A Unified Framework for the Transportability of Population-Level Causal Measures
Authors: Ahmed BOUGHDIRI
Clément Berenfeld
Julie Josse
Erwan Scornet
Venue: NeurIPS 2025 poster ()
Keywords: Causal inference
Generalisation
Causal measure
Primary area: probabilistic_methods

Abstract: Generalization methods offer a powerful solution to one of the key drawbacks of randomized controlled trials (RCTs): their limited representativeness. By enabling the transport of treatment effect estimates to target populations subject to distributional shifts, these methods are increasingly recognized as the future of meta-analysis, the current gold standard in evidence-based medicine.
Yet most existing approaches focus on the risk difference, overlooking the diverse range of causal measures routinely reported in clinical research. Reporting multiple effect measures—both absolute (e.g., risk difference, number needed to treat) and relative (e.g., risk ratio, odds ratio)—is essential to ensure clinical relevance, policy utility, and interpretability across contexts.
To address this gap, we propose a unified framework for transporting a broad class of first-moment population causal effect measures under covariate shift. We provide identification results under two conditional exchangeability assumptions, derive both classical and semiparametric estimators, and evaluate their performance through theoretical analysis, simulations, and real-world applications. Our analysis shows the specificity of different causal measures and thus the interest of studying them all: for instance, two common approaches (one-step, estimating equation) lead to similar estimators for the risk difference but to two distinct estimators for the odds ratio.

A Unified Framework for the Transportability of Population-Level Causal Measures
Ahmed Boughdiri Premedical Inria
Clément Berenfeld
Premedical Inria Univ. Montpellier, France cement.errent elddinria..f.
Univ. Montpellier, France
ahmed. boughdiri iettsdirr@iiniii... fr
Julie Josse Premedical Inria Univ. Montpellier, France j josseeinria.
Erwan Scornet
Sorbonne Université and Université Paris Cité, CNRS Laboratoire de Probabilités, Statistique Modélisation
F-75005 Paris, France
erwan. orw.a...correeeeebooooounutvttte
Abstract
Generalization methods offer a powerful solution to one of the key drawbacks of randomized controlled trials (RCTs): their limited representativeness. By enabling the transport of treatment effect estimates to target populations subject to distributional shifts, these methods are increasingly recognized as the future of mnta-anaayssss, the current gold standard evidence-basee medicine. Yet most existing approaches focus on the risk difference, overlooking the diverse range causal measures routinely reported in clinical research. Reporting multiple effect measures- both absolute (c.g.. risk difference, number needed treat) and relative (c.g., risk ratio, odds ratio)-is essential to ensure clinical relevance, policy utility, and interpretability across contexts. To address this gap, propose unified framework for transporting broad class of first- first-moment population causal effect measures under covariate shift. We provide identification results for both continuous and binary outcome under two conditional exchangeability assumptions, derive both classical and semiparametric estimators, and evaluate their performance through theoretical analysis, simulations, and real-world applications. Our analysis shows the specificity of different causal measures and thus the interest of studying them all: for instance, two common approaches (one-step. estimating equation) lead similar estimators for the risk difference but distinct estimators for the odds ratio.
Introduction
10 4 have emerged a powerful response the restricted ex- ternal validity (36] of RCTs: due to stringent inclusion and exclusion criteria, RCT populations often exclude key segments of the real-world patient population- such individuals with comorbidities, pregnant women, or other vulnerable groups reesuliing in trial samples that are poorly representative. Consequently, the findings of many RCTs may lack relevance for broader clinical policy applica- tions. Generalization techniques address this gap by exploiting treatment effect heterogeneity- is, the fact that treatment efficacy can vary systematically with patient characteristics. By adjusting for
Generalization methods 33 4 12
Code & Python library: http://////////oooooooo///ooo/aa
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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differences in the distribution of these covariates between the trial population and a target population. these methods can estimate treatment effects beyond the original study population. This is especially valuable given the high costs, long timelines, and operational complexity of conducting new trials. As such, generalization approaches are increasingly viewed as a pivotal step toward rethinking the role of clinical trials in modern evidence generation. The implications are far-reaching. For instance, when drug reimbursement decisions are partly tied to estimated real-world efficacy [17], the ability to predict treatment benefits across diverse populations could influence pricing, access, and healthcare policy. Moreover, recent works suggest that generalization methods may also redefine the role of traditionally viewed as the top of the pyramid of evidence-basse medicine.
meta-analysis [T 37 20
Most generalization methods are dedicated to estimating the average treatment effect (ATE) via the risk difference (RD), owing to its linearity and analytical convenience. Yet this focus is incomplete. Clinical guidelines and regulatory bodies explicitly recommend reporting both absolute and relative effect measures [39 31. as they capture complementary aspects of treatment impact. Among absolute measures, the RD remains standard, but the number needed to treat (NNT), direct, clinically intuitive transformation of the RD, offers additional interpretability [29] On the relative scale, measures such as the risk ratio (RR), widely used in public health research [28], and the odds ratio (OR), very popular in epidemiology [8 play a critical role in framing treatment efficacy. Presenting multiple causal estimands is not simply recommended- it is essential. A treatment effect that appears homogeneous another, phenomenon that may seem counterintuitive
on one scale may reveal heterogeneity but carries significant implications for personalized decision decision-making. Furthermore, the perceived magnitude of benefit can shift dramatically depending on the baseline risk and the effect measure used. Consider a treatment that reduces mortality from 3% to 1%: the RD of 0.02 may suggest a minor effect, yet the RR reveals threefold increase in risk for the untreated, reframing the impact as clinically substantial. This striking contrast illustrates how the choice of causal measure directly shapes interpretation and ultimately, policy and clinical decisions.
Contributions. In Section 2 we present unified framework for transporting a broad class of first-moment population causal effect measures, defined as functionals of the expectations of potential outcomes. This class, composed of more than dozen (RD, RR, etc.) and non-collapsible measures (odds ratio-OR, NNT etc.) the latter posing unique generalization challenges. Building on this formalism, we develop generic identification strategies under two key assumptions: (i) exchangeability in mean (Section 3. and (ii) exchangeability in effect measure (Section 4. The latter assumption, though weaker, requires access to control outcomes in the target population a condition met, for instance, when treatment has not yet been deployed. Crucially, our identification results extend non-collapsible measures even under this weaker condition, which to our knowledge has not been previously derived. Within each setting, we derive two broad families of estimators. The first approach adapts classical methods- as weighting HHorviz-TThompson and regression-basee strategies (G-formula)-Fe for which we establish asymptotic properties and derive closed-foree variance expressions. To the best of our knowledge, no prior work has formally studied these properties using density ratio estimation. The second leverages semiparametric efficiency theory to construct new doubly-robust estimators using either one-step or estimating equation approaches. While these often coincide in the linear case (e.g., RD), we highlight that for nonlinear measures such as the RR or OR, they enverge-leadinn to fundamentally different estimators. Finally, in Section [ we conduct an extensive empirical evaluation of all estimators on synthetic and real-world dataset.
estimands, includes collapsible
Related work. Most generalization work focused on RD. While the idea of weighting randomized controlled trials (RCTs) is not new, using external data can be traced back to the foundational work of [5) 14 10 provided a comprehensive survey of generalization techniques, and (6] contributed additional consistency results for the main classes of estimators: regression-basee
rrightnng-bssed and hybrid approaches. [8] further investigated the inverse probability of sampling weighting (IPSW) estimator, deriving finite-sample bias and variance, as well as an upper bound on its risk under the assumption that the covariates X are categorical. Other studies have addressed the generalization problem by modeling the outcome directly 12 13. I1 extended this line of research to settings involving multiple source datasets (i.e., several RCTs) and set of covariates drawn from a distinct target distribution. They position this framework as a natural evolution of traditional meta-analysis, highlighting its potential to unify evidence across studies. Although the present work focuses on a single RCT and a single target population, all of our results seamlessly extend to the muT case. This generality allows us to offer a compelling alternative to classical meta-an satat-aaallss-ssee
2

