Title: Optimal Treatment Regimes for Proximal Causal Learning
Venue: NeurIPS 2023
Accepted: yes
Keywords: Optimal treatment regimes

Policy-making

Proximal causal inference

Unmeasured confounding

Value function
Abstract: A common concern when a policymaker draws causal inferences from and makes decisions based on observational data is that the measured covariates are insufficiently rich to account for all sources of confounding, i.e., the standard no confoundedness assumption fails to hold. The recently proposed proximal causal inference framework shows that proxy variables that abound in real-life scenarios can be leveraged to identify causal effects and therefore facilitate decision-making. Building upon this line of work, we propose a novel optimal individualized treatment regime based on so-called outcome and treatment confounding bridges. We then show that the value function of this new optimal treatment regime is superior to that of existing ones in the literature. Theoretical guarantees, including identification, superiority, excess value bound, and consistency of the estimated regime, are established. Furthermore, we demonstrate the proposed optimal regime via numerical experiments and a real data application.

Optimal Treatment Regimes for Proximal Causal Learning

Abstract

A common concern when a policymaker draws causal inferences from and makes decisions based on observational data is that the measured covariates are insufficiently rich to account for all sources of confounding, i.e., the standard no confoundedness assumption fails to hold. The recently proposed proximal causal inference framework shows that proxy variables that abound in real-life scenarios can be leveraged to identify causal effects and therefore facilitate decisionmaking. Building upon this line of work, we propose a novel optimal individualized treatment regime based on so-called outcome and treatment confounding bridges. We then show that the value function of this new optimal treatment regime is superior to that of existing ones in the literature. Theoretical guarantees, including identification, superiority, excess value bound, and consistency of the estimated regime, are established. Furthermore, we demonstrate the proposed optimal regime via numerical experiments and a real data application.

1 Introduction

Data-driven individualized decision-making has received tremendous attention nowadays due to its applications in healthcare, economics, marketing, etc. A large branch of work has focused on maximizing the expected utility of implementing the estimated optimal policy over a target population based on randomized controlled trials or observational studies, e.g., Athey and Wager (2021); Chakraborty and Moodie (2013); Jiang et al. (2019); Kitagawa and Tetenov (2018); Kosorok and Laber (2019); Murphy (2003); Qian and Murphy (2011); Robins (1986, 1994, 1997); Tsiatis et al. (2019); Wu et al. (2019); Zhao et al. (2012, 2019).

A critical assumption commonly made in these studies, known as unconfoundedness or exchangeability, precludes the existence of unmeasured confounding. Relying on an assumed ability of the decision-maker to accurately measure covariates relevant to a variety of confounding mechanisms present in a given observational study, causal effects, value functions, and other relevant quantities can be nonparametrically identified. However, such an assumption might not always be realistic in observational studies or randomized trials subject to non-compliance (Robins, 1994, 1997). Therefore, it is of great interest in recovering confounding mechanisms from measured covariates to infer causal effects and facilitate decision-making. A prevailing strand of work has been devoted to using instrumental variable (Angrist et al., 1996; Imbens and Angrist, 1994) as a proxy variable in dynamic treatment regimes and reinforcement learning settings (Cui, 2021; Cui and Tchetgen Tchetgen, 2021b,a; Han, 2023; Liao et al., 2021; Pu and Zhang, 2021; Qiu et al., 2021; Stensrud and Sarvet, 2022).

Recently, Tchetgen Tchetgen et al. proposed the so-called proximal causal inference framework, a formal potential outcome framework for proximal causal learning, which while explicitly acknowledging covariate measurements as imperfect proxies of confounding mechanisms, establishes causal

identification in settings where exchangeability on the basis of measured covariates fails. Rather than as current practice dictates, assuming that adjusting for all measured covariates, unconfoundedness can be attained, proximal causal inference essentially requires that the investigator can correctly classify a subset of measured covariates L ∈ L into three types: i) variables X ∈ X that may be common causes of the treatment and outcome variables; ii) treatment-inducing confounding proxies Z ∈ Z ; and iii) outcome-inducing confounding proxies W ∈ W .

There is a fast-growing literature on proximal causal inference since it has been proposed (Cui et al., 2023; Dukes et al., 2023; Ghassami et al., 2023; Kompa et al., 2022; Li et al., 2023; Mastouri et al., 2021; Miao et al., 2018b; Shi et al., 2020b, 2021; Shpitser et al., 2023; Singh, 2020; Tchetgen Tchetgen et al., 2020; Ying et al., 2023, 2022 and many others). In particular, Miao et al. (2018a); Tchetgen Tchetgen et al. (2020) propose identification of causal effects through an outcome confounding bridge and Cui et al. (2023) propose identification through a treatment confounding bridge. Adoubly robust estimation strategy (Chernozhukov et al., 2018; Robins et al., 1994; Rotnitzky et al., 1998; Scharfstein et al., 1999) is further proposed in Cui et al. (2023). In addition, Ghassami et al. (2022) and Kallus et al. (2021) propose a nonparametric estimation of causal effects through a minmax approach. Moreover, by adopting the proximal causal inference framework, Qi et al. (2023) consider optimal individualized treatment regimes (ITRs) estimation, Sverdrup and Cui (2023) consider learning heterogeneous treatment effects, and Bennett and Kallus (2023) consider off-policy evaluation in partially observed Markov decision processes.

In this paper, we aim to estimate optimal ITRs under the framework of proximal causal inference. We start with reviewing two in-class ITRs that map from X ×W to A and X ×Z to A , respectively, where A denotes the binary treatment space. The identification of value function and the learning strategy for these two optimal in-class ITRs are proposed in Qi et al. (2023). In addition, Qi et al. (2023) also consider a maximum proximal learning optimal ITR that maps from X × W × Z to A with the ITRs being restricted to either X ×W → A or X ×Z → A . In contrast to their maximum proximal learning ITR, in this paper, we propose a brand new policy class whose ITRs map from measured covariates X × W × Z to A , which incorporates the predilection between these two inclass ITRs. Identification and superiority of the proposed optimal ITRs compared to existing ones are further established.

