Estimating individual treatment effects under unobserved confounding using binary instruments
Authors: Dennis Frauen
Stefan Feuerriegel
Venue: ICLR 2023 poster (2023)
Keywords: Causal machine learning
treatment effect estimation
instrumental variables

Abstract: Estimating conditional average treatment effects (CATEs) from observational data is relevant in many fields such as personalized medicine. However, in practice, the treatment assignment is usually confounded by unobserved variables and thus introduces bias. A remedy to remove the bias is the use of instrumental variables (IVs). Such settings are widespread in medicine (e.g., trials where the treatment assignment is used as binary IV). In this paper, we propose a novel, multiply robust machine learning framework, called MRIV, for estimating CATEs using binary IVs and thus yield an unbiased CATE estimator. Different from previous work for binary IVs, our framework estimates the CATE directly via a pseudo-outcome regression. (1)~We provide a theoretical analysis where we show that our framework yields multiple robust convergence rates: our CATE estimator achieves fast convergence even if several nuisance estimators converge slowly. (2)~We further show that our framework asymptotically outperforms state-of-the-art plug-in IV methods for CATE estimation, in the sense that it achieves a faster rate of convergence if the CATE is smoother than the individual outcome surfaces. (3)~We build upon our theoretical results and propose a tailored deep neural network architecture called MRIV-Net for CATE estimation using binary IVs. Across various computational experiments, we demonstrate empirically that our MRIV-Net achieves state-of-the-art performance. To the best of our knowledge, our MRIV is the first multiply robust machine learning framework tailored to estimating CATEs in the binary IV setting. 

Published as a conference paper at ICLR 2023
ESTIMATING INDIVIDUAL TREATMENT EFFECTS UNDER UNOBSERVED CONFOUNDING USING BINARY INSTRU- MENTS
Dennis F Stefan Feuerriegel Munich Center for Machine Learning
2 LMU Munich
ABSTRACT
Estimating conditional average treatment effects (CATEs) from observational data is relevant in many fields such as personalized medicine. However, in practice, the treatment assignment is usually confounded by unobserved variables and thus introduces bias. A remedy to remove the bias is the use of instrumental variables (IVs). Such settings are widespread in medicine (e.g., trials where the treatment assignment is used as binary IV). In this paper, propose novel, multiply robust machine learning framework, called MRIV, for estimating CATEs using binary IVs and thus yield unbiased CATE estimator. Different from previous work for binary IVs, our framework estimates the CATE directly via a pseudo-outcone regression. (1) We provide theoretical analysis where we show that our framework yields multiple robust convergence rates: our CATE estimator achieves fast convergence even if several nuisance estimators converge slowly. (2) We further show that our framework asymptotically outperforms state-of-the-ar plug-in IV methods for CATE estimation, in the sense that achieves faster rate of convergence if the CATE is smoother than the individual outcome surfaces.
(3) We build upon our theoretical results and propose tailored deep neural network architecture called MRIV-Net for CATE estimation using binary IVs. Across various computational experiments, we demonstrate empirically that our MRIV-Net achieves state-of-the- art performance. To the best of our knowledge, our MRIV is the first multiply robust machine learning framework tailored to estimating CATES in the binary IV setting.
INTRODUCTION
Conditional average treatment effects (CATEs) are relevant across many disciplines such marketing (Varian 2016) and personalized medicine (Yazdani & Boerwinkle, 2015). Knowledge about CATES provides insights into the heterogeneity of treatment effects, and thus helps in making potentially better treatment decisions (Frauen et al. 2023).
Many recent works that use machine learning to estimate causal effects, particular CATEs, are based on the assumption of unconfoundedness (Alaa & van der Schaar [017 Lim et al. 2018 Melnychuk al. 2022a b). In practice, however, this assumption is often violated because is common that some confounders are not reported in the data. Typical examples are income or the socioeconomic status of patients, which are not stored in medical files. the confounding is sufficiently strong, standard methods for estimating CATES suffer from confounding bias (Pearl 2009), which may lead to inferior treatment decisions.
To handle unobserved confounders, instrumental variables (IVs) can be leveraged to relax the assumption of unconfoundedness and still compute reliable CATE estimates. IV methods were originally developed in economics (Wright 1928), but, only recently, there is growing interest in combining IV methods with machine learning (see Sec. 3). Importantly, IV methods outperform classical CATE estimators if a sufficient amount of confounding is not observed (Hartford et al.) 2017). We thus aim at estimating CATEs from observational data under unobserved confounding using IVs.

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In this paper, consider the setting where single binary instrument is available. This setting is widespread in personalized medicine (and other applications such as marketing or public policy) (Bloom et al. [197). In fact, the setting is encountered in essentially all observational or randomized studies with observed non-compliance (Imbens & Angrist, 1994). As an example, consider a randomized controlled trial (RCT), where treatments randomly assigned patients and their outcomes are observed. Due to some potentially unobserved confounders (c.g., income, education), some patients refuse take the treatment initially assigned to them. Here, the treatment assignment e.g., to
serves as a binary IV. Moreover, such RCTs have been widely used by public decision- analyze the effect of health insurance on health outcome (see the so-called Oregon health insurance experiment) (Finkelstein et al. 2012) or the effect of military service on lifetime earnings (Angrist 1990).
We propose a novel machine learning framework (called MRIV) for estimating CATES using binary Ivs. Our framework takes an initial CATE estimator and nuisance parameter estimators as input to perform pscudo-outcome regression. Different to existing literature, our framework is multiply robus i.e., we show that it consistent in the union of three different model specifications. This is different from existing methods for CATE estimation using IVs such Okui et al. 2012). Syrgkanis
jet al.. (019), or plug-in estimators (Bargagli-Stofi et al. 2021 Imbens & Angrist 1994).
We provide theoretical analysis. where we tools from Kennedy (2022) to show that our framework achieves a multiply robust convergence rate, i.e., our MRIV converges with a fast rate even if several nuisance parameters converge slowly. We further show that, compared to existing plug-in IV methods, the performance of our framework is asymptotically superior. Finally, we leverage our framework and, on top of it, build a tailored deep neural network called MRIV-Net.
Contributions: (1) We propose novel, multiply robust machine learning framework (called MRIV) to learn the CATE using the binary IV setting. To the best of our knowledge. ours is the first that is shown to be multiply robust, i.e.. consistent in the union of three model specifications. For comparison, existing works for CATE estimation only show double robustness (Wang & Tchetgen Tchetgen| 2018 Syrgkanis et al. 2019). (2) We prove that MRIV achieves a multiply robust convergence rate. This is different to methods for IV settings which do not provide robust convergence rates (Syrgkanis al.) 2019). We further show that our MRIV is asymptotically superior to existing plug-in estimators. (3) We propose a tailored deep neural network, called MRIV-Net, which builds upon our framework to estimate CATEs We demonstrate that MRIV-Net achieves state-of of-the-art performance.
2 PROBLEM SETUP
Data generating process: We observe data D = (Ii, 21. 0j, W. consisting of n N observations of the tuple (X, Z, A,Y.Y. Here, X E X are observed confounders, Z E {0, 1} is a binary instrument, assume the
A E {0, }} is binary treatment, and Y R is an outcome of interest. Furthermore, existence of unobserved confounders U E 44 which affect both the treatment and the outcome Y. The causal graph is shown in Fig.
Our proposed framework is widely applicable in practice, namely all settings with the above data generating process. This includes both (1) observational data and (2) RCTs with non-compliance. For (1), observational data is commonly encountered in, e-g. personalized medicine. Here, modeling treatments as binary variables is consistent with previous literature on causal effect estimation and standard in medical practice (Robins et al. 2000). For (2), our setting is further encountered in RCTs when the instrument Z is a randomized treatment assignment but individuals do not comply with their treatment assignment. Such RCTs have been extensively used by public decision- makers,, e.g., to analyze the effect of health insurance on health outcome (Finkelstein al. 2012) or the effect of military service on lifetime earnings (Angrist 1990).
Applicability:
Underlying causal graph. The instru- ment Z has direct influ- ence on the treatment A, but does not have a direct effect on the outcome Y. Note that we allow for un- observed confounders for both Z-A (dashed line) and A-Y (given by U). For a detailed introduction to multiple robustness and its importance in treatment effect estimation, we refer (Wang& & Tchetgen Tchetgen 2018). Section 4.5.
Figure 1:
We build upon the potential outcomes framework (Rubin) 1974) for modeling causal effects. Let (a, denote the potential outcome that
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would have been observed under A = a and Z Following previous literature on IV estimation (Wang & Tchetgen Tchetgen 2018), we impose the following standard IV assumptions on the data generating process.
Assumption (Standard IV assumptions (Wang & Tchetgen Tchetgen 2018: Wooldridge 2013)
We assume: (1) Exclusion: (a, = Y()) for all a, 2,z {0, 1]. i.e., the instrument has direct effect on the patient outcome; (2) Independence: Z 1l U | X; (3) Relevance: Z M. A X, (iv) The
model includes all A-Y confounder: (() II (A,Z) (X, U) for all a E (0,
Assumption 1 is standard for IV methods and fulfilled in practical settings where IV methods are applied (Angrist 1990 Angrist & Krueger 1991 Imbens & Angrist 1994). Note that Assumption [ does not prohibit the existence of unobserved Z-A confounders, On the contrary, it merely prohibits the existence of unobserved counfounders that affect all Z, A. and Y simultancously, as it is standard in IV settings (Wooldndge 2013). A practical and widespread example where Assumption I is satisfied are randomized controlled trials (RCTs) with non-compliance (lmbens & Angrist, 1994. Here, the treatment assignment Z is randomized, but the actual relationship between treatment A and outcome Y may still be confounded. For instance, in the Oregon health insurance experiment (Finkelstein et al. 2012), people were given access to health insurance (Z) by a lottery with aim to study the effect of health insurance (AA) on health outcome (Y) (Finkelstein et al., 2012). Here, the lottery winners needed to sign up for health insurance and thus both Z and A are observed. Objective: In this paper, we are interested in estimating the conditional average treatment effect (CATE)
Txx) = (1) (()) | = x].
(1)
If there is no unobserved confounding (U = 0). the CATE is identifiable from observational data (Shalit et al. 2017). However, in practice, it is often unlikely that all confounders are observable. To account for this, we leverage the instrument Z to identify the CATE. We state the following assumption for identifiability.
Assumption 2 (Identifiability of the CATE Wang & Tchetgen Tchetgen 2018)). At least one of the following statements holds true: (1) E Z U, E 2 0, X,U = E[A
X| - |[A|| Z = 0, X]: or (2) B (1) (0) X,U = E - - [X]
Example: Assumption 1 holds when the function f(a, X)U) (a) | X, X, is additive with respect to a and U, e.g., (a, U) = g(a, X) + hU) for measurable functions h and g. This implies that no unobserved confounder affects the outcome through a path which is also affected by the treatment. For example, with patient income as unobserved confounder, the treatment should not affect the (future) patient income.