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that supports the generalization of both absolute and relative treatment effect measures in a unified framework. We also highlight recent works on the analysis of externaly controlled single-arm trials and hybrid trials [2 which aim to enhance the precision of RCT-based estimands by incorporating external information. Similarly, [155] incorporate observational data to RCT's using the prediction- powered inference framework. While these approaches address distinct inferential questions, they share structural similarities with ours in using auxiliary data to improve estimation. Another line of work has focused on estimating alternative causal effect measures beyond the risk difference. For instance, ( proposes several strategies for estimating the risk ratio (RR) in observational settings, without addressing the generalization to target population. [7] examined several key properties (e.g.. collapsibilityy of different causal effect measures (e.g.. RD, RR, and OR) to identify which estimands less sensitive to distributional shifts. While their work emphasizes which causal estimands are easier to identify and generalize, ours provides a unified framework that generalizes all first- moment causal measures, focusing on estimation strategies and efficiency for practical deployment with theoretical guarantees. Their focus is on distinguishing treatment effects from baselines; ours highlights the shared structure among estimands, enabling unified identification and estimation. 34 44 [4]] introduce methods for estimating conditional risk ratios, including approaches based on causal forests. Yet, is crucial to emphasize that due to the non-linearity of relative measures, estimating the CATE does not directly yield the ATE.
2 Problem setting: notation and identification assumptions
Following the potential outcomes framework 38
we consider random variables (X, S, A, A..AA (,))) y()), where X R" denotes patient covariates, S {0. indicates sample membership (S = for the source population and S = 0 for the target population), and A E {0, 1) denotes treatment assignment (A 1 if treatment is administered and A = otherwise). Potential outcomes Y (0) and (1) represent outcomes under control and treatment respectively, of which only one is observed per individual depending on the assigned treatment, yielding the observed outcome
AY(") + (1 ()
We observe data from two distinct populations: a Randomized Controlled Trial (RCT) dataset (X;, A, Ee) from the source population, where treatment assignments and outcomes are recorded, and a target dataset (Xi)se(m). where only covariates are observed. This reflects the practical constraint that treatment and outcome data are collected only for individuals enrolled in the trial, while covariate information is available for both populations. We model this two datasets as stemming from one probability distribution Pote and we observe a N N-sample:
(Zi)sEIM) == (S;, X,, S,A,, S,Y,) SE[N) ~ PPN
Let a be the Berouuli parameter of the random variable S. Thus, (n. m) follow a multinomial distribution with parameter N and probability (o, a). Let Ps and Pr denote the conditional distributions of Z given S = and S = 0, respectively, with corresponding expectations Es[-] and ET[.]
2.1 Causal estimands of interest
Causal effect measures, as formalized by Pearl 332 are expressed in terms of the joint distribution of potential outcomes. Individual-level causal effects rely this joint distribution, which is generally not identifiable from observed data. Therefore, in this paper, we focus on a specific subclass: the first- moment population causal measures [16]. This class includes commonly employed estimands- such as RD, RR, and OR-and and less frequently used quantities, including the Switch Relative Risk [22). Excess Risk Ratio (ERR) 44, Survival Ratio (SR), and Relative Susceptibility (RS), and Log Odds Ratio (log- (R)g see Appendix AarendixAAooo for a detailed enumeration of measures falling into this class.
Definition 1 (First moment population causal measures). Let P denote the joint distribution of potential outcomes (0 (()). The quantity TP is a first moment population causal measure if there exists an effect measure c D+ + R, with Do C IR, such that for all distributions P with (Y()) (y(()) E De.
T = $ [Eply(U),,Bp/P Bp(Y(1)))
(1)
3

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We require that for all do E R. the map th + D(111 vb) is injective on its definition domain. It's inverse, when exists, denoted by vg) and is called the effect function. This definition extends to subpopulations by conditioning on covariates X, yielding, when it exists, the conditional effect
(x) = : (EplY (xp|
Example For these well-known causal measures, the effect measures and functions are given by:
Measure
Effect Measure
Effect Function r
Risk Difference (RD)
( tha) 1/1 4oD
[(r, tho) tho
Risk Ratio (RR)
( tho) = Mi
T(T, 16) = T v0
Wn
Odds Ratio (OR)
( vbo) =
2/1
1-6l
[(T, 100) I+F:
7-10
I-V1
Va
The existence of ensures that for a fixed baseline, distinct treatment responses yield different causal measures. In the following, we use Th to denote any first moment population causal measure evaluated under distribution P. Our objective is to estimate 7 + (ET[Y((), BT BT(()))), for all effect measures 4 where the expectations are taken with respect to the target population distribution.
2.2 Identification assumptions
Because the effect Th defined in terms of potential outcomes in the target population, it is not directly identifiable from trial data alone. The core difficulty lies in the fact that while treatment assignment the trial randomized, the trial sample may not be representative of the broader target population. To ensure identification of the estimand from observed data, we introduce standard causal inference assumptions (Assumption [
and assume covariate overlap (Assumption 2
Assumption (Trial's Internal Validity). The RCT is assumed to be internally valid, that is:
A. lgnorability: (Y(), y(o)) A S 11$=1, 1;
B. Stable Unit Treatment Value Assumption (SUTVA): Y = AA + (1 A)Y
C. Positivity and Randomized Assignment: A B(x), where < 1 (typically 0.5).
Assumption (Overlap). For all x E supp(PT), we have P(S X x) 0.
Generalization under exchangeability in mean
In addition to internal validity and covariate overlap, identification under this setting requires a transportability condition that links the distribution of potential outcomes between the trial and target populations. This condition can be expressed as conditional exchangeability in mean, which states that, conditional covariates, the average potential outcomes the same across populations.
Assumption 3 (Exchangeability in mean). For all supp(Ps) n supp(Pr) and for all {0, 1}. we have 1 (a) (a) = (([,(a) (x) where u (x) = E (a) X xr.
Leveraging observations from the randomized controlled trial and under Assumption to three
identification formulas that express Er (Y in terms of source population quantities can be derived:
Er Er (a)) = ET
Es
Y
(a) [
(transporting conditional outcomes)
(3)
Pr( (a)
Ess
(weighting outcomes)
(4)
Ps(X)
(X)
= Es
Es [x)
(weighting conditional outcomes),
(5)
(x)
see Appendix B.T for details. the next section, we present several estimators any first moment population causal measure, based on the three identification formulas above.

