Estimating Multi-cause Treatment Effects via Single-cause Perturbation
Authors: Zhaozhi Qian
Alicia Curth
Mihaela van der Schaar
Venue: NeurIPS 2021 Poster (2021)
Keywords: Causal Inference
Treatment Effect

Abstract: Most existing methods for conditional average treatment effect estimation are designed to estimate the effect of a single cause - only one variable can be intervened on at one time. However, many applications involve simultaneous intervention on multiple variables, which leads to multi-cause treatment effect problems. The multi-cause problem is challenging because one needs to overcome the confounding bias for a large number of treatment groups, each with a different cause combination. The combinatorial nature of the problem also leads to severe data scarcity - we only observe one factual outcome out of many potential outcomes. In this work, we propose Single-cause Perturbation (SCP), a novel two-step procedure to estimate the multi-cause treatment effect. SCP starts by augmenting the observational dataset with the estimated potential outcomes under single-cause interventions. It then performs covariate adjustment on the augmented dataset to obtain the estimator. SCP is agnostic to the exact choice of algorithm in either step. We show formally that the procedure is valid under standard assumptions in causal inference. We demonstrate the performance gain of SCP on extensive synthetic and semi-synthetic experiments.

Estimating Multi-cause Treatment Effects via Single-cause Perturbation
Zhaozhi Qian University of Cambridge zhaozhi qi anomaths cam ac uk
Alicia Curth University of Cambridge amc2530cam ac.
Mihaela van der Schaar University of Cambridge
UCLA The Alan Turing Institute mv4720cam. ac uk
Abstract
Most existing methods for conditional average treatment effect estimation are designed to estimate the effect of single cause only variable be intervened on at one time. However, many applications involve simultaneous intervention multiple variables, which leads to multi-cause treatment effect problems. The multi-cause problem is challenging because one needs to overcome the confounding bias for large number of treatment groups, each with different cause combination. The combinatorial nature of the problem also leads to severe data scarcity we only observe one factual outcome out of many potential out- comes. In this work, we propose Single-cause Perturbation (SCP), a novel two-step procedure to estimate the multi-cause treatment effect. SCP starts by augmenting the observational dataset with the estimated potential outcomes under single-caus cause interventions. It then performs covariate adjustment on the augmented dataset obtain the estimator. SCP is agnostic to the exact choice of algorithm in either step. We show formally that the procedure valid under standard assumptions in causal inference. We demonstrate the performance gain of SCP on extensive synthetic and semi-synthetic experiments.
Introduction
Estimating treatment effects from observational data a central problem in causal inference and has many applications such as precision medicine [11] In this work, we focus estimating conditional average treatment effects (CATE) reflect the heterogeneity within population (.. The vast majority of the CATE estimation methods consider the single-cusse setting, where only one variable can be intervened on, e.g. the decision to give (or not to give) a particular drug. However, in many applications it is necessary to intervene multiple variables simultaneously achieve the desired outcome (the multi-caute setting). For example, multiple drugs are needed to treat patients with comorbid chronic diseases systemic discases such cancer (21). However, finding the best drug combination for each patient is very challenging and the current clinical practice clearly sub-optimal [30]; studies have shown that nearly 50% of the elderly population in developed countries take one or more drugs that are not medically necessary [39]. Similar examples are abundant the methodology estimate the combined effect of multiple causes (drugs), a challenge we undertake in this work.
medical literature and beyond (Appendix A.5). which calls for
We make distinction between the terminology cause and treatment. We refer to cause as atomic variable that can be intervened on, and treatment as configuration of all causes. Therefore, if the problem involves K causes and each cause is binary variable, there will be 2 possible treatments. The key objective causal inference to overcome the confounding bias treatment assignment. This is challenging in the multi-cause setting because large number of treatment groups need to be
35th Conference on Neural Information Processing Systems (NcurIPS 2021).

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(A1)
(A2)
(81): Observational data
[B2):Interveningg on both
(8te:eenegggning on A
Yaa
Causes
A1
A
A
Aa
Outcome
The aingle-cuuse setting: out 2 observed
Only out 2 potential cutcomes observed.
Observations
Final objective
Aualiary
causes and A2: the single-cause setting. Each row contains one observation. Three green cells in each row will filled in by SCP's first step to form augmented dataset. (B) Interventions on an illustrative DAG. Bl: observational data (no intervention), B2: intervening both causes, B3: intervening on A only. In B3, the intervention on A generates an effect on the outcome and the cause A2. The covariate X greyed for visual clarity.
Figure 1: (A) Iustration of the data scarcity challenge. A1:
balanced. (In contrast, one only needs to balance two treatment groups in the single-cause setting.) Furthermore, the combinatorial treatment also leads data scarcity outcome under the treatment that was given (factual outcome), but not the potential outcomes (PO) under all other treatments (2 - in total, as illustrated Figure A). As the number of causes increases, the fraction of observed outcomes decreases exponentially, which challenges the reliable estimation of CATE.
can only observe the
Most single-cause methods consider only two treatments (treated or untreated). In fact, many popular architectures and regularization methods aimed to overcome confounding do not scale computationally to large treatment spaces 757 58) 38]. As a remedy, one may make additional assumptions on the data generating process (DGP), for instance, assuming linear model generates the outcome (29 or a low-dimensional latent variable generates the treatment 75 However, such
assumptions may limit the scope of application.
In this work, we take a different direction: instead of making additional assumptions on the functional form latent variable structure, we exploit the connection between single-cause intervention and multi-cause intervention [ We establish that, under standard assumptions in causal inference, the single and multi-caus potential outcomes equal in expectation under appropriate conditioning.
Based on this finding, we propose single-cause perturbation (SCP), novel two-step procedure to estimate CATE in the multi-cause setting. In the first step, SCP generates K additional datasets by predicting the potential outcomes resulting from perturbing each of the K causes to their opposite value. then performs covariate adjustment on the combined dataset. By deliberately perturbing the causes, the treatment assignment in the augmented dataset would be more balanced than the observational data, thereby reducing the confounding bias. SCP agnostic to the exact choice of algorithm in either step and the user can choose the algorithms based on the application.
Contributions. We present SCP, two-step multi-cause CATE estimator that leverages the connection between single and multi-cause cause interventions. SCP overcomes confounding bias by using single-cause CATE estimators to augment the observational data with the estimated potential outcomes. Compared with existing works, SCP does not make assumptions about the distributional or functional form of the DGP, making it suitable for complex problems healthcare. We demonstrate and analyze the performance gain of SCP via extensive experiments.
2 Problem formulation and notations
In this work, focus the CATE estimation problem with K binary causes Let the causes A = (Ai,..., Ag) ) be a multi-dimensional random variable with sample space Ω {0, 1) where A the 1h cause. Let A & Ω {0, 1) jK- -1 be the collection of all but the tthh cause. Let X E P and Y E IR be the covariates and observed outcomes respectively. The causal relationship
'SCP also applics multi-level categorical causes, i.c. A E (0, 10.,,..., L.J, E N and multi-dimensional outcomes. i.e. Y MM Here, use the current setting for illustration.

