Accounting for Missing Covariates in Heterogeneous Treatment Estimation
Authors: Khurram Yamin
Vibhhu Sharma
Edward Kennedy
Bryan Wilder
Venue: Accepted by TMLR ()

Abstract: Many applications of causal inference require using treatment effects estimated on a study population to then make decisions for a separate target population that lacks treatment and outcome data. We consider the challenging setting where there are important covariates that are observed in the target population but are missing from the original study. Our goal is to estimate the tightest possible bounds on heterogeneous treatment effects conditioned on such newly observed covariates. We introduce a novel partial identification strategy based on ideas from ecological inference; the main idea is that estimates of conditional treatment effects for the full covariate set must marginalize correctly when restricted to only the covariates observed in both populations. Furthermore, we introduce a bias-corrected estimator for these bounds and prove that it enjoys fast convergence rates and statistical guarantees (e.g., asymptotic normality). Experimental results on both real and synthetic data demonstrate that our framework can produce bounds that are much tighter than would otherwise be possible.

Accounting for Missing Covariates in Heterogeneous Treatment Estimation
Khurram Yamin
kyamin@cmu.cde edu
Deportment of Machine Learning Cornegie Mellon University
Vibbhu Sharma
mibhhus@andreee www.:awwwwwwwwwww.w...mmmmmmmm.. cmar, eduu.
Department of Computer Science
Cornell University
Edward Kennedy
edward@stat. crvuu edu
Department of Statistics
Carnegie Mellon University
Bryan Wilder
bwilder@oma..o edu
Deportment of Machine Learning Cornegie Mellon Universily
Abstract
Many applications of causal inference require using treatment effects estimated on study population to then make decisions for separate target population that lacks treatment and outcome data. We consider the challenging setting where there are important covariates that are observed in the target population but are missing from the original study. Our goal is to estimate the tightest possible bounds on heterogencous treatment effects conditioned on such newly observed covariates. We introduce a novel partial identification strategy based on ideas from ecological inference; the main idea that estimates of conditional treatment effects for the full covariate set must marginalize correctly when restricted to only the covariates introduce bias-corrected estimator for
observed in both populations. Furthermore, these bounds and prove that it enjoys fast convergence rates and statistical guarantees (e.g., asymptotic normality). Experimental results on both real and synthetic data demonstrate that our framework can produce bounds that are much tighter than would otherwise be possible.
1 Introduction
Many applications of causal inference require using treatment effects estimated on a study population to then make decisions for a separate target population that lacks treatment and outcome data. As a motivating example, consider a health system that wishes to deploy a new intervention in its population while making use of existing study data, that was used estimate heterogeneous treatment effects. The health system will almost certainly have access features that were not measured in the original study, due to differences in institutional settings. For example, if the initial study
RCT (Randomized Control Trial), it may have failed to measure practically important covariates such social determinants of health (Kahan et al. 2014 Huang et al. 2024). Since the intervention has not previously been used by the health system, no oulcome data linked lo these new covariates is available. However, trealment decisions would ideally reflect whether the intervention likely to be beneficial to a patient conditional on all information available, not just covariates that happened be in the original study. This paper studies the question: how precisely can we identify treatment effects conditional on such new covariate?? precise estimates are available, the decision maker can proceed confidently with deployment. Conversely, considerable uncertainty remains about

