and the CATT estimator is given by (cid:98) η ( x )
=
T ( x ; (cid:98) Q , (cid:98) µ (1) ( (cid:98) Q ))
− T ( x ; (cid:98) Q , (cid:98) µ (0) ( (cid:98) Q )) .
We remark that there exists a trade-off between data ﬁdelity of the current source and transportabilityof historical sources for the choice of λ .
When λ → ∞ , (cid:98) η ( · ) is reduced to the naive estimatorproposed in (11), ignoring the information of current data.
In contrast, λ = 0 means that we onlyuse current data to learn CATT, which, under the unconfoundedness condition, is similar in spirit tocausal tree estimators proposed in [47] and [3].
For the optimal choice of λ , our method minimizesthe confounding bias introduced by unmeasured confounders that deteriorate the performance ofconventional causal trees, and thus the name “debiased causal tree” follows.
2.3 Ensemble of Debiased Causal Trees by Gradient Boosting
Ensemble is a commonly used method to improve the performance of tree-based models.
Boostingand bagging are two representative methods, where the former is designed to reduce bias, and thelatter focuses on variance reduction.
Existing tree-based methods developed for treatment effectestimation combine multiple trees to make a random forest, since unregularized decision trees areprone to overﬁtting, especially when the tree in employment is deep.
In our case, the objectivefunction (12) includes a regularization term.
As a consequence, our estimator tends to be underﬁttinginstead of overﬁtting, particularly when the working densities that deﬁne (8) are misspeciﬁed and/orthe tuning parameter λ is relatively large.
This motivates us to choose the boosting frameworkto improve the performance of single debiased causal tree.
We deﬁne the boosting of k trees as
F ( x ; Q k , M k ) =
(cid:80) ki =1 T ( x ; Q i , µ i ) , where Q k = { Q 1 , . . .
, Q k } ≡ Q k − 1 ∪{ Q k } denotes the setof partitions of each tree, and M k = { µ 1 , . . . , µ k } ≡ M k − 1 ∪{ µ k } denotes the set of correspondingparameters.
To highlight the difference, we use Θ k = { θ 1 , . .
, θ k } to denote parameters of auxiliarytree models that measure confounding entropy.
At the k -th iteration, we have already obtained k − 1 trees, F ( d ) ( x ; (cid:98) Q k − 1 , (cid:99) M ( d ) k − 1 ) , to learn E ( Y ( d ) t m |
D = d, X = x ) , d = 0 , 1 , whose difference yields CATT, and k − 1 trees, F ( d ) bias ( x ; (cid:98) Q k − 1 , (cid:98) Θ ( d ) k − 1 ) ,to learn E ( Y t | D = d, X = x ) , t < t m , whose difference yields confounding bias.
Notice that thetrees for confounding bias share the same splitting regions with trees for CATT.
In the following, weillustrate how to obtain the k -th tree.
Parallel to a single debiased causal tree, we ﬁrst show a reﬁned representation of confoundingentropy by integrating multiple trees.
Following the gradient boosting algorithm [16], the k -th tree
T ( x ; Q k , θ ( d ) k ) manages to ﬁt the residual, Y i,t
( X i ) − F ( d ) bias ( X i ; (cid:98) Q k − 1 , (cid:98) Θ ( d ) k − 1 ) with D i =
d for eachd = 0 , 1 .
Denote by (cid:98) θ ( d ) k
( Q k ) =
( (cid:98) θ ( d ) k
( Q k,j ))
the local mean for any partition Q k , where (cid:98) θ ( d ) k
( Q k,j )is deﬁned similarly to (7) with Y i,t being replaced by residuals.
Similar to (8), we use the cross meansquared error to represent the confounding entropy
[Equation]
13) where (cid:101) Q k
= (cid:98) Q k − 1 ∪ { Q k } , (cid:98) Θ ( d ) k
= (cid:98) Θ ( d ) k − 1 ∪ { (cid:98) θ ( d ) k
( (cid:98) Q k ) } , and Y ( d ) i,t denotes Y i,t with D i = d .