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(A)
(B)
X
X
Aggregate
X
Y
Ak
Y
A
Y
Illustrative causal graphs. (A) Intervention all causes A. (B) Intervention the single cause As- The other causes are partitioned into descendants A e and non-descendants A Al, k.- Purple edges: confounding to treatment assignment. Brown edges: effects on the (combined) outcomes. Some less important edges are greyed out for visual clarity.
Figure 2:
between these variables is illustrated in Figure A, which is direct generalization of the single cause setting [56]. We have access to an observational dataset Dn = [x, Vt, a ke[NN with No independent samples from the random variables defined above. Throughout the text we capital letters for random variables and lower case letters for fixed constants. We use boldface for vectors or dillii-dmmenionaal random variables. When the context clear, we will simplify the conditional expressions, e.g. P(Y == P(Y|X = x).
2.1 Multi-cause intervention
We formulate the CATE estimation problem using the potential outcome (PO) framework [56 Let (a) E R denote the potential outcome in a world where the treatment a E Ω was given. We would like to estimate the CATE between any treatments given the covariates i.e. r(a, a',
x). Va, a' E Q, E R.D. We can estimate CATE by estimating all potential outcomes EE (a) X). Va Z
(a) Y(a')
The following three assumptions have been proposed to identify the multi-cause PO [66 24 (I) Consistency Va E f if A (a) Y. (2) Weak unconfoundedness: (a) Il A X, Ya E Ω. (3) Overlap: P(A = a > 0, va E Ω if P(X) > 0. The assumptions stated above allow the expectation of multi-cause PO be estimated from observational data: Va Ω Vx E Rº:
[[V(a)|X = x] Y = x, A =a
(1)
2.2 Single-cause intervention
Here we consider the intervention a single-cause, e.g. adding drug A the existing medications. Such intervention may affect the outcome and the other causes. For example, the inclusion of drug A₁ may promote the usage of another drug A2 because A₂ mitigate the side effects of A [88]
We denote Y (ax) E R as the potential outcome where the cause A& is set to be ag- We refer to Y (ax) as the single-cause PO. Note that the single-cause PO Y (ax) ) is different from the multi-cause PO Y(a) because the latter refers to potential world where all causes are intervened on. We sometimes denote the multi-cause PO as (a) == Y(a,, a)).
We assume that, based domain knowledge, can partition the rest of the causes A into A₁ 's causal descendants AL and its non-descendants A illustrated in Figure 2
B 45]. We
denote A (ax),, A &. (ax) and A (ax) as their potential outcomes respectively. By definition, the non-descendants should be unaffected by the intervention:
A A = (1) A
As shown in Figure it is convenient to aggregate all the variables affected by A into combined outcome Y and aggregate all the variables confounding A₁ as a combined confounder X
(Y. A ));
Y((a)) = (Y( A (ak)) X
(X
XX
In Appendix A.4 we present an alternative formalism using do- do-operation algorithm be derived using either formalism.
46
We show that the same SCP

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Table 1: Summary of the data augmentation task in SCP's first step.
Equation
Target
Input Covariates
Estimated Value
Algorithm
Eq.
A (af)
a B (ak) a
Eq.
A (a(())
X X A
at ( ((A , Ad) y(aa) E(Y X A Ax)
DR-CFR DR-CFR
Eq.
Y(ak)
To identify the combined PO Y (ax),, we make the standard assumptions using AR. Y and X (4) Single-cause Consistency: vk S. K, Va {0, 13 if A ag. (a()))) Yg (5) Single-cause Unconfoundeddess: Y (ax) IL A& X Vag E [0,1}, vk S K. The multi-cause overlap (Section 2.1p implies single-cause overlap, but the multi-cause consistency and unconfoundedness do not imply the single-cause counterparts (Appendix ).3).. Appendix AA.] Proposition [rpposition]]hooee shows that, under these assumptions, we can identify Y/(ax) from observational data as: Ak S K, Vag {0, 1}.
((Y(())) = PAA A X A& = ox) X A+ As az))
(4)
Discussion on partitioning the causes. We can always partition the causes into descendants and non-decadents as long as the structure between the causes follows DAG (hence no cycles). In practice, such structural knowledge is often available, e.g. we can use the clinical guidelines to identify the drugs whose prescription will be influenced by the usage of another drug. Note that we do not need to specify the causal graph of all individual variables (e.g. the link between two covariates X,, X))). However, when the full causal graph available, we can adapt SCP to make use of the additional structural knowledge as discussed in Appendix [A.6] On the other hand, we show empirically that SCP is not sensitive to misspecified partitioning (Section 5.1. Appendix A.3 contains an extended discussion on all our assumptions.
3 Single Cause Perturbation
3.1 The algorithm
In this section, we introduce our proposed method - single cause perturbation (SCP). Given an observational dataset Do with No data points: Dn {xi, Mi, a: JejN)) SCP proceeds in two steps: first fits set of models that can predict the effects of changing a single cause, and uses them to No for & E [K] each corresponding to the potential
create K additional data sets Dj- = {x,, D a scenario of perturbing a single cause. It then fits a final model on this enlarged dataset, which is used to estimate the multi-cause CATE. The pseudocode is detailed in Appendix A.7 Algorithm[] [
Training single-cause models. Based on Equation 4 we will train two separate models to estimate the combined PO Y' (ax): one for A (a*) and one for Y(x)). Note that for CATE estimation, we only need to estimate the expectation E(Y|X, A+ Ax) rather than the full probability distribution. The models trained the observational data Dy. We can any single-cause CATE estimator for this purpose since only one cause is intervened on.
We choose to use the state of the art single-cause CATE estimator, Disentangled Representations for Counterfactual Regression algorithm (DR-CFR) [22). DR-CFR achieves higher estimation accuracy by learning to distinguish between true confounders, adjustment variables and instruments contained X We provide a self-contained description of DR-CFR in Appendix A.8
Data augmentation. As illustrated in Table D once the single-cause models are fitted, sampling perturbed data points from observations (x, a) E Do involves three steps: (1) obtain (a())
directly from the observations, (2) sample a (a() ~ PP(A (af)) and (3) obtain y(ak)) ==
E(Y (a((a()) (xx)) (a())) Here a' = 0g corresponds to perturbing the cause A& (recall that as E (0, 1)). Note that in step two we sample the new causes at (a() from the distribution in order to keep them binary variables. To generate new data point (x, j at), we define ÿ :- y((()) and a = (a', a (a{)). Denote DR = (x;, 5/t " as the perturbed data for Ag. We combine all perturbed datasets DR. &s E [K] and the original dataset Do to create the augmented training data