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important subgroup, decision maker may exercise more caution invest more resources in monitoring additional data collection.
Formally, we aim to derive bounds on conditional average treatment effects (CATEs) when novel covariates are observed in the target population. We refer to the CATE conditional on both the common and new covariates as the fully conditional CATE and the CATE conditional on only the common covariates (which is what can be estimated from the original study) as the restricted CATE. Intuitively, what makes informative bounds possible is that the fully conditional CATE must be consistent with the restricted CATE when marginalized to only the common covariates. This idea is reminiscent of the ecological inference literature, which focuses on inferring a joint distribution from its marginals. Ecological inference has long been used in the quantitative social sciences, e.g. for election analysis (Glynn & Wakefield 2010; King et al.] 2004). However, almost no previous work uses ideas from ecological inference in causal settings. We provide a partial identification strategy new to causal inference by connecting ideas from ecological inference to causality. The resulting bounds on treatment effects use the joint distribution of the common and new covariates to link the fully conditional and restricted CATEs.
We make the following contributions. First, we formally provide provable bounds on conditional treatment effects by leveraging ideas from ecological inference. These bounds contain novel muisance functions that must be estimated. Our second contribution is a bias-corrected estimator that exhibits favorable istiis properties such as allowing for the use of non- oon-aaamettric and/or slow converging machine learning models to estimate these nuisance functions without sacrificing fast Op(7x) rates convergence. We also prove that our estimator is asymptotically normal, facilitating the construction of confidence intervals. Finally, we demonstrate these propertics empirically through the use of simulation and application to data from a real RCT.
Additional related work: There is a great deal of work that focuses on combining experimental and obscrvational data to estimate treatment effects. This paper does not focus on the distinction between experimental and observational data: our setup is agnostic as to whether the study population is experimental or observational as long as it satisfies standard identification assumptions. Our focus is on incorporating covariates that are newly obscrved in the target population and not present in the study. One major line of previous work attempts to use outcome data from both an RCT and observational study to jointly estimate treatment effects (on common covariates). (Chen et al. 2021 Hatt et al.) 2022; Demirel et al. 2024 Guo let al. 2022 Schweisthal et al.) 22244) Often, this involves fitting a model for the confounding bias present in the observational study (Kallus et al. 2018 Yang et al.) 2020 Wu & Yang. 2022). By contrast, estimate the CATE in the target population where outcome data is available and new significant covariates are present that were not present in the study population (we do not only study common covariates between data sources). A second line of work focuses on the case where the observational study has no outcomes by correcting for shift in the covariate distribution (Lesko et al. 2017 Lee et al. 2022). These methods deal only with covariates found in common, and focus on average effects (as opposed to our focus on conditional effects).
Recently, a few papers have tried to quantify the uncertainty from covariates missing in the RCT. Colnet et al (2024) focus quantifying how estimate of the ATE might be biased when covariate is missing in one or both populations. In contrast, our focus is on bounding conditional treatment effects. They also require distributional assumptions (e.g., Gaussianity) on the missing covariate, as opposed to our nonparametric approach. Similarly, Nguyen et al. (2016) propose a sensitivity analysis framework for estimation of the ATE in a target population when a treatment effect mediator is unobserved in the target population (as opposed to our focus on covariates unobserved in the study population). Andrews & Oster (2019) propose framework to assess the external validity of RCTs when missing covariate induces selection bias into the trial.
Several works make strong assumptions that allow for point identification when covariates are missing, Pearl (2012) studies noisy measurements W that causally depend on latent covariates Z, enabling point identification by assuming W contains no additional necessary information beyond Z. In contrast, we consider novel covariates unique to the target domain that cannot be reconstructed from study data, thus precluding point identification and requiring bounds. Similarly, [Bareinboim & Pearl (2014) achieves point identification by assuming access to a transportability graph that models causal links between all variables across domains
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Table 1:
Notation and Definitions (Not Exhaustive)
Symbol
Definition
V
Common covariates observed in both study and target populations.
W
Additional (discrete) covariates observed only in the target population.
T e (0.1)
Treatment assignment indicator in the study population.
Y
Observed outcome in the study population.
ylyo
Potential outcomes under treatment. (1) and control (0).
EE (0.1)
Population indicator: E = for study, E = for target. Observed data tuple (V, E, W(1 E), E (T.Y)).
Z
X
Shorthand for the full covariate vector (V, W).
pe(w)
Y V v, T t, F. 1), Le {0,1).
((e
(P(W w V v, E 0).
(x()) (a)
Un-truncated lower/upper bound on CATE numerator: (m) 440 + (b - a) 3/v. Truncated lower lower/upper /upper bounds on E[ v, w. (Thm.
we(,,w), To (u( w)
m(
model class for Parametric projection of the bounds.
B
Parameters of the chosen model class for m)
((X)
Analyst-chosen weight. function in the projection. (0m/03) h(X) in the moment conditions.
(())
(e(
Joint propensity P(T E = V / u), te {0,1). Selection probability PP(E = 0 V =). v). Indicator (x)(XX +
Polu)
f(X)
2 0} (or analogously for upper).
a,
Known bounds on Y: Y [a, b] = y - y E [a b, b- a].
Sensitivity parameter: E[Y" y W| - E|y - v 5.
assumption we do not make. Berrevoets et al. (0233) enables point identification by making an overlap assumption where all covariates and sets of covariates have observed outcome values with non-zero probability. In our setting however, we have certain covariates in our target population that we never have outcome data for. To our knowledge, our paper is the first work to ever pon-paramtrricllyy estimate bounds on the CATE in this setting, where the target population has no observed data and has covariates not observed in the study population, without making the kind of strong assumptions that allow point- conntiidetiificttion while compromising real-world applicability.
2 Problem Setup
We examine the situation where we are allempting to transport estimates of heterogeneous treat effects between two populations. In the first population, which we refer to the study population, we observe covariates V, treatment assignments T f0, 1), and outcomes Y. Each individual has potential outcomes Y and Y that would be realized if they were (respectively, were not) treated. We assume that Y y [a, b] with probability 1 for some constants and b, i.e., the outcomes are bounded between known values. We observe Yf corresponding to the treatment assignment. In this population, we impose standard identifying assumptions (most prominently, no unobserved confounding) that allow estimation of the conditional average treatment effect (CATE) E(y1- - Y of the study population which we refer to as the restricted CATE. This study population could represent a randomized experiment or an unconfounded observational sctting. This assumption is formalized T IL y given the context of the study population.
In the second population, which we refer to as the target population, we do not observe the treatment outcome variables. Instead, we observe just the covariates, which consist of both V and new set of covariates W which were not observed in the study. We assume that W consists only of discrete covariates (although V may be either discrete or continuous). This holds naturally in many settings of practical interest (e.g., social determinants of health are very often discrete variables (Sarkar) 2014)). Furthermore, this assumption not overly restrictive can otherwise be ensured via discretization of continuous values (c.g., many clinical risk scales already discretized into fixed set of levels). (Ustun & Rudin 2019) This assumption is technically
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required so the probability of specific realization of W is well-defined. We use an indicator variable E to indicate whether the subject is in the study population (E = or the target population (F = 0). Our observed data consists of samples
Z = V, W, --VEEWWI-DD,,,,
V.W : = V,T.Y =E=1
Our goal is to estimate IE - V W], or the CATE conditioned on both V and W. We refer to this quantity as the fully conditional CATE We are going to use the standard assumptions of consistency (Y=Y²
whenever T = 1), positivity (P(T = V/V = v, W = w) > 0 for all combinations of v, w)). We also assume that (1) P(W = w V v. = 1) = P(W =
1) (1) the covariates missing in the study population have the same conditional distribution those in the target population, and (2) the fully conditional CATE (Bjr²- Y VI) is the same in the study and target population. Many notations and
u, E = 0), and (2) E(Y" y V. V,W,E
E(yl- - y V, V,W, E 0). These two assumptions could be restated
symbols are summarized in Table !
Furthermore, is true that the treatment effects given are unconfounded based on data from the study population. However, unconfoundedness given V simply means that V gives sufficient information understand treatment choices in the study population. It does not mean that V contains all variables that impact treatment effects for specific individuals. As such, while we can successfully estimate the CATE conditions just on V different subgroups wilhin a stratum of V may still have heterogeneous treatment responses. In our model, W represents important additional covariates that create smaller subgroups within strata of V. We wish to bound the treatment effects for these smaller subgroups as they can better allow us to tailor treatments to individuals at a more personalized level.
3 Methodology
In order to estimate treatment effect bounds in the presence of covariates observed only in the target population, our approach proceeds in three key stages. First, we invoke to derive sharp, nonparametric bounds the fully conditional CATE E[ - y V, W] by leveraging the fact that these effects must aggregate to the known restricted CATE given V alone. Next, recognizing that these bounds depend on several unknown nuisance functions (e.g. outcome regressions, propensity scores, and the conditional distribution of W), we develop a bias-corrocted estimator based on inffuence-function mdjustments: by carefully correcting the usual plug-in estimator, we allain fast O(n-1/2) convergence rates even when the nuisances are learned flexibly at slower, nouparametric rates. Finally, to further tailor our inferences to scttings where one believes the additional covariates W have quantifiable limited impact (by domain knowledge), we introduce a simple sensitivity-anallsis model that restricts the deviation between the restricted CATE (using only V) and fully-condittonal CATE (using V, W) by user-specified parameter 5, yielding strictly tighter bounds when plausible. These steps are summarized in Figure [ We now describe each of these steps in turn.
moological-infeeene argument.
3.1 Partial identification bounds
Our goal is to provide as much information as possible about the fully conditional CATE (i.c., conditional on both V and W) when outcome data is linked only to V. Clearly, in this setting it is not possible to exactly identify the fully conditional CATE. However, we can use ideas from ecological inference to partially identify it. Specifically, if we average the fully conditional CATE over values of W, we must obtain the CATE conditioned only on V, which is identified. Formally, long line of work in ecological inference (Jiang et al. 2019 Plescia & De Sio. 2017 Manski 2016 uses marginal consistency conditions of the following form, for a single outcome Y:
BY v) = Σ E(Y v, w)p(w v).
(1)
For the sake of brevity, we use capital letters (eg V, W) to represent random variables and lowercase versions (e.g. v, w) to represent specific values of these letters (such that V u, W w). If p(w v) is known
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Study Population Limited covariates with treatments/ /outcomes (V. (V.T.))
Target Population Full covariates without treatments/ outtcmees (V, W)
Ecological Inference Derive bounds on fully-conditional CATE (1)
V, W], Sec 3.1
Bias-corrected Estimation Use influence function to estimate bounds, Sec 3.3
Sensitivity Model
Limit how much unseen covariates W can change our effect estimate, Sec 3.4
Figure Roadmap for bounding and estimating the fully-conditional CATE
be estimated, we can then rearrange this expression to obtain bounds on E(Y u, w). We apply similar strategy in the context of the CATE (using also the causal assumptions that link to the observable data) and obtain the following provable bounds on the fully conditional CATE:
Theorem 1. Assuming the conditions in Section and that Y is real-valued outcome bounded in [a.b]
7e(u, w) S EE y N. E 1) S 70(,,w)
such that
(1(1) mo(u)) - (b a1 v) (u(()) - mo(v) - (a b)(( - v) b - a
Te(U, w) = max
(a w) = min
where (4) = E( V u. =t = 1), u(u, = PP( = w V = U, =
This bound utilizes the worst-case scenario that if Y E [a, bounds (our estimand of interest) are a function of V and W. We introduce a framework that estimates the projection of these functions onto a parameterized model class m chosen by the analyst. Different use cases may call for different parameterizations. E.g., an analyst may prefer a lincar model for simplicity and ease of interpretation, or use more intricate nonlinear model such a neural network if they wish for greater flexibility to capture the underlying function. Regardless of the choice, we assume that the model is parameterized by some 3 of fixed dimension with respect to Note that we do not assume that the model m is correctly specified; i.e. we estimate the projection of y onto model class instead of setting equal to some model.
b), then y2- yo E -, a]. Note that these
For simplicity of notation, we will use interchangeably with the joint tuple (V, W). We attempt to find the best approximation of (x( and (x))) in the form of model m. We use y to represent the bound of interest, which could be either (x(x)) 7x(x) desired. We seek model parameters B which minimize the
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mean squared error between m and y::
B arg min E (+(X))((X) -m(X:0)3²
DERP
This formulation is weighted by an analyst-chosen function h which allows a degrec of choice on what observations to place emphasis on (for example, we could choose emphasize certain covariate regions of interest). Differentiating then gives us the moment condition (3)
Tmm(X: h(X)(1(X) - m(X; {)} = (
0
0
Our goal becomes to find the 3 such that M(3) = 0. To keep notation concise, define
Tπ(x) mi (v) mo(v) - (b - a)(1 -v)
V
so that 7x(z) = max(r=(z), - b}. and similarly for (x)) Letting g(X) = 8m(X (0
h(X), this moment
08
condition is equivalent to
Mr(3e) =
(2)
(a.).)) = (lg(())((((() -a-50)
=0
(3)
3.2
Estimation
If 7 were known, we could simply solve the resulting least-squares problem. However, in fact depends number of nuisance functions that are not known and must be estimated from the data: fi (V)),, (o and P(W w V =, then plug the estimates into the moment condition.
u. E 0). A naive plug-in strategy would be to estimate each of the muisance functions and
However, the quality of the resulting solution will depend sharply on how well the nuisances are estimated. In general we will not obtain consistent estimates for even the projection onto the parametric model class unless the nuisances are estimated consistently. As the nuisances class, consistent estimation will require the use of nonparametric methods that converge only slowly (slower than On)= Conversely, if the true values of the nuisance functions were known, f could be estimated at the (faster) root-n parametric rate (Vaart 2000). We draw on techniques from the semiparametric statistics/ double ML literature to propose Bins-Corrected estimator that attains the parametric rate for B even when the nuisance functions are estimated at slower nonparametric rates.
unlikely to lie exactly in any specifie parametric
3.3
Estimator
In this section, we will focus the derivation of Blas-Corrected estimator for Te(z); 7a(x) follows a similar form, shown in the Appendix. Full proofs of all claims can be found in the Appendix. The starting point is to derive an influence function for our estimand of interest. Intuitively, influence functions approximate how errors muisance function estimation impact the quantity of interest, in this case 7. A common strategy in semiparametrie statist to use the influence function for the target quantity to provide a first-order correction for the bias introduced by nuisance estimation. This dampens the scnsitivity of the estimator to errors in the nuisances and will allow to derive fast convergence rales for 3 even when the nuisances converge more slowly.
Unfortunately, influence functions typically only exist for quantities that are pathwise differentiable. The expression for 7t contains non-differentiable max, which shows up as an indicator function in the moment (3r) 2 That is, the moment condition discontinuous on the margin 7 ((x) +b b- a = 0 to Te(x)+ + +b-a a < O. We employ margin condition strategy used by Kennedy et al. (2019), who analyzed a nondifferentiable instrumental variable model, and others (Kootffe et al. [022 Vigogna et al., (022). Specifically, we can hope for fast estimation rates when the classification problem of estimating the indicator for given X is not.