It sufﬁces to consider the integration of trees to learn δ t m ( x ) .
Again, the k -th tree T ( x ; Q k , µ ( d ) k )manages to ﬁt the residual, (cid:98) R ( d ) i
= Y ( d ) i,t m − F ( d ) ( X i ; (cid:98) Q k − 1 , (cid:99) M ( d ) k − 1 ) with D i =
d .
Let (cid:98) µ ( d ) k
( Q k ) =( (cid:98) µ ( d ) k
( Q k,j )) be the local mean for any partition Q k , where (cid:98) µ ( d ) k
( Q k,j ) is deﬁned similarly to (11)with Y ( d ) i,t m being replaced by (cid:98) R ( d ) i .
The objective function for the k -th tree is displayed as follows:
[Equation]
Update (cid:98) Q k = (cid:98) Q k − 1 ∪ { (cid:98) Q k } , and (cid:99) M ( d ) k
= (cid:99) M
( d ) k − 1 ∪ { (cid:98) µ ( d ) k
( (cid:98) Q k ) } , d = 0 , 1 .
The CATT estimator atthe k -th iteration is (cid:98) η
( x ) =
F (1) ( x ; (cid:98) Q k , (cid:99) M (1) k ) − F (0) ( x ; (cid:98) Q k , (cid:99) M (0) k ) .
We refer to our proposed method as Gradient Boosting Causal Tree ( GBCT ).
We remark that theobjective function (14) is greedily minimized by gradually adding the tree.
Also, to speed up thecomputation and stabilize the algorithm, we follow XGBoost to approximate the objective functionwith a second-order Taylor expansion and exploit a squared (cid:96) 2 regularization.
See [11] for details.
Applicability of Debiased Causal Trees Without Confounders
In this section, we discuss the applicability of debiased causal trees without confounders.
Despite thatthe confounding bias b t ( x ) ≡ 0 for any t and x without unmeasured confounding, numerically, theterm (cid:98) b t ( Q j ) in (7) may not vanish, especially when the minimal sample size is small.
This issue issubstantive when treated and untreated units are imbalanced.
Notably, the imbalance can signiﬁcantly degrade the performance of T-learners and S-learners [e.g., see 27].
Given the historical trainingset { ( X i , Y i,t , D i ) } ni =1 , the smaller empirical confounding entropy is for the partition Q , the morebalanced samples can be within each small region.
Here, for simplicity, we consider m = 2 such thatonly one historical data source is used.
Let n d ( Q j )
=
|{ i : D i =
d, X i ∈ Q j }| be the number ofsamples in Q j with D i =
d .
We formalize the above intuition with the empirical confounding bias(cid:98) H( Q ) deﬁned in (10) with working distribution being Gaussian.
Proposition 1.
Suppose there are no unobserved confounders, and if Y i,t ∼ Y t with mean µ t andvariance σ 2 t independently for a given t < t m , then, under the Gaussian speciﬁcation of workingdensities, it holds that E (cid:2)(cid:98) H( Q ) (cid:3)
= 2 σ 2 t + σ 2 t (cid:80) qj =1 ( n 0 ( Q j ) + n 1 ( Q j )) /
( n 0 ( Q j ) n 1 ( Q j )) .
By Proposition 1 and the harmonic mean inequality, to reduce (cid:98) H( Q ) , it sufﬁces to ﬁnd a Q j such thatn
( Q j ) ≈ n 1 ( Q j ) , especially for Q j such that max { n 0 ( Q j ) , n 1 ( Q j ) } is small.
Numerical analysiscan be found in Figure 1 in the Appendix, which indicates that, our proposed debiased causal tree isrobust to imbalance of treatments.
Experiments
We conduct experiments with both simulated and real data for a better understanding of our proposedstrategy.
We investigate two scenarios in simulation studies: one with and one without unobservedconfounding.