Under Assumptions and the CATE is identifiable (Wang & Tchetgen Tchetgen 2018 It can be written
(2x (x)x) 1/ (x) (x)x)
(2)
(z((x) (z) (z(2)) (z) 5Λ(T)*
where 2u (z)= = E|y = = x] and (z) = 1 11
=,x=ix==0 2]. Even if Assumption
does not hold, all our results in this paper still hold for the quantity the right-hand side of Eq. (2)
In certain cases, this quantity still allows for interpretation: If no unobserved Z-A confounders
conditional version of the local average treatment effect (LATE) [Imbens & Angrist 1994) under a monotonicity assumption. Furthermore, under a no-current- assumption, it can be interpreted conditional trealment effect on the treated (ETT) /Wang& & Tchetgen Tchetgen 2018 This has an important implication for our results: If Assumption 2 does not hold in practice, our estimates still provide conditional LATE or ETT estimates under the respective assumptions because they are based on Eq. (2). If Assumption does hold, all three i.e., CATE, conditional LATE, and ETT coincide (Wang & Tchetgen Tehetgen 2018
exist, it can be interpreted
peaament-valuueinteeaa
3 RELATED WORK
Machine learning methods for IV: Only recently, machine learning has been integrated into IV methods. These are: (Singh et al. (2019) and Xu al. (2021a) generalize 2SLS by learning complex
The conditional LATE measures the CATE for individuals which are part of the complier subpopulation, i.c., for whom A(Z 1) A(Z 0). The conditional ETT measures the CATE for treated individuals.
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feature maps using kernel methods and deep learning, respectively. Hartford et al. (2017) adopts two-stage neural network architecture that performs the first stage via conditional density estimation. Bennett al. (2019) leverages moment conditions for IV estimation. However, the aforementioned methods are not specifically designed for the binary IV setting but, rather, for multiple IVs or treatment scenarios. In particular, they impose stronger assumptions such as additive confounding in order to identify the CATE. Note that additive confounding is a special case of our Assumption 2 Moreover, they do not have robustness properties. In the binary IV setting, current methods proceed by estimating M. (x) and (xr separately, before plugging them in Eq. 2 (Imbens & Angrist 1994 Angrist et al., 1996 Bargagli-Stoth al. 202I). As a result, these sufter from plug-in bias and do
not offer robustness properties.
Table 1: Key methods for causal effect estimation with IVs and associated robustness properties
Doubly robust IV methods: Recently, doubly robust methods have been proposed for IV set-
tings: Jesnedy et al.] (2019) propose a pseudo
Robustiess
ITF
Doubly
Synakanis
2019
regression estimator for the local average treat- ment effect using continuous instruments, which has been extended to conditional effects by Semen- ova & Chernozhukov (202T). Furthermore, Singh & Sun (2019) use a doubly robust approach to esti- mate average compiler parameters. Finally, Ogburn et al. (2015) and (Syrgkaniss et al.) (2019) propose doubly robust CATE estimators in the IV setting which both rely on doubly robust parametrizations of the uncentered efficient influence function (Okui et al. 2012). However, none of these estimators has been shown to be multiply robust in the sense that they are consistent in the union of more than two model specifications (Wang& & Tchetgen Tchetgen 2018).
Muttiply robust
Wing
2014
Multiply robust IV methods: Multiply robust estimators for IV settings have been proposed only for average treatment effects (ATES) (Wang & Tchetgen Tchetgen 2018) and optimal treatment regimes (Cui & Tehetgen 2021) but not for CATES In particular, Wang & Tchetgen Tchetgen (2018) derive a multiply robust parametrization of the efficient influence function for the ATE. However, there exists no method that leverages this result for CATE estimation. We provide a detailed, technical
comparison of existing methods and our framework in Appendix G
Doubly robust rates for CATE estimation: Kennedy (2022) analyzed the doubly robust learner in the standard (non-IV) setting and derived doubly robust convergence rates. However, Kennedy's s result not applicable in the IV setting, because we use the multiply robust parametrization of the efficient influence function from Wang & Tchetgen Tchetgen 2018). In our paper, rely certain results from Kennedy, but use these to derive of multiply robust rate. In particular, this required the derivation of the bias term for larger number of nuisance parameters (see Appendix B.
Research gap: To the best of our knowledge, there exists no method for CATE estimation under unobserved confounding that has been shown to be multiply robust. To fill this gap, we propose MRIV: a multiply robust machine learning framework tailored to the binary IV setting. For this, we build upon the approach by Kennedy (2022) derive robust convergence rates, yet this approach has not been adapted to IV settings, which is our contribution.
MRIV FOR ESTIMATING CATES USING BINARY INSTRUMENTS
In the following, we present our MRIV framework for estimating CATEs under unobserved con- founding (Sec. 4. We then derive asymptotic convergence rate for MRIV (Sec. (42) and finally use our framework to develop a tailored deep neural network called MRIV-Net (Sec. 4.4).
4.1 FRAMEWORK
Motivation: A naïve approach to estimate the CATE is to leverage the identification result in Eq. (2). Assuming that we have estimated the nuisance components jo and Áı for i E {0. 1). we can simply plug them into Eq. (2) to obtain the so-called (plug-in) Wald estimator (x Wald 1940).
However, in practice, the true CATE curve r(x) is often simpler (e.g., smoother, more sparse) than its complements u. (z) or μe (x) (Künzel et al. 2019). In this case, Áw(x) is inefficient because it models all components separately, and, to address this, our proposed framework estimates 7 directly using a pseudo-outcome regression.
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We now propose MRIV. MRIV is two-stage meta learner that takes any base method for CATE estimation as input. For instance, the base method could be the Wald estimator from Eq. (2). any other IV method such as 2SLS, or a deep neural network (as we propose in our MRIV-Net later
Overview:
in Sec. 4.4). In Stage I, MRIV produces nuisance estimators P. (z), Ag (a), 8x()), and R (x), where i (a) is an estimator of the propensity score π(x) = P(Z = 1 | X = x). In Stage 2, MRIV estimates (ar) directly using pseudo outcome YMR as a regression target.
Given arbitrary initial CATE estimator Pinit (x) and nuisance estimates Po (2), P (2), 8Λ(Z), and f (z). we define the pseudo outcome
(2-11-2) (2(x)) ("
Y- (XA (8004012)00
PMRR=
(x)))(
The pseudo outcome YAR in Eq. (3) is multi- ply robust parameterization of the (uncentered) efficient influence function for the average treat- ment effect Ex [T()] (see the derivation in (Wang & Tchetgen Tchetgen (2018). Once we have obtained the pseudo outcome Y MMR, we regress it on X to obtain the Stage 2 MRIV esti- mator AMRIV (x) for (z)). The pseudocode for MRIV is given in Algorithm [ MRIV can be interpreted as a way to remove plug-in bias from Aiiit (z) (Curth et al. 2020). Using the fact that YMR multiply robust parametrization of the efficient influence function, we derive multiple robustness property of (z))v(Z))
Algorithm 1: MRIV
Input: data (X, Z, A, Y), initial CATE estimator Fua
// Stage 1: Estimate nuisance componenta
*(x) + E[2 X = s/.
(](x)) (x) - B|y ==
A. (z) + B|A
X ==
(a + B[a == = =
Stage 2:
regression
YAR +
(2-1-2)
(x(X)
0000 ((x)
(X)
(x) + B|SMR | =x
Theorem 1 (multiple robustness property). Let po (z). (z).. 8L(2), f(x), and (x) denote estimators of no (z), pá (z)) (z(z). T()), and T(x), respectively. Then, for all x X, it holds that E[ X 2] if)). least one of the following conditions is satisfied: (1) p.ù = pp Pg = pg.
and Aiii = r; or (2) = π and A = SA or (3) ने = x and finit =.
The equalities in Theorem ! are meant to hold almost surely. Consistency of fMRIV (2) is direct consequence: If either the nuisance estimators in (1), (2), or (3) converge to their oracle estimands, fxrav will converge to the true CATE As result, our MRIV framework is multiply robust in the sense that our estimator, ÊMRIV(2)) is consistent in the union of three different model specifications. Importantly, this is different from doubly robust estimators which are only consistent in the union of two model specifications (Wang& & Tchetgen Tchetgen 2018). Our MRIV is directly applicable to RCTs with non-compliance: Here, the treatment assignment randomized and the propensity score (x) is known. Our MRIV framework can be thus adopted by plugging in the known π (2r) into the pseudo outcome in Eq. Moreover, fMRIV (I) is already consistent if either (x) (x) are.
4.2 THEORETICAL ANALYSIS
We derive the asymptotic bound on the convergence rate of MRIV under smoothness assumptions. For this, we define s-smooth functions as functions contained in the Hölder class Hss), associated with Stone's minimax rate (Stone 1980) of n (2)(pt) where is the dimension of X'
Assumption 3 (Smoothness). We assume that (1) the nuisance component ná (-) is a-smooth, H&(() is 3-smooth, 7T-) is y-smooth, and (-(-) is 5-smooth; (2) all nuisance components are estimated with their respective minimax rate of , where k E {a, 3,,,,, o); and (3) the oracle CATE T(-) is sy-smooth and the initial CATE estimator Ait converges with rate r, (n:. We provide a rigorous definition in Appendix D
Assumption 3 for smoothness provides us with a way to quantify the difficulty of the underlying nonparametric regression problems. Similar assumptions have been imposed for asymptotic analysis of previous CATE estimators in (Kennedy 2022 Curth & van der Schaar (021). They can be replaced with other assumptions such assumptions the level of sparsity of the CATE components. We also provide an asymptotic analysis under sparsity assumptions (see Appendix C).
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Assumption 4 (Boundedness). We assumee that there exist constants p. p. K > 0 such that for all X' it holds that: (1) p (x)) S C; (2) (6x(z)) (2((2) n6 (x)) 2 p and [zx(2) 2 p
(3) S (x) x - c; and (4) Pimit (x) & K.
Assumptions 4 1. 3. and 4 are standard and line with previous works on theoretical analyses of CATE estimators (Curth & van der Schaar 2021; Kennedy (022). Assumption 4 ensures that both the oracle CATE and the estimator are bounded. Violations of Assumption 4] may occur when working with "weak" instruments, which are IVs that are only weakly correlated with the treatment. Using IV methods with weak instruments should generally be avoided (Li al., 2022). However, in many applications such RCTs with non-compliance, weak instruments are unlikely occur as patients' compliance decisions are generally correlated with the initial treatment assignments.
We state now our main theoretical result: an upper bound on the oracle risk of the MRIV estimator. To derive our bound, we leverage the sample splitting approach from (Kennedy (022). The approach
in (Kennedy) 2022) has been initially used to analyze the DR-learner for CATE estimation under unconfoundedness and allows for the derivation of robust convergence rates. It has later been adapted to several other meta learners (Curth van der Schaar, (02T), yet not for IV methods.