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3.1 Weighting and regression strategies under exchangeability in mean
The Horvitz- -himmpoo estimator [1 is probably one of the most simple estimators to use in RCT. Based on equation (4), we construct the weighted Morvitz-Thomper estimator. The following assumption is also required for technical reasons.
Assumption For a E {0, 1), we assume that y(o) and Pr(()y(a)/Ps) (X)(X) are square-integrable Horvitz-Thompson, Definition (Weighted Horvitz- Thompson). For any first moment population causal measure 4
define the weighted Horvitz- Sorvit--Thompon estimator wHT as follows:
(1 - n2 9,=1
A.Y, π
--)YY, π
To wIII
+
7X.))
Σ 5,=1
F(X,)
where F(X) is any estimator of the density ratio between the target and source covariate distributions.
When F is replaced by the oracle quantity show that To. wirr is asymptotically normal and an unbiased estimator of Tig (see Proposition 8 Appendix B.2 Since
Pr( P(S = 1) P(S = 0| =x)
r(x)= = Px(X) P(S 0) P(S =
one can estimate r(X) by estimating P(S = X = x) via a logistic regression model. By doing so, avoid imposing parametric model, such as a Gaussian distribution, the distribution of X [see also (24]. In this context, the M -estimation framework [42] can then be applied to derive asymptotic variances for this estimator.
Logistic model. We assume EXXX" is positive definite, X is Sub-Gaussian, and 38x0 = (B 3.. E IR++ s.t. P(S = 1|X) Bxx) (1 exp( B --2))-1 a.s.
Proposition (asymptotic normality of weighted Norvitz-Thomper estimator). Grant Assumption 7 to Let By denote the maximum likelihood estimate (MLE) obtained from logistic regression of the selection indicator S on covariates X. Define the estimated density ratio
(x) = n 1 00, 3x) N-n Oxx, Bx)
with n Σ> St
Under the logistic model, the Horvitz-Thompson estimator T0. weighted by the estimated ratio
r(x, 3x) is asymptotically unbiased and satisfies VN (f T 4 N (0, Vorrr).
Appendix B.3 provides full proofs and asymptotic variances for all first moment population causal measures, including results for the Neyman estimator with estimated treatment probabilities-ress that, to knowledge, are novel even for the RD.
Alternatively, using the identification results in equations and ( one can adapt the regression- based approach of Robins [35] to derive weighted or transported G-formula estimators. While the transported version appears Dahabreh et [12 for the RD, the weighted version, to the best of our knowledge, has not been previously presented.
Definition WWeighted/Transpon
G-formula). For any first moment population causal measure 4 we define the weighted G-formula estimator Fo. and the transported G-formula estimator To as:
f(X.) 1S (X.))
F4. +
F(X)) P (X.),
(8)
8,=1
5,=1
Fo.GG Φ
(X.); N n
Σ
pio ( (Xi)
N n
5,=0
5.=0
where F, P.y (1): Poy are any estimators of f Poy
Using the M-estimation framework 42 one can derive asymptotic variance estimates for both G- formula estimators. Furthermore, assuming linear model for the outcomes, we prove that regression adjustment leads to a lower asymptotic variance compared to the weighted Horvitz-Thie Thompson.
5

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Linear model 1. For all a, {0,1}. let y/a) = VT 3/00 + +el where V = [1, X with Ble!") (a) X] = O and Var(es (a) |x]= -².
Proposition (Asymptotic Normality weighted and Transported G-formula Estimators). Let fs. G and wG denote respectively the weighted G-formula where F is a logistic regression estimator 7 are ordinary least squares estimator. Then, under Assumptions
and Pay P()
VN (Fo. -60) NN VONGG)
VN (Fo.G - 4 N (0, VO.GG.
Besides,
VOLS 0.1G
G.wG S Vo.IT, where all variances are explicit Appendix B.4
Given these results, might wonder whether it's better to avoid estimating the density ratio altogether, as the transported G-formula is more efficient under correct specification. However, when models are misspecified, weighting the outcomes may perform better. This motivates doubly robust estimators, which retain consistency if either model is correctly specified.
3.2 Semiparametric efficient estimators under exchangeability in mean
common approach to constructing doubly robust estimators relies on semiparametric efficiency theory. By deriving the efficient influence function (EIF) of the target parameter [see, e.g. 22], one can build estimators that are not only robust to mispecification, but also achieve the lowest possible asymptotic variance among unbiased estimators. We denote by so (Z, 17, o vi)) the EIF of Tø. which depends on (i) the nuisance parameters lj = (110))P(1);;)) and (ii) the values of wi and vig
Based on estimators f). 2T. %0 one can use the EIF framework via one of the following two techniques to build new estimators of T whose properties are described below:
One-step estimators 70s consist in applying first-order bias correction estimator 7 ( vo) resulting in F = + + x ΣN so (Z, 77 v 7
initial plug-in
(ii) Estimating equation estimators are obtained by setting the empirical mean of the EIF to zero 2T v0 are solutions to
This amounts to finding an estimators FE . v0 such that of.
EN Pp (Z,, 17, 11 5%
(Estimating Equation).
In practice and in both case, it is usual to resort to crossfitted techniques 3 by estimating Te and f and evaluating the EIF yog superscript the rest of this section for the sake of clarity. Both above approaches require knowing the EIF so. As To ( rh)) letting y (resp. (Po) be the EIF of t/1 (resp. tho), standard calculations on EIF functions (see e.g. 126 Sec 3.4.3)) show that
different datasets to enforce independence. We drop the T
pe (2 n, 1/. vo) = 0 ( do)) (Z, 11) + 00 4o po)ppo( (Z, v0).
With this equality, and the following proposition, one can compute the influence function f4. Proposition 3. Grant Assumption | Too For all a E {0, 1), we have
- S La (Z, 1 (M))(X) vaa a aP(A=
S1 ( A a}
The exact expressions of one-step and estimating equation estimators depend on pa, which in turns depends on the effect measure Q. While the first approach leads to explicit expressions, estimating equation estimators are defined implicitly through the estimating equation in (id), which writes
N 9 (Z. 31) + 2 Jo)
N
0 ( 3o) N
Po (o 7 vo) 0.
(10)
One way - though not the only to satisfy (00) is to find two estimators 1 wo that cancels the empirical mean of their EIF (second and fourth term in (TO)), that is the corresponding EE estimator for vi and v/n- We end up with a plug-in estimator of the form (E v0 precise expression is given below.
TEE), whose