When implementing GBCT , we subtract the empirical pre-treatment confoundingbias from effect estimates as a trick to improve performance, i.e., (cid:98) η ( x ) =
F (1) ( x ; (cid:98) Q k , (cid:99) M (1) k ) −
F (0) ( x ; (cid:98) Q k , (cid:99) M (0) k ) − 1 m − 1 (cid:80) m − 1 u =1 (cid:80) Q ∈ (cid:98) Q k (cid:98) b
t u ( Q ) .
We evaluate GBCT against state-of-the-art causalinference algorithms: (i) meta learners [27], which adopt machine learning models to ﬁt the responsesurfaces on the treated and control populations separately, including TLearner , SLearner and Xlearner ; (ii) causal forests, which are forest-based methods to model the treatment effects, including generalized causal forest [ GRF , 4], doubly robust forest [ DR-RF , 5] and double machine learningforest [ DML-RF , 12].
These benchmarks are implemented by the third-party library E COML [24].
To make a fair experimental comparison, we set all hyperparameters using the same strategy.
Thenumber of trees in ensemble models (including boosting and bagging) is 200 , the sub-sample ratios ofinstance and feature are 0 .
8 and 0 .
6 respectively, and the learning rate is 0 .
The maximum depth ofeach tree in forest-based ( GRF, DML-RF and DR-RF ) and boosting-based (meta learners and GBCT )methods is 10 and 3 , respectively, where it should be noted that due to the respective characteristics ofthe bagging and boosting frameworks, the trees in random forests are generally deeper.
Meta learnersemploy LightGBM as base models and causal forests utilize sub-sampled honest random forest [3].
Both covariates X and pre-treatment outcomes are used as input features for benchmark methods.
However, we only use covariates X to build GBCT , and the pre-treatment outcomes are treated aslabels to de-bias in training phrases.
4.1 Simulation Studies
Data Generating Process.
We start by generating a p -dimensional confounder vector X from [-=
-]269 (cid:80)
Ss =1 I( G = s ) N ( µ s , Σ s ) , where G ∈ { 1 , 2 , · · · , S } is a hidden variable indicating which latentcluster each subject belongs to.
The treatment-independent feature vector W is developed similarly.
We then generate the potential outcomes by the following factor augmented auto regressive processof order one, for each 1 ≤ k ≤ m ,
[Equation]
where ρ ∈ [0 , 1] is the individual-speciﬁc auto-correlation coefﬁcient following a truncated normaldistribution ¯ N ( µ ρ , σ 2 ρ ; [0 , 1]) , λ t is the time-varying factor, the initial value Y (0) t= β
( X, W ) ,{ α ( · ) , β ( · ) , f ( · ) , g ( · ) } are speciﬁed functions, and ε, ε 1 are mutually independent noise items subjectto N (0 , 1) .
To ensure the samples are drawn from the stationary distribution of the data generatingprocess, we use outcomes with time larger than a threshold t T as observations.
We assign treatment for each unit according to its latent cluster, such that P ( D = 1 | G = s ) − P
( D =
| G = s ) =
( − 1) s φ for s = 1 , 2 , · · · , S , where φ ∈ [0 , 1] is a speciﬁed value.
The parameter φmeasures the degree of balance between the treated and control populations; as φ rises from zero to one, the imbalance gradually increases.
In particular, φ = 0 means the treated group has thesame population as the control group (i.e., RCT), whereas φ = 1 means all treated group’s unitsare from different latent clusters than the control group’s.
In following experiments, we set S = 2 ,µ ρ = 0 .
7 , σ ρ = 0 . 2 , and the dimension of feature space 2 p = 20 .
A total of 20000 samples aregenerated and randomly split into training and validation sets 10 times.
No Unmeasured Confounding Scenario.
The confounding vector X blocks the backdoor pathfrom treatment to potential outcomes.
Thus, given X , potential outcomes Y ( d ) t m is independentof D for d ∈ { 0 , 1 } .
Suppose that we observe all confounders X , and apply GBCT as well asbenchmarks to estimate the conditional treatment effects.
We compare our proposed method withbenchmark algorithms on the mean absolute error (MAE) of conditional average treatment effects, i.e., MAE
[Equation]
i | , where η i is the true value of the i th individual’s treatment effect.