Theorem 2 (Oracle upper bound under smoothness). Let Dr,for E (1, be independent samples of size n. Let finst (x), P (x), and i (x) be trained on DDi, and let (x(z) and (xx) be trained on D2. We denote YMR as the pseudo outcome from Eq. (3) and fMRIV (x) = E, MMR X = x]| as the pseudo-outcome regression on D3 for some generic estimator E. X x] of El
X x])
We assume that the second-stage estimator En yields the minimax rate n and satisfies the stability assumption from Kennedy 20022, Proposition (see Appendix B). Then, under Assumptions 1-4 the oracle risk is upper bounded by
2n
E T(x)))
-
S
2n n 29+ +
(r+
-26 (n 27+7 + n 28
+n 2( 20 + 2o +n 2(
25
Proof. See Appendix The proof provides a more general bound which depends on the pointwise mean squared errors of the nuisance parameters (Lemma 2)
Recall that the first summand of the lower bound in Eq. 2 is the minimax rate for the oracle CATE 7(x:) which cannot be improved upon. Hence, for fast convergence rate of AMRRY (z), it is sufficient if either: (1) r- (n) decreases fast and a, .8 are large; (2) y and 8 are large; or (3) r- (n) decreases fast and y large. This is in line with the multiply robustness property of MRIV (Theorem I and means that MRIV achieves a fast rate even if the initial or several nuisance estimators converge slowly.
Improvement over Amit (x): From the bound in Eq. (2), it follows that ÊM improves on the convergence rate of the initial CATE estimator âmi (x) if its rate r, (n) is lower bounded by
T- (n) 2 2y1P
21.) + (अी
Hence, our MRIV estimator more likely to improve on the initial estimator Amit (x) if either (1) ? is large or (2) a and B are large. Note that the margin of improvement depends also on the size of y and 8, i.e., on the smoothness of I (x) and 8A In fact, this is widely fulfilled in practice. For example, the former is fulfilled for RCTs with non-compliance, where ((x) is often some known, fixed number p E (0, 1).
4.3 MRIV vs. WALD ESTIMATOR
We compare +MRIV (z) to the Wald estimator (z). First, we derive an asymptotic upper bound.
Theorem 3 (Wald oracle upper bound). Assume that μo (x), Mo (x) are a amoohh,, μ1 (z), m (z) are 3-smooth, and are estimated with their respective minimax rate, Let d po ) A (2) satisfy Assumption 4 Then, the oracle risk of the Wald estimator îw (x)) is bounded by
E [((w(()) - -(x)) S
20 20 + n2 28FF
28
(5)
Proof. See Appendix B.

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We now consider the MRIV estimator 'MRIV (z) with Auit (2( ie.. initialized with the Wald estimator (under sample splitting). Plugging the Wald rate from Eq. (5) into the Eq. (2) yields
E (fMRIv(() -(=))² nn + n 2 mny t
2(ale + (hi))) 2n -n-2( Paddo mip))
F B 7 6, the rates of AMRIV (1)) and Fw (a) reduce to R (funnv (1) (x))) x 2040
20+p and IE (fw(z) ((x))) I n 2 Hence, (z) outperforms Åw (w( asymptotically for nj > i.e. when the CATE 7 (xx) is smoother than its components, which is usually the case in
practice (Künzel et al. 2019). For 1) = a, the rates of both estimators coincide. Hence, we should expect MRIV to improve on the Wald estimator in real-world settings with large sample size.
4.4 MRIV-NET
Based on our MRIV framwork, we develop tailored deep neural network called MRIV-Net for CATE estimation using IVs. Our MRIV-Net produces both an initial CATE estimator fi (r) and nuisance estimators Dn (z), Ad(x)) (z). (1))), and #xx).
For MRIV-Net, we choose deep neural networks for the nuisance com- ponents due to their predictive power and their ability to learn complex shared representations for several nuisance components. Sharing repre- sentations between nuisance components has been exploited previously for CATE estimation, yet only under unconfoundedness (Shalit et al. 2017, Curth & van der Schaar, 202T). Building shared representations is more efficient in finite sample regimes than estimating all nuisance components separately as they usually share some common structure.
Figure 2: Architecture of MRIV-Net.
In MRIV-Net, not all nuisance components should share a representation.
Recall that, in Theorem 2 we assumed that (1) Amit (I), P (Ir), and (ar); and (2) (x) and (a:) trained on two independent samples in order derive the upper bound on the oracle risk. Hence, we propose to build two separate representations $ and 2, so that (i) $ is used to learn the parameters (1), and (ii) 22 used to learn the parameters (2). This ensures that the nuisance estimators (1) share minimal information with nuisance estimators (2) even though they are estimated on the same data (cf. (Curth & van der Schaar 2021
The urchitecture of MRIV Net is shown in Fig. MRIV-Net takes the observed covariates X as input to build the two representations 1 and 22. The first representation 4 is used to output estimates it (x), Po (x), jf(x)) and ig (ar) of the CATE components. The second representation 4 is used to output estimates 1 (x). Mo (x:). and म (x:). MRIV-Net is trained by minimizing an overall loss
-(0== = -T (AY, (a.)
);)² BCE (ac) (x))..) a BCE (id) (x:)..: + BCE (+(s)).)))
(6)
where 0 denotes the neural network parameters and BCE is the binary cross entropy loss. After training MRIV-Net, we obtain the Amit (x)
a (x)- (x) and obtain the nuisance estimators Ai (x),
A (=)
An (x), (z))) = Af (z) 76 (x) and (xx). Then, we perform, we perform the pseudo regression (Stage 2) of MRIV to obtain (MRIV(*)) (z)).
Details on the implementation, the network architecture and hyperparameter tuning are Appendix We perform both the training MRIV-Net and the pseudo-outcome regression on the full training data. Needless to say, MRIV-Net can be easily adopted for sample splitting or cross-fitting procedures as in Chernozhukov et al. (2018), namely, by learning separate networks for each representation . and 42 In our experiments, as this can affect the performance in finite sample regimes. Of note, our choice is consistent with
Implementation:
do not sample splitting or cross- fotts-fitiig,
previous work (Curth & van der Schaar 2021). In Appendix K we report results using cross- cross-fitting fitting.
COMPUTATIONAL EXPERIMENTS
5.1 SIMULATED DATA
In causal inference literature, it is common practice to use simulated data for performance evaluations (Bica et al. 2020a Curth & van der Schaar 2021 Hartford et al. 2017). Simulated data offers the
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crucial benefit that it provides ground-truth information the counterfactual outcomes and thus allows for direct benchmarking against the oracle CATE.
Data generation: We generate simulated data by sampling the oracle CATE T(x) and the nuisance components n (x). (z), and T T(x) from Gaussian process priors. Using Gaussian processes has the following advantages: (1) It allows for fair method comparison, as there is no need to explicitly specify the nuisance components, which could lead to unwanted inductive biases favoring a specific method; (2) the sampled nuisance components are non-linear and thus resemble real-world scenarios where machine learning methods would be applied; and, (3) by sampling from the prior induced by the Matérn kernel (Rasmussen Williams 2008), we can control the smoothness of the nuisance components, which allows us to confirm our theoretical results from Sec. 4.2) For a detailed description of our data generating process, we refer to Appendix E.
We compare our MRIV-Net with state-of-the-art IV baselines. Details regarding baselines and nuisance parameter estimation are in Appendix G. Note that many of the baselines do not directly aim at CATE estimation but rather at counterfactual outcome prediction. We nevertheless use these methods as baselines and, for this, obtain the CATE by taking the difference between the predictions of the factual and counterfactual outcomes.
Baselines:
Performance evaluation: For all exper- iments, we use 80/20 split train- ing/test set. We calcalute the root mean squared errors (RMSE) between the CATE estimates and the oracle CATE on the test set. We report the mean RMSE and the standard deviation over five data sets generated from random seeds.
Table 2: Performance comparison: our MRIV-Net vs. ex- isting baselines.
STANNNN
JAKNei (Sampee AIT
0.76 U.34
0.70 0.32
SUD9 di U.37 0.70 0.10
TARNer VOR
MI7 Seenedy 2037
0.78 0.
0.06 0.00
(2) Gentraa IV
35.. Manalaad
5:22
0.79 1.37
Des 10EER
0.45 0.31
0.40 11.21
0.46
Decpth mecuroiia 10 A01
0.06 0.30
0.28 2: 11.00
0.25 2 0.24
Semplaaneiiiiieee MMM
0.38
0:37 0.00
Table 2 shows the results for all baselines. Here, the DR-Icarner does not improve the performance of TAR- Net, which is reasonable as both the DR- learner and TARNet assume unconfoundedness and are thus biased in our setting. Our MRIV-Net outperforms all baselines. Our MRIV-Net also achieves a smaller standard deviation. For additional results, we refer to Appendix J.
Results:
DMILI
0.92
(3) WALD 35TMATON
Wata [940
Linar
1.05 9.53
0.82 4: 0:22
2.81 2: 0.54
BAKI
0.95 0.30
0.05 0.33
3.85
MHUV-Ne Toam)
0.26 0.11
0.15
0.00
0.13 1 0.03
Repornt RMSR for Now wutherk Dreas 2 condinlldrviaion devistion Lower
hatee Obesr is twid)
We further compare the performance of two different meta-learner frameworks - DRIV (Syrgka- nis et al. 2019) and our MRIV- across different base methods. The results are in Table 3 The nuisance parameters are estimated using feed forward neural networks (DRIV) or TAR-
Table 3: Base model with different meta-learners (i.e., none, DRIV, and our MRIV).
3000
5000
8000
Base methods
Meta leorzers
None
DRIV
MRIV (ouns)
Nove
DRIV
MRVV (cuEs)
None
DRIV
MRIV (ours)
STANDARI
0.76 0.14
0.51
0.05
0.70
0.17
0.06
0.17
0.17
0.11
Gaaceet TV
25L. Moydihoog
=.10: 0.22
0.31 :: 0.05
0.73
0.17 :: 0.09
0.19::0.05
0.29
0.21 2:
0.16 0.02
KIV
151:20..5
0:40 0010
0.59 2 0.11
1.18 1:14
0.20 1000500000 0.08
0.17:2:00 0.06
5.00
990--11.1
4.71
0:31 5 0-0.1
0.28 2: 0.19
DETV
0.43::0.11
0.26
= 0.05
0.27 6.67
0.31:
0.18 1: 0.00
0.18 0.04
0.05 0.54
0.21 0.00
0.18 0.05
Deepth
2013
0.96 0:30
022 2010
0.26 0.05
0.26 0000
0.18 -: 0.08
0.18 1: 0.00
0/23 0:04
1121 0.00
U.1G 0.03
Desp (oeerer
0.36
0.10
0.15
0.00
0.10::
0.14
0.21
DMLJV (alp St
0.11
1.14
0121
WALD ESTMAAAK WALD 1940
Lancar
1.06 2
0.12
0.38
0.14
0.02::0:22
0.19 +: 0.09
0.25::00.0
0.81 0.54
0.19 0..0
0.18 0.06
BART
0.96
0.30
0.14
0.40
0.112
0.03
0.20
0.20 0.07
0.88 0.28
0.91 0.00
0.29 0.04
MRIV Net setwnck only louns)
0,20 0.13
03622012
0.26
226:000
0.11
0,01
2004
0.19
0.15 0.03
0.06
D.18 0.38
0.13 0.03
Reportal RMSE lncas manded devinson). Lower bese (bal
mote in bali
Nets with either binary or continuous outputs (MRIV). Our MRIV improves over the vari-
ant without any meta-learner framework across all base methods (both in terms of RMSE and standard deviation). Furthermore, MRIV is clearly superior over DRIV. This demon- strates the effectiveness of our MRIV across different base methods (note: MRIV with an arbitrary base model is typically superior to DRIV with our custom network from above). Table 4: Ablation study.