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too hard, in the sense that not too much probability mass is concentrated near the boundary. Specifically, we assume that.
Assumption (Margin Condition): ((( + b - al 1 {) - CÇ
(4)
for some constant C and some a 2 0. Similar assumptions are often imposed in the context of classification problems (Audibert & Tsybakov] 2007). We have a 0 with no further assumptions, and a 1 holds if ((x) has a bounded density, A bounded density is a relatively weak assumption, so 2 1 is likely to hold in many cases of interest. Under this margin condition, we employ a two-part strategy. First, we derive an influence function for the moment condition under the assumption that the true value of the indicator function is known, rendering the expression diffcrentiable in the estimated nuisances. Second, our final estimator replaces the indicator function with plug-in estimate; the margin condition entails that this step introduces relatively small bias compared to if the true indicator were known. Specifically, we prove that
(f + b a) (Te(T)(X) + b -a]] - CX) (x))1
When a 2 1, this term now depends only on squared errors in the estimation of t and will be negligible o(n-+) +) rate. The proof for this is contained in the appendix. For simplicity of notation, we call the indicator f(X) such that f(X) = ((x(x) + - a 2 0). Next, we turn to deriving the influence function for Y under the assumption that the indicator f known (where our eventual estimator will use the analysis above to justify replacing f with its plugin estimate). Given this strategy, we derive the influence function cx(X, n) (full derivation in Appendix) to obtain:
asymptotically so long as êr converges at a o(n
Lemma 1. (xX, B, 3) is given by (assuming conditions from Section
ET (Y - -i(V)) E(1-T) T) (Y Ho(V))) (b a)(1 - E) Po(V)
Σ
f(V, { (V)
vo(V)
(Xf(x) Σ (V, 1- E PO( (xf(X)t((X) Σ w
(f (f*
+ (T(T + (b (x) b m(X, where Te(V) = p(T =, == (V) p(E [
Note that π and p appear as new muisance functions that help compensate for errors in the original plug-in estimate. From this influence function, we can construct our bias-corrected estimator. The key idea is to find 3 that solves an estimating equation implied by the influence function:
B such that Pa (x, (((((X,,))) B)) =
where PR is an empirical average. Formally, in order to construct the estimator, we employ a sample splitting procedure detailed in Algorithm [ This follows the strategy, common in semiparametric inference Kennedy et al.] (2023), of splitting the dataset into two independent halves. The first is used to estimate the nuisance functions (including the indicator f). Then, we fix the muisances and construct (XX, 3, f) for the points in the second half, which are used to estimate 3 via the above moment condition. The computational approach to solving the moment condition will depend on the model family chosen. For linear models m(v, 3, = BT v + 88 w, we give closed-form solution in the appendix that can be computed as standard OLS problem. For more general differentiable model classes, one strategy would be to minimize (o 9, } using gradient-based methods.
We now turn to analyzing the convergence properties of the bias-corrected estimator, with the goal of showing that 3 + 3 at a fast rate even when the muisances are estimated slowly. We start by examining the bias R, of our influence function. Let % be the true values of the nuisance functions while f is our estimate. The bias Rn quantifies the difference between the expected influence function at no and f and plays key role in controlling the convergence rate of 3. Formally, we decompose the bias as: Theorem 2. Let Rt = ( 3, f)-
((x;; 8, mm)}. Assuming that all nuisance functions and their estimates are bounded below by a constant larger than 0 and that all probabilities are bounded above by 1,
Rr, - ||- 2llz 10 2|12 + 110 v112 + || - F ll2ll - -P1l2
+ 1l Tollzllão soll2 1|mm
اا||2|10 xl12 + -1ãñ0 Hollalle 2112

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sensitivity model imposes the assumption that Roughly, estimated 3 will converge quickly if R. = (a statement formalized below). For this to occur, all of the products above must be or( (.) It is sufficient but not necessary condition that p, D, A, and j converge to their true functions at or(n=²) rates for R. to be orl (t)) Note that n is substantially slower than the parametric root-n rate, and is satisfied by even many nonparametric methods. It becomes clear from the bias structure that D P(W =VV v, E = 0) would need to be correctly specified for 3 to be a consistent estimator of B. This is due to the nature of the ecological inference setting where & is the only thing linking the target population to the study. Thus, our estimator is not doubly robust in the sense that no other nuisance can compensate for errors in D. However, the error is still second-order as it involves only squared errors for v. Additionally, there a mixed- property with respect to all of the other nuisances, where each nuisance can converge at a slower rate individually if others converge faster (e.g., lr can converge more slowly if and y converge faster). Therefore, our estimator exhibits robustness misspecification in all nuisances except v. Putting these pieces logether, we oblain a convergence guarantee for 3: Theorem 3. If R. (x) and assuminn the condilions in Section 2 and that the margin condition holds for a 2 1:
||3 B112 Op (+) and vT(B- 3) 2 NO..M
where M- (X( and V is the variance of 3, no)
03
Theorem 3 follows from Theorem 2 combined with standard analysis of M-estimators under misspecification
(c.f. Theorem 5.2.1 of Vaart [2000] and Lemma 3 of Kennedy et al. (023)). ). Asymptotic normality is valuable
as it allows for the construction of confidence intervals, e.g. with the usual sandwich estimator (or potentially easier in practice, the bootstrap).
A Algorithm 1: Algorithm for Constructing Bias-Corrected Estimator
Given imput samples D, split uniformly at random into D2 and D2
Use Dj to estimate nuisance functions: (X),,, (x))),, (X))), A1 (X), *o (x(X), ((X)
Use the estimated muisances estimates to construct the estimated indicator: f(X) = 1(f(x)())))))))))))())) + 2 0) using the first part of the data. Let f be the set of all nuisance estimates, including now f(X).
4: Find 3 that solves the following estimating equation on D2: D2 ExeD, ((X, 3,, 3) = where ((X,3,1) 3,,
is constructed using f following the expression in Theorem
5: Output? B
3.4 Bounds in a sensitivity model
In some cases, we may be willing to impose additional assumptions limiting the deviation between the restricted and fully conditional CATEs. This may be justified based either on domain knowledge, or taken in the spirit of scnsitivity analysis where the analyst varies a parameter controlling the strength of such assumptions to see how much variation across levels of W their conclusions are robust. to. In this section, we propose such sensitivity analysis model to formalize the case where W is believed to have a limited impact on treatment effects, after conditioning V. Specifically,
|E - y V v, W = w, 1) E(Y - y v= E 1) 2
(5)
or that the fully conditional effects cannot differ from partly conditional effects by more than 8. 5 here is user- chosen parameter, that may be varied to test the robustness of estimates to an increasingly strong effect of W. At 5 55=00 0, the fully conditional CATE is equal to the restricted CATE, and at 5 = b-bb. we recover our previous bounds (that use only the boundedness of the outcome to [a, b)). At any intermediate level of 5, we obtain partial identification bounds that are of a similar form to those in Theorem [ (but stronger), simply replacing the terms (a-b) or (b-a) by (Y y V = v, E 1) We estimate these bounds using the same stratogy as for the original model, plugging in the identified quantity B(Y Y using data from the study population.
V = v, 1) = p1(v) - =o(
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4 Experiments
Setup: We start our experimentation with experiments on simulated data so that the ground-truth CATEs are known, and afterwards give an application on data from a real RCT. Full details of the simulation are given in Appendix. The process samples 10,000 observations of 3 continuous covariates for V and discrete (binary) covariates for W. W is simulated as P(W; = 1) = logit(a V) for a coefficient. vector di, where the choice of a allows us control the degree of dependence between and W. E and T are generated similarly as functions of V, producing covariate shift between the populations and nonuniform assignment to treatment across levels of V within the study population. Finally, the CATE is specified function of both V and W and we sample observed outcomes from the study population matching the CATE.
To our knowledge, ours is the first paper to present an algorithm to identify and estimate CATE bounds for covariates unobserved in the original study. We compare to three baselines. First, we compare the performance of our bias-correct estimator to the plug-in estimator discussed in Section 3.2. which uses the same nuisance function estimates as our model but estimates 3 directly by plugging these estimates into the moment condition, without the influence function-based correction. Second, we compare the informativeness of our bounds to those which use all of our assumptions except the key ecological inference component in Equation 1 to test whether using this information results in tighter identification (more details on this below). Third, we compare the frequency with which our bounds cover the true treatment elfects for specific subgroups of (V, W) with the coverage of confidence intervals for the restricted CATE (which is measured on V and not W).
Baselines:
In order to set the sensitivity parameter d, we can use benchmarking methods similar to other sensitivity models in causal inference. (Byun et al. 2024 Hosman et al. 2010: McClean et al. 2024). For specific instance, the analyst holds out n variables from V to form "simulated" set of additional covariates W' mimicking W where V represents the original V excluding W', They then use the difference | - ,'IV", as proxy for [y- -[ - V'V,,W in order to set 8. This procedure is performed on all subsets of n variables in V and results are averaged. As in sensitivity analyses for unmeasured confounding, this provides an interpretation (based on domain knowledge) that the analysis is robust to variables "at least as important' as W'.
Benchmarking:
Simulation: Figure 2a demonstrates the difference in estimation error between the plug in and bias corrected estimators across 200 random seeds where we inject varying amounts of error into estimates of outcome and propensity models. We find that the bias-corrected model produces more accurate estimates of the bounds (known in simulation) than the plug-in by large margins in the vast majority of situations, especially when there is a high amount of error in the outcome regression. The plug-in model performs better only when there is little error in outcome regression modeling and high error in propensity modeling. Intuitively, we might expect ecological bounds to be tighter when the V is more informative about W. To test this, we vary the distribution of v and plot its entropy against the width of our bias-corrected bounds. We observe that as the entropy decreases, the average bound size also decreases (Figure 2bb. This shows that when the known covariates in the study population become more predictive of the new covariates in the target population, our bounds are tighter.
In Figure 3a we apply the sensitivity model discussed in section 3.4 The left plot shows the average value of the bounds as a function of the sensitivity parameter 8. As expected, the CATE is point-identified at 8 8= O, with progressively greater uncertainty as 6 grows. For reference, we also plot bounds (in black) which use only the sensitivity assumption in Equation 5 By itself, this assumption implies bounds of width 28 for the CATE. The bounds output by our method are substantially stronger (less than half of the width), indicating that our ecological inference framework which uses the joint covariate distribution provides substantially more informative inferences than would otherwi be possible. The right figure shows the rate at which our bounds cover the true CATE as a function of ô. As expected, if is close to its true value (known in the simulation), we observe close to nominal coverage levels. Additionally, Figure 3b) shows that the 95% CI of the restricted CATE (measured on V and not W) baseline only correctly covers 43.7 percent of the true treatment effects, while at both the true and estimated values of our sensitivity parameter 8 (8 comes from Benchmarking distribution in Figure 2g] estimated and true 8 are very close- within 10% of each other), roughly 98 percent of the true treatment effects are covered by our method. We can also observe that at.
9