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D (D*tRejo, For each unique x, D contains K + different treatments a ä" and their corresponding outcomes.
Covariate adjustment on augmented data. We can estimate CATE by learning the conditional expectation in Equation 1 using the augmented data DJr. We use standard feed-forward neural network, fe IR x Ω + R with trainable weights 0.
3.2 Validity of SCP: linking single and multi-cause PO
One may wonder why the augmented data points (single-cause POs) would help estimate the multi- cause PO: they correspond to different interventions, i.e. intervention on a single cause versus intervention on all causes simultaneously. Proposition shows that given our assumptions the single and multi- cause POs are equal in expectation under appropriate conditioning therefore, (imputed) single cause POs can be used for multi-cause estimation. The proof is shown in A.1
Proposition 1 (Equivalence of the single and multi-cause PO's conditional expectation). Under the sequential ignorability assumption (53), Ak S K,
EYY (ax, a x = E(Y (ax)]x, A- (ax) - a
(5)
Note that the (ax) and (=x(az) on the right hand side (RHS) is precisely what we estimated and added the augmented dataset D; in the first step. Thus if we train supervised learning model on Dj to estimate the RHS, the trained model can also estimate the multi-cause PO on the LHS. Moreover, since the relationship in Equation holld for all k. we can pool all the augmented datasets into one training dataset DTT, which is K times the size of the observational data i.e.
[D (K + 1)) The increased sample size mitigates the data scarcity issue and allows the estimator to generalize better.
Proposition 1 also highlights the necessity of estimating (- in addition to Y(as) in the first step. This is because Equation isi conditioned on A A~x(ap) rather than the observed cause A Note that A (aa)) A.
k. Vax {0, 1} only when Ag has no descendants.
3.3 SCP creates a more balanced dataset via data augmentation
In addition increased sample size, there is also less obvious (but equally important) reason why SCP would achieve performance gain: the augmented data tend to be more balanced than the observational data. This is because SCP perturbs every single cause of all the observations. For instance, by combining Do and Dj, the empirical distribution P(A,|X = x;) 0.5, Vx, E Do- Balancing is important because prior research has shown that CATE estimators trained on balanced dataset tend to generalize better [57]. In fact, many existing causal inference methods employ balancing techniques to improve performance (see Section In Section 5.1 we demonstrate experimentally that SCP consistently improves the balancing of the observational dataset.
3.4 Trade off between sample size, balancing, and first step error
SCP's data augmentation increases sample size and improves balancing, both of which are beneficial to CATE estimation. However, there is caveat: the augmented dataset will also carry the finite- sample estimation error made in the first step. There is a risk that this additional source of noise will reduce or even cancel out the benefits of data augmentation.
In the simulation study in Section 5.1 we investigate this empirically, and observe that SCP's actual error in the first step is usually much smaller than the error required to offset the benefits of data augmentation. We conjecture that this is because SCP only perturbs one cause at a time. The effect of such localized perturbation can be efficiently estimated by the existing methods tailored for the single-cause setting.
One can envision an alternative way where we bundle together any two (or even more) causes A, and A₁ and perturb both of them simultaneously. This will further increase the sample size and improve the balancing, but the first step error will also increase because the effect of a joint perturbation is harder estimate. After all, if we were able do this well, there is no need for data augmentation in the first place.
5

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Table 2: Comparison with the related works. The ATE methods are listed for completeness.
Method
Ref
Estimand
Balancing method
Sample size
Intermediate estimand
SCP
This work
CATE
Data augmentation
E(Y, (a)))) P(Y A) P(A P(Z X)
Cov. Adjustment
3
CATE
None
VSR
75
CATE
Weighting
Deconfounder
72
CATE
None
Z, A)
Weighting
34
ATE
Weighting
X P(A A)Y
Matching
37
ATE
Matching
J.
G computation
5
ATE
Marginalization
NA
A complete theoretical analysis of the trade off is challenging because all three interacting factors contribute to the overall estimation error. Moreover, an important feature of SCP is that does not make any assumption about the DGP (functional form error distribution). However, such assumptions are usually necessary to establish statistical efficiency bounds (44. For these reasons, we will defer the theoretical analysis of the trade off future works.
4 Related works
4.1 Multi-cause and single-cause CATE estimation
Table summarizee the causal inference methods related to SCP. The covariate adjustment method uses supervised learning to estimate the PO from the "feature vector" (x, a) by Equation [ [60 (26
In the single-cause setting, recent works have proposed various architectures and regularization methods [57 38 73 58 74, 22]. Unfortunately, these methods often fail to scale with the number of treatments. For instance, the popular multi-head neural network architecture requires one output head for each of the 2 treatment levels [57). which will be infcasible even with moderate sizeateee-iedd K.
In the multi-cause setting, Variational Sample Re-weighting (VSR) 75 and Deconfounder 772 improve estimation accuracy under additional assumptions about the DGP. Both methods assume that the propensity score (PS) is determined by low-dimensional latent variables Z, i.e. PPA Ez (AA IZ) (Z X). This assumption also makes Deconfounder robust to certain type hidden confounders 72. In comparison, SCP does not make this assumption and it improves balancing by data augmentation as discussed in Section 3.3 Other existing methods assume the outcome is generated from linear model [43 SCP does not make any assumption the function form.
4.2
Multi-cause average treatment effect (ATE) estimation
The methods for multi-cause ATE estimation broadly fall into two categories: weighting and matching 25 37..
The weighting methods assign an importance weight each data point in order create
a balanced dataset for ATE estimation [ 34 To adapt these methods for CATE estimation, we could perform covariate adjustment on the weighted data. In comparison, matching methods achieve balancing by removing unmatched data points and will end up with a smaller dataset [37] 7 62). Since CATE is much more complex estimand than ATE (and thus requires more samples), matching methods designed for ATE are unlikely to achieve good performance for multi-cause CATE estimation.
G- T-Computation is also technique for ATE estimation G-computation marginalizes over the confounders X. The standard implementation estimates the covariate distribution P(X) and uses Monte Carlo sampling for marginalization [52 [3] This makes G-computation conceptually very different from SCP because SCP's s data augmentation is unrelated marginalization its purpose is increase sample size and balaneing for covariate adjustment. We discuss several other less related works in Appendix A.9
54
To compute the average effect,
4.3
Causal data augmentation
Causal data augmentation uses known or learned causal structure to generate augmented datasets (in contrast heuristic data augmentation [59 36)). Several recent works apply this approach to domain