MRIV-Net is overall best. We also per-
formed additional experiments where
we used semi-synthetic data and cross- fitting approaches for both meta-learners (see Appendix Sannndixann]]k [K.
Method
== 3000
12 5000
f 8000
MRRV-Nerr netwoek only
0.31
0.26 0.06
MRIV-Net single rope.
0.28 = 0.12 0.26
0.21 2: 0.04
0.32 2 0.10
MRIV-Net (ours)
- 0.11
0.15 4 0.03
0.13 2 0.03
Reported: RMSE (mean =: standard deviation), Lower = betier (best in bold)
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Ablation study: Table 4 compares different variants of our MRIV-Net. These are: (1) MRIV but network only; (2) MRIV-Net with a single representation for all nuisance estimators; and (3) our MRIV-Net from above. We observe that MRIV-Net is best. This justifies proposed network architecture for MRIV-Net. Hence, combing the result from above, our performance gain must be attributed to both framework and the architecture of our deep neural network.
Robustness checks for unobserved confounding and smoothness: Here, we demonstrate the importance of handling unobserved confounding (as we do in our MRIV framework). For this, Fig. B] plots the results for our MRIV-Net vs. standard CATE without customization for confounding (i.e.. TARNet with and without the DR-learner) over over different levels of unobserved confounding. The RMSE of both TARNet variants increase almost linearly with increasing confounding. In contrast, the RMSE of our MRIV-Net only marginally. Even for low confounding regimes, our MRIV-Net performs competitively.
5603
VIE- Sunt
Grrtundig has C.
Conturdeg leve 6
Contunding ievl
Figure 3: Results over different levels of confounding ar. Shaded area shows standard deviation.
Fig. 4 varies the smoothness level. This is given by a of ()) (controlled by the Matérn kernel prior). Here, the performance decreases for the baselines, i.c., DeepIV and our network with- out MRIV framework. In contrast, the peformance of our MRIV-Net remains robust and outperforms the baselines. This confirms our theoretical results from above. thus indicates that our MRIV framework works best when the oracle CATE 7 (ar) is smoother than the nuisance parameters juy (z)).
0.25
Mothod
Decepi/
0.20
0.15
0:30
Figure 4: Results over different lev- els of smoothness a of μu (.), sam- ple size n 8000. Larger a smoother. Shaded areas show stan- dard deviation.
5.2
CASE STUDY WITH REAL-WORLD DATA
Setting: We demonstrate effectiveness of our framework using a case study with real- world, medical data. Here, we use medi- cal data from the so-called Oregon health insurance experiment (OHIE) (Finkelstein et al.) 2012). It provides data for an RCT with non-compliance: In 2008, 300,000 low-income, uninsured adults in Oregon were offered participation in health insurance program by lottery. Individuals whose names were drawn could decide to sign up for health insurance. After period of 12 months, in in-person interviews took place to evaluate the health condition of the respective participant.
In our analysis, the lottery assignment is the in- strument Z, the decision to sign up for health insurance is treatment A, and an overall health score is the outcome Y. We also include five covariates X, including age and gender. For de- tails, we refer to Appendix F. We first estimate the CATE function and then report the treatment effect heterogeneity w.r.t. age and gender, while
Mahid
Wilodaa busi
Figure 5: Results on real-world medical data. fixing the other covariates. The results for MRIV-Net, our neural network architecture without the MRIV framework, and TARNet are in Fig.
Our MRIV-Net estimates larger causal effects for an older age. In contrast, TARNet does not estimate positive CATES even for an older age. Even though we cannot evaluate the estimation quality on real-world data, our estimates seem reasonable in light of the medical literature: the benefit of health insurance should increase with older age. This showcases that TARNet may suffer from bias induced by unobserved confounders. We also report the results for DRIV with DMLIV as base method, and observe that in contrast to MRIV-Net, the corresponding CATE does not vary much between ages. Interestingly, both our MRIV-Net estimate a somewhat smaller CATE for middle ages (around 30-50 yrs). In sum, the findings from our case study are of direct relevance for mecision-makers in public health (Imbens& & Angrist 1994), and highlight the practical value of our framework. We performed further experiments real-world data which are reported in Appendix L
Results:

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Reproducibility: The codes for reproducing the experimental results can be found https
7/github. letgithub.com/Deee www/Dennnssaaauue///aaaaaaaaa MRRVVVV
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A EXTENDED RELATED WORK
CATE methods without unconfoundedness: Various machine learning methods for estimating
CATES without unobserved confounding have been proposed in recent literature (Alaa van der Schaar 2017 Curth & van der Schaar [021 Künzel et al. 2019; Lim et al.) 2018:) Shalit et al. [017]]
Wager Athey 2018 Yoon al., 2018 Zhang et al. 2020). To remove plug in bias, the DR leamnee performs a second stage regression on the uncentered influence function of the average treatment effect (Kennedy 2022). However, under unobserved confounding, all of these methods are biased (see Appendix As a result, this hampers their performance in our setting.
Non-IV methods for unobserved confounding: There a rich literature for causal effect estimation under unobserved confounding, which does not assume the existence of instrumental variables. Methods include deconfounding methods (Wang & Blei 2019) Bica al. 22020b), proxy learning
methods (Cui et al. 2020; Xu et al. 2021b), and causal sensitivity analysis (Kallus et al. 2019 Jesson
ct al. 2021)..
Classical IV methods: IV methods address the problem of unobserved confounding by exploiting the variance in treatment and outcome induced by the instruments. Traditionally, two-stage least squares (2SLS) has been used for estimating causal effects (Wright 1928| Angrist & Krueger 1991). 2SLS was originally developed in economics, and follows a two-stage procedure: it performs a first stage regression of treatment A the instrument Z, and then uses the fitted values for second stage regression to predict the outcome Y. Several nonparametric methods have been developed in econometric to generalize 2SLS in order account for non-linearities within the data (Newey & Powell 2003
Wang et al. 2021 yet these are limited to low- diwdeemnsionll settings.
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B PROOFS
We start by deriving auxiliary Lemma. That is, derive explicit expression for the Stage oracle pseudo-outcome regression B[YMR
X =x] of MRIV.
Lemma 1.
E[SMR 1x=a0 = x] (Φ)
(1 - x(x))
(x)) (11 (x) 141 (1) Ainit()))
+
(1(1) - -i(x)) (x)
(2) Fmit(z) - 1O (x))
(7)
Ao (x) (x) i0 (x)
(x)
1- -x)
+ (m)
81(z)
a(x)
- $ (xx)
Proof.
E|YMR X = xr]
(8)
A Amit - 11 (X) (X) +
(X) (X)
=
x= = x. Z =
6A(X) (XX)
Y Amit (X) - A/ (X) + A6 (X) (X)
+ (1 - x()))
X z, Z = 0
+ fimi (x)
8x(X) (1 - -x()))
(9)
T(z)
(NY (x) - (x)xx) Ami - PD (x) + p (x) (x))
(x( (z))
1 1-)(x)
(z) (1-)(()) (1)(2)) (x) m (m) - 1%(0)) (x) + ( (x) Amit(()) + Aiii (10)
Rearranging the terms yields the desired result.
B. PROOF OF THEOREM I (MULTIPLE ROBUSTNESS PROPERTY)
We use Lemma to show that under each of the three conditions it follows that E|Y X = x]
(ir).
1.
E[YMR X x]
(11)
((x)
(x)- (())-
(x) ((x))
- T()))
(x)) + +(x) (z))
(12)
( (1 +(x))
T(x)
(x)x))(()) (8r ((()) r(x).
(13)
2.
B|YMR X =x] (uY (z) - (x) Rm))((x)) (N6 (x) Amit - no (x))
+ Ami (x)
(x)x)
(2(2)
(14)
8y (x) - Amit (x) 5
Aiii (x) T(x).
(15)
(z)
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3.
B/YMR X x| (x)(z) (z) - (x) (1) r(x) (x))
+ T(x)
(16)
(Λ(x)
81(x)
By (x)
(xx) 5Λ(Ø)
+ -())= T(x)
(17)
(z) (z)
B.2 PROOF OF THEOREM 2 (CONVERGENCE RATE OF MRIV)
To prove Theorem 2, we need an additional assumption on the second stage regression estimator É., We refer Kennedy (2022) (Proposition 1) for detailed discussion on this assumption.
Assumption 5 (From Proposition Kennedy (022 ). Let YMR be the corresponding oracle the estimated pseudo-outcome YMR- We assume that the pseudo-regression estimator E,, satisfies
E, YMR X 2) Em MMR X = x) En, MR YMR X x
P O
(18)
(Bn (YRR. X x] E[YMR X u1) 2
and
[(a(((()) x = =P]== =E [F(x)))
(19)
where r(x) E[ x] T(x)
To prove Theorem 2] we derive a more general bound the depends on the pointwise mean squared enrors of the nuisance estimators. Theorem 2 follows immediately by applying Assumption 3.
Lemma 2. Consider the setting described in Theorem 2
Then,
E (finie( (x) - =(x)))]
(20)
$ R(x) + ((finie - - (8) (5a((o) + + { ((f(o)=>(z)))) -(0))))
+ +E (A(()) (=(x)) (H (10
(x) (10
wo (z)) - ((0)()))(0(z)) r6(=))
(21)
Proof. Let YMR be the corresponding oracle YMR and define M (x) = E, (YMR X = x].