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Proposition 4 (Estimating equation estimators.). Given estimators P(oa) (resp. F) of P(a) (resp. r),, an estimating equation estimator FEE of Tap given by 7EB
( TE
40 TEE) where for all {0, 1}
EE V
1 Σ (Xi) +
1-a
1{A a]
(X (Xi))
(11)
771
mI S,=1
P(A a)
S,=0
A similar estimator already appears in [1]. We prove below that FEE is (weakly) doubly-robust. Proposition 5. Let Plaa and F be two estimators independent from Zq...., Zn Then, under bound-
edness of ( F and Y, and assuming that & is continuous, the estimator F ( [ FBF) is consistent soon as either ((a) for all E {0, f
40
It would be easy to extend this result to strong robustness property, meaning that v VN - -Tp) (O,, Var ( under mild convergence requirements of ((a) and f towards P(a) and r. Concerning the OS estimators, needs to be assessed on a case- case-by-case basis. For instance, [1 shows that the OS estimator for the the estimation of the nuisance parameters and some extra assumptions on 10g light of these observations, we recommend using EE estimators rather than OS estimators, when possible, they also usually lead to better results empirically, see Figure D
cannot derive any formal robustness results in full generality, and this property
RR is doubly robust based on usual requirements
When + a linear functional (e.g., RD), the two approaches, one-step and estimating equations yields the same estimators, up to a scaling term depending on a and N. For nonlinear functionals however (such as the RR or OR), the estimators generally differ (see below and Appendix B.6
(RD) For the risk difference, the estimating equation approach yields FFF T... 40 The one step approach, based on initial estimators v and wo yields an estimate of the form:
EE + (1 N(1-a)
m
N)1 ARD-
m
RRD
N(1
In particular, estimators Lh of the form G-formula weighted Horwitz-Thomee yield final one-step estimator that has the same structure as the estimating equation estimator. T
(RR) For the risk ratio, the estimating equation approach yield 'RR TF one-step approach, based on initial estimators & and d yields
In contrast, the
OS TRR
11
EE a)
EE Vo).
40
20
N(1
N(1
②))
In general, *SS *
SE RR.
unless course we initially picked Vo = JEE
Generalization under exchangeability in effect measure
Estimators in Section were derived under the exchangeability in mean assumption (Assumption 3) Under this assumption, generalizing necessitates full access to all prognostic covariates whose
distributions is shifted between the source and target populations [8] A weaker assumption consists in transportability of conditional treatment effects.
Assumption (Exchangeability in effect measure). Vx E supp(Pr) n supp((Ps). -f(()) = - (x).
Under Assumption 5 the effect of the treatment depends on the patient's 's features in the same way in the source and target population. While exchangeability in mean implies exchangeability of the effect measure for any Φ, the reverse does not generally hold.
4.1 Transporting causal measures under exchangeability in effect measure
To ensure identification, we typically require a relationship between the conditional average treat- ment effect (CATE) and the average treatment effect (ATE). A classical concept that supports this
7

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relationship is collapsibility: a measure is said to be collapsible if it can be expressed as a weighted average of conditional effects [see, e.g., 19 23 However, some measure (like OR) are not col- lapsible, thus questioning their transportability under this Assumption 5 However, it turns out that under Assumption any first-moment causal measure is identifiable assuming access the control potential outcomes Y (0) for all individuals in the target population:
T T
+ Er r
Tx()) (4) (x)] Brr (x
(transporting)
(12)
x) (x) ps(
pr
T (X), (1) (l)) (x)) EET [r((0]
Es
(weighting)
(13)
where I is the effect function (see Definition D.. Note that for collapsible measures such as RD or RR, expressions (12) and (3) reduce to the identification results derived by (7]. Thus, framework can be viewed as a natural extension of their approach to any first moment population causal measures even non-collapsible measures such the OR. The detailed identification results for RD, OR, and RR are provided in Appendix C Using the identification formula (12). we derive I'-formula estimators, which, to the best of our knowledge, are novel contributions.
Definition 4 (Transported I' I'-formula). For any first moment population causal measure 4 define the transported I -formula estimator far as follows:
Σ S
Σ (x.)
(14)
N
n
N n
S,=0
We can also define, as before, a reweighted version of the I'-formula (see Definition 3).
4.2
Semiparametric efficient estimators under exchangeability of treatment effect
Under Assumption 5] the identifiability formula (12) serves as the basis for constructing the EIF, Given access the target baseline distribution, M() and oi are known. To construct one-step and estimating equation estimators (see Section 3.2). we require EIF of 7 denoted yo (Z, 1,, Th) which is related to 9i, the EIF of of. via the chain rule:
Pa [Z,n, + = :p (Z, 1 1 ( v
where 7) (m) r) the nuisance parameters.
Proposition 6. The inffuence function of oF at Pass is given by
4P (2, vT) = Sr(X)
A (A - I(To - (T))
Sr(X) 1 (Y- (X))
1-5 + 1-a (T(ro(X), wF)
As in Section
3.2, we can either find the estimating equation estimator of ti and plug it into to obtain +
TEE wit",, or apply a one step correction to initial estimator ক of the form -OS- ( vo) + (1/N) Σ P (Zi, T Like in Section 3.2 the estimating equation +4
estimators and one-step estimators have no reason coincide general. Exact computations of the RD, RR and OR are provided in Appendix C.3
5 Simulations
5.1 Synthetic data
We generate data (S, A, ()) y()) using the following binary outcome model: for all a, E {0, 1). (a) = = X,S = 8) = p. (V) where VT = [1, TT] and X = s (V,,,, We set d 5 and S ~ B(0.3) to reflect limited RCT data relative to the target population, and 5 = B(0.5). We evaluate the estimators from Section 3 and 4 estimating nuisance paaponents-rrgresssee surfaces and density ratios-using parametric methods (linear/logistic
8