The main contributions of our work are four-fold. Firstly, by leveraging treatment and outcome confounding bridges under the recently proposed proximal causal inference framework, identification results regarding the proposed class D Π ZW of ITRs that map X × W × Z to A are established. The proposed ITR class can be viewed as a generalization of existing ITR classes proposed in the literature. Secondly, an optimal subclass of D Π ZW is further introduced. Learning optimal treatment regimes within this subclass leads to a superior value function. Thirdly, we propose a learning approach to estimating the proposed optimal ITR. Our learning pipeline begins with the estimation of confounding bridges adopting the deep neural network method proposed by Kompa et al. (2022). Then we use optimal treatment regimes proposed in Qi et al. (2023) as preliminary regimes to estimate our optimal ITR. Lastly, we establish an excess value bound for the value difference between the estimated treatment regime and existing ones in the literature, and the consistency of the estimated regime is also demonstrated.

2 Methodology

2.1 Optimal individualized treatment regimes

We briefly introduce some conventional notation for learning optimal ITRs. Suppose A is a binary variable representing a treatment option that takes values in the treatment space A = {-1 , 1 } . Let L ∈ L be a vector of observed covariates, and Y be the outcome of interest. Let Y (1) and Y ( -1) be the potential outcomes under an intervention that sets the treatment to values 1 and -1 , respectively. Without loss of generality, we assume that larger values of Y are preferred.

Suppose the following standard causal assumptions hold: (1) Consistency: Y = Y ( A ) . That is, the observed outcome matches the potential outcome under the realized treatment. (2) Positivity: P ( A = a | L ) > 0 for a ∈ A almost surely, i.e., both treatments are possible to be assigned.

Weconsider an ITR class D containing ITRs that are measurable functions mapping from the covariate space L onto the treatment space A . For any d ∈ D , the potential outcome under a hypothetical

intervention that assigns treatment according to d is defined as

where I {·} denotes the indicator function. The value function of ITR d is defined as the expectation of the potential outcome, i.e.,

It can be easily seen that an optimal ITR can be expressed as

There are many ways to identify optimal ITRs under different sets of assumptions. The most commonly seen assumption is the unconfoundedness: Y ( a ) ⊥ A | L for a = ± 1 , i.e., upon conditioning on L , there is no unmeasured confounder affecting both A and Y . Under this unconfoundedness assumption, the value function of a given regime d can be identified by (Qian and Murphy, 2011)

where f ( A | L ) denotes the propensity score (Rosenbaum and Rubin, 1983), and the optimal ITR is identified by

We refer to Qian and Murphy (2011); Zhang et al. (2012); Zhao et al. (2012) for more details of learning optimal ITRs in this unconfounded setting.

Because confounding by unmeasured factors cannot generally be ruled out with certainty in observational studies or randomized experiments subject to non-compliance, skepticism about the unconfoundedness assumption in observational studies is often warranted. To estimate optimal ITRs subject to potential unmeasured confounding, Cui and Tchetgen Tchetgen (2021b) propose instrumental variable approaches to learning optimal ITRs. Under certain instrumental variable assumptions, the optimal ITR can be identified by

where M denotes a valid binary instrumental variable. Other works including Cui (2021); Cui and Tchetgen Tchetgen (2021a); Han (2023); Pu and Zhang (2021) consider a sign or partial identification of causal effects to estimate suboptimal ITRs using instrumental variables.

2.2 Existing optimal ITRs for proximal causal inference

Another line of research in causal inference considers negative control variables as proxies to mitigate confounding bias (Kuroki and Pearl, 2014; Miao et al., 2018a; Shi et al., 2020a; Tchetgen Tchetgen, 2014). Recently, a formal potential outcome framework, namely proximal causal inference, has been developed by Tchetgen Tchetgen et al. (2020), which has attracted tremendous attention since proposed.

Following the proximal causal inference framework proposed in Tchetgen Tchetgen et al. (2020), suppose that the measured covariate L can be decomposed into three buckets L = ( X,W,Z ) , where X ∈ X affects both A and Y , W ∈ W denotes an outcome-inducing confounding proxy that is a potential cause of the outcome which is related with the treatment only through ( U, X ) , and Z ∈ Z is a treatment-inducing confounding proxy that is a potential cause of the treatment which is related with the outcome Y through ( U, X, A ) . We now summarize several basic assumptions of the proximal causal inference framework.

or

Assumption 1. We make the following assumptions:

(1) Consistency: Y = Y ( A,Z ) , W = W ( A,Z ) .

(2) Positivity: P ( A = a | U, X ) > 0 , ∀ a ∈ A .

(3) Latent unconfoundedness:

( Z, A ) ⊥ ( Y ( a ) , W ) | U, X, ∀ a ∈ A .

The consistency and positivity assumptions are conventional in the causal inference literature. The latent unconfoundedness essentially states that Z cannot directly affect the outcome Y , and W is not directly affected by either A or Z . Figure 1 depicts a classical setting that satisfies Assumption 1. We refer to Shi et al. (2020b); Tchetgen Tchetgen et al. (2020) for other realistic settings for proximal causal inference.

X

Z

W

U

A

Y

Wefirst consider two in-class optimal ITRs that map from X×Z to A and X×W to A , respectively. To identify optimal ITRs that map from X × Z to A , we make the following assumptions.

Assumption 2. Completeness: For any a ∈ A , x ∈ X and square-integrable function g , E [ g ( U ) | Z, A = a, X = x ] = 0 almost surely if and only if g ( U ) = 0 almost surely.