Using Assumption 3
apply Proposition of Kennedy (2022) and obtain
B (finit
- (x) (x))) S R(x) + [=
(22)
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where R(x) = E
- (Tm))
is the oracle risk of the second stage regression. We can apply
Lemma to obtain
f(x) (1) ((x) (uY (2r) --1 (z) fmn))(1)) + (1 - T T())
(x))
(x( (1 - -x(x)) (16) (x) (2) Fmii((x) - 16 (a:)
P6 (z) Fiii )n)((()) - Ab (x)
[())
1 1-T(z)
+
+ ffiii (2) - =(x)
(23)
(1())
f(z)
1 #(x)
uf (x) uá (2)
x(x)
nĀ (I) - id (x)
(Tx)
1- (-T(xx)
+ (fiit (2) - Txx))
(1(Z)
a(x)
(z
+(x))
1- f(x))
(16 (x) 11 (a:)) Pinit (x)
T(x)
(AP 11((x)) Ainit (x)
TTx))
1 - πT (2r)
(1(z)
(())
8イ(2)
#(x)
(xx))
(24)
(y(x) + (ub (x) io (x)) (π(अ) - +x))) (fmit(z)- - T(x))
(Z(Z) (1x)
(1(x) [(x) (1-#(z))
(z) (x) (z)
(16 -
(x) 1z(())) Pinit ( -x(x)
(25)
(1( #(x)
(x(x) fxx) (1 - =xx))
(T(x) - (x)))
(1) R(x) (1 (xx) [w6 (x) - (x)) + (16 (x) N mx)))) (r)
+ (fiit(() - -xx)) + =()))1(2) (r(x) Am))()))
(26)
x(z)81(x)
(T(x) (())) A()))
3A(x) (x)))((x)( (1 - f(x)) [116 (x) - Ab (x)) + (R6' (x) - H (x)) (m))
+ (T(zz) - (z)( (5a(z)- - -da(z)) x(x) + (T(z) (x)) (T(Z) ((z)))a(())
(27)
Applying the inequality (a+ < 2(a² +b²) together with Assumption 4 and the fact that (xx) S 1 yields
F(x)2 S
dpp (F(2) - =(x))² (w) (1) (z)))= + (K)(x) - má(x)) 2 x²]
4
+ (T(xx) - (x)) (5((a)- - + +4(r(x) - (=)) (T((T) - =x))) (28)
By setting K- max{K, 1), we obtain
f(0)2 - 4R²
etp2 (IT(z) - =(x))² (1²
(x) - (x())) + (Ad(2) - (x))) + (f - [(x))²]
-(x)))
+ (7(z) (x)) (Ba(z)- -8a(=)))
(29)
Applying expectations on both sides yields the results, because (x()),8,(x)) (i0 (z), (x),, (x)) due sample splitting.
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B.3 PROOF OF THEOREM 3 (CONVERGENCE RATE OF THE WALD ESTIMATOR)
Proof. We define C = max (C, 1 and obtain the upper bound
(fw(x) - -xx)) -M - (y(x)
(30)
(31)
(A(2)
4
[naY (x) - (x)) + (A) (z) - u (z)) + - -
(32)
80
[aa
(x) (x)
(x))² + (PO (x) (x)) (A1(x) 2x))))
ppp
+(60 nO¹ (z))²]
(33)
where we used the inequality + b) < 2(a² + b²) several times. Taking expectations and applying the smoothness assumptions yields the result.
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C THEORETICAL ANALYSIS UNDER SPARSITY ASSUMPTIONS
In Sec. we analyzed MRIV theoretically by imposing smoothness assumptions on the underlying data generating process. In particular, we derived multiple robust convergence rate and showed that MRIV outperforms the Wald estimator if the oracle CATE is smoother than its components, In this section, we derive similar results by relying on different set of assumptions. Instead of using smoothness, we make assumptions on the level of sparsity of the CATE components. This assumption often imposed in high-dimensional settings (n < p) and in line with previous literature on analyzing CATE estimators (Curth & van der Schaar 2021 Kennedy 2022).
In the following, we say a function f(x) k-sparse, if linear in x RP and only depends on k < min (n. p} predictors. (Yang & Tokdar 2015) showed, that in this case the minimax rate of f(x) given by &log(2) The linearity assumption can be relaxed to an additive structural assumption. which we omit here for simplicity. In the following, we replace the smoothness conditions in Assumption 3 with sparsity conditions.
Assumption 6 (Sparsity). We assume that (1) the nuisance components nu (+) a-sparse, (.) and (-)-) (-) are 8-sparse, and T--) is 8-sparse; (2) all nuisance components are estimated with their respective minimax rate of klog(p) where & E {a, 9, 8); and (3) the oracle CATE r(-) is 7-sparse and the initial CATE estimator Tinit converges with rate r- (m)).
2
We restate now our result from Theorem 3 for MRIV using the sparsity assumption.
Theorem 4 (MRIV upper bound under sparsity). We consider the same setting as in Theorem under the sparsity assumption 6 If the second-stage estimator En yields the minimax rate " log(p) and satisfies Assumption
5 the oracle risk is upper bounded by
E (fARIIV (x) r(x))²] 7 7log(p)
(3 log(p) + (a + +3)5 (p)
(r
n
n
Proof. Follows immediately from Lemma by applying Ass-
Again, obtain multiple robust convergence rate for MRIV in the sense that MRIV achieves fast rate even if the initial estimator or several nuisance estimators converge slowly. More precisely, for a fast convergence rate of fMRIV (z), it is sufficient if either: (1) r-(n) decreases fast and 5 is small; (2) r- (n) decreases fast and a and 3 are small; or (3) all a. 3, and are small.
We derive now the corresponding rate for the Wald estimator.
Theorem 5 (Wald oracle upper bound). Given estimators ji (x) and in (z). Let
P (x) satisfy Assumption Then, under Assumption 6 the oracle risk of the Wald estimator fw (x) is bounded by
(a + log(()
(fw((z) - [(x))²]
S
(34)
Proof. Follows immediately from the proof of Theorem i.e., from (0) by applying Ass- 6
If aaa= 3= 6, we obtain the rates
log(p)
a² log)(p)
log(p)
E (()))) $
and
[T)))]] (()())-(()))]] E. (fw(c)) -
n
2
n
(35)
which means that ÁMRIV (z) outperforms (w(x) for < a. i.e., if the oracle CATE is more sparse than its components.
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D MATHEMATICAL DETAILS REGARDING ASSUMPTION 3
In this section, we briefly state the formal definitions of the convergence ratesin Assumption 3 We follow Stone (180). Let 0 be a parameter and T()) some larget functional that want estimate.
Definition 1. A sequence of estimators Ta (0) of a functional T(0) converges with rate (r(n) if
lim lim inf sup (T, (0) T(0) cro(n)) 0.
(36)
wEN 8Ee
Definition 2. A rate rr (n) called an upper bound the rate of convergence for all estimators (0) it holds for that
lim inf sup Po Po( - T(0) > cra(n)) 0.
(37)
MEN AE
and
lim lim inf sup (0) > cra(n)) 1.
(38)
c-0
WEN
lEe
(r(n) is called optimal it is both achievable and an upper bound.
Stone (1980) showed that for nonparametric regression problem with an 7-smooth regression function, the optimal rate of convergence is 26+p, where is the dimension of the covariate space.
2n
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E SIMULATED DATA
In the following, we describe how we simulate synthetic data for the experiments in Sec. 5.1 from the main paper. As mentioned therein, simulate the CATE components from Gaussian processes using the prior induced by the Matern kernel (Rasmussen & Williams, (0008)
V2v ar -2,l2
12v ( - xz
(2i, xz))
Kr
(39)
((v)2u-T
where r(.) is the Gamma function and (-)+) is the modified Bessel function of second kind. Here, & is the length scale of the kernel and controls the smoothness of the sampled functions.
We é and sample functions öy GP(O, Ke,)), mn (P(0, fi ~ GP(O, Ke,s). So CP(O, Ke,s) and g GP(O, Ke,z). Then, define pi by + H uú 0.3 ao fi + 0.7,
N4 0.3 for da p¹ po 10 cadA, and o g. Finally, the oracle CATE to
" Pa y
dy
(40)
"1 1 8A
Note that we can create a setup where the CATE r smoother than its components by using a small a/B ratio. An example is shown in Fig. 6
11
YIZ-1
-2
Y
Figure 6: Gaussian process simulation for a 1.5 and / 50.
In the following, we describe how we generate data the (X, 2, A, Y) using the CATE components NY (x), 1 (x), and π (z). We begin by sampling observed confounder X NO, 1), unobserved confounders U ~ N (0, 0.29), and instruments Z ~ Bernoulli(T(X)). Then, we obtain treatments via
A = Z 1QU + > (x))) + (1 (1 Z) with indicator function 1, noise 6Λ ~ V (0, 0.1... and o; (X) -- (1-)(()) 20.12
ax())}
(41)
where Φ denotes the quantile function of the standard normal distribution. Finally, generate the outcomes via
Y = A (KT(X) 1)23 (X) - (X) (X) pt PX
(42)
(X)
+ +11 - A) ( (X) u (X) )) (X)
+ anU +er,
(43)
(x)
where ey N (0, 0.3²) is noise and av > is a parameter indicating the level of unobserved confounding. This choice of and in Eq. (41) and Eq.
(42) respectively, implies that r(x) is indeed the CATE, i. i.e.. it holds that 7(x: ( Y X xr).
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Lemma 3. Let (x, A,Y) be sampled from the the previously described procedure. Then, it holds
that
(= ||A| Z = = and (x) E Z = ii =x).
x/)
(44)
Proof. The first claim follows from
[ 2 2 = i, X = x] PU + (x) = - -1-0(0 (1- (1-)(t)))
= 14 (2)),
(45)
because U EΛ ~ (O, v0.13 The second claim follows from
E|y 2 -, x x| - =4 (uf((z)
1) (x) (z) (2)
(46)
(x)
+ (1 - (x)) ( p (2) (x) - PO (z) (x)
(47)
(x(Z)
(o)6
(x).
(48)
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F OREGON HEALTH INSURANCE EXPERIMENT
The so-called Oregon health insurance experiment (OII Finkelstein ct al. 2012) was an important RCT with non-compliance, It was intentionally conducted large-scale effort among public health to assess the effect of health insurance on several outcomes such as health or economic status. In 2008, lottery draw offered low-income, uninsured adults in Oregon participation in Medicaid program, providing health insurance. Individuals whose names were drawn could decide to sign up for the program.
In our analysis, the lottery assignment the instrument Z, the decision to sign up for the Medicaid program is the treatment A, and an overall health score is the outcome Y. The outcome was obtained after a period of 12 months during in-person interviews. We use the following covariates X: age, gender, language, the number of emergency visits before the experiment, and the number of people the individual signed up with. The latter is used to control for peer effects, and it is important to include this variable in our analysis as is the only variable influencing the propensity score (see below). We extract 10,000 observations from the OHIE data and plot the histograms of all variables Fig. 7 We can clearly observe the presence of non-compliance within the data, because the ratio of treated / untreated individuals is much lower than the corresponding ratio for the treatment assignment.
Forpanty
20
68
50
Age
Frongancy
Purrdo
Ote
Engiih
Nim egrad
Num wibe
Langugge
Figure 7: Histograms of each variable in sample from OHIE.