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regression). The red (resp. gray, when displayed) dotted line represents the treatment effect in the target (resp. source) population. basic linear setting, in which all estimators perform well, is presented in Appendix D
Experiment 1 (Exchangeability in mean and non-linear/non-loiiste response): We consider setting under which Assumption holds and Va, 8 E {0,1}, p. (V) = ( V (VT 3))"). estimators exhibit substantial bias across all evaluation metrics, which is
Both G-formula-basse expected given that the non-linear response surfaces are misspecified by using linear regression. In contrast, the estimating samation-based estimators remain unbiased across all measures, benefiting from their double robustness property. Among the one-step estimators, only the RD variant is accurate in this setting. This is also anticipated, one-step estimators generally do not retain double robustness-exee except in cases where they coincide with the corresponding estimating equation estimator, which holds true for the RD variant.
Hervit-Thompson
+ H
W fommalla
G Bormul
Esimating equation
One-dip
Risk
Figure I: Comparison of estimators across different causal measures under non-linear outcome model with sample size of N 50,000 and 3,000 repetitions. Source values are 0.45 / 3.2 7.5.
Experiment 2 (Exchangeability in effect and linear/logistic response): We now consider setting for which Assumption 5 holds and such that, depending on the causal measure: (RD) Model [ and
a(a) =335 + 0 (RR) ps (a) (X) a (xx
7) (OR) ps (a) (x) o + 7)). In
this setting, Assumption 3 no longer satisfied, the nuisance functions depend on S. However, one can verify that Assumption [S] holds for each model. Estimators introduced in Section 3 still
converge for the RD, despite the violation of strong transportability, thanks to the linearity of the RD. In contrast, the RR and OR estimators fail to converge, since these measures are nonlinear. On the other hand, all estimators introduced Section iisoonorentt remain unbiased, as expected.
Horvite- Thompson
Extimaiing equation
T TT-oomaaaa
T-Cormula
I' ceatmorin--euuatoon
I Oce-step
-0.15
-0.10
-0.05
0.00
0.55
0.5
0.4
0.5
0.5
0.7
0.8
0.0
1.0
1.3
1.2
3.4
1.5
Risk Difference
Risk Ratio
Oddla Ratio
Figure 2: Comparison of estimators across different causal measures under a linear outcome model with sample size of N 50,000 and 3,000 repetitions.
5.2 Real- World Experiment
evaluate estimators using a case study on the effectiveness tranexamic acid on mortality for brain injury patients, combining data from the CRASH-3 trial and the Traumabase registry CRASH-3, is RCT with over 9,000 TBI patients from 29 countries, while Traumabase provides detailed clinical data 8,000+ patients from 23 French trauma centers. Following 9 we consider

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six covariates (age, sex, injury time, systolic BP, GCS score, pupil reactivity). Since this is real dataset, the true treatment effect unknown. Results are displayed in Figure All estimators (except one) indicate positive treatment effect; however, the confidence intervals are wide and include the null value (zero or one), preventing any definitive conclusions about the treatment's effectiveness.
Horvite-Thempoon
G-formula
G-fermala
Esimating equatinn
I-fermala
W T-formula
Esiimotingeequtioo
One-step
-0.06
-0.94
-0.02
0.00
D.62
0.04
0.7
0.8
0.9
1.0
1.1
1.2
0.6
0.7
0.8
0.9
1.0
1.1
1.2
Risk Difference
Risk Ratio
Oddde Rotio
Figure 3: Comparison of estimators across different causal measures on the combined CRASH-3 and
Traumabase dataset. Confidence intervals were estimated using stratified bootstrap resampling.
Conclusion
This article introduces general framework for the generalization of first-moment, population- level causal estimands from RCT's to broader target populations. We propose different estimators (weighting based on density ratios, outcome regression methods, and approaches based on EIF) and analyze their statistical properties. A central source of complexity lies in the inherent nonlinearity of many causal estimands, which disrupts the alignment between two classical EIF techniques: one-step and estimating equations. This divergence gives rise to diverse landscape of possible estimators, which differ depending on the underlying identification assumptions. While assuming exchangeability in effect measure is less restrictive than exchangeability in mean, few estimating methods available in the former case, as most them are based on CATE estimation, which is difficult for non- non-linear recommend resort to EE estimators, as they provably doubly robust and behave well our controlled experimental setting. Further layers of complexity arise from the selection of appropriate nuisance estimators, whether parametric nonparametric, each with its own trade-off'. Nevertheless, this work represents a step forward in enabling the computation of both absolute and relative causal measures in target populations, thereby contributing more informed clinical maciing and external validity in causal research.
measure. In practice,
7
Acknowledgments
This work has been done in the frame of the PEPR SN SMATCH project and has benefited from governmental grant managed by the Agence Nationale de la Recherche un- der the France 2030 programme, reference ANR-22-PESN- 0003.
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13

---

Technical Appendices and Supplementary Material
Contents
Introduction
2
Problem setting: notation and identification assumptions
Causal estimands of interest
2.2
Identification assumptions
3
Generalization under exchangeability in mean
Weighting and regression stralegies under exchangeability mean
3.2
Semiparametric efficient estimators under exchangeability in mean
4
Generalization under exchangeability in effect measure
Transporting causal measures under exchangeability in effect measure
4.2
Semiparametric efficient estimators under exchangeability of treatment effect
5
Simulations
5.1
Synthetic data
8
5.2
Real-World Experiment
Conclusion
10
Acknowledgments
10
First Moment Causal Measures
15
TransportingRReveigtting a causal effect under exchangeability of conditional outcome
15
B.I
Identification under exchangeability of conditional outcome
15
B.2
Oracle weighted Horvitz-Thompson
15
B.3
Logistic weighted Horvitz-Thomson Estimator
18
B.4
Weighted and transported G-formula
24
B.5
Semiparametric Efficient Estimators under Exchangeability of conditional outcome
35
B.6
Computations of one-step estimators and estimating equation estimators for the OR,
36
Transportig/Reeeiigttngg a causal effect under exchangeability of CATE
37
Identification under exchangcability of conditional outcome
37
Semiparametric efficient estimators under exchangeability of treatment effect
37
Computations of one-step estimators and estimating equation estimators under ex- changeability of CATE
38
Simulation
38
14