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identity output function
D
B-M
76
10
20
30
40
430 700
1400
2000
0.05
0.50
0.35
0.20
0.3
0.3
0.8
Sigmoid output function
K
No
RMMSE
31
67
400
100
1400
2000
9.05
0.12
0.35
0.20
0.1
0.5
08
Figure 3:
Simulation Results (best viewed in color). RMSE is plotted with the 95% confidence interval shaded (the lower the better). Algorithms include NN, NN-IPW, OP, DEC, VSR and SCP. CFR and DR-CFR'S RMSE is an order of magnitude bigger and is shown in Appendix A.10 separatlly
adaptation 664 27,, robustness 35 65 and reinforcement learning 47.. To our knowledge, SCP the first method that applies causal data augmentation to multi-cause CATE estimation.
5 Experiments
5.1 Simulation study
We created range of synthetic datasets to examine the performance of SCP under different scenarios. Each dataset contains No samples for training, 200 samples for validation and 4000 for testing. The training and validation sets contain observations (X,, Vi, ax) whereas the testing set contains (x,, (a)), ( Va f2. To generate an observation, we first sample D covariates independently: Vd < NN 1). Then we obtain the causes dak, vk < K and the outcome y::
Dataset B
k-1
0sk B B(a Σ
Um Tim
Σ
Unain + Exk]]; y4 Φl Σ staaa
ΣΣ 1=1
disighely + +e), (6)
where x/ (x,, 1) RR, s, are weights, B[-] denotes Bernoulli random variable, denotes the sigmoid function, is either identity or the sigmoid function depending on the simulation setting. To generate various response surfaces, only fraction ps of the weights sampled i.i.d from N(0, 1). resulting in not all covariates and causes contributing to the outcome. The weights d are generated in the same way with the sparsity controlled by Pa, resulting in varying degrees of interaction between covariates and causes. The weights u's are obtained similarly with sparsity Pe Pu 0.3. and are white noises sampled from (O,, 0.01). We evaluate the models using the Root Mean Squared Error (RMSE) on all potential outcomes, which is defined
non-zero and
as ΣN
(a:) X.] (a,))² The simulation parameters of all the experiments below are listed in Appendix Taloe
Benchmarks. We included seven benchmarks to compare with SCP. As baseline, we used covariate adjustment with feed-forward neural networks (NN). We compared with VSR and Deconfounder (DEC), the SOTA methods multi-cause CATE estimation [75 72). For completeness, we also included Counterfactual Regression (CFR) and DR-CFR from the single-cause CATE literature [57 22 as well as the propensity score (NN-IPW) and overlap score (OP) methods from the ATE literature 25 34]. Appendix A. 1. training and hyper- parameter tuning procedure in detail.
Main results. In total, we performed 168 simulations with different sets parameters. The main results are presented in Figure 3 (additional results in Appendix 1..1). In each panel, one simulation (additoonai
parameter is varied while the rest are fixed (see Appendix 10) SCP consistently outperforms
'The implementation of SCP and the experiment code are available at hhttps://gitie com/ ZhaozhiqIAAV/Snggle- P Neur 2021 or https://github com/ orgs/ Vanderschharrabb/rppsttttre.

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0.25
NN En-Eneemble
RMSE
Ablation
SCP
0.15
Oracle PO
0.10
0
2
3
5 6 7
8 9
10
Number augmented datasets or ensembles
Figure 4: The inclusion of augmented data points reduces error. RMSE as more datasets Dr are added to D or more models are added the NN ensemble. In total, there are K 10 causes this simulation.
(A)
Balancing of the dataset (smaller better)
(B)
SCP Final Prediction Error (RMSE)
1.5
Observational
0.30
SCP Augmented
Baseline
Distance:
RMSE
0.25
0.5
0.20
Actual step one error
0.0
0.3
0.5 0.6
Confounding level [Vml
Simulated Step One Error E
Figure 5: (A): SCP consistently improves the balancing of the observational data. Error bars represent the standard deviation of five runs. (B): Relationship between the step one final prediction error. A first step error of 0.4 will degrade SCP's overall performance to the NN baseline (dotted horizontal line). However, the actual step one error is only half of that value (around 0.2).
the benchmarks across different number of causes K, covariate dimensionality D, sample sizes Pa- The performance gain becomes more pronounced the number of causes increase, e.g. K 10. Note that VSR and DEC's DGP assumption is approximately valid here because the um and un that govern treatment assignment are sparse vectors (Equation
No. and sparsity of the causal structure Ps: Ps.Pa-
Why is SCP working? SCP's performance gain roots from the increase in sample size and the improvement in balancing. In Figure 4 we show that SCP's prediction accuracy improves consistently as each augmented dataset Dx. E 00 K] added to the training data Dr (this simulation involves K 10 causes). The benchmark NN ensemble refers to ensemble of NN models trained using bootstrapped observational data D, [50]. The performance improvements of NN ensemble much slower and smaller than SCP because it only bootstraps Do without augmenting it with new data points. The other benchmarks in the figure will be discussed later.
To measure the improvements in balancing, we use the sum of the distributional distances between the treatment groups, i.e. b Eaeo MMD(PP
a), PX A ب a)), where MMD is the maximum mean discrepancy 4 The value appears in the generalization bound of a CATE estimator [57] (also see Appendix A.2 Hence, achieving smaller (more balancing) is highly desirable. We generated range of observational datasets with varying confounding levels, and use SCP to augment each dataset (the confounding level is controlled by the u. Equation (6) Figure (A) shows that SCP's augmented data is consistently more balanced than the observational data (the improvements RMSE shown Appendix [2.1
Relationship between step one error and overall error. Next, we study how the step one error affects the overall error. We the augmented data points to be the true expected PO corrupted by Gaussian noise: V (aY( A+ &. The standard deviation of & a proxy for step one enror. As expected, Figure 5 shows that the overall error increases with the step one emror. SCP's performance becomes simiTar to the NN baseline (black line) when the step one error reaches 0.4,
which is twice as much SCP's actual step one error 0.2 (dotted orange line).