Assumption 3. Existence of outcome confounding bridge: There exists an outcome confounding bridge function h ( w,a, x ) that solves the following equation

almost surely.

The completeness Assumption 2 is a technical condition central to the study of sufficiency in foundational theory of statistical inference. It essentially assumes that Z has sufficient variability with respect to the variability of U . Werefer to Tchetgen Tchetgen et al. (2020) and Miao et al. (2022) for further discussions regarding the completeness condition. Assumption 3 defines a so-called inverse problem known as a Fredholm integral equation of the first kind through an outcome confounding bridge. The technical conditions for the existence of a solution to a Fredholm integral equation can be found in Kress et al. (1989).

Let D Z be an ITR class that includes all measurable functions mapping from X ×Z to A . As shown in Qi et al. (2023), under Assumptions 1, 2 and 3, for any d z ∈ D Z , the value function V ( d z ) can be nonparametrically identified by

Furthermore, the in-class optimal treatment regime d ∗ z ∈ arg max d z ∈D Z V ( d z ) is given by

On the other hand, to identify optimal ITRs that map from X × W to A , we make the following assumptions.

Assumption 4. Completeness: For any a ∈ A , x ∈ X and square-integrable function g , E [ g ( U ) | W,A = a, X = x ] = 0 almost surely if and only if g ( U ) = 0 almost surely.

Assumption 5. Existence of treatment confounding bridge: There exists a treatment confounding bridge function q ( z, a, x ) that solves the following equation

almost surely.

Similar to Assumptions 2 and 3, Assumption 4 assumes that W has sufficient variability relative to the variability of U , and Assumption 5 defines another Fredholm integral equation of the first kind through a treatment confounding bridge q .

Let D W be another ITR class that includes all measurable functions mapping from X × W to A . As shown in Qi et al. (2023), under Assumptions 1, 4 and 5, for any d w ∈ D W , the value function V ( d w ) can be nonparametrically identified by

The in-class optimal treatment regime d ∗ w ∈ arg max d w ∈D W V ( d w ) is given by

Moreover, Qi et al. (2023) consider the ITR class D Z ∪ D W and propose a maximum proximal learning optimal regime based on this ITR class. For any d z ∪ w ∈ D Z ∪D W , under Assumptions 15, the value function V ( d z ∪ w ) for any d z ∪ w ∈ D Z ∪ D W can be identified by

The optimal ITR within this class is given by d ∗ z ∪ w ∈ arg max d z ∪ w ∈D Z ∪D W V ( d z ∪ w ) , and they show that the corresponding optimal value function takes the maximum value between two optimal in-class ITRs, i.e.,

2.3 Optimal decision-making based on two confounding bridges

As discussed in the previous section, given that neither E [ Y ( a ) | X,U ] nor E [ Y ( a ) | X,W,Z ] for any a ∈ A may be identifiable under the proximal causal inference setting, one might nevertheless consider ITRs mapping from X ×W to A , from X ×Z to A , from X ×W×Z to A as well as from X to A . Intuitively, policy-makers might want to use as much information as they can to facilitate their decision-making. Therefore, ITRs mapping from X × W × Z to A are of great interest if information regarding ( X,W,Z ) is available.

As a result, a natural question arises: is there an ITR mapping from X × W × Z to A which dominates existing ITRs proposed in the literature? In this section, we answer this question by proposing a novel optimal ITR and showing its superiority in terms of global welfare.

We first consider the following class of ITRs that map from X × W × Z to A ,

where Π is the policy class containing all measurable functions π : X → { 0 , 1 } that indicate the individualized predilection between d z and d w .

Remark 1. Note that D Z , D W and D Z ∪D W are subsets of D Π ZW with a particular choice of π . For example, D Z is D Π ZW with restriction on π ( X ) = 1 ; D W is D Π ZW with restriction on π ( X ) = 0 ; D Z ∪ D W is D Π ZW with restriction on π ( X ) = 1 or π ( X ) = 0 .

In the following theorem, we demonstrate that by leveraging the treatment and outcome confounding bridge functions, we can nonparametrically identify the value function over the policy class D Π ZW , i.e., V ( d π zw ) for d π zw ∈ D Π ZW .

Theorem 1. Under Assumptions 1-5, for any d π zw ∈ D Π ZW , the value function V ( d π zw ) can be nonparametrically identified by

One of the key ingredients of our constructed new policy class D Π ZW is the choice of π ( · ) . It suggests an individualized strategy for treatment decisions between the two given treatment regimes. Because we are interested in policy learning, a suitable choice of π ( · ) that leads to a larger value function is more desirable. Therefore, we construct the following ¯ π ( X ; d z , d w ) ,

In addition, given any d z ∈ D Z and d w ∈ D W , we define

We then obtain the following result which justifies the superiority of ¯ π .

Theorem 2. Under Assumptions 1-5, for any d z ∈ D Z and d w ∈ D W ,

Theorem 2 establishes that for the particular choice of ¯ π given in (5), the value function of d ¯ π zw is no smaller than that of d z and d w for any d z ∈ D Z , and d w ∈ D W . Consequently, Theorem 2 holds for d ∗ z and d ∗ w . Hence, we propose the following optimal ITR d ¯ π ∗ zw ,

and we have the following corollary.

Corollary 1. Under Assumptions 1-5, we have that

Corollary 1 essentially states that the value of d ¯ π ∗ zw dominates that of d ∗ z , d ∗ w , as well as d ∗ z ∪ w . Moreover, the proposition below demonstrates the optimality of d ¯ π ∗ zw within the proposed class.

Proposition 1. Under Assumptions 1-5, we have that

Therefore, d ¯ π ∗ zw is an optimal ITR of policymakers' interest.