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Plap in Losa Sas Conwoted Losa
Benchmarking
10
Lebere
sagena
Ero Prosrrs))
2.0
Diciors
Bound Width
Frequency
2.5
9.5
130
Ero Chasann
9. 15
1.3 14 15 Entropy (b)
14
3 Delta Subsets
Estimated Delta Subset
(a)
(c)
(d)
Figure 2: (a) Difference in estimation error between the plug in and bias corrected estimators under varying errors in outcome and propensity modeling. Red is higher loss for the plug-in. All errors measured in mean absolute deviation. (b) Average worst-case bound width
function of the entropy of y where the range of outcomes is 40. (c,d) Benchmarking distribution for (c): Simulation Data, (d) RCT Data. Final represents the mean of each distribution and is within ~ 10% of both true 6. We consider all potential subsets of V and W that can be made from V given the number of variables in W (3 in Simulation and in RCT). Full range of possible ö for x-axis is (0,16).
both the true and estimated values of 6, the bounds are very informative as the CI of the mean lower bound of the treatment effect (n, w)) exceeds 0. More specifically, we have that (w, w) ~E (1.1, at 8.
Simulation Data Bounds
(b)
Coweage Teatment Elleet
Simulation: Treatment Effect Coverage
Treatment Effect
b:a 30-
1.00
0.75
0.50
-00
0.25
Delta
12
16
0.0g
2.5
Delta
7.5
10.0
15.0
Stan-Corr Drundd (Dur Meehhol
Deba
(123)-Y1201113 deite (Saee
Detimeted Data
Rectreted (hoosl
RCT Data Bounds
(d)
Coverage of Thue Thearmam Etleccs
RCT: Treatment Effect Coverage
Treatment Effect
30
ba
1.00
0.50
0.25
-15
Delta
12
0.00
Delta
7.5
10.0
Bis Corr. Brunds (Dur Meshed)
Thu Detaa Estimated Dula
1... AlVl)) V duka (sseline)
Biy Con. beand Methiet
..0 Estvattdd bet
Restrcted CATE (0oceee)
The Dello
Figure 3: Left (a,c): Bounds in the sensitivity model a function of 8, averaged over units. The blue lines are bounds output by our method (Bias Correction), with a 95% CT. The black lines are bounds that use only the sensitivity assumption in Equation without our ecological inference framework. Right (b,d): Blue line is the fraction of times our bounds cover the true CATE compared to green line which is the baseline coverage resulting from a Doubly Robust estimate of the Restricted CATE's 95 % CL. Top(a,b): Simulation Data. Bbt): RCT Data.
Finally, we illustrate our method by estimating conditional treatment effects for a real-life RCT that measured the effect of a throat treatment (gargling with licorice solution) prior thoracic surgery on post-operative swallowing pain (Ruetzler et al. 2013). We split the dataset into a "study' and 'target" population by holding out one of the covariates in the study population to form W. All of
Real World Data:
10

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the variables in V are standard measurements such pain index, while the variable separated for W represents an indicator to whether the patient experienced coughing after the ventilation tube was removed from their airway. Although this indicator is considered by many physicians to be a valuable metric (and would be visible to physicians working with the target population), it is frequently not recorded (Duan et al. 2021h and hence might not be available a sludy dataset. Further details about variables and the setting of the RCT can be found in the Appendix. Figure 3e shows the average width of bounds we obtain as a function of 6, and we once again observe that the ecological framework provides substantially improved identifying power compared to naive bounds that only use the sensitivity assumption itself.
We again see very informative bounds in Figure 3c In particular, the CI of the mean upper bound of the treatment effect lies entirely below 0 (nu(e, w) ~ (-2.44-3.00 at å where å comes from the Benchmarking process shown in Figure [d] indicating a provable reduction in postoperative throat pain due to the treatment. At the estimated value of 6, our bounds have almost exactly the desired 95% coverage level (Figure [d.. By contrast, the bounds output by a DR estimator for the restricted CATE have only 50% coverage. This indicates that the held-out covariate has significant impact treatment effects, such that the expected effect for many patients moves outside the original Cl after seeing the new covariate. Our bounds properly account for this uncertainty while remaining informative about effects. Similarly to the simulated data, even if we select a value of 5 that matches the average width of CIs for the restricted CATE, our bounds have 25% better coverage. This shows how leveraging the joint covariate distribution allows us to provide strictly more informative uncertainty quantification about treatment effects than otherwise would be possible.
5 Discussion
In this paper, we give the first formal presentation of an identification and estimation strategy for the CATE when generalizing to a larget population that has covariates not observed an earlier study. We develop bias-corrected estimator that retains fast O₂( Vn convergence rates even when nuisances are estimated nonparametrically, and is asymptotically normal under standard conditions. We also introduce a sensitivity model to bound impact of the new covariates on treatment effects. Empirically, we find that our method is often able to substantially reduce uncertainty about heterogencous treatment effects, even in this challenging setting where no outcome data directly linked to the new covariates is observed. We caution that, like all causal inference methods, our framework requires domain expertise to assess the plausibility of assumptions, c.g. that W and V follow a consistent joint distribution between the populations. However, when used appropriately, our framework gives users one way to assess the generalizability of effect estimates to newly identified subpopulations before committing to a treatment assignment policy, helping to avert unintended negative consequences.
11