---

A
First Moment Causal Measures
Measure
Effect Measure p(111, vo)
Effect Function ['T,, v/o)
Risk Difference (RD)
(1h1, s/o) 1/1
4/0
[(T, vho) = vho +
Risk Ratio (RR)
( v/o)
1/1
F(T, v0) - 150
Win
Odds Ratio (OR)
(111, vho)
4b1
1-slu
['T, vo)
T-p
I-W-
14T-80--8h
Number Needed to Treat (NNT)
+ ( v/o)
[(T, 10) 4.0
Wi Via
if 4/1 400
- r)(1 10o)
Vo
Switch Relative Risk (GRRR)
@(1/1; 1/))
if 1/1 Vo
T(r, vo)
vo
1 +
14
if 9/1
3/0
7(1 + +
if 4o 0
Un
Excess Risk Ratio (ERR)
(1/1, who)
11
T(r, (/u)
(1 + 3/0
P2p
Survival Ratio (SR)
(1/1, 4Poo
1 1h
I(T, v/)) = 1 - T(1 2/0)
1-
Relative Susceptibility (RS)
( vho
1-
[[T, 1/o)
1-1
I- 14
100
Log Odds Ratio (log-OR)
(vV1, sho) log
6b 6b (11--
[(r, r/o)) exp(1io)
1-r+exp(wo)
V
Odds Product
(v/1, wo) = 1-101
Wo
P(T, v/o)
I- 4o
14 wq
Aresine Difference
(1/1, v0) arcsin( (v
arcsin(y
[(7, sho) sin(1/ + arrssi((F))*
Relative Risk Reduction (RRR)
@ v/bo =
vi Wo
F(T, vo) = -(1 (1))))
B Transportin/Reweighhin a causal effect under exchangeability of conditional outcome
B.1 Identification under exchangeability of conditional outcome
ET
(a)
Er
ET
(a)
|
=Er
Es
(e)
x]]
(Transportability)
Px()
=Rs
Es
(()
x
(Overlap)
Ps(X)
Pr(X)
= =Es
(a)
(x(
B.2 Oracle weighted Morvitz-Thompeet
Proposition 7. For ease of notation, we let := Ps(AP Ps 71 ( 1 Let tha = [y(a denote the target population mean potential outcome under trealment a {0, 13. Define the oracle estimator
1(A, a)Y
Ti Σ S. r(X.)
To
where r(X r(X) denotes the density ratio between the target and source covariate distributions and
n = Eim St is the number of units in the source sample. Then, under Assumption to we have,
VN (HG vb) NN VawHr))
15

---

where the asymptotic variance V is given by
1 1 ET
- -(5a)2 -
V ",,HT
(15)
n
Proof. Define
1( = a.Y
Zi Srrr(())))
so that
N N
1
7-
We first compute the expectations of Z, and S, By the definition of Z, and using that S Bernoulli(a).
S rX)) 1(4= a
E|Z E
Ta r(x) 1 1fA Y
Es
Ta
a Es
((X)) y(()
= a (EEr((())
aata.
where we used the consistency assumption y y(A), the randomization of A, and the density ratio property. Since (Z,, Si) for i = 1,..., N are i...d,, one can apply the multivariate central limit theorem to
N
N
1
Zi
S
N
N
1=1
1=1
which leads
VN
N
1 Σ 1=1 Si
1
Z
E
E[S.
)
4 3 E),
where
vIZi| Cov(CCv.)) Zi)
Cov(Z,, V[S.]
Σ =
Noting that war can be written as
Σ 2 N
one can apply the Delta method to the map h (I, y) + x y,, whose gradient evaluated at (anka, a)
d/a
Vhlara a) -
a
Thus,
VN (VS tha 4 N(0, Vi Vi
where
v* 0. = Vh(ana, a) EVh(aqaa a))
(16)
1
(2a)²
2 Cov
V[Z
V[S:]
(17)
a²
a2
remains to compute each term. Since S, ~ Bernouli(aa),
V[S.] a(1 a).
16

---

By direct computation,
T(())(((a))²)
VIZi| a Er
(a
Moreover, since S₁ x Z; Zi we have
Cov(Zi, 8:) = (11))=11aaaatt
Substituting into the expression of V a,wHr, we find
V a,wlfT a 1(-EA Er [ - -(0a)))
(18)
claimed.
Proposition 8 (Asymptotic Normality Oracle weighted Horvitz- Horvitz-Thompssee Thompson). Let T wir denote the
oracle weighted Horvitz-Thon TTrompso estimator. Then under Assumption we have:
VN WHT T 4 N (0, Vā
Proof. Noting that wHE i WHT) and using Proposition we know that (HII,TT wHT))) are jointly asymptotically normal. Specifically,
VN wir v1 O. v0
where Σg is the asymptotic covariance matrice. Applying the delta method to the smooth function : R2 + R, we have
VN WIT + (
14
where 4 denotes the gradient of & evaluated at (w1, v/). Moreover, because the treatment as- signment A is binary (A E {0, 1}), we have that for each unit, A; A,) = 0. so the covariance between w and v/g
Cov(vi WIT) - tho
Expanding the delta method variances and fully factorizing we get:
(
1
UT Oiú
2
(())((())2
2
(())
V WIIT
Er
+
Er
a
Ps(A=
=
awo
Ps(A= 0)
ET (1)
24 Ouoo
ET
(x100)
2
Oupi
Example 2.
Measure
Variance
1
Ex (+(X))()())
Er [r(X)(V(0)2)
Risk Difference (RD)
(TRD)
(TRR)²
Ey [+(X)(Y())2)
Er [r(X)
Risk Ratio (RR)
T (a)
(1 (1- (-T))E (
Ey [T
Er [r(x)(X(0))²
Odds Ratio (OR)
a क
(1 7 (Y
17