The data collection in the OHIE was done follows: After excluding individuals below the age of 19, above the age of 64, and individuals with residence outside of Oregon, 74,922 individuals considered for the lottery. Among those, 29,834 participation in the program. However, the probability of selection depended on the number of household members on the waiting list: for instance, an individual who signed up with another person was twice as likely to be selected. From the 74,922 individuals, 57,528 signed up alone, 17,236 signed up with another person, and 158 signed up with two more people on the waiting list. Thus, the probability of being selected conditional on the number of household members on the waiting list follows the multivariate version of Wallenius' noncentral hypergeometric distribution (Chesson 1976).
selected randomly and were offered
Propensity score: We computed the propensity score as follows. To account for the Wallenius' noncentral hypergeometric distribution,
the R package BiasedUrn to calculate the propensity score π (<x) P(Z X z). We obtained
0..345,
if individual signed up alone,
क (xx
0.571,
if indiviidual signed up with one more person,
(49)
0.719,
if indiviidul signed up with two more people.
Data available here: http:///wwwwrroogeeeeeeeeeeeeeee experiment
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During the training of both MRIV and DRIV, we use the calculated values from Eq. 49 for the propensity score.
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G DETAILS FOR BASELINE METHODS
In this section, we give a brief overview on the baselines which we used in our experiments. We implemented: (1) CATE methods for unconfoundedness: TARNet (Shalit et al.
2017) and TARNet combined with the DR-learner (Kennedy, 2022); (2) general IV methods, i.e., IV methods developed for IV settings with multiple or continuous instruments and treatments: 2SLS (Wright 1928), kemel IV (KIV) (Singh et al. 2019), DFIV (Xu et al., 2021a), DeepIV (Hartford et al. 2017). De (Bennett et al. 2019). DMLIV (Syrgkanis
al. 2019), and DMLIV combined with DRIV (as described in (Syrgkanis et al. 2019)); (3) the (plug-in) Wald estimator using linear models and Bayesian additive regression trees (BART) (Chipman et al., 2010). Of note, the DR-learner assumes unconfoundedness, which is why we only combine it TARNet in our experiments. In the following, we provide details regarding methods and implementation.
G.1 CATE METHODS FOR UNCONFOUNDEINNESS
Many CATE methods assume unconfoundeduesss. i.e., that all confounders are observed in the data. Formally, the unconfoundedness assumption can be expressed in the potential outcomes framework as
(1),, A A X
(50)
Under unconfoundedness, the CATE is identified as
r(I) = (z (o))) with [4(2) = E| A- x²)
(51)
Methods that assume unconfoundedness proceed by estimating (.) E|Y AA i, X Eq. (51) However, if unobserved confounders U exist, it follows that
2] from
T(x) E A 1, X x, U] Y A 0, X x, UJ # m1(x) Ho
(52)
which means that estimators that assume unconfoundedness. are generally biased. Nevertheless, include two baselines that assume unconfoundedness into our experiments: TARNet (Shalit et al. 2017) and the DR-learner (Kennedy) (022)).
TARNet (Shalit et al., 2017): TARNet (Shalit et al. 2017) is a neural network that estimates the CATE components jo (xr) from Eq. 5 learning shared representation ((x) and two potential
outcome heads (+(()) We train TARNet by minimizing the loss
12
[(0) = Σ (ha, (B(2,, 0x), 9:)).
(53)
where : 0o) denotes the model parameters and denotes squared loss if Y is continuous or binary cross entropy loss if Y is binary.
Note regarding balanced representations: In (Shalit et al., 2017). the authors propose to add an additional regularization term inspired from domain adaptation literature, which forces TARNet to learn a balanced representation ((r), i.e., that minimizes the distance the treatment and control group in the feature space. They showed that this approach leads to minimization of a generalization bound on the CATE estimation error the representation is invertible.
In experiments, we refrained from learning balanced representations because minimizing the regularized loss from (Shalit et al. 2017) does not necessarily result in an invertible representation and thus may even harm the estimation performance. For detailed discussion, der Schaar 2021). Furthermore, by leaving out the regularization, we ensure comparability between the different baselines. If balanced representations could also be extended to MRIV-Net, as we also build MRIV-Net on learning shared representations.
refer to (Curth & van
desired, the balanced representation approach
DR-learner (Kennedy) 2022): The DR-learner (Kennedy 2022) is meta learner that takes arbitrary
estimators of the CATE componenets μ4 and the propensity score (2x) = PPA- = 1 X x) as input and performs pseudo-outcome regression by using the pseudo outcome
MMR
A
1- xx)
1-
Y
=(X) (x(X)
A
1-A A x(x) (1
1-
(X) (54)
In our experiments, use TARNet as base method to provide initial estimators A. (X). We further learn propensity score estimates f (X) by adding a seperate representation to TARNet as done in (Shalit et al. (017).
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G.2 GENERAL IV METHODS
2SLS [Wright 1928): 2SLS (Wright 1928) is a linear two-stage approach. First, the treatments A are regressed on the instruments Z and fitted values A are obtained. In the second stage, the outcome Y is regressed on A. We implement 2SLS using the scikit-learn package.
KIV (Singh et al. 2019: ; Kerne IV (Singh al. 2019) generalizes 2SLS nonlinear settings. KIV assumes that the data is generated by
f(A)+ U,
(55)
where U is an additive unobserved confounder and f is some unknown (potentially nonlinear) structural function. KIV then models the structural function via
f(a) p² (a) and B(p(44) Z 2] Vń(z),
(56)
where vp andØ are feature maps. Here, kernel ridge regressions instead of linear regressions are used in both stages to estimate y and V.
Following (Singh et al. 2019) we use the exponential kernel (Rasmussen & Williams, 2008) and set the length scale the median inter-point distance. KIV does not provide a direct way to incorporate the observed confounders X. Hence, we augment both the instrument and the treatment with X, which consistent with previous work (Bennett al., 2019. Xu et al., 2021a). We also use two different samples for each stage recommended in (Singh et al. 2019).
DFIV (Xu et al., 2021ap: DFIV (Xu et al. 2021a) similar approach KIV in generalizing 2SLS to nonlinear setting by assuming Eq. (55) and Eq. (56). However, instead of using kernel methods, DFIV models the features maps V0,, and d0g as neural networks with parameters 0 and oz. respectively. DFIV is trained by iteratively updating the parameters A and fz.. The authors also provide a training algorithm that incorporates observed confounders X, which we implemented for our experiments. During training, we used two different datasets for each of the two stages as described in in the paper.
DeepIV (Hartford et al. 20177: DeepIV (Hartford et al. 2017) also assumes additive unobserved confounding as in Eq. (5). but leverages the identification result (Newey & Powell 2003)
Y
X Z
h(a, x) dF x,
(57)
where lıaa, x) = fa, .²) + EIU X = x] is the target counterfactual prediction function. DeepIV estimates F(a| x,, 2), i.e. the conditional distribution function of the treatment A given observed covariates X and instruments Z. by using neural networks. Because we consider only binary treatments, simply implement (tunable) feed-forward neural network with sigmoid activation function. Then, DeepIV proceeds by learning a second stage neural network to solve the inverse problem defined by Eq. (57.
DecPGMM (Bennett et al. 2019): DeepGMM (Bennett et al. 2019) adopts neural networks for IV estimation inspired by the (optimally weighted) Generalized Method of Moments. The DeepGMM estimator is defined as the solution of the following minimax game:
1 Σ f(z, r) - g(a;, 0))
4n Σ P(a, - 0))² (58)
1
E arg min sup DE TET
where f(z. and g(a;,)) -) are parameterized by neural networks. As recommended in (Bennett et al. (2019). we solve this optimization via adversarial training with the Optimistic Adam optimizer
Daskalakis al. 2018), where we the parameter 0 to the previous value of 0.
DMLIV (Syrgkanis et al. 2019): DMLIV SSrgkannis et al. 2019) assumes that the data is generated
via
== ==r(X)a TX)) f(X)+ + +U,
(59)
where the CATE f some function of the observed covariates. First, DMLIV estimates the functions x(X) = EEy X]. 12,2,(,, X)- E,A Z, X). and p(X) = E[A X). Then, the CATE is
learned by minimizing the loss
(()) = Σ (y. - ((x;) - fxx, 0) (h(s), x2.) - -(xx)))
(60)
1=1
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where +(X,.) is some model for 7 7(X). In our experiments, (tunable) feed-forve forward neural networks for all estimators.
DRIV (Syrgkanis et al., 2019h: DRIV (Syrgkanis et al. 2019) is meta learner, originally proposed in combination with DMLIV. requires initial estimators for ((X), p(X), (X) = EiZ XX = x). and f(X) E[A Z | z well initial CATE estimatior (. (e.g., from DMLIV).
The CATE is then estimated by a pseudo regression on the following doubly robust pseudo outcome:
DR = (X) + (Y (X) Aiii (x)
x (A - - (())
(61)
f(X) (XX)
We implement all regressions using (tunable) feed-forward neural networks.
Comparison between DRIV vs. MRIV: There are two key differences between our paper and (Syrgka- nis et 2019): (i) In contrast to DRIV, we showed that our MRIV is multiply robust. (ii) We derive a multiple robust convergence rate, while the rate (Syrgkanis et al. 2019) is not robust with respect to the nuisance rates.
Ad (i): Both MRIV and DRIV perform a pseudo-outcome regression on the efficient influence function (EIF) of the ATE. The key difference: DRIV uses the doubly robust parametrization of the EIF from (Okui et al. 2012), whereas we use the multiply robust parametrization of the EIF from (Wang& & Tchetgen Tchetgen 2018 Hence, our MRIV frameworks extends DRIV in non-trivial way to achieve multiple robustness. Thus, our estimator consistent in the union of three different model specifications
Ad (ii): Here, we compare the convergence rates from DRIV and our MRIV and, thereby, show the strengths of our MRIV. To this end, let us assume that the pseudo regression function is 7-smooth and
2y that we use the same second-stage estimator E,, with minimax rate n 2 for both DRIV and MRIV. If the nuisance parameters (XX), p(X). f(X). and (XX) are a-smooth and further are estimated -2n
with minimax rate 204F, Corollary 4 from Syrgkanis et al. 2019) states that DRIV converges with rate
(fDRIV (x) (((()==))))()))) (x:))² S n + n
In contrast, MRIV assumes estimation of the nuisance parameters HĀ (x) with rate n n (z) -20
2i
and (x) with rate and r((x) with rate n ,. the initial estimator Ami (r) converges with
rate (n)), our Theorem 2 yields the rate
E ((())
28 S n SYPP (n) n 2844
-24 204p
+ zo-p 2640 + n2 -2 20-p zm1p))
If all nuisance parameters converge with the same minimax rate of n n the rates of DRIV and our MRIV coincide. However, different to DRIV, our rate is additionally multiple robust in spirit of Theorem T This presents a crucial strength of our MRIV over DRIV: For example, if is small (slow convergence of (.))), our MRIV still with fast rate long as and B are large (i.e., if the other nuisance parameters are sufficiently smooth).