3 Statistical Learning and Optimization

3.1 Estimation of the optimal ITR d ¯ π ∗ zw

The estimation of d ¯ π ∗ zw consists of four steps: (i) estimation of confounding bridges h and q ; (ii) estimation of preliminary ITRs d ∗ z and d ∗ w ; (iii) estimation of ¯ π ( X ; d ∗ z , d ∗ w ) ; and (iv) learning d ¯ π ∗ zw based on (ii) and (iii). The estimation problem (i) has been developed by Cui et al. (2023); Miao et al. (2018b) using the generalized method of moments, Ghassami et al. (2022); Kallus et al. (2021) by a min-max estimation (Dikkala et al., 2020) using kernels, and Kompa et al. (2022) using deep learning; and (ii) has been developed by Qi et al. (2023). We restate estimation of (i) and (ii) for completeness. With regard to (i), recall that Assumptions 3 and 5 imply the following conditional moment restrictions

respectively. Kompa et al. (2022) propose a deep neural network approach to estimating bridge functions which avoids the reliance on kernel methods. We adopt this approach in our simulation and details can be found in the Appendix.

To estimate d ∗ z , we consider classification-based approaches according to Zhang et al. (2012); Zhao et al. (2012). Under Assumptions 1, 2 and 3, maximizing the value function in (1) is equivalent to minimizing the following classification error

̸

over d z ∈ D Z . By choosing some measurable decision function g z ∈ G Z : X × Z → R , we let d z ( X,Z ) = sign ( g z ( X,Z )) . We consider the following empirical version of (6),

Due to the non-convexity and non-smoothness of the sign operator, we replace the sign operator with a smooth surrogate function and adopt the hinge loss ϕ ( x ) = max { 1 -x, 0 } . By adding a penalty term ρ z || g z || 2 G Z to avoid overfitting, we solve

where ρ z > 0 is a tuning parameter. The estimated ITR then follows ˆ d z ( X,Z ) = sign (ˆ g z ( X,Z )) . Similarly, under Assumptions 1, 4 and 5, maximizing the value function in (2) is equivalent to minimizing the following classification error

̸

over d w ∈ D W . By the same token, the problem is transformed into minimizing the following empirical error

The estimated ITR is obtained via ˆ d w ( X,W ) = sign (ˆ g w ( X,W )) .

For problem (iii), given two preliminary ITRs, we construct an estimator ˆ π ( X ; ˆ d z , ˆ d w ) , that is, for x ∈ X ,

where ˆ δ ( x ; ˆ d z , ˆ d w ) denotes a generic estimator of

where the expectation is taken with respect to everything except ˆ d z and ˆ d w . For example, the Nadaraya-Watson kernel regression estimator (Nadaraya, 1964) can be used, i.e., ˆ δ ( x ; ˆ d z , ˆ d w ) is expressed as

where K : R → R is a kernel function such as Gaussian kernel, || · || 2 denotes the L 2 -norm, and γ denotes the bandwidth.

Finally, given ˆ d z , ˆ d w and ˆ π ( X ; ˆ d z , ˆ d w ) , ˆ d ˆ π zw is estimated by the following plug-in regime,

3.2 Theoretical guarantees for ˆ d ˆ π zw

In this subsection, we first present an optimality guarantee for the estimated ITR ˆ d ˆ π zw in terms of its value function

where the expectation is taken with respect to everything except ˆ π , ˆ d z and ˆ d w .

We define an oracle optimal ITR which assumes ¯ π ( X ; ˆ d z , ˆ d w ) is known,

The corresponding value function of this oracle optimal ITR is given by

where the expectation is taken with respect to everything except ˆ d z and ˆ d w .

Then the approximation error incurred by estimating ˆ π ( X ; ˆ d z , ˆ d w ) is given by

Moreover, we define the following gain

It is clear that this gain G (¯ π ) by introducing ¯ π is always non-negative as indicated by Theorem 2. Then we have the following excess value bound for the value of ˆ d ˆ π zw compared to existing ones in the literature.

Proposition 2. Under Assumptions 1-5,

Proposition 2 establishes a link between the value function of the estimated ITR ˆ d ˆ π zw , and that of ˆ d z , ˆ d w , and ˆ d z ∪ w . As shown in Appendix G, K (ˆ π ) diminishes as the sample size increases, therefore, ˆ d ˆ π zw has a significant improvement compared to other optimal ITRs depending on the magnitude of G (¯ π ) .

Furthermore, we establish the consistency of the proposed regime based on the following assumption, which holds for example when ˆ d z and ˆ d w are estimated using indirect methods.

Assumption 6. For ˆ d z , ˆ d w , E [ h ( W, ˆ d z ( X,Z ) , X ) | X ] -E [ h ( W,d ∗ z ( X,Z ) , X ) | X ] = o p ( n -ξ ) almost surely and E [ Y q ( Z, A, X ) I { ˆ d w ( X,W ) = A }| X ] -E [ Y q ( Z, A, X ) I { d ∗ w ( X,W ) = A }| X ] = o p ( n -φ ) almost surely.

Proposition 3. Under Assumptions 1-6, we have V ( ˆ d ˆ π zw ) p - → V ( d ¯ π ∗ zw ) .

4 Numerical Experiments

The data generating mechanism for ( X,A,Z,W,U ) follows the setup proposed in Cui et al. (2023) and is summarized in Appendix I. To evaluate the performance of the proposed framework, we vary b 1 ( X ) , b 2 ( X ) , b 3 ( X ) , b a and b w in E [ Y | X,A,Z,W,U ] to incorporate heterogeneous treatment effects including the settings considered in Qi et al. (2023). The adopted data generating mechanism is compatible with the following h and q ,

where t 0 = 0 . 25 , t z = -0 . 5 , t a = -0 . 125 , and t x = (0 . 25 , 0 . 25) T . We derive preliminary optimal ITRs d ∗ z and d ∗ w in Appendix J, from which we can see that X,Z,W are relevant variables for individualized decision-making.