---

B.3 Logistic weighted Horvitz-Thomson Estimator
Proposition 9. Let tha (r( denote the target population mean potential outcome under treatment E {0,1) Define the estimator
1[A=a) = aa
n 1 1-
ox, 3N)
PaNHT S BN)
with B(z
Na
N (Ir
where n = -E= St and By the maximum likelihood estimate (MLE) from logistic regression of the selection indicator S on covariates X. and (xx (1+ (1+e= ee = the logistic function. Then, under Assumption 1 3.
VN 1I 11 4 N (0, Ewtr),
VV2 CwUTT
Cuttr Vo
with Ewit
VO,HHT - 10
where the asymptotic variance Va,HH is given by
VHT (x)
ET (2(
(19)
OTa
1 a
(20)
ET BT
and Cwnr
(21)
1
+ Er 1=
(22)
+ E =
(23)
Er - [
(24)
with V = (1 and Q = (1- - ar 3VVT
Proof. La XS, ( A, x Y' and define the parameter vector (0o, 01, 0g 3). We define the estimating function X(2, 0) and the estimator O as follows:
1-0(X.8) S1{A=0JY Og) 1- a(X Ps(A=0)
UH. V I.TT
ST ST(A-1jy 01 (x,)
A(2,0)
I- 02 a(x,3) Ps(A=1)
0N =
N Σ Si
82
V(S (X, B))
BN
Note that the reweighting function can be expressed as:
r(X, o(xi, ā
3(X,,3x) 1 1-6 -
Thus, we can rewrite the estimator as:
1 N
1(A,- a)y
1 a WHH
Σ
S) BN)
(25)
Nā
Na
j=l
1
S 1- o (X,
1(4, = oY8,
(26)
N
a(X,, (1
na
Furthermore, the log- likelihood function of 3 is:
II = Σ
8: log σ (X:; 3) + (1 so)) - (X,;;11 B))
18

---

where (X; = (1 exp( 1opp--"aa B1 - - -! Simple calculations show that
Oln Ln
N
Oln LN (B) - Σ X1(S; -X(;;))
0 (g) =
Σ (S: - -xxx( and
0,
Recalling that V = (1, by definition of the MLE BN. we get
N
Ja(Zz, 0N) = Σ Vi(S, oxx;; O.
1=1
Gathering the previous equality and Equation 26 obtain
A(Z = 0,
(27)
which proves that ON is an M-estimator of type X. (Er [r] Er [r(]]] a, 3o0), we compute the following quantities:
Furthermore, letting 0xc
1 (X) S1ÇA =
rxX) S1 =
E
E
na
a
Na
1
P(S = 1)B ((( ((x)1(A=a)x10)
Taa
1 -E ((x)){A==))+) a}Y(
Ta
Es ( since r(X) = (x))/P(X)
Er [r(u]
Thus, we have E (2. E (2, = O. Besides,
E[,a(Z,0z.)) = E [V(SS - oX))]
E E[V -ESS (X) X]]
(Law Total Probability)
E V (E X| - [X])]
(a(X) is function of X) (Definition of oxX))
Therefore, we have
E 22 = 0.
(28)
Now, show that fxx defined above is the unique value that satisfies 8g2 = a. Let
28
We directly have that
L(3) - E In(o(X,33)) In((x))(((X,00)) + (1 (1 In (xx
A direct calculation shows that
VgL(B) = () ss] and V2L(3)= - =
Since Eis X] oXx, Boo), and V =11
vLL(3x) = E v making 3o0 stationary point. Furthermore, using overlap we have o(X, (X (1- 2 n² therefore Vo E R
=0
VVL(330 = =|IIV T o1 8) (1 [xx)
2 2n² 21
2 n T
19

---

Since we assumed that B[X X is positive-defintt, the Hessian V2L(3) is positive-definic so L(8) is strictly convex. Hence there is unique global minimizer of L(3); since Aox is critical point, must be that unique minimizer. Consequently, any solution to
v (o 5) =0
must equal 8xc. Since 3, 02 are now fixed and since the first two components of d are linear with respect to 0g and 01, 0xx is the only value satisfying (28). We want to show that for every 0 in neighborhood of doc, all the components of the second derivatives are integrable for all 30....,33.
82
(z, 0)
820
While this holds trivially for most components, the integrability of the following specific terms requires closer attention:
02
0² As 0)
da (2,
0)
Az 0)
for all a E {0,
00,03
05a3
0308
First, consider the mixed partial derivative with respect to 02 and B:
8²
1- 1xx -1 VT.
(z(2
00,03
(1 02)
(Xx
Na
For each coordinate i [I,...,d].. d]. the expectation bounded as follows:
((X
- Vi
E
da (2, 8)
00,03
(1 02)
(X
(xxx3)) (1
2 (x, ["'
ST = a}
E
E[V?
(1 - 02)
T
using the Schwarz inequality. The first expectation is finite under the assumption that Y square-integrable, and due the exponential tail behavior of the logistic function. The sub- Gaussianity of X implies that all moments of V are finite, ensuring integrability of E[V?3).
Second, consider the pure second derivative with respect 8:
82
1 (XX S VVT. (2, 0) = 02
0
x(X,3)
Na
Each entry of this matrix takes the form C. V.V, for some random coefficient C, and the integrability of these entries follows from the same reasoning as above.
Third, the second derivative of # given by
0²
0333 Aa(z, 0) = IV& V.V. o(X, oXX 2o SVAVV.
By applying Hölder's inequality, the following bound is obtained:
E [IVV < EIV/)"".BBVV""''' E/V
which is finite due to the sub- sub-Gaussianity of X. Consequently, each second derivative component
8
{x(2,
820
integrable for all (O....,3),
33, in the neighborhood of 9.c. Define
A (8xz)= = E
and B(0x) = (Z2
00 0=0m
20

---

Next, we verify the conditions of Theorem 7.2 in [2]. To do so, we compute A (fc) and B (0xc). Since
1
0
1-0(X,0 S1(A=0JY
-1 1- SHAA=OYY V
22)²
x(
Pe)A=0)
T- o(x,9) BelA=0)
0A
-- ST VT 1-0g o(x,8) Fs(d=1) 0 ox (1 (xx,3))
2,0)
0
-1
1-
(( Fs(1=1)
00
0 0
0
0
(29)
We obtain with xx = (Er [r[0]]
Er [y]]] 3rx)
Y [0)]
-1
Er [T
A (0o0) =
0 -1
Ey
ET
[Y(vT]]
0
0
0
-Q
where Q = E [o(X, 3x0) (X,
Boo)) VV which using Schur complement leads to:
0
Er
Br
A(0xc)-
0
-1
Er (r
1-a
Br
0
0
0
Q-1
Regarding B (f20), elementary calculations show that
Br (=[1(000110)) Er [x0 (
Er [x
Er [
Rr [r(0]
By [a 11- X(X))Ty)
[ [ Er [r
By (1))
arg(d-1) Er [r
- Br [r00]
Er (1 - O(X)) you
B(0x))=
(1 a)Er [
(1 (1- [xa)
a(1 - a) (1 o1- (x(V)
(1 a1- [[(x)]
Er [1 -1- (X))V
Br (1- [
Based on the previous calculations, have:
A(z, and its first two partial derivatives with respect to 0 exist for all and for all the neighborhood of 0x.
For each in the neighborhood of 6oo. we have for all & {0.3) 020 Ao 0) is integrable.
A(9.o) exists and is nonsingular.
B(0x²) exists and is finite.
We also have
=0 and 0N & 0xc.
Then the conditions of Theorem 7.2 Stefanski and Boos 42 are satisfied, and we can conclude that
Vr (Ōx 0x)) 4 N (0, (O.,( B(0x) BO0A B(O))
Letting vo and uJ be respectively the first and second row of 100x0)-11
[y
v/ -1, 0. - 1-a E, VVT
and
0, -1, Er
ET BT
21