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A Appendix
In this appendix we provide full derivations of the bounds, moment conditions, and seltennc-functiion-beeee expansions used in the main text. We also add explanatory commentary to make the main algebraic steps more explicit.
Throughout, we keep the same notation as in the main body. In particular, we use:
B[|] to denote expectations.
P(.) to denole probabilities.
E E {0, 1} exclusively as the indicator the study/target population (not as an expectation operator).
A.1 Fully Conditional Bounds
We begin by deriving bounds on the conditional mcan E( | W w. V v) when we only know that Y lies in bounded interval and we observe the distribution of (Y, V).
Assume that Y E [a 6] with probability one. We first apply the law of total expectation by conditioning the event {W = / vs. {W * w
B(Y V v) E(Y W * w, V PW # w V = o)
B(Y W w. V
P(W = w = v)
The expression above is an identity that simply rewrites E(Y V = v) in terms of E(Y W = w. V= v) and E(Y W * w. = v); and then solves for E[ | I = w, V = =v).
Because Y |a, almost surely, we know that IE(Y | W + w, V u) must lie in [a, b). Minimizing and maximizing the right-hand side over all such values yields the sharp interval:
B(Y
V v) 6P(W + V v) )
E(Y W w. V u)
max
ag. E(Y V = aP w V v) b b)]
P(W = w V = v)
min
P(W = w V = v)
We now apply the same idea to the conditional average treatment effect y1 yo instead of Y. Under the assumptions laid out in the Problem Setup (in particular, that treatment is randomized within the experimental sample), we can write:
E(Y - y V v W w. E=
E(Y' - 0 V E 1) E(Y! Y W = # V ==
P(W = w V = v, v,E=0
E(Y V v. = 1, 1) E(Y V V,T
==
* w V = E =0
P(W V =, E
The first equality above again uses the law of iterated expectations, now applied to y - y across the partition {W = w) vs. {W * w). The second equality uses randomization of in the experiment to replace E(Y' V = v. E 1) by the observed regression E(Y V = =,T = E, = 1), and defines
alv, w) B(y' - Y W + w. V u, F=
In this representation the only unknown quantity at (v, w) the conditional effect
r(1, w) == Eey--111-0 y V u. W w. E 1).
If all that is known is that Y Ja, b), then (by monotonicity) we have Y y E - a] almost surely. This implies that the nuisance term alv, w) is only known to lie in (a b, b a] pointwise in (v, w).
15

---

Plugging the extremal values of a(v, w) into the identity above, and combining this with the bounds for B(Y W = w, V w), yields sharp bounds
7e(u, w) - EE - r V W = w, E, S wv)
where
(1(0) Mo(v) (b (W + = U, 0)
Ve(u, w) = max
(a b)
PP(W = w V = v, U =
M/ (v) (o(w)) - (a - PV * V F= = 0) (b a) }
Fu(U, w) min
P(W = w V = v, =
and where have defined the experimental outcome regressions
(e(0) = E(Y V = =,
A.2 Moment Condition
We next make explicit how the projection moment condition (B8) looks when we plug in the truncated lower and upper bound functions. The idea is simply that we regress a (possibly truncated) bound function onto a working model m(u, w; 3) using instruments g(x).
For the lower bound, recall that define 7e(v, w) max (Tx)),)()),cc a-b), where (u, w). Then the population moment condition is
M(3) = (1 m(v,
(max
b} m(v wz 8))]
We can rewrite the maximum explicitly in terms of indicator functions:
M(8) ((g(()((T((x)) + a) (x) +b- 0) -a b- b-m(v, we 8)
+ 1(T(())))))))))))))))))))))))))) 2 (a b)) (a- (x) (a- m(u, wx 8)1
The last line is just reparameterization of the indicator in terms of Te(x) + a, which will be convenient when we apply the margin condition.
For the upper bound, we truncate at - a instead of - b. Let ru(u, w) = min(ra(2), a}. Then
M(3) = (l w) - m(v, w: B))]
(gg(()) (r. a) m(v, ())
(lgg())((T.)()) + 6) ((T.(z) + b
mm,, un 8)}]
Thus, for both lower and upper bounds, the moment condition has the same form: weighted residual of truncated function of (2) or (z) against the working model m(v, B).
A.3 Use of Margin Condition
We now record the basic incquality that underlies the use of the margin condition. This incquality controls the error incurred when we replace the unknown sign of r(x) by the sign of an estinator f(x)) inside indicator functions.
Consider the difference
B[r1(f +b- a) - -T + =f/r))) - -
We can bound the absolute value of this difference using the triangle inequality:
Elr1(f) r (1 < (1(f) 1(7)l]-
16

---

The indicator difference is only nonzero where & and have different signs; this in turn implics that |r| S | that event. Using this observation, we obtain
- / 88- - 7/1([T] - I 7))dP(2)
E| (1()) 1(7)ll
/ 1t - 7|(([]] P 7))dP(x))
I sup 7 - P - x $ C - -
- 7
The third line follows from the fact just mentioned: if r and f are of opposite signs, then |7| l 7/. The last inequality is exactly where the margin condition is invoked; it controls the probability mass near zero and yields a (1 + order bound in the sup-norm error.
A.4 Influence Function Derivation
We now derive the influence function used in the main text. The strategy is to start from basic, well-known influence functions and then apply the usual "influence function calculus" (e.g,, product and quolient rules) to obtain the influence function of the more complicated functional of interest.
Basic facts. We first collect several standard influence functions and identities that we will use as building blocks; see Section 3.4.3 of Kennedy 2023) for more background and proofs:
IF(B(Y X x)) 1(X x) (Y B[y X = =xx}
P(X = x)
P(W w X = =x B[[(W = =w) X x].
p(V = V,T 1,== = p(W V=
p(x)
p(T = L IF(p(z)) = (1(x=))=x0 =
Here IF(.) denotes the influence function of the corresponding functional, and p(.) is the density mass function of X.
Derivation. We now apply these basic components to our particular ratio functional. For clarity, we introduce explicit notation for the numerator and denominator of the scalar quantity r(x).
We say that Taum is the numerator of T,
Tuum (I) (i(u) - po(u) (b ao-aaP(WWa # w V v, 0).
and that Taen is the denominator of T,
Tden (x) = P(W = w. = c, F= 0),
(xumn()) so that T(x) Toum (z)/Taeu (x). We use "num/den" here to avoid the subscript n being confused with sample size.
As mentioned in the paper, we also define the indicator
f(z) = ((Te(x) +b 0).
Using the quotient rule for influence functions, we obtain
Faum (x))
TF(((x))= = IF
Telen (x))
17

---

IF [Tnum (x))
Toum (x)
I (2))
Tden (x)
Taen (xr)
We now spell out the inftuence functions of the numcrator and denominator in terms of the nuisance functions involved:
B T) * 1(V = pVV = = =,E= (Y -i(0)) P(V = v, = 0- (Y po(v))
ET * 1(V =)
a)l(V =.,
p(V =v,' (1(W = =w) - p(W = w V =v,,==0)
1(V v, F= IF (Tden (2))
= u, (1( = P(W V v, F= = 0))
Here ETI(V= = u) is the indicator of the event = 1, v). and the expressions follow from the standard influence functions of conditional means and conditional probabilities as in the "facts" section above.
We now consider the functional
[otw)ttter -()))((z)
where we call the known portion ((x) such that
<(x) ((0==0000(((()aa-mmmml) m)=;3))
+
Applying the product and chain rules for influence functions to this functional yields
IF Σ ((x)((())(())
Σ (o(z)) IF( (nmmm (1))
(F (x)) } + +xxx)(((X))(xx o(()))
Tden(x)
Tden (x)
For convenience, we decompose this influence function into three pieces:
(X; E { I IF (x)) )
(d))(x)) } [
(2 = Eg(x) ((x))F(Taan(z) f(z)p(x),
(2)a(
(x( = (zx)() - q(1)}.
Then the overall influence function is
<(x;3) = +xxx)) (x(;3)).
We now simplify s and 12 by substituting the concrete forms of IF (Tnum (x)) and [F(Teeu and using the assumptions on p(W (V) (e.g., that it common across E). After straightforward algebra, we obtain:
(x;
pT = E, = 1 V) ( (l()) Σ 9(V, wV, w)p(w) V) * Fdon (V, w)
ET
E(1 TT)
p(T = 0, E 11 (X Ao(V)) (X-0(())) Σ WVV, V)*
Adon (V, w)
(b - 6) E)
p(W V)
OVV,W)/(K,W)
ME (E=0|1)
Pdon (V, W)
18