Table 1 shows the results.
As we can see, GBCT outperforms the benchmark algorithms in most cases,especially when φ is larger, suggesting that it is beneﬁcial to reduce the empirical confounding biasin case of no unmeasured confounding.
As φ increases towards one, except for GBCT and GRF , othermethods generally lead to severe bias and signiﬁcant performance degradation due to low overlap.
This conﬁrms the robustness of GBCT against imbalance between the treated and control populations. Moreover, we perform ablation experiments to compare GBCT with its variants without using de-biasloss.
The results presented in the Appendix demonstrate the important role of the strategies to reduceconfounding bias in making the estimates less sensitive to the degree of imbalance. , whereas II represents the presence of unmeasured confounding.
In addition [Table] • / ◦ indicates whether GBCT is statistically superior/inferior to the benchmark algorithms (pairwise t-test at 0.05 signiﬁcance level)
[Table]
Unmeasured Confounding Scenario.
We consider the scenario where the confounding vector
[Table]
X are not fully observed.
For estimation and evaluation, we retain only the ﬁrst dimension of theconfounding vector and discard the rest.
When φ = 0 , the treatment is assigned randomly, resultingin no confounding bias.
However, as φ approaches one, the impact of confounding on treatment [Table] assignment grows stronger.
Therefore, a large φ suggests a substantial confounding bias due tounmeasured confounding.
In this case, most traditional methods may fail to adjust for bias withoutfurther assumptions, while in contrast, GBCT can approximate CATT well.
To support the superiorityof our approach, we use MAE CATT =
[Equation]
| [Equation] ) − η i | as the criterion toevaluate the performance of algorithms, and compare GBCT with the benchmarks in numerical studies. We report the results in Table 1.
As expected, GBCT yields smaller mean absolute errors than othermethods, and the gap becomes more pronounced as φ increases, indicating that GBCT can offer amore robust and compelling solution for estimating conditional treatment effects in the presence ofunmeasured confounding.
We also compare the performance of GBCT with GRF at the single-tree scaleon removing confounding bias, showing that our method can be more effective, see the Appendix.
4.2 Real Data: Credit Card Balance Dataset
The dataset comes from a randomized controlled trial (RCT) by a commercial ﬁnance company aimedat assessing users’ heterogeneous responses to increasing credit line of credit card.
The trial employs astratiﬁed random assignment design with strata based on risk, dividing users into two groups: low-riskand medium-risk.
Within each group, users are randomly assigned to one of four treatment groups: increasing credit line by 0 , 2000 , 3000 or 6000 converted to some currency.
In this study, the outcomeof interest is the balance of the credit card account in the following month after treatment, i.e., thetotal amount of debt owed to the credit card.
In addition, we observe 87 -dimensional covariates at theindividual level, as well as outcome values over the last eight months.
In real-world ﬁnancial scenarios, however, increasing credit line always favors low-risk users;such preference can lead to confounding bias in estimation.
In order to assess the performance ofdifferent methods in the real-world scenario, we artiﬁcially construct a biased observational datasetby increasing credit line for the low-risk group but not for the medium-risk group.
See Table 3 in the Appendix for details of the trial dataset and the biased observational dataset.
We are interested in the causal effects of increasing credit line on balance within each risk group.
Unlike simulated data, we do not know all potential outcomes for each user, so we cannot use
MAE CATE for evaluation.
Fortunately, due to the randomness of treatment allocation, the groundtruth of the average treatment effect is approximately the mean difference between any two treatmentgroups in the trial dataset, denoted as η ATE .
Note that in RCT, the average treatment effect isequal to the average treatment effect on the treated group.
Therefore, we compare our proposedmethod GBCT with the benchmarks on the mean absolute error of the average treatment effect, MAE [Table]
[Equation]
[Table]
The results are reported in Table.2.
It is impressive to observe that on both trial and biased data, GBCTsigniﬁcantly outperforms the benchmarks in terms of mean error and standard deviation.