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Table 3: Results of the semi-synthetic data experiment using different data sizes No-
RMSE
Ranking Error
Method
No =500
1000
1500
No =500
1000
1500
NN
1.257 (.004)
1.383 (.006)
1.116 (.004)
282.3 (0.9)
321.6 (1.0)
228.1 (1.5)
VSR
1.246 (.004)
1.186 (.004)
1.140 (.005)
270.3 (1.2)
253,4(1.4)
233.6 (1.6)
DEC
1.268 (.004)
1.200 (.004)
1.118 (.005)
283.9 (0.8)
259.1 (1.3)
236.4 (1.5)
CFR
2.028 (.006)
1.924 (.007)
1.856 (.008)
393.2 (1.0)
380.8 (1.1)
335.4 (1.3)
DR-CFR
2.118 (.006)
2.005 (.0D8)
1.929 (.008)
401.1 (1.0)
(1.12
379.6 (1.4)
NN-IPW
1.354 (.005)
1.244 (.003)
1.123 (.004)
295.4
21.00
225.9 (1.4)
OP
1.365 (.005)
1.426 (.006)
1.215 (.005)
287.8 (0.8)
316.1
238.1 (1.4)
SCP
1.117 (.004)
1.098 (.004)
1.044 (.004)
230.5 (1.3)
221.3 (1.4)
217.9 (1.4)
Sensitivity to mis-specified partitioning and step one error. To better understand the sensitivity, we compare the SCP with an ablated version (Ablation) where there is no prior knowledge about the non-descendants of single cause, i.e. A Ø. As reference, we also consider Oracle PO, SCP with error-free data augmentation step. Figure 4 shows that the correct partitioning of causes is indeed important because the ablation incurred noticeable performance loss compared with other SCP versions. However, even the ablated version consistently outperforms the ensemble of NN. This suggests that the increase in sample size and balancing tend to bring more benefit than the noise introduced in the first step. fact, the Oracle PO achieves more than 60% performance improvement over the NN, which gives wide "safety margin" for step error.
present additional simulation studies that further illustrate SCP's source of performance gain under different settings. Our results consistently suggest that the increase in sample size and the improvement balancing are the two key drivers of the gain.
Further experiments. In Appendix A.12
5.2 Semi-synthetic experiment
Dataset. We used the de-identified COVID-19 Hospitalization in England Surveillance System (CHESS) data, which contains individual-level risk factors, treatments and outcomes of N 3, 090 ICU patients admitted during the first peak of the pandemic. Based on the prior research on COVID-
19 [9],
we extracted D = 17 covariates X (e.g. age and multi-morbidity) and K = 5 causes A (e.g. ventilation and anti-viral treatments). The full list of covariates, causes and the assumed causal structure are shown in Appendix A1 The outcome of interest is the patient's length of stay (LoS) in ICU [5 Achieving shorter LoS is crucial for handling the large influx of patients during the peak of pandemic. We simulate the potential LoS for all treatments based on the sfte-oo-thhear LoS model proposed in (70]. which is generalized linear model with interactions:
(ag
Σ
B +
where x' = (x, 1) is the concatenation of the covariates, causes and vector of ones, Bij is the coefficient sampled from (O, 0.5) and & is white noise NO0,0.1)
Training and evaluation. We use the same benchmarks as in the simulation study. After sorting the data chronologically according to the date of admission, we train and tune the algorithms on the first No patients, and perform evaluation on the rest of the patients. Compared with random splitting, this evaluation strategy preserves the temporality of the data and better mimics the actual training and deployment of the algorithm. For decision support, we would like the CATE estimator to rank higher the treatments that lead to better potential outcomes. Therefore, addition to RMSE, we also report the ranking error, measured by the Spearman's 's Footrule distance between the treatment rankings induced by the true and the estimated POs [3T]. A detailed explanation of the distance is given in Appendix A.II
Results. The experimental results are presented in Table We find that SCP consistently outperforms the benchmarks in both evaluation metrics. Achieving smaller ranking error means that SCP is better at creating a short list of plausible treatment plans for the clinicians to choose from. In practice, narrowing down the large number of treatments into short list might help streamline the clinician's 's decision process and improve efficiency. Moreover, SCP also consistently achieves the best accuracy in terms of RMSE and its performance is relatively stable and improving when No increases.

---

It is worth highlighting that SCP is more data efficient than the benchmarks: it achieves better RMSE with Ng = 500 samples than the benchmarks trained with Ng 1500 samples. Being data efficient is crucial for urgent applications such as pandemic control, where the practitioners would like to perform inference with limited amount of data.
6 Discussion on failure modes and usage guidelines
From our analysis and experiments, we have identified two major failure modes of SCP: (1) the violation of the causal assumptions, and (2) the failure of the step one CATE estimator. However, the causal assumptions as well as the step one CATE estimation error are not empirically verifiable measurable. Hence, as a practical guideline, we would recommend the analyst to start from the single-cause estimation problem (this is essentially the one-factor-at-a-tine principle for scientific discovery [16 10
Suppose the analyst believes that a certain single-cause CATE estimator would perform well in the application. Then, SCP can build upon that estimator to tackle the multi-cause problem (given that the causal assumptions hold). If the analyst believes that no single-cause estimator can perform well in the application, SCP (or other algorithms)
not expected perform well in the (more challenging) multi-cause setting. If the analyst instantiates SCP with poorly performing single-cause estimator (Figure B), the procedure may fail due excessive step one error.
7 Conclusion and future works
SCP is a principled way to leverage existing single cause CATE estimation algorithms in the multi- cause setting. It increases sample size and balancing by augmenting the observational dataset with the estimated potential outcomes. In principle, SCP may be used jointly with other data augmentation procedures in the first step produce even richer training dataset [67]. Although
make the unconfoundedness assumption in this work, it may also be possible to modify SCP to overcome certain types of hidden confounders 772. We will leave these extensions to future works.
Acknowledgments and Disclosure of Funding
We thank anonymous reviewers as well as members of the vanderschaar-lab for many insightful comments and suggestions. This work is supported by the US Office of Naval Research (ONR), the National Science Foundation (NSF Grant number: 1722516), and AstraZeneca.
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14