---

Ea(V, ((V, V) p(W= = w V, 0)
idon (V,
ET
p(T = 1,E= (Y (V() E9(l oV,,,)
E(1 -T)
Ao(V)) E Lg(V.w)f(V..) D(T = -0, E, - [V] 1
-X--
(b (b- (1 - E.
-E= ((V, - E (VV 1
m).
To go from the first expression for Φ1 to the second, we use that den W) = P(W V) (by construction) and that, under our assumptions, p(W| V) does not depend on E.
Similarly, for y22 we obtain
E-E (x = D(F 0 [V { (VV, IV)+
f(X)
dd (X)
+(V, W)W = w V, E = 0)
Σ (VV (V)*
iden (V, w)
1- E
i(E 0 v) (VV,
Σ (WW -IV -V, }
Putting these pieces together, we have
((: = (xx () + (X) ())
and the result stated in the paper follows upon specifying the choice g(x) = =luT, w T Writing 3) explicitly, we obtain
ET
(x
p(T = I,E - (V - (V)) Σ o(V, 1)) (VV,v)
E(1-3 T))
p(T 0,E 1 1 (Y -o(V)) (X--)()))
E [VVV,
(b (b-a) a0 [E) E)]
((V.W)((K.W) Σ 9(V, w) fVV, w))( ms}
1-E E
D[E = =0 (LV.W))(V
E
= V,E=00) 0)}
- + +((X))((X)((X) + -
The influence function for the upper bound Tu (x) has an identical structure, with the truncation endpoints swapped. Specifically,
ET
x(X:3)=
p(T = 1, E 1| V) (Y - (v)) (M2((V))) * E9(V,
9(V, (VV,
w
E(1
0, 01 (Y - Ao(V)) * Σ [
19

---

(b- (1-)) - 15]
g(V g(V, -V w) vi}
ME = -
M = 01 V] { (SL,W))(V,W)(C)
1-
Eg(V, (F), -, IV, 0) xX(()(((XX)1(x) + -- b--b)m(X,)),,,
+
where, for the upper bound, we use f(X) = ((T (x) - b 0) and flip and b a wherever they occur.
A.4.1 Bias Derivation
We now turn to the bias calculation underlying the second-order remainder term. In this subsection we keep the algebra as originally written, and insert commentary describing the main steps and tools (iterated expectation, algebraic rearrangement, and grouping of terms into products of nuisance estimation errors).
In the steps below, we primarily use iterated expectation and algebraic manipulation. For notation purposes we say P(W| V) 1 - P(W V). We will eventually convert some of the nuisance functions to the notation used in the paper (e.g. wV, = p(W = w V, F = 0) and po = p F = 0 V)), but for now we leave them in their original form as it makes them easier to visualize and manipulate. For brevity we use Ts = pTT = x, E 1 V) in these proofs. It is also important to recall that faon (V. W) = p(W | V), so we can interchange these as we like.
Our general strategy is to write the bias as an expectation of the difference between the estimated and "true" influence functions, and then algebraically rearrange this expression so that every term is either second order or can be decomposed into a sum of first-order and second-order term. We focus first on rewriting each candidate first-order term as a sum of a second-order term plus a remainder, and then handle any remaining first-order pieces.
The bias is given by:
R₂ = P()(P m))
We start by examining the contribution from 91
()) = E
D(T = - (Y -
ET
(Y-(v)) (1(1)) Σ DgVV w)
E(1 - T)
P(T = 0, E =1 v (V -oo())) * Σ (VV
D(E = 0VV) (o(X))(X) Σ 9VV, (VV FK
a)=a)(1 E)
E
T - -V)))) (41(V)-A1(1)
Eo(V,
#1
To (vo(V))AooV) (V)) [o(V, w)
no
(6 a)p(E = 0 X)
i(E V) ((x)
(b a)p(E= VV
EV(V w)p(w [V)].
In the last equality, we have taken conditional expectations given V and used the definitions of π. and the various nuisance functions. We then expand each ratio, such as πΙ n1/#1, around 1 separate first-order and second-order parts.
20

---

Contimuing this expansion, write:
11 #1 (HI(V) - R((V)) Σ (VVV))
#1
Ru (vO(V) An(V)) Σ (VV V)
+ (vi(V) - (V(V)) E D(VV
(Vo(VV - Ao(V)) " Lg(V,w)f(V.
(-))ppEE =0 -) D(E ([E=0VV) ((X)s(X) (6 (
Σ s(V, WVV, w))( v)].
ME -
The terms involving (T,- - - (us - P.) are already second order. The remaining two terms involving p(E 0 -) are the most delicate and are handled next.
We now isolate and manipulate those latter terms:
P = x) i(E = t)
E[(b- - a)
Σ
[())(( w)p(()]
E = 0 v)
- (b 0
Σ
{(x)(( ( (V)(V)
E = 0 v)
P
= B[(b- Σ
F(E = 0 gf (1)p(())))()))
P v)
8) Σ
((x)\(())(()))) [)p(()]
M(F = 0 m))
B[(b Σ (z))f(=)p(U) p(w 10)p(E== x)
P(W
V, E 0)p(E = 0 (8) H]
-(E=0
D(E = 0 v)
B[(b- a) Σ (z)))())p(()) p v)
1}*
M(E= =
(p(W = vv p(W w V U, E 0)}
+ (b a) Σ g(z))(=)<((W- w V v E 0) P(W w V == = =.
= B[[(b- a) Σ gV.. w) [V, w p(E 0
DF==0 H)p[W = w V,E = -0) M(W = E. 0)}
"
a E a) Σ g(V. w) (r( = V, F = 0) - p(W = w: V, F. = 0))
This algebra rearranges the terms so that we can clearly see (i) a second-order product of errors in p(E = 0 V) and p(W = w V, E = 0), and (ii) a leftover first-order term in p(W = w V,E = 0). The leftover term will later be paired with pieces from F2 and (ps and turned into a second-order term as well.
Next, handle the remaining first-order terms in Φ together with the contributions from sou and pp- For bookkeeping, it is convenient to rewrite as
x = PiX; B) + +2 B) + g(x) Faoo (X)
Fnum (X)
f - ((x))
-x()
Taum 1 ((x))
Taln (x)n()))))))
frum (X) ʃ - s-((x) - xx(X)
Tnum (X) / - ((x))
=x2(X; + +X)
faen
Tden
+ +x;(8; + g(x)
fnum (X)
Taen (X) / o(x)) g(x)
Taum (X)
Tden (X) f - g(x))
21

---

The first line collects everything: the second line groups together the pieces that will form second -odder products in Iden and the pieces that involve fumm
Tmum-
We first examine the contribution from the first group:
Faum fdon
f(x) - -x(((x)
Frum S(X) o(x)
P2 + 1
Tden
= [ p(E= V) (x(X) 10) ((X)
1-E E
Tdeu (X)
Σ g(V, w)f (X)(( V)
P(V, P(V, w) (W = w
V. E 0)
Tden (V, w)
mum
f(X) o(x))
Fuum f(X) g(x)
+
Fdem
Teln
p(E X)
(. = V) (X))((X)( Fnum (X)
E
iden (X)
p 0 V)
nuum (Vww)
fnum
fuum
+
P E = 0 V) Σ (o(V, ()) (VV, V)
+ gl
Pden V, w)
iden
Jden
10
fmmmg(V V, w) SV w)
00E=0 =0 V)
E
DE = 0| 1 1HB(W = w V,E = 0) p(W = w V,E=
idon [V, w)
Fumm 9 (V. w) wV. w) Fden V))
Fmmm
Am
(p(W = w V,E = 0) p(W = w V,E = -}}} + gf
iden
Tden
The first line in the last display is again second order (a product of errors in pl F = 0 V) and p(W V. F
0)). The second line is made socond-order by combining the factor p(W = V, E = 0) pp(W-WWWW-- = w V. E 0) with the difference in denominators, as we now show.
We examine this remaining part:
fnumg(V V, w)f(V, w) (o(W = w V,E = 0) p(W = = V, = 0)})
Fdon (V, w)
fumm
Fdcm
Teen
E
Σ
fnumg(V (V, (w) WWV((V w)
[fden (V,w) ) Tden (V, [V)] + g fmom
(
1
Pden (V, w)
Aden
Tdou
on
numgg( w) fV, w)
1
E
{fden Tden + faumg VV, w) wVV (el,((V,W))) w)p(w
V)
)]
Aden V, w)
Fden
Tden
w
1
E
f gumg (V, (V,)) w)(
(fden (V, Tden W])] +
Tdea (V. w) )) wn)))
iden (V, (V,w) w)
iden (V, w)
Tden (V,w)
1
1
1
E
i num (V, w)(
+
Adlm V,
Fdon (V, w)
7tenn (V, w)
Taon (V,
E Σ
uum g (V, w) Taen (V, w))fdn (V, w)
1
(aen (V, w) - Taen ( (V, =
This shows that these terms contribute only a second-order remainder in the denominator estimation error. We then turn to:
Faum (X)
TTuu (X) o(x))
9 (X; + g(X)
Falen (X) f o(x)) 9(x)
Taon (X)
and carry out a similar decomposition:
P +
Faum
f 1-2(2 x(x))
Toum
f -x())
Tden
Taon
22