We consider six scenarios in total, and the setups of varying parameters are deferred to Appendix I. For each scenario, training datasets { Y i , A i , X i , Z i , W i } n i =1 are generated following the above mechanism with a sample size n = 1000 . For each training dataset, we then apply the aforementioned methods to learn the optimal ITR. In particular, the preliminary ITRs ˆ d z and ˆ d w are estimated using a linear decision rule, and ˆ π ( x ; ˆ d z , ˆ d w ) is estimated using a Gaussian kernel. More details can be found in the Appendix K.

To evaluate the estimated treatment regimes, we consider the following generating mechanism for testing datasets: X ∼ N (Γ x , Σ x ) ,

where the parameter settings can be found in Appendix I. The testing dataset is generated with a size 10000 , and the empirical value function for the estimated ITR is used as a performance measure. The simulations are replicated 200 times. To validate our approach and demonstrate its superiority, we have also computed empirical values for other optimal policies, including existing optimal ITRs for proximal causal inference, as discussed in Section 2.2, along with optimal ITRs generated through causal forest (Athey and Wager, 2019) and outcome weighted learning (Zhao et al., 2012).

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Figure 2 presents the empirical value functions of different optimal ITRs for the six scenarios. As expected, ˆ d z , ˆ d w , ˆ d z ∪ w , and ˆ d ˆ π zw consistently outperform ˆ d cf and ˆ d owl , which highlights their effectiveness in addressing unmeasured confounding. Meanwhile, across all scenarios, ˆ d ˆ π zw yields superior or comparable performance compared to the other estimated treatment regimes, which justifies the statements made in Sections 2 and 3. In addition, as can be seen in Scenario 5, all ITRs relying on the proximal causal inference framework perform similarly, which is not surprising as ˆ d z ( X,Z ) and ˆ d w ( X,W ) agree for most subjects. To further underscore the robust performance of our proposed approach, we include additional results with a changed sample size and a modified behavior policy in Appendix L.

5 Real Data Application

In this section, we demonstrate the proposed optimal ITR via a real dataset originally designed to measure the effectiveness of right heart catheterization (RHC) for ill patients in intensive care units (ICU), under the Study to Understand Prognoses and Preferences for Outcomes and Risks of Treatments (SUPPORT, Connors et al. (1996)). These data have been re-analyzed in a number of papers in both causal inference and survival analysis literature with assuming unconfoundednss (Cui and Tchetgen Tchetgen, 2023; Tan, 2006, 2020, 2019; Vermeulen and Vansteelandt, 2015) or accounting for unmeasured confounding (Cui et al., 2023; Lin et al., 1998; Qi et al., 2023; Tchetgen Tchetgen et al., 2020; Ying et al., 2022).

There are 5735 subjects included in the dataset, in which 2184 were treated (with A = 1 ) and 3551 were untreated (with A = -1 ). The outcome Y is the duration from admission to death or censoring. Overall, 3817 patients survived and 1918 died within 30 days. Following Tchetgen Tchetgen et al. (2020), we collect 71 covariates including demographic factors, diagnostic information, estimated survival probability, comorbidity, vital signs, physiological status, and functional status (see

Hirano and Imbens (2001) for additional discussion on covariates). Confounding in this study stems from the fact that ten physiological status measures obtained from blood tests conducted at the initial phase of admission may be susceptible to significant measurement errors. Furthermore, besides the lab measurement errors, whether other unmeasured confounding factors exist is unknown to the data analyst. Because variables measured from these tests offer only a single snapshot of the underlying physiological condition, they have the potential to act as confounding proxies. We consider a total of four settings, varying the number of selected proxies from 4 to 10. Within each setting, treatment-inducing proxies are first selected based on their strength of association with the treatment (determined through logistic regression of A on L ), and outcome-inducing proxies are then chosen based on their association with the outcome (determined through linear regression of Y on A and L ). Excluding the selected proxy variables, other measured covariates are included in X . We then estimate ˆ d z , ˆ d w , ˆ d z ∪ w , and ˆ d ˆ π zw using the SUPPORT dataset in a manner similar to that described in Section 4, with the goal optimizing the patients' 30-day survival after their entrance into the ICU.

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The estimated value functions of our proposed ITR, alongside existing ones, are summarized in Appendix M. As can be seen, our proposed regime has the largest value among all settings. For a visual representation of the concordance between the estimated optimal ITRs, we refer to Figure 3 (results from Setting 1). The horizontal ordinate represents the 50 selected subjects and the vertical axis denotes the decisions made from corresponding ITRs. The purple and yellow blocks stand for being recommended treatment values of -1 and 1 respectively. For the subjects with purple or yellow columns, ˆ d z ( X,Z ) = ˆ d w ( X,W ) , which leads to the same treatment decision for the other two ITRs. For columns with mixed colors, ˆ d z ( X,Z ) and ˆ d w ( X,W ) disagree. We see that in this case ˆ d z ∪ w ( X,W,Z ) always agree with ˆ d z ( X,Z ) , while ˆ d ˆ π zw ( X,W,Z ) take values from ˆ d z ( X,Z ) or ˆ d w ( X,W ) depending on the individual criteria of the subjects as indicated by ˆ π . In addition to the quantitative analysis, we have also conducted a qualitative assessment of the estimated regime to validate its performance. For further details, please refer to Appendix M.