Reviews:
[{"review_id": "elVl-vgbwD", "reviewer": "Reviewer_yyUp", "summary": "The authors propose a data augmentation method for CATE estimation in the setting of multiple causes. In this setting, there is sparsity in the observed treatment assignments, which makes estimation harder. The authors address this sparsity by aggregating multiple datasets together, generated by estimating single cause potential outcomes, which are theoretically shown to be equivalent in expectation to the multi-cause PO's of interest. The method is agnostic to the methods employed for CATE estimation and covariate adjustment on the adjusted data.", "questions": "", "limitations": "", "rating": 9, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "**Originality**: Novel method for causal estimation in multi-cause settings. Dispenses with certain assumptions made by past works on multi-cause CATE estimation. In theory, multi-cause methods for ATE estimation could be adapted, but this is discussed. To the best of my knowledge, related work is well cited. \n\n**Quality**: Submission was technically sound and overall of a very high quality. I was particularly impressed by how well the method was explored: discussion of assumptions, possible limitations, avenues for future work outlined, investigation into the inner workings of the method, etc. There is empirical support for the method, as well as theoretical support for why estimating single-cause PO's and using them to augment data is a valid thing to do. \n\n**Clarity**: I thought the paper was very well written and organized. \n\n**Significance**: CATE estimation in multi-cause settings is undoubtedly an important problem. I find this work very compelling because it is a data augmentation scheme that can work in tandem with a wide variety of CATE estimation / covariate adjustment methods, which greatly extends its applicability. The method is scalable but also seems to work well on relatively small datasets. High significance overall. ", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Yes", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "2"}, "extra_text": {}}, {"review_id": "_KEsOVxK4uZ", "reviewer": "Reviewer_YNQn", "summary": "In this paper, the authors aim to the estimation of CATE when there are multiple treatment variables and the causal relations between them are known in advance. The main innovation in methodology is an additional step of flipping the value of each variable and updating the corresponding descendants according to the estimated causal mechanisms. This step leads to that more data could be obtained and the data are more balanced, which two steps could benefit the CATE estimation. ", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "After the rebuttal, the authors have addressed my main questions. I am willing to increase my score.\n\n------\nThis paper is well written and organized well. The problem the authors tackle is common in reality. And for the notations or concepts that possibly cause ambiguity, the authors provide detailed illustrations. I enjoyed reading this paper. \n\nMy main concern is about the novelty. Although I do not see it as a drawback, I correspondingly pay more attention to details. For example, as the discussions on Line 177-181, I am curious about the experimental result if we perturb two variables at once. In that way we could have more data than the current method. I think it will be more persuasive if the authors could provide the support from the theoretical perspective or empirical perspective for the balance between the first step error and the sample size. \n\nIn addition, I have one more question. What method did the authors adopt to estimate the first term on the right hand of Eq. 4?\n\nI am willing to increase my score if the authors could provide more details about the balance between the first step error and the sample size, or some further theoretical guarantee for the effectiveness of the proposed method. I also possibly change my score according to other reviewers' comments.\n\nLine 136: It should be ``a, \\tilde{a}^1, \\ldots, \\tilde{a}^K''. If $k$ denotes an index from [K], there should be an ellipsis ahead of \\tilde{a}^k.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Yes.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "3"}, "extra_text": {}}, {"review_id": "US0SM01iXhI", "reviewer": "Reviewer_1Xus", "summary": "This paper presents a novel method for estimating conditional average treatment effect (CATE) of multiple treatments (or a multi-cause treatment). The research is motivated by data scarcity in multi-cause CATE estimation, and  a single-cause perturbation (SCP) procedure is proposed to augment the observational dataset to improve the estimation.  Detailed theoretical analysis has been provided in the paper (main text and appendices) to support the development of SCP. A comprehensive evaluation of SCP with synthetic data has been conducted, and an application of SCP to a real-world COVID-19 dataset (with simulated outcomes) is presented too. The experiment results have demonstrated the effectiveness and superiority of SCP in comparison with the benchmark methods.  ", "questions": "", "limitations": "", "rating": 8, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Please find below the detailed comments on the originality, technical quality, clarity and significance of the paper:\n1. Originality:\n\nIt is a very interesting and effective idea to make use of single cause CATE estimation for multi-cause CATE estimation. The authors have done a great job in establishing the theoretical link between single cause CATE estimation and multi-cause CATE estimation, and then utilizing this link and the idea of perturbing individual causes to create augmented and more balanced datasets for better multi-cause CATE estimation. The proposed approach has also provided the opportunity to use off-the-shelf single cause CATE estimation algorithms for the multi-cause problem.\n\nFrom the paper, the originality of the work is clear overall, but to strength the originality further, it would be helpful to clarify the following points:\n\n(a) For single-cause CATE estimation and ATE estimation, is there any existing research which employs data augmentation? If yes, how is the existing research related to SCP proposed in this paper?\n\n(b) Are there any existing research which links single cause CATE or ATE estimation  and multi-cause CATE or ATE estimation? If yes, how is the existing research related to the idea proposed in this paper?\n\n(c) Although in the Appendix, the authors have clarified that the setting of their work is different from mediation analysis, I have a feeling that the analysis in Section 2.2 is related to mediation analysis (or direct vs indirect causal effect), although the overall approach and aim of SCP is different from mediation analysis. It would be good if authors could elaborate on this a bit more.\n\n(d) Related t multi-cause ATE estimation, the joint-IDA method should be mentioned:\nNandy, P., Maathuis, M., & Richardson, T. (2017). ESTIMATING THE EFFECT OF JOINT INTERVENTIONS FROM OBSERVATIONAL DATA IN SPARSE HIGH-DIMENSIONAL SETTINGS. The Annals of Statistics, 45(2), 647-674. \n\n2. (Technical) quality\n\nThe proposed method is sound, and detailed discussions (and proofs) have been presented to support the correctness of the SCP algorithm. The experiments are comprehensive and the results are promising. I appreciate the authors' thoroughness in developing, evaluating and presenting the proposed method. \n\nBelow are some points that may need clarification to improve the paper:\n(a) In terms of partitioning the causes, what about if A_k and some of the other causes are siblings? That is, unlike what's illustrated in Figure 2(B), some other causes are neither A_k's parents/ancestors or children/descendants? I don't think this would change the results the authors have presented, but it would be good to include this case more explicitly.