G.3 WALD ESTIMATOR
Finally, we consider the Wald estimator (Wald) 1940) for the binary IV setting. More precisely, we estimate the CATE components ux (z) and μ (x) seperately and plug them into
ii (x) i. (x)
r(x)
(62)
7 (x)
We consider two versions of the Wald estimator:
'For a detailed discussion on multiple robustness and the importance of the EIF parametrization, we refer to
Wang & Blei 2019). Section 4.5.
On a related note, a similar, important contribution of developing multiply robust method was recently made for the average treatment effect. Here, the estimator of (Okui et al.) 2012) was extended by the estimator of (Wagg& & Tchetgen Tchetgen 2018) to allow for multi robustness. Yet, this different from our work in that it focuses the average treatment effect, while we study the conditional average treatment effect in our paper.
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Linear: We use linear regressions to estimate the ( and logistic regressions to estimate the 1 (x).
We use Bayesian additive regression trees (Chipman et al. 2010) trees to estimate the n (x) and random forest classifier to estimate the μ (x).
BART:
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H VISUALIZATION OF PREDICTED CATES
We plot the predicted CATES for the different baselines and MRIV-Net in Fig. ( n. 3000). As expected, the linear methods (2SLS and linear Wald) are flexible enough provide accurate CATE estimates. We also observe that the curve of MRIV-Net without MRIV is quite wiggly, i.e. the estimator has relatively large variance. This variance is reduced when the full MRIV-Net is applied. As result, curve is much smoother. This is reasonable because MRIV does not estimate the CATE components individually, but estimates the CATE directly via the Stage 2 pseudo-outcome regression. Overall, this confirms the superiority of our proposed framework.
Nethod TARbiet
Methed 2515
Nethod KN
Nethed DIV
Nethod - DecplV
Method - GWM
Mothod DMUV
Nethod wold (Inear)
Method Wold
Method MRIV thetwork only)
Methed MRIV
Figure 8: Predicted CATEs (blue) and oracle CATE (red) for different baselines.
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1 IMPLEMENTATION DETAILS AND HYPERPARAMETER TUNING
Implementation details for deep learning models: To make the performance of the deep learning models comparable, we implemented all feed-forward neural networks (including MRIV-Net) as follows: We use two hidden layers with RELU activation functions. We also incorporated a dropout layer for each hidden layer. We trained all models with the Adam optimizer (Kingma & Ba 2013) using 100 epochs. Exceptions are only DFIV and DeepGMM, where we used 200 epochs for training, accounting for slower convergence of the respective (adversarial) training algorithms. For DeepGMM, we further used Optimistic Adam (Daskalakis et al. 2018 as in the original paper.
Training times:
We report the approximate times needed to train the deep learning models on our simulated data with n = 5000 in Table 5 For training, we used an AMD Ryzen Pro 7 CPU. Compared to DMLIV and DRIV, the training of MRIV-Net is faster because only a single neural network is trained.
Table 5: Training times for deep learning models (in seconds).
TARNet
TARNet + DR
DFIV
DeepIV
DeepGMM
DMLIV
DMLIV DRIV
MRIV-Net
10.62
~28.57
164.98
~30.21
17.31
~74998
~91.1
~>2220
We performed hyperparameter tuning for all deep learning models (including MRIV-Net), KIV, and the BART Wald estimator on all datasets. For all methods except KIV and DFIV, we split the data into training set (80%) and a validation set (20%). We then performed 40 random grid search iterations and chose the set of parameters that minimized the respective training loss on the validation set. particular, the tuning procedure was the same for all baselines, which ensures that the performance gain of MRIV-Net is due to the method itself and not due to larger flcxibility. Exceptions are only KIV and DFIV, for which we implemented the customized hyperparameter tuning algorithms proposed in (Singh et al. 2019) and (Xu et al. 2021a to ensure consistency with prior literature. For the meta learners (DR- learner, DRIV, and MRIV), we first performed hyperparameter tuning for the base methods and nuisance models, before tuning the pseudo-outcome regression neural network by using the input from the tuned models. The tuning ranges for the hyperparameter
Hyperparameter tuning:
shown in Table These include both the hyperparameter rangers shared across all neural networks and the model-specific hyperparameters. For reproducibility purposes, we publish the selected hy perparrameeers in our GitHub project as .aml files.
Table 6: Hyperparameter tuning ranges.
MODEL
HyperpARAMETER
TUNING RANGE
Feed-forward neural networks (Shared parameter ranges for all deep learning baselines)
Hidden layer sizefes)
p. Sp. 10p. 20p. 30p (simulated data)
p. 3p. 5p. Sp, 10p (OHIE)
Learning rate
0.0001, 0.0005, 0.001, 0.005, 0.01
Butch size
64, 128, 256
Drupout probability
0, 0.1.1, 0.2, 0.3
KIV
A (Ridge penalty first stage)
5, 6. 9, 10, 12
{ (Ridge penalty second stage)
5, 6, 7, 8, 9, 10, 12
DFIV
X (Ridge penalty first stage)
0.0001, 0.001, 0.01, 0.1 (simulated data)
0.01, 0.05, 0.1 (OHIE)
Aa (Ridge penalty second stage)
0.0001, 0.001, 0.01, 0.1 (simulated data)
0.05, 0.1 (OHIE)
DeepGMM Wald (BART)
(learning rate multiplier) Number of trees (BART)
0.5. 1, 1.5, 2,
20, 30, 40, 50
Number of trees (Random forest classifier)
20, 30, 40, 50
p network input size
Hyperparameter robustness checks: We also investigate the robustness of MRIV-Net with respect to hyperparameter choice. To to this, we fix the optimal hyperparameter constellation for our simulated data for n = 3000 and perturb the hidden layer sizes, learning rate, dropout probability, and batch size. The results are shown in Fig. We observe that the RMSE only changes marginally when perturbing the different hyperparameters, indicating that our method is to a certain degree robust against hyperparameter misspecification. Furthermore, results indicate that the performance improvement
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of MRIV-Net over the baselines observed in experiments is not due to hyperparameter tuning, but to our method itself.
Parameter Hidden size
Parameter Learning rate
0.4
RMSE
0.3
0.0
20
0.0025 0.0050 0.0075 0.0100
Value
Vallue
Parameter -Dropout prob.
Parameter = Batch size
RMSE
0.3
0.2
0.0
0.2
50
100
150
200
250
Value
Figure 9: Robustness checks for different hyperparameters of MRIV-Net.
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J RESULTS FOR SEMI-SYNTHETIC DATA
In the main paper, we evaluated MRIV-Net both on synthetic and real-world data. Here, we provide additional results by constructing semi-synthetic dataset on the basis of OHIE. it common practice in causal inference literature to use semi-synthetic data for evaluation, because it combines advantages of both synthetic and real-world data. On the one hand, the real-world data part ensures that the data distribution is realistic and matches those in practice. On the other hand, the counterfactual ground-trume truth is still available, which makes possible to measure the performance of CATE methods.
We construct our semi-synthetic data as follows: First, we extract the covariates X IR and instru- ments Z {0, 1] of our OHIE dataset from Sec. Then, we construct the treatment components 14 (x) via
" (X) = 0.3 o (X1)) + 0.7 and no (X) 0.3 o (Xi),
(63)
where X1 is the (standardized) age and (-) is the sigmoid function. The outcome components are constructed via
( - 0.5x,
x? and μ (X) 0.5x,
Σ x?-
(64)
2=
We then sample treatments A and outcomes Y as in Eq. 41 and Eq. 42 Lemma ensures that
1. (X) E Z X] and (x) E[A Z
Given the above, the oracle CATE becomes
Xi 0.7
(X)
(65)
Note that r(X) is sparse in the sense that it only depends on age, while the outcome components depend all five covariates. Following our theoretical analysis in Sec. MRIV-Net should thus outperform methods that aim at estimating the components directly. This is confirmed in Table 7 where we show the results for all baselines and MRIV-Net on the semi-synthetic data. Indeed, we observe that MRIV-Net outperforms all other baselines, confirming both the superiority of our method as well as our theoretical results under sparsity assumptions from Sec.
Table Results for semi synthetic data.
Method
7 = 3000
n 5000
8000
(1) STANDARD ITE
TARNet (Shalit et al. 2017
1.66 1 0.11
1.58 1 0.07
1.57 1 0.11
TARNet + DR (Sallt
2017
Kennedy 2022
1.31 1. 0.28
1.22 1 0.37
1.12 1 0.15
(2) GENERAL IV
2SLS WWoolridge 2013
1.34 + 0.06
1.31 0.03
1.32 1 0.02
KIV (Singh el al.
2019
1.97 0.10
1.92 1 0.05
1.9340.00 0.05
DFIV (Xu al. 2021a
1.67 0.44
1.63 0.47
1.17
DeepIV (Hariford
2017
1.26
0.99
0.11
DeepGMM (Bennett et al. 2019
1.39+0.00
1.16
1.18 0.16
DMLIV (Syrgkanis et al. 2019)
0.10
2.09
2,02 - 0.11
DMLIV FDRIV (Syrgkanis et al. 2019
1.22 0.10
1.19
1.00 0.08
(3) WALD ESTIMATOR (WALD.
1940
Linear
1.42 + 0.24
1.28 0.07
1.32 + 0.07 1.06 0.13
BART
1.48 + 0.24
1.29 0.04
MRIV-Net (network only)
1.11 + 0.15
0.84 0.14
0.95 0.21
MRIV-Net (ours)
0.71 + 0.24
0.75 + 0.18
0.78 + 0.26
Reported: RMSE (mean + standard deviation). Lower better (best in bold)
31

---

Published as a conference paper at ICLR 2023
K RESULTS FOR CROSS-FITTING
Here, we repeat our experiments from the main paper but now make use of cross-fitring. Recall that, Theorem 2 we assume that the nuisance parameter estimation and the pseudo-outcome regression are performed on three independent samples. We now address this through cross-fitting. To this end, our aim is show that our proposed MRIV framework is again superior.
For MRIV, we proceeded as follows: We split the sample D into three equally sized samples D1. D2. and D3. We then trained finit (z), P (I)) and A (a:) on D1, & (z) and fi (z) on D2, and performed the pseudo-outcome regression on D3. Then, we repeated the same training procedure two times, but performed the pseudo-outcome regression on D2 and D1. Finally, averaged the resulting three CATE estimators. For DRIV, we implemented the cross-fitting procedure described in (Syrgkanis) ct al. 2019). For the DR-learer, followed [Kennedy] (2022).
The results are in Table S Importantly, the results confirm the effectiveness of our proposed MRIV. Overall, we find that our proposed MRIV outperforms DRIV for the vast majority of base methods when performing cross pross-fitting. Furthermore, MRIV-Net is highly competitive even when comparing it with the cross-fitted estimators. This shows that heuristic to learn separate representations instead of performing sample splits works in practice. In sum, the results confirm empirically that our MRIV is superior.