---

E
#1
(VI(V) -ii(V)) Eg(h L (V,
To
to (Vo(V) - Ao(V))
_ P =0 v) - 1KP(W = w V, = 0) Bw== E 0
9V, w) WV w)
+ + a) E 9(V, WVV,, w)
AE = 0 v) + - Aa(()) * E (VV,
(vo(V) Aoo( DV(V.
(b a) Σ (VV, = V.E P(W = w V, = 0))
Fnum (X)
Tden (X) f -(x)) -9( -0( Tnum (X)
f f 5 ((
+ (X()
Idea (X)
As before, the first three lines of the expectation are already second order. The remaining terms are combined to yield products of (me - AA), (p(W = w V, = 0) -0- p(-)), and (Taen - fden), as shown in the algebra that follows.
After carrying out these rearrangements (which we keep as in the original derivation), we ultimately obtain the final bias expression:
Fn (V, w)f(V, w)
p(E = =0 0VV) - HHk(W = V.E = 0) p(W = w V.E E -)}+ p(E 0
E
fdeal (Vw))
w
mumg (V, (V,)).(V,w) * iden (V, w)² (Aden (dlen(,, (V.)) -den V w)) 2+
T
म1 (VI(V) (V)) Σ 9(V, w) WVV(V
To
Ao (HO(V) - Po(V)) Σ g(V, w) w)((V w)
40
((E = 0 V)
- (b. 10-00 a) Σ
w))
H)p(W = w V,E = 0) P(W = w V, E = 0)}
P(F = 0 vj
1
(XX))(X))(m (V) (V))
Fden (X)
Telen X)
-x(X))(()(((xOV)) - (V)(
Fden (X)
Telen (x)
1
(b- (6-aa)) (x-x))(X))XXXX)() V, 0) r(W V, |V,E= 0)]{ Pden (X)
Tdem (X)
Equivalently, regrouping some terms, we can write
Σ s(V, w) (V, w) - a + F(V, p( = 0 V) - 1H5( = V, -
E
=) 0) p(W- = = V. = =0)
w
Auumg(V. (V..g((.w))))) w)F(V w)
iden (V, w2 (Fden (V, w) Tden (V, w)) +
1
#1 (WI(V) - (Ar(V)) Σg(V,
11
To
(vo(V)- - -oo(V)) * Σ [o(V,w)((VV. toe
w
23

---

1
(I(V)) (I))) + +x(x)f(XX((((V)
}}
Tdon (X)
Tdon (X)
1
- -(XX) Idon (X)
Tdon (X) 3}
(b (6-=))(X)((())6 V 0) p(W | V,E =]){
Tden (X)
Tilen (X)
Bounding the Bias. Finally, we bound the magnitude of the bias using the Cauchy-Schwarz Schwarz inequality. The point of the following display is to make explicit that each term is a product (or square) of nuisance estimation crrois:
E Σ (V
2
11 p = 0 12 = w V, = 0) - p(W = w V, = 0))l2
P 0
1
f numm (V, [V, w
iden (V, w2 (fden (V, w) Idea ( (V, w2) 11
11
(v) (1))) Σ wVV,,)
A1
11 Fu
No (22l((o(V) [V)))) Σ 9V, w w)((V,))
Au
1
+ +x(X)((x))(mm (V) (2))
)|12
Fden (X)
Tden (X)
1
-x(x))(()]((xx(V) - - Pden (X)
)1/2
Tden (X)
1
(b (b-=))(x)/((X1II IV, = 0) PP V. E = 0)]12]1
11
fden (
Tden
(X)
Setting wVV p(W V,E=O0 0) and (V p(E V), it becomes clear that the bias is second order and bounded by products of L2-norms of nuisance estimation errors.
Thus our theorem, where we write the second-order remainder as
Rn = Pp 8, n) ((X;
(6)
||0 2||2|l 0- /1[2 + ||0 v1l2 + ||
ग :lllll #1l12
+ Ao oool2 ||00 moll2 + || f.l12110 w112 + Ilão moll2l10 vl12,
(8)
makes explicit that R2 is of second order in the nuisance errors, claimed in the main text.
A.5 Simulation Setup Details
We follow the following procedure to create our synthetic dataset for simulation:
We start by sampling n = 10, 000 observations covariates V. Three of these covariates are continuous and three are discrete.
1. Covariates Generation
V continuous
N(H 1, 0.53)
V discrete
Bernoulli(p = 0.5)
V | continuous: V ddiscrete]
24

---

The simulations do not require heavy compute or anything beyond a personal machine. iteration of the 2. Generating Discrete Variables W
W. Wa, W2, Wa
where each Wiy is generated using a logistic function:
Wij Bernoulli (a (V:-83)
Here, (xx 14e- is the logistic function, and Bg random coefficients:
B3 Uniform(-1,1)
3. Generating E and We generate the binary variable E' based on the covariates V:
Vix YER) ))
E Bernoulli
where YEk Uniform(-1,1) are random coefficients and o(x) = 1 is the logistic function.
Similarly, we generate the binary variable T:
6
A ~ Bernoulli
Va 7Ak
where YAk ~ Uniform(-1,1) are random coefficients.
4. True Conditional Treatment Effect: To generate the true conditional treatment effect for each observation, we compute BA as lincar combination of V and W:
BAi Σ
Va oVR + Σ Wiy awj
k=1
where avk Uniform(O, 1.5) and awj ~ Uniform(O, 1.5) are random coefficients.
5. Outcome Variable Y: The outcome variable Y is computed as:
Y = BAi A; +
Σ j=1
Wij Bwi +
Σ
Va BVR +
k=1
where Swj Uniform(1,3) and Pyr Uniform (1, 3) are random effect sizes, and e; N(0, 1) is random noise component.
simulation may take 1-2 minutes run on a personal laptop.
A.6 Application: Licorice Gargle
This dataset from a study conducted by (Ruetzler et al. 2013) which tested the hypothesis that gargling with licorice solution immediately before induction of anesthesia prevents sore throat and postextubation coughing in patients intubated with double-lumen tubes. In this study, total of 236 adult patients who were scheduled for elective thoracic surgery that required the use of double-lumen endotracheal tube were recruited. The dataset includes information various patient characteristics such gender, physical status, body mass index (BMJ), ago, Mallampati score (a mcasure of airway visibility), smoking status, preoperative pain, and the size of the surgery. The intervention received by each patient. is also recorded. The outcomes of interest, which include the presence of cough, sore throat, and pain during swallowing, were assessed at different time points. The dataset has been thoroughly cleaned and is complete, with only two patients missing outcome data. We manually remove these two patients from consideration. Therefore, finally, number of subjects N 234. No outliers other data issues were identified in the dataset.
25

---

Patients randomly allocated to of two groups: the first group reccived 0.5 grams of licorice, while the second group received grams of sugar, which was chosen to match the sweetness of the licorice solution. The patients' condition was assessed at multiple time points. At each assessment, the severity of sore throat was measured using an 11-point Likert scale, where indicated no pain and 10 represented the worst possible pain. The presence and severily of cough were also evaluated at these time points. idnalln, pain during swallowing was assessed using the same 11-point Likert scale 30 minutes after the patients' arrival in the PACU (post-anethesia
unit). For the purpose of this study, sore throat was considered present when the patient reported score greater than 0 on the visual analog scale.
For our experiment, we consider the following variables as W (present only in the target population data):
"extubation cough': This binary variable represents the presence or absence of coughing right after the removal of the endotracheal tube (extubation).
We consider the following variables as (present both study and target population data):
:aleB_'a This variable represents the calculated Body Mass Index (BMI) of the patients before the surgery (preoperative).
'preOp_age': This variable represents the age of the patients before the surgery.
'preOp_pain': This variable represents the presence or severity of pain experienced by the patients before the surgery on an point Likert scale from 10.
aheOO__mallampati This variable represents the Mallampati score of the patients before the surgery. The Mallampati score is a measure of airway visibility and is used to predict the difficulty of intubation.
'preOp_ae :sa"' This variable represents the American Society of Anesthesiologists (ASA) physical status classification of the patients before the surgery. The ASA physical status is measure of the patient's overall health and is used to assess the risk of anesthesia and surgery.
"preOp smoking': This binary variable represents the smoking status of the patients before the surgery.
We use 'pacu30min swallowPain', the pain in swallowing after minutes, a measurement on a Likert scale from 0 10, the outcome variable Y. Logistic regression used for model fitting.
26