\n\n(b) Lines 120- 121 on page 4: Are the two models trained definitely the predictors for the \"potential\" treatments or outcome? That is, there is no formal analysis showing that the two factors on the RHS of equation (4) are the potential treatments (A^{\\downarrow}_{_k}(a_k)) and the potential outcome (Y(a_k)) respectively. I think you can still get the potential treatments or outcome from data based on (4), but the statement here needs clarification/correction or analysis to support.\n\n(c) Section 3.2: It sounds ok to pool the augmented datasets, but what about if the estimations of the expected outcome (i.e. the RHS of (5) estimated from data) for two different k values are not consistent? Have you tried estimating the RHS for different k values and check their consistency?\n\n(d) Page 7, about the benchmarks: It would be good if more details could be added on the benchmarks, in particular, how those single cause CATE methods are used as benchmarks (i.e. how are they applied for multi-cause CATE estimation)?\n\n3. Clarity/presentation\n\nAlthough as mentioned above, some technical details might need some clarification/improvement, this paper is well written and very easy to follow. It is enjoyable to read it.\n\nOne point though (on Appendix A.1): the sequential ignorability assumption does not seem to appear in reference [50]? Also, it would be good to provide some intuition on this assumption. \n\n4. Significance\n\nMulti-cause CATE estimation is an important and challenging task, and data scarcity is a practical issue for the task. The proposed SCP method could a very useful tool for tackling the challenge. Although strong assumptions are required by SCP, comparing to existing methods (which make similar assumptions), from the experiments, SCP and causal data augmentation look like a promising direction to follow. \n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "The authors have addressed some specific limitations of their work (e.g. in Section 3.4). It would be ideal if the authors could elaborate a bit more on the restrictions created by the assumptions, e.g. unconfoundedness and sequential ignorability. ", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "6"}, "extra_text": {}}, {"review_id": "EYnw7nsV4k2", "reviewer": "Reviewer_f9co", "summary": "This paper proposes a data augmentation approach for conditional average treatment effect (CATE) estimation with multiple treatments. Under the assumption that the multiple treatments follow a DAG, sequential ignorability, and sequential overlap, the conditional average of each (multi-cause) potential outcome can be identified through the conditional average of the corresponding single-cause potential outcome. The proposed method starts by fitting models for (1) how each treatment affects its causal descendants, and (2) how covariates, each treatment and its causal descendants affect the outcome. Then it generates $K$ pseudo-samples, where $K$ is the number of treatments, from the estimated data generating process, each flipping one of the treatment. Finally, it fits a regression model on the augmented data. The proposed method outperforms a handful of competing methods on extensive numerical experiments. ", "questions": "", "limitations": "", "rating": 7, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "The single-cause perturbation-based data augmentation is novel and interesting for causal inference with complex and high-dimensional treatments. Generally speaking, I feel that data augmentation should be used more in causal inference, because it helps regularizing the estimation under data scarcity. On the other hand, the proposed single-cause perturbation appears to be an effective augmentation scheme because it exploits the DAG structure. \n\nFurthermore, the paper has a very impressive experimental assessment --- it not only shows that the proposed method has a superior performance (i.e., RMSE), but also presents various robustness checks to investigate why it works. For example, Figure 4 and Figure 5 are very informative. They exactly answer some of my questions when I read the previous sections. ", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "(1) The paper claims that \"instead of making additional assumptions on the DGP, we exploit the connection between a single-cause intervention and a multi-cause intervention\". This is an overstatement. The proposed method does impose more restrictions on the DGP that is not required in other works (e.g., deconfounder). It requires a DAG structure among the treatment, which is a big assumption. In many applications involving multiple treatments, such as those in two-sided markets, it is unclear whether the acyclic assumption holds among the treatments. Even if it holds, it is not always easy to draw a reliable DAG. \n\n(2) Line 120-123: what does \"the model for $A_{-k}^{\\downarrow}(a_k)$\" and \"the model for $Y(a_k)$\" mean? In order to generate the pseudo-samples, it seems that they should be the whole distribution instead of just the conditional mean. The algorithm 1 in Appendix does not seem to clarify this. \n\n(3) Line 130-132: similarly, the word \"obtain\" is confusing. Do you mean sampling from the estimated distribution or plugging in the estimated conditional mean or something else?\n\n(4) Does the proposed method have any connection with the marginal structural models (MSM)? \n\n(5) Line 235: should the metric be $\\sqrt{\\frac{1}{N_t}\\sum_{i=1}^{N_t}\\sum_{a\\in \\Omega}(y_i(a) - \\hat{y}_i(a))^2}$? Why do you use $y(a)$ instead of $E[y(a)|x]$?", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "5"}, "extra_text": {}}, {"review_id": "5ZciNgeZQgF", "reviewer": "Reviewer_sokR", "summary": "The paper presents a data augmentation method for learning the simultaneous effects of many causes in settings where treatments manipulate many causes at once. Specifically, the authors consider a case where the treatment manipulates a K-dimensional binary vector of variables that cause a single outcome Y. The problem being addressed is sparsity in observed treatments: because there are $2^K$ possible treatments, as $K$ grows, it is less and less realistic that we will observe all levels of the treatment.\n\nThe proposed SCP method suggests an approach where the potential outcome distributions for single-cause interventions are learned first, then these are used to generate synthetic data with single-cause perturbations to augment the dataset. The augmented dataset is then used to estimate the potential outcome distributions for all treatments. The authors make several heuristic arguments for why this might work, referencing sample size and covariate balance. They then demonstrate on fully simulated data, and semi-synthetic data using covaraites and treatment indicators for COVID-19 treatment with simulated length-of-stay outcomes.", "questions": "", "limitations": "", "rating": 7, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "**Per discussion below, authors' addition of failure cases and general guidance, as well as agreement to switch some framing has moved my recommendation to \"Accept\"**\n\n# General Feedback\n\nThe paper is written clearly, and the problem is well-introduced. However, I remain skeptical of the approach because it is not clear where the information about sparsely-observed treatment combinations is coming from. Because there is no theoretical treatment in the paper or even demonstrations on simple, transparent examples, it is difficult for me to endorse the approach as presented. However, I do think there are some interesting possibilities here that the authors should do more to explore. In my opinion, the current \"reasons\" for SCP working that are considered are not sufficient to justify the approach, and there are more interesting potential justifications that would be more compelling.