Table 8: Results for base methods with different meta-learners (i.e., DRIV, and our MRIV) using cross-fitting and results for MRIV-Net without cross- cross-fitting..
3000
2000
8000
Meta:
MRIV
MRIV
MRIV
STANDARD
TARNet (Shalit ot al
2017
0.30
0.02
0.36 + 0.16
0.18 = 0.0G
0.1G + 0.03
0.21 0.08
0.13 # 0.04
TARNet - DRCISsne Smilit ot a1. 2017 Kennedy 2022
0.80 = 0.11
1L06 0.08
0.67 +: f..12
(2) GENTRAL TV
Wonklddee 2015
0.42
0.33
0.07
0.23
S11 5039
DRIV
0.05
0.28 0.09
0.22 072-1:000
0.18 0.08
0.251:0.12
0.16::00 0.04
DesplV
HHatole od 2017
0.38 0.09
0.41 0.16
0.20 0.07
0.19 1: 0.07
0.20 0.08
0.12
0.02
DeepGMor Tecanelt et
2019
0.42 0.09
0.42 0.16
0.19 2: 0.04
0.19
1: 0.07
0.22 1 0.06
0.13
1: 0.02
DMLIV (Syrpkanis
0.44 0.09
0.16 0.116
0.21 1.004
0.19 1: 0.07
0.21 1.005
0.14
0.02
(WALD 1940
0.22
0.12
BART
0.12
0.39 -: 0.12
0.14 0.06
+ 0.05
0.08
MKIV-Net network only lours)
0.30 0.12
0.26 0.11
0.19
0.15 4: 0.03
0.18 4: 0.08
0.13 0.03
Reported:
deviation). Lower
32

---

Published as a conference paper at ICLR 2023
L FURTHER EXPERIMENTAL RESULTS ON REAL-WORLD DATA
we estimated the ITE on the OHIE data and visualized the treatment heterogeneity with respect to age and gender. In this section, we provide additional results and also visualize the heterogeneity with respect to age as well as additional covariates. These are: (I) the number of emergency visits a patient has in its history before signing up for the lottery and (2) the language spoken by the patient (English or other). We fixed the gender to "female".
In Section 5.2
For (I), we plot the estimated ITE for three different age groups over the number of emergency visits.
The results are shown in Fig. 10
age 50
ago - 60
age 70
ITS
Method
TARNet
0.5
network only
MRLV-Net (ours)
0.0
10.0 Number of emergency visits
2.5
5.0
7.5
12.5
2.5
7.5
10.0
12.5 0.0
2.5
5.0
1.5
10.0
32.5
Number of emergency visits
Numbber emergency visits
Figure 10: Estimated treatment effects for different age and number of emergency visits.
We observe that all methods tend to estimate a larger effect for individuals who had more emergency visits in their patient history. However, the IV methods (in particular MRIV-Net) estimate much larger effect for patients with many visits. In contrast to the other methods, MRIV-Net also estimates larger effects for older than for younger patients. The results provided by MRIV-Net seem intuitive, as older patients with history of emergency visits should be exposed to higher health-related risks, thus benefiting from health insurance. The fact that TARNet consistently estimates small (and even negative) effects could be an indicator of bias due to unobserved confounding.
For (2), we plot the estimated ITE for three different age groups over the spoken language. The
results are shown in Fig.
3g0 50
60
age 70
0.8
0.6
Method MRIV-Net (ours)
0.2
0.0
English
English
English
Language
Language
Language
Figure 11: Estimated treatment effects for different age and language
For patients of age 50, our MRIV-Net estimates a higher effect for the English-speaking patients. Interestingly, for older patients, the estimated effect increases also for non-English speaking patients.
33


Reviews:
[{"review_id": "uuBOXwykMH", "reviewer": "Reviewer_BewB", "summary": "", "questions": "", "limitations": "", "rating": 6, "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": "", "summary_of_the_paper": "The authors present a method for estimating the CATE in the presence of unobserved confounding when a binary IV is available.", "summary_of_the_review": "Overall I found this work to be a reasonable, if straightforward extension of CATE estimation methods to the IV model. I have the following concerns, but most are relatively minor and I recommend acceptance.\n\nAbstract:\n\n1. In partial compliance settings, treatment assignment (not compliance) is used as the IV for actual drug usage.\n\n2. I recommend being more specific \"asymptotically outperforms.\"\n\nSection 1:\n\n1. I recommend replacing the term \"Individual treatment effect\" with \"Conditional average treatment effect since \"individual treatment effect\" is used to mean something different in much of the causal inference literature.\n\n2. help in potentially --> helps in making potentially\n\n3. , --> .\n\n4. Race and gender are standard parts of intake forms and are both typically stored in medical records.\n\n5. \"Our framework...\": I found these two sentences hard to follow. I would focus on giving intuition here rather than jargon. For example, the meaning of \"multiply robust\" depends on the context. You should give intuition for what it is robust to why that is important.\n\n6. Registering for Medicaid should not be referred to as \"compliance\" even if the statistical model is the same as a partial compliance model. There is no mandate that someone needs to register if eligible. \n\n7. Assumption 2 is a *major* assumption that requires more discussion than it is given here. For example, if $U$ includes socioeconomic status, as suggested by the authors then this assumption implies that SES cannot moderate the treatment effect. \n\nSection 2:\n\n1. In Assumption 4: do these conditions need to hold everywhere on $\\mathcal{X}$ or is it sufficient for them to hold almost everywhere w.r.t. $P(X)$?\n\n2. Need to explicitly define $r_\\tau(n)$ or at least give intuition.\n\nSection 5:\n\n1. \"One explanation...\": I would refrain from making this hypothesis without including explicit citations to back up the underlying theoretical model.", "strength_and_weaknesses": "See below.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "sZ2pEmej6X", "reviewer": "Reviewer_ozYZ", "summary": "", "questions": "", "limitations": "", "rating": 6, "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": "", "summary_of_the_paper": "The paper studies the problem of estimating individual treatment effects (ITE) under unobserved confounding using binary instruments. The paper starts with discussing conditions under which the ITE is identifiable. It then proceeds with proposing a two-stage meta learner (MRIV) that estimates the ITE. In the first stage, nuisance components are estimated; and in the second stage, a pseudo outcome is regressed on covariates. The authors then establish theoretical properties of the proposed MRIV estimator and develop a tailored deep neural network using this framework. Finally, the authors demonstrate the performance of the proposed estimator through simulations and real data analysis. ", "summary_of_the_review": "Overall, I find this paper a good contribution to the literature. See the above sections for more detailed comments.  ", "strength_and_weaknesses": "Strengths\n\n-\tThe paper is very well-written. The proposed method is motivated and explained in a very nice way. \n\n-\tThe paper addresses an interesting and important question: estimating individual treatment effects under unobserved confounding.\n\n-\tThe proposed method is very elegant conceptually. \n\n-\tThe proposed method appears to have good performance both theoretically and empirically. \n\nWeaknesses\n\n-\tThere are a few grammar mistakes in the paper. \n\n-\tThe assumptions are not stated clearly in the theorems. \n\n-\tThe paper appears like a nice combination of the Wang & Tchetgen Tchetgen, 2018 paper and the Kennedy, 2022 paper, but there aren’t too many new ideas beyond what’s already in the two papers. It will be extremely helpful if the authors can emphasize a bit more on the new ideas. ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "YNQNLREKvvV", "reviewer": "Reviewer_B9xa", "summary": "", "questions": "", "limitations": "", "rating": 6, "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": "", "summary_of_the_paper": "This paper introduces a method of estimating individualized treatment effect (of binary treatments) with a multiple robust estimator. To this end, the authors build on previous work on ITE estimation as well as multiple robust models and extend the latter which results in a two-stage algorithm  (MRIV) which outputs a consistent multiple robust estimator of the ITE. In the first step of MRIV the nuisance parameters and the propensity score are estimated; in the second step the ITE estimator is computed by regressing the pseudo outcome on the confounders. The authors, building on existing approaches, show what the convergence rate of MRIV is, and compare it to the Wald estimator. Subsequently, a TARnet-like architecture of a deep network (but with two distinct representations) is proposed in order to estimate the nuisance parameters (this can be plugged in to step one of the MRIV procedure). The paper concludes with experiments on simulated and real data.", "summary_of_the_review": "This paper fills a small but relevant gap in causal inference by proposing a multiply robust estimator for the ITE. While the authors rely extensively on existing results, the proposed method is novel and and constitutes a valuable contribution to the field.", "strength_and_weaknesses": "The method presented in the paper is novel and relevant. The paper is intuitive and easy to follow.\n\nThe experiment on real data is limited and somewhat inconclusive.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "KmIG-H6055", "reviewer": "Reviewer_r18k", "summary": "", "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": "", "summary_of_the_paper": "This paper proposes an estimator for the conditional average treatment effect (CATE) when there are unobserved confounders. The estimator uses binary IV. This estimator is multiply robust (a more general property than the double robustness property), implying that the mean squared error of the estimated CATE decays with the sample size $n$ when one of the three conditions holds true. \n\n", "summary_of_the_review": "This paper provides a multiply robust estimator for CATE, and shows an upper bound for the MSE of the estimated CATE. The proposed estimator can be broadly applicable. I think the paper can be strengthened by addressing the questions listed in \"weaknesses\".", "strength_and_weaknesses": "Strength: \n\nThis paper provides a nice solution to the important problem of estimating CATE in the presence of unobserved confounders. The proposed method is easy-to-implement, enjoys the nice multiple robustness property, and is demonstrated to have low estimation error in the estimation of CATE as compared to a number of benchmarks. \n\nWeakness:\n\n1. The target estimand in this paper, $\\tau(x) = E[Y(1) - Y(0) \\mid X = x]$, is generally referred to as the conditional average treatment effect (CATE) in the literature. Calling it ITE is a bit confusing. \n\n2. In Theorem 1, the three conditions do not look correct to me. If $\\hat{\\mu}^Y_0$ refers to an estimator of $\\mu^Y_0$, $\\hat{\\mu}^Y_0$ should have some estimation error, and therefore it is impossible that $\\hat{\\mu}^Y_0= \\mu^Y_0$. Similarly for $\\hat{\\tau}$, $\\hat{\\tau}_{init}$, etc. For these three conditions, do the authors refer to the consistent or unbiased estimation of $\\mu^Y_0$ (and other parameters)?\n\n3. Given that the technical results are largely built on Kennedy (2022), it is useful to clarify the technical and other contributions as compared to Kennedy (2022).\n\n4. Theoretical results are about the MSE of $\\hat\\tau(x)$. Can the authors say something about whether $\\tau(x)$ can be consistently estimated for every $x$ (i.e., pointwise result)?", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Decision: Accept: poster