Reviews:
[{"review_id": "8kfAVEzhTn", "reviewer": "Reviewer_EinB", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "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": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper addresses the challenge in causal inference of estimating heterogeneous treatment effects (CATEs) in a target population where certain covariates are newly observed but not available in the original study population. Traditional methods fail when these new covariates are unlinked to treatment or outcome data. To bridge this gap, the authors propose a partial identification framework inspired by ecological inference, which derives sharp bounds on the fully-conditional CATE by requiring consistency with marginal distributions observed in the study. Additionally, they introduce a bias-corrected estimator based on influence functions, allowing fast convergence even with nonparametric nuisance estimators. Experimental results on both simulated and real RCT data demonstrate tighter bounds and better coverage compared to conventional approaches.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The claims are supported by a theoretical identification guarantee and experimental results on different datasets.", "audience_yn": "Yes", "audience_explanation": "This paper studies the causal inference problem, which is a highly related topic to the TMLR community.", "requested_changes": "1.\tThe assumption that the distribution of new covariates W conditioned on common covariates V is the same across study and target populations seems strong and may not hold in practice, especially when there are structural differences between populations. The paper could benefit from discussing diagnostics or empirical checks for this assumption.\n2.\tWhile the method assumes W is discrete, the implications of discretizing continuous covariates (especially in high dimensions) are not thoroughly analyzed. How sensitive are the bounds to the binning scheme?\n3.\tThe estimator lacks robustness to errors in estimating ν, which is critical to the ecological linking. Although this is acknowledged, it may limit practical deployment unless ν is modeled very accurately.\n4.\tThe plug-in estimator used as a baseline lacks robustness and is expected to perform poorly under misestimation. Including a stronger baseline, such as doubly robust learners adapted to the restricted CATE, or recent deep learning based approaches, would provide more meaningful comparisons.\n5.\t(minor) The framework focuses on binary treatment and bounded continuous outcomes. Extensions to continuous treatments or multi-valued treatments are not discussed, which limits generalizability.", "broader_impact_concerns": "No ethical concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "uAUwxdxXtq", "reviewer": "Reviewer_WQUR", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "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": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper examines the situation where a clinical study was used to estimate the CATE in a first step. Then in a second step, clinicians have access to a new set of covariates that was not available in the previous study. The authors derive estimators of bounds of the CATE with additional covariates, assuming that the CATE was estimated for a reduced set of covariates on a first step. They use double machine learning and sample splitting for a (potentially misspecified) semiparametric model. This allows them to derive a semi-parametric estimator with parametric convergence rate and asymptotic normality. The problem is motivated well and the solution with double machine learning appears elegant and useful.  I also very much like how it is explained what proof techniques are involved!", "claims_and_evidence_yn": "Yes", "claims_explanation": "The explanations in the paper are easy to follow, as the very natural \"ecological\" inference idea is combined with double machine learning techniques based on earlier papers. The proofs, while in need of cleaning, appear reasonable and the experiments are convincing for me.", "audience_yn": "Yes", "audience_explanation": "I feel like the paper addresses an important problem in practice. As such this is a very interesting paper for people in CATE estimation, such as Machine Learning researchers working in healthcare applications. In addition, the fact that general machine learning methods can be used to estimate the nuisance functions in a first step make the paper even more interesting for a general audience.", "requested_changes": "I think this is an important paper and that the main part of the paper is well written and explained. However, I think it is very important that the authors improve writing in their proofs. Since the techniques are quite important, I think it would be very nice to have well-written proofs in the paper. I recommend going through the proofs as a whole carefully and improve writing, but in particular:\n\n- Formulas seem to come out of nowhere, such as on page 15. Please preface them with text as you would do in the main part of the paper.\n- Its probably not the best notation to use $\\tau_n$ as the numerator of $\\tau$ (since this usually means an estimator with $n$ samples).\n- While throughout the paper $ \\mathbb{E}[]$ is used to denote expectations, it suddenly seems that throughout the proofs $E() $ is used to denote expectations. If this is indeed true, this needs to be carefully changed, as this is very confusing with the variable E involved.\n- There is an \"A\" that suddenly appears to come out of nowhere, is this supposed to be T? Please check this carefully.\n\nGeneral comments:\n- Roadmap is very helpful!\n- In Theorem 1, $\\nu$ should be $\\nu(v,w)$\n- When introducing f(X), $\\tau(x)$ should have capital X.\n- Could you maybe add more detail on how the nuisance functions are estimated in the application?\n- For the Benchmarking explanation on page 9, please use another symbol than n in \"n variables from V\".\n- The bibliography doesn't appear consistent, sometimes with URL/ISSN/DOI, sometimes not. \n\nIf the proofs are adequately improved, I would very much like to see this paper published in TMLR.", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "qNhPrBepwK", "reviewer": "Reviewer_nnC3", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "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": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper addresses the transport of heterogeneous treatment effects when the target population contains covariates not measured in the original study. It introduces a partial-identification framework inspired by ecological inference to bound fully conditional CATEs, ensuring consistency with restricted CATEs when marginalizing over the new covariates. The authors develop a bias-corrected estimator with influence-function adjustments that achieves $\\sqrt{n}$ convergence and asymptotic normality. Simulations and an RCT example demonstrate that the method yields substantially tighter and more informative bounds than baseline approaches.\n\n**Strengths**\n1. Conceptual novelty. The connection between ecological inference and causal partial identification is both creative and technically interesting.\n2. Practical relevance. The setting tackled is relevant to real-world applications, such as healthcare interventions where additional covariates are often observed in deployment populations but absent in study data.\n3. Empirical validation. Both simulated and real data experiments highlight the informativeness of the proposed bounds compared to natural baselines.\n\n**Weaknesses**\n1. Strong assumptions. Transport assumptions (equal distribution of W across populations and equal CATEs across domains) are restrictive and difficult to validate in practice.\n2. Discrete-W requirement. The framework requires W to be discrete (or discretized), potentially limiting applicability or inducing discretization bias for inherently continuous covariates.\n3. Estimator vulnerability. Performance depends critically on estimating P(W|V) in the target population; the estimator is not fully doubly robust.\n4. Clarity and accessibility. Some proofs and derivations are dense, and key results could be stated more clearly in the main text for readability.\n5. Minor presentation issues. Inconsistencies in notation, missing proof of Theorem 3, and figure labeling errors affect polish.", "claims_and_evidence_yn": "No", "claims_explanation": "The authors didn't provide the proof of Theorem 3.", "audience_yn": "Yes", "audience_explanation": "At least some members of the TMLR audience will be interested in this paper because it tackles a practical challenge in causal inference: transporting treatment effect estimates when new covariates appear in the target population. This setting arises often in healthcare, policy, and other applied domains where study data omit important variables. The paper’s use of ecological inference ideas for partial identification, coupled with a bias-corrected estimator, provides novel methodological contributions and practical tools for generalization and fairness—topics of broad relevance in the machine learning community.", "requested_changes": "1. Discuss the strength and restrictions of the assumptions in Section 2, and compare them with assumptions used in related work.\n2. Discuss whether the framework can be adapted to handle continuous W without discretization.\n3. Discuss potential strategies to achieve robustness, or at least mitigate the impact, when $\\hat{v}$ is misspecified.\n4. The appendix is difficult to navigate; it would improve readability if theorems were stated clearly in the main text, with only the proofs deferred to the appendix and cited appropriately.\n5. In Section 3.3, the presentation of the Bias-Corrected Estimator, together with Assumption (4) and the subsequent inequality, would benefit from being stated formally as a theorem in the main body. Doing so would enhance clarity, provide a clear reference point for readers, and align with the presentation style of other main theoretical results.\n6. Provide the proof of Theorem 3.3.\n7. Ensure consistent notation for $\\phi(X, \\beta, \\eta)$ versus $\\phi(X; \\beta, \\eta)$.\n8. Correct the caption of Figure 3 to read Bottom (c, d) instead of Bottom (b, c).", "broader_impact_concerns": "No ethical concerns were identified that would require a broader impact statement.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)