6 Discussion

We acknowledge several limitations of our work. Firstly, the proximal causal inference framework relies on the validity of treatment- and outcome-inducing confounding proxies. When the assumptions are violated, the proximal causal inference estimators can be biased even if unconfoundedness on the basis of measured covariates in fact holds. Therefore, one needs to carefully sort out proxies especially when domain knowledge is lacking. Secondly, while the proposed regime significantly improves upon existing methods both theoretically and numerically, it is not yet shown to be the sharpest under our considered model. It is still an open question to figure out if a more general policy class could be considered. Thirdly, our established theory provides consistency and superiority of our estimated regime. It is of great interest to derive convergence rates for Propositions 2 and 3 following Jiang (2017). In addition, it may be challenging to develop inference results for the value function of the estimated optimal treatment regimes, and further studies are warranted.

Acknowledgments and Disclosure of Funding

Yifan Cui was supported by the National Natural Science Foundation of China.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors present a new optimal individual treatment regime (ITR) within the proximal causal inference framework, which avoids the strong assumption of no unmeasured confounding. Instead, one assumes the effect of the unmeasured confounders flows exclusively through proxy variables, as defined through outcome-inducing and treatment-inducing confounding bridges. Compared to prior work, this optimal ITR that is defined with respect to a more flexible function class that depends on known confounders X, treatment-inducing confounding proxies Z, and outcome-inducing confounding proxies W.\n\nSTRENGTHS:\nThe proposed ITR is a natural extension of existing ITRs by using a function \\pi(x) that selectively chooses between two existing ITRs based on known confounders x. Under the proximal causal inference frameowrk, the proposed ITR is proven to be superior to existing ITRs in the literature. (Existing ITRs from Qi 2023 can be viewed as special cases of the proposed ITR.) The authors introduce a simple plugin estimator for the proposed ITR and show that the value of the resulting estimator is determined by approximation error of \\pi and the gain from using \\pi. Simulation studies show that the proposed ITR is either superior or comparable to existing ITRs. The manuscript is clearly written. The authors provide a nice review of prior work in this area and clearly describe how their work builds on existing work.\n\nWEAKNESSES:\n1. The proposed extension of the ITR function class appears quite incremental. The value of the proposed ITR follows directly from application of the tower rule. The paper would be greatly strengthened if the authors can show that this is the best one can do, e.g. showing that the value of a more complex ITR function class would be unidentifiable without much stronger assumptions.\n\n2. In the simulation studies, the improvement in mean value when using the proposed optimal ITR over existing ITRs is large only in scenario 2. In all other scenarios, the improvement is small. Can the authors explain the behavior in this simulation study? Also, can the authors explain settings in which the proposed ITR is expected to substantially improve over existing ITRs? My guess is that the gain is biggest when (i) there are large differences between expected value at each X for the ITR with domain (X,W) and the ITR with domain (X,Z) and (ii) the optimal pi function has high variance (e.g. pi(X) is equal to 1 half of the time). Does this correspond to scenario 2?\n\n3. The authors perform a real-data analysis in Section 5, which illustrates how the proposed ITR is different from existing ITRs. However, the authors do not calculate the values of the estimated ITR, so readers cannot compare the performance of the proposed ITR against existing ITRs. Do the authors have estimates of the values of the estimated ITRs?\n\n4. The number of treatment-inducing confounding proxies and outcome-inducing confounding proxies were small in both the simulation studies and real-world data analysis. However, the practical appeal of the proximal causal inference framework is its use of proxies for unmeasured confounders, which would suggest the use of many variables as potential proxies. Can the authors include simulations that reflect more realistic settings where more proxy variables are used? How does the proposed method perform as the number of these proxies increases?\n\nQUESTIONS:\n1. How does one do statistical inference for the value of the estimated ITR?\n\nLIMITATIONS:\nThe authors have not discussed limitations of the work.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 5, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2023-07-02"}

comments: SUMMARY:
The goal is to learn an optimal individual treatment rule (ITR) where the data suffer from unobserved confounding but where the researcher has a treatment proxy and an outcome proxy. 

While the general problem has been studied before by Qi et al (JASA 2023), this paper’s contribution is to broaden the class of ITRs. For a broader class of ITRs, the authors identify the value function and show that it exceeds the value function of the narrower class.

STRENGTHS:
My comments are brief because this is a strong paper.

Originality: The essence of the improvement is that, for different covariate values x, one may either use the “outcome” ITR or the “treatment” ITR. This departs from previous work, where either the “outcome” ITR or the “treatment” ITR is used across covariate values.

Quality: The proofs look correct, and the results are easy to interpret. Rates for the objects in Proposition 1 would be an improvement; see the question below.

Clarity: The paper is well written and well referenced.

Significance: The paper contributes to two popular literatures: proxies and ITRs. While its theoretical contribution is modest, it does appear to have practical relevance.

WEAKNESSES:
The theoretical contribution is somewhat incremental.

Take a pass to fix typos, e.g. “netwrok” on line 73.

An extra sentence in Remark 1 would be welcome, that explains the point summarized as “originality” above.

QUESTIONS:
Is it possible to derive rates for K(pi_hat) and G(pi_bar) in Proposition 1? Or can the authors at least pose this question for future work and give citations of where similar results are derived?

LIMITATIONS:
Yes
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 4
DATE: 2023-07-05

comments: SUMMARY:
The paper discusses the optimization of treatment rules in the context of observational data and under assumptions of proximal inference. Various theorems are introduced, and a real data analysis performed using a healthcare example.

STRENGTHS:
Below is a list of perceived weaknesses. 

The paper is overall sound and the topic of importance. I appreciate the presence of the real data application. Assumptions and results clearly stated.

WEAKNESSES:
Below is a list of perceived weaknesses. 

It was not clear to me how the empirical results compare to competing methodological baselines from other approaches (I don't believe the different values presented in the figure represent different algorithmic approaches). 