\n\nThe problem of combinatorially exploding cause combinations has a long history in the statistical experimental design literature, going back to Fisher (see, e.g., https://digital.library.adelaide.edu.au/dspace/bitstream/2440/15191/1/48.pdf). The main issue with so-called factorial experiments is figuring out how to estimate the effects of interactions between causes. For this reason, analysts will often make assumptions about certain orders of interaction being zero (e.g., assuming that there are no 3-way or higher interactions between causes). There are then experimental designs (so-called fractional factorial designs) that can identify the remaining low-order interaction effects (see, e.g., Box, Hunter, and Hunter Chapter 6, hosted here: http://pages.stat.wisc.edu/~yxu/Teaching/16%20spring%20Stat602/%5BGeorge_E._P._Box,_J._Stuart_Hunter,_William_G._Hu(BookZZ.org).pdf).\nAn extreme case of this is assuming that on a certain scale there are no interactions between causes (e.g., the expectation of the treatment is just a linear combination of the causes). In this case, one could estimate the potential outcome distribution for all 2^K treatments by simply observing K treatment levels, so long as those treatment levels are linearly independent of each other.\n\nThis paper seems to aim at addressing a similar issue (at least per the introduction), but now in a more challenging observational context. However, it is not clear how this method addresses the fundamental question of having an exploding number of interaction effects to estimate. The empirical exploration is no help here. Note, for example, that the synthetic experiment only includes first-order effects of causes (there are no interactions between causes, only between causes and covariates), and the simulated outcomes in the \"real data\" experiment only include, at maximum, second-order interactions between causes. If we knew at the outset that there were only second-order interactions between causes, then we could estimate the effects of all treatment levels with many fewer observed treatment levels (see https://link.springer.com/chapter/10.1007%2Fb105081_8). Thus, SCP could succeed on these experiments by simply implicitly regularizing the estimated treatment effects toward low-order interactions between causes. This is not necessarily a bad thing, but is very different from the current framing of the paper that repeatedly emphasizes that SCP makes no assumptions about the DGP.\n\nThere are some ways that I could imagine SCP could help. For example, in cases where there is non-trivial causal structure between the causes, SCP might enforce certain marginalization constraints between causal effects, i.e., that the causal effects of downstream variables should average out in a particular way to match the causal effect of intervening on an upstream variable. These kind of moment constraints could be helpful regularizers for CATE estimators.\n\nThere is also a possibility that SCP does nothing that is actually causally interesting, but applies useful regularization to the neural networks used to estimate P(Y | X, A) by self-distillation (see, e.g., https://arxiv.org/abs/2002.05715).\n\nTo make their analysis of SCP clearer, I would suggest that the authors examine the method in the following contexts:\n* Consider an ideal experimental setting where there is no confounding, and we are just interested in the average treatment effect at each level $\\bm a$. This separates out the treatment sparsity problem (the main motivation of this work) from the confounder adjustment problem.\n* In the experimental context, show how SCP would work with a simple model that is not a neural network (e.g., a linear model with higher-order interactions with K relatively small). A transparent toy example worked out analytically would be extremely useful, even if it does not prove general properties of the approach. If SCP does very little here, it would suggest that there is something special about the self-distillation.\n * Consider a range of settings with different orders of important interactions. I suspect that SCP will perform worse when higher-order interactions play a major role.\n * Consider a range of cases where the causal structure between the causes is changed systematically. It may be the case that SCP performs better when there is detailed causal structure, as this could correspond to a system of moment constraints that regularize the CATE estimator.\n\nImportantly, when suggesting a causal procedure that relies on unverifiable assumptions, or for which there is no general theoretical justification, it is important to characterize cases where the method breaks, as well as those cases where it succeeds. I suspect that the above analyses would reveal such cases, and this is fine. There is no free lunch when it comes to sparsely observed data, so the method must have limitations. The authors should articulate them clearly.\n\nI consider these to be more essential explantations for SCP's performance than the \"balance\" and \"sample size\" considerations listed in the paper, because these would also be reasons that SCP would help with a single binary treatment; they are not unique to the multi-cause context. (As an aside, it would be interesting to see whether SCP also seems to help with estimation in the standard setting. If it does help, then that would again suggest an alternative explanation for why SCP might help with many treatments as well.)\n\n# Detailed Comments\n\n## Notes on Related Work\n\nIn Table 2, G-computation and covariate adjustment should have their intermediate estimands listed as P(Y | A, X) since they both both estimate outcome models as nuisance functions. G-computation is also usually done by simply taking the empirical measure of X to be P(X) and P(Y | A, X) is just averaged over the observed values of X. Deconfounder methods generally do not rely on weighting, but instead involve an outcome model P(Y | A, Z).\n\n## Careful with the \"Real Data\" Example\nSection 5.2 uses a dataset with simulated outcomes, so should not be called a real data experiment, especially for such a sensitive topic (COVID-19 Treatment). This should be called a \"semi-synthetic\" experiment.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "As stated above, given that this is a method without any theoretical characterization, it is important to highlight cases where the method would fail. As of now, there are only two simulated settings where the method works well; I would encourage the authors to do more stress testing. The ablation study is a useful first step.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "3 hours"}, "extra_text": {"ethical_concerns": "As noted above, the \"real data\" example is designed around data from COVID-19 treatment, which demands some care in how those results are presented. As suggested above, this section should be re-titled \"semi-synthetic example\" to emphasize that the outcomes were simulated."}}]

Author rebuttal:

Meta-review: The authors propose a data augmentation method for learning conditional average treatment effects with multiple treatments. Assuming sequential ignorability and sequential overlap, the authors propose to first fit models that describe how changing a single cause affects causal descendants; then to augment the data set with new observations; and finally to fit a regression model on the augmented data. This is a strong paper. All reviewers agree that it should be accepted. Reviewer yyUp praises how well the method was explored, including discussion of limitations. Reviewer f9co states that the paper has "a very impressive experimental assessment" that not only shows superiority to competing methods but also investigates why the method works. Reviewer 1Xus states that the paper is “enjoyable to read”. Reviewers 1Xus, sokR and YNQn had some reservations about the method, in particular how well it performs under higher-order interactions. The authors have provided thorough responses, with additional simulations and high-level guidelines to understand failure modes. The authors state that they will expand the discussion with the additional insights.
Decision: Accept (Poster)