Althoughthe benchmarks DR-RF and DML-RF do not perform poorly on the trial data, they suffer severely frombias regarding the observational data; a possible reason is the low overlap between the low-risk andmedium-risk groups.
As a comparison, GBCT suffers substantially less from bias than others.
Theseﬁndings are in line with those of the prior simulation studies.
Therefore, we recommend using GBCTin practice to gain a more stable and robust estimate of causal effects.
Discussion
In this work, we propose a simple yet powerful method, debiased causal tree, to remove confoundingbias caused by unmeasured confounding in the estimation of causal effects (e.g., CATT).
Our methodonly hinges on the availability and transportability of historical controls, i.e., outcomes of units beforethe treatment, to ensure CATT to be partially identiﬁable.
This allows our method to be readilyapplicable in a wide variety of scenarios, especially in clinical trials [45].
Moreover, our methodcan be generalized to integrate observational data and external control [30], with a much weakertransportability condition than existing work [40].
Theoretically, our identiﬁability result providesthe possibility of statistical justiﬁcation for the proposed estimators with ﬁnite samples. For example,provided that the optimal partition Q ∗∗ is estimated accurately, the convergence rate of (cid:12)(cid:12)(cid:98) η ( · ) − η ∗ ( · ) (cid:12)(cid:12)can be derived using the adaptive concentration results developed in [48], which will be left for futurework.
A Appendix
In this section, we include related proofs, additional simulation plots that demonstrate the necessityof our proposed debiased causal tree, and a brief introductive table of the real-data.
A.1 Related Proofs
Proof of Theorem 1.
By the triangular inequality, we obtain
[Equation]
By the Cauchy–Schwarz inequality, we further obtain
[Equation]
Next, we bound (cid:80) qj =1 (cid:12)(cid:12)
b t m ( Q ∗∗ j ) (cid:12)(cid:12) 2 = exp { B ( t m ) } .
By Assumption 1, it holds that
[Equation]
L | t m − t k | , i.e.,
[Equation]
(cid:88)
[Equation]
which implies that
[Equation]
Choosing C L ≡ e L ( t m − t 1 ) , and combining (15)–(17) givesup
[Equation]
Proof of Proposition 1.
By (10), under the Gaussian speciﬁcation of working densities,(cid:98) H( Q ) =
q (cid:88) j = (cid:26)(cid:0)(cid:98)
[Equation]
(cid:88)
i : X i ∈
Q j (cid:26)
D i
( Y i,t − (cid:98) θ (1) t,j )|{ i : X i
[Equation]
D
[Equation]
1 }| + (1 − D i )
( Y i,t − (cid:98) θ (0) t,j )|{ i : X i ∈
Q j , D i = 0 }| (cid:27)(cid:27) .
Under the unconfoundedness condition, if Y i,t ∼ Y t with mean µ t and variance σ 2 t independently,for each given Q j ,(cid:98) θ ( d ) t,
[Equation]
= n (cid:88) i =
Y i,t I { D i =
d, X i ∈ Q j } n d ( Q j ) , where n d ( Q j ) =
|{ i : X i ∈
Q
j , D i =
d }| , is the mean of i.i.d random variables with ﬁnite second moment and (cid:98) θ (0) t,j is independent of (cid:98) θ (1) t,j .
Thus,
E (cid:2)(cid:0)(cid:98) θ (0) t [Equation] [Equation] 2 (cid:3)
[Equation]
(cid:18)
[Equation]
(cid:19) , and
E   (cid:88) i : X
i ∈ Q j (cid:26)
[Equation] (1) t [Equation] )|{ i : X
[Equation]
|
[Equation] [Equation] [Equation] | (cid:27)
[Equation]
n 1 ( Q j ) + n
[Equation]
We then conclude that
[Equation]
5 depth 0.
c o n f o und i n g b i a s
GRF GBCT (a) φ =0.
5 depth 0.
c o n f o und i n g b i a s
GRF GBCT (b) φ =
[Figure]
[Table]