The paper is quite heavy on notation and, at least to me, light on intuitive explanation for findings as they are discussed, limiting insight into the inner workings of why the method works.

I don't know the proximal causal inference literature well so am not well-positioned to discuss the contribution in that subfield of causal inference. 

I don't see a discussion of uncertainty estimation in the theoretical or empirical results. Uncertainty estimation in optimized treatment effect regimes can be difficult (e.g., the bootstrap may not be appropriate or may have poor coverage) but may be important to usefulness in practice.

QUESTIONS:
Is there quantitative evidence that "proximal causes abound in real life scenarios"? The matter would seem dependent on substance area, or knowing this would seem to require access to nature's true DAG.

LIMITATIONS:
I see no ethnical limitations here.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2023-07-06

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nMost estimation methods for individualized treatment rules (ITRs) assume no unmeasured confounders for valid causal inference. However, such an assumption can be unreasonable, such as when estimating ITRs from observational data. Previous work has applied proximal causal inference to estimate ITRs when this assumption is violated, but is restricted to policy classes that either exclude treatment-inducing confounding proxies or exclude outcome-inducing confounding proxies [1]. To this end, the authors propose estimating a stochastic mixture of both policy classes from [1] to yield a more flexible ITR. Theoretical and simulation results demonstrate the superiority of the proposed method.\n\nSTRENGTHS:\nThe assumption of no unmeasured confounders is nearly ubiquitous across ITR estimation methods, despite frequent violations when dealing with observational data. This makes the problem the authors are trying to solve - estimating ITRs when this assumption is violated - very significant. Moreover, the theoretical and simulation results are of sufficient quality to convince me that the method outperforms [1], the existing state-of-the-art in proximal learning.\n\nWEAKNESSES:\nWhile I believe the merits and potential contribution of the paper outweigh its limitations, the theoretical and empirical results of the paper are weaker than that of previous work, and the clarity of the paper can be improved. I go into more detail below:\n\n1. **Theoretical guarantees are much weaker than those of previous methods.** Convergence rates, finite-sample error bounds or asymptotic normality is often derived for ITR methods [2-5], including the method this work seeks to improve over [1]. However, while the authors prove that the proposed method will converge to a policy with better value than that of [1] asymptotically, they do not derive a rate of convergence or establish any finite-sample error bounds. Moreover, the asymptotic analysis from Appendix G assumes convergence of several estimators in $L\\_\\infty$ space. This is a much stronger assumption than the assumptions made in previous work, which only assumes convergence of estimators in $L\\_2$ space (e.g. see Assumption 12 from [1] or Condition B5 from [3]).\n\n2. **The real data analysis does not strengthen the validity of the proposed method.** When applying ITR estimators to real datasets, it is common practice to assess performance by using either (1) an estimator of the value [2-6] or (2) arguments based on domain knowledge that support the validity of the proposed method [1,6]. For example, in [1], the authors argue that the estimated policy is accurate by demonstrating that the estimated coefficients and interpretation of the policy is consistent with findings from previous literature. In contrast, the only conclusions that these authors draw from their real data analysis is that the proposed estimator gives different results than previously proposed methods. Such a conclusion says little about the validity or superiority of the proposed method. One way this analysis could be greatly improved is to look at the patients from Figure 3 where the recommended treatment differs between the proposed method and that of [1], and use domain knowledge or previous literature to argue that the recommendations given by the proposed method is more sound. Alternatively, the authors could make this conclusion by comparing the coefficients of the proposed policy with that of [1]. \n\n3. **Empirical comparisons were relatively limited.** The authors only benchmark the proposed method against those of a single previous work, [1]. To conclude that the proposed method achieves state-of-the-art performance, the authors should benchmark the proposed method against additional baselines as well. For example, there are many methods that assume no unmeasured confounders which the authors could evaluate to demonstrate the utility of using a proximal causal inference framework (e.g. [7,8]). There are also other methods that relax the no unmeasured confounders assumption or have shown robustness when such assumptions are violated, such as instrument variable (IV) methods [3,9] and M-learning [4]. How are the assumptions made by proximal learning less restrictive than those made by IV approaches, and can such methods be applied to the simulated datasets? If so, the authors should benchmark the proposed method against these method. And if these methods are not applicable, the authors should explain why in the paper. In addition to the number of competing baselines being limited, the simulated datasets from this work all have the same sample size and behavior policy. When deriving new ITR methods, it is common practice to evaluate the method on datasets of different sample sizes (and if observational data is of interest, varying behavior policies) so as to demonstrate robust performance [2-5,7-11]. \n\n4. **The implementation uses very simple estimators.** In theory, $d_z,d_w$ and $\\delta$ could be estimated by any weighted classification and regression algorithm. However, in their empirical experiments, the authors only explore estimating $d_z$ and $d_w$ with linear policies and estimating $\\delta$ with a Nadaraya-Watson estimator where the bandwidth is chosen based on a heuristic. Moreover, while $h$ and $g$ were estimated using neural networks, the architecture and number of training iterations was fixed a priori. This contrasts to previous works which use more cutting-edge machine learning approaches to estimate the ITR, such as kernel machines, random forests or neural networks, and adopts hyperparameters to the data at-hand using cross-validation [8]. Such works are especially prevalent in top-tier ML conferences [10,11], and better demonstrate broad applicability and flexibility of the proposed method. \n\n5. **Many parts of the paper need to be better written to avoid confusion and address some open questions.** For example, for the real data analysis, it is not clear what assumptions proximal learning is making and how it is useful for the analyzed dataset. It is mentioned that patients were arranged in a \"control group\" on line 289. Were patients randomized to receive a treatment? If so, wouldn't the no unmeasured confounders assumption hold, as treatment assignment is not being affected by any unmeasured covariates? Also, what is the logic behind the choice of $Z$ and $W$ on line 296 (e.g. why are the variables in $Z$ expected to affect treatment but not outcome), and what specifically are the unmeasured confounders $U$ that we are trying to adjust for? Finally, it is stated that the outcome is censored. Does this mean that 30-day survival rate is censored for some of the patients? It is well-known that optimizing censored outcomes without adjusting for the censoring mechanism can yield bias [5]. \n\n>> Here are some other suggestions to reduce points of confusion and improve readability:  \na. The explanation of how the proposed method improves upon [1] in the Introduction section (lines 56-65) is confusing. For example, it is stated on line 69-61 that [7] maps an ITR with domain $\\mathcal X\\times\\mathcal W\\times\\mathcal Z$ with the domain being restricted to $\\mathcal X\\times\\mathcal W$ to $\\mathcal X\\times\\mathcal Z$. While this makes more sense after reading section 2.2, these statements initially appear contradictory. Also on lines 59 and 63 \"two optimal in-class ITRs\" should be changed to \"these two optimal in-class ITRs\" to clarify that the authors are referring to the classes mentioned on line 57.  \nb. The paper has many typos. For example, \"Tchetgen Tchetgen et al\" in line 34 should include the year and a link to the reference, \"netwrok\" on line 73 should read \"network\", and on line 169 the authors should add $V(d_{z\\cup w})$ to the argmax.  \nc. On line 142-143 the authors state that Assumption 3 assumes \"Z has sufficient variability with respect to the variability of U\". But isn't assumption 3 actually assuming that U has sufficient variability with respect to the variability of Z?  \nd. On line 173 it is stated that $\\mathbb E[Y(a)|X,U]$ may not be identifiable under proximal causal inference. But doesn't assumptions 1-5 allow for such identifiability?  \ne. Remark 1 is not true. For example if $\\pi(X)=0.5$ then $\\pi$ is constant but the policy class will not be in $\\mathcal D_{\\mathcal Z}\\cup \\mathcal D_{\\mathcal W}$.   We actually need the restriction that $\\pi$ is both constant and in the set $\\\\{0,1\\\\}$.  \nf. The results of Appendices E and G should appear in the main paper as propositions, theorems or corollaries.  \ng. For sections where over 10 references are cited back-to-back, I think readability would be improved if these citations appeared in chronological order.  \nh. The authors should add a Discussion section that summarizes the results of the paper and proposes important avenues for future work (also see my comments on the Limitations section).\n\n\n\n\nReferences:\n\n1. Qi Z, Miao R and Zhang X. Proximal Learning for Individualized Treatment Regimes Under Unmeasured Confounding. JASA. 2023.\n\n2. Zhao Y, Zeng D, Rush AJ and Kosorok MR. Estimating Individualized Treatment Rules Using Outcome Weighted Learning. JASA 107 (499): 1106-1118. 2012. \n\n3. Qiu H et al. Optimal Individualized Decision Rules Using Instrumental Variable Methods. JASA 116 (533): 174-191. 2021.\n\n4. Wu P, Zeng D and Wang Y. Matched Learning for Optimizing Individualized Treatment Strategies Using Electronic Health Records. JASA 115 (529): 380-392. 2020.\n\n5. Zhao YQ, Zeng D, Laber EB, Song R, Yuan M and Kosorok MR. Doubly robust learning for estimating individualized treatment with censored data. Biometrika 102 (1): 151-168. 2015. \n\n6. Raghu et. al. Continuous State-Space Models for Optimal Sepsis Treatment: a Deep Reinforcement Learning Approach. MLHC 2017.\n\n7. Zhao YQ, Laber EB, Ning Y, Saha S and Sands BE. Efficient Augmentation and Relaxation Learning for Individualized Treatment Rules using Observational Data. JMLR 20: 1-23. 2019.\n\n8. Zhou X, Mayer-Hamblett N, Khan U and Kosorok MR. Residual Weighted Learning for Estimating Individualized Treatment Rules. JASA 112 (517): 169-187. 2017.\n\n9. Pu H and Zhang B. Estimating optimal treatment rules with an instrumental variable: A partial identification learning approach. JRSS-B 83 (2): 318-345. 2021. \n\n10. Yoon J, Jordon J and van der Shaar M. GANITE: Estimation of Individualized Treatment Effects using Generative Adversarial Nets. ICLR 2018. \n\n11. Chen Y, Zeng D, Xu T and Wang Y. Representation Learning for Integrating Multi-domain Outcomes to Optimize Individualized Treatment. NeurIPS 2020.\n\nQUESTIONS:\nQuestions that I feel the paper should address and general weaknesses of the paper can be found in the Weaknesses section. As to my suggestions, it may not be feasible to address weaknesses (1) and (4) prior to the camera-ready. However, I would suggest the authors at least acknowledge weakness (1) as a limitation, and discuss how to implement their method in a more data-adaptive manner, even if they don't change the implementation for the empirical experiments. Such a discussion should include how to tune hyperparameters for all functional classes, including estimators of $d_z$, $d_w$, $\\delta$, $h$ and $g$.  As to weaknesses (2), (3) and (5), I think it would be feasible to mostly address these weaknesses prior to the camera-ready, and my suggestions on how to address them can be found in the Weaknesses section.\n\nLIMITATIONS:\nI did not notice the authors acknowledge any limitations of the proposed work. I would recommend adding a \"Discussion\" section that summarizes the results of the work, and addresses limitations by discussing important avenues for future work. In addition to what was discussed in the Weaknesses and Questions sections, another important limitation is that the proposed method assumes that assumptions (1)-(5) hold and that both $h$ and $g$ are correctly specified, while $d_z$ and $d_w$ appear to only require some of these assumptions to hold.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2023-07-07"}