---

Expanding the quadratic form explicitly, and using Lemma
1
we get:
aTa (x (Y
ET [r(
B(0s))u B(000
a "u
and:
Canl = V/ B(0 M1-0 = ET (T(P())T((P()) + ]
1-a
(X))
Lemma 1. We have (x
"d(+))" I-a
and Q = - [ ()
Proof.
IPSS 1)E5
(1 = (1 - as (00
-11 aTT
=IT 1o 3VVV
Therefore using using the block inverse matrix formula:
0' 1-a ( -S-1p-1tyll(x)x - (
S-1
B p-1 -
where p= P= and (x(a(X)) - -x
Expanding [BT[o(X)] (1- we get:
=1
and -B-16(X))5511B116(xxx1111"1111 B "ropr//"""...odooooor"""ooxxx"""
Hence:
() 1-a
Er(o(()V
Proposition 10 (asymptotic normality of weighted Korvitz-Thomppe estimator). Let o B) (1 exp(-x" 31 Bo)) denote the logistic function, where AN the maximum likelihood esti- mate (MLE) obtained from logistic regression of the selection indicator S on covariates X. Define the estimated density ratio as:
(x,
N
r(x, N -
n
with n
S,
(xx Bx)
Let wHT denote the weighted Horvitz- Horvitz-Thompenn Thompson estimator constructed using the estimated ratio
r(x, 3x). Then, under Assumption to 3.1
(f T 4 N (0, Va,wHr)
22

---

Proof. We begin by observing that the estimator of interest can be written as smooth transformations of its vector-valued estimator:
f WHT & (1//1. who.
By Propositions 9 the pair (61 wHT. D wHT) jointly asymptotically normal. Specifically,
VN (I. 4/1 و N (0, EW,TT, 4u
where EwIT the asymptotic covariance matrices, with entries determined by the variances and covariances of the components wHT and wH, Since & : R + R is assumed to be a smooth function, can apply the delta method each estimator. Let 4 denote the gradient of 4, evaluated at the true parameter vector (4/1 4o). Then:
VN T م N(0 WHT)
where the asymptotic variance given by the quadratic form:
WH,
V V4.
V EWHT
Proposition 11. Let s/a [a)) denote the targel population mean potential outcome under treatment a E {0,1}. Define the estimator
N
11A, a
(2, 3N) with r(z,BN) - n (x
WA,N Σ
8 r( r(x, 3N)
where
N
Σ S,=1
1(A = a}.
n Σ S
By is the MLE from logistic regression of S on X, and o(x, B) = (1 + -- -= By-1 is the logistic function. Then, under regularity conditions,
VN
$I,N
151 V
d N EO,E,),
with EN =
VI.N
ON
ON
Vo.N
where
Er [r(X) (2)
(2 1
Ve,N
Ey (r a a(1 - a) am
a Ta
- [x + 2 2T @ (E
1 - 2a
O2 Er EE
a(1 - a)
1 1Q-1E,1(1 E a (X))
Er(y(1-o) VTQ ( ((X))x((l1)
+ V
with V (1,X) E B and
Q= (1 Eo VVT].
Proof. We follow the same initial setup and notation in the proof of the Re-weighted Horvitz- Thomson estimator. The key difference is the empirical estimation of n.. introducing an additional
23

---

estimating equation for 83. The remainder of the proof existence, uniqueness, M-estimation structure, and regularity conditions-follet identically. Define here:
1-0
S1(A=0)Y
80
02
(X.3)
Bg
0g
1-a (1,8)
$1(A-1) 81
1-02
o(X,3)
01
á : N 5 S
0)
S 0
ON
S1{A 0} 03
Σ N
5 0)
S1 A = 1) 04
i=1 5:1 =
N
V(S oXX, B))
AN
We now rewrite the estimator using this notation. Recall
-ox 3N)
r(X,,Bn)=
o(Xj, 3xx)
1
Then:
1
N
1(A,= :
Gas =
Syr(Xy, 3x)
Nā
Aaa
and
1
a.
S,1{A a}.
Ná
Substituting in gives:
1 - o(X,, BN)
à
1fA, = = 1 N
L.
Sj
N
o Bx)
1-à
EM S,1/A: = a
Ø=1
Let A(0xe) the Jacobian of X(Z, 0) at the population limit 8x., and B(fxa) the corresponding covariance matrix where
0 = [BT ET (Y()), a, amo am, 300)
Let
Q = Er,o(, B10) (X,
The inverse Jacobian block A(0x0) block lower-triangular, with expressions for rows v/ pT given by:
v = -1f = 0}
(y (1-oja
Er
-fa = 1}
O3
We then compute the asymptotic variance as
Va,4 = v. B(xx))a C+=1i
B(0c))m
where the expansion of v ( )os yields the desired closed-form expressions for Vo,N and ON stated in the proposition.
B.4 Weighted and transported G-formula
Proposition 12. Let V.a (Y)) denote the target population mean potential outcome under treatment a E {0, Define the oracle estimators
1 N Σ
N S. rxx)) (X,,, and V..G N
Wg,wG
Σ S.) 1=1
( (X,)
where r(X) denotes the density ratio between the target and source covariate distributions and
Em Sy is the number of units in the source sumple. Then, under Assumption to
VN (Vā,G .ba) N Vana),
24

---

Furthermore under Section 3.7
VN (Vawa - -ba) 4 NN(0, Vi,wa);
where the asymptolic variances are given by
Vawa = 1 (Er (xX))xx(x (Br(Y()))?
a
and
1 ET (x(((x)))) Er. (1(op))))) -1)()))
V B.G
Proof. Transported G-formula. Define
2i = (1 (:) v1 (X),
so that
+ EΣ Zi
V/ G,G 1
E 1-
N
We first compute the expectations of Ze and S.. By the definition of Z, and using that S Bernouli(aa..
B|Z E (1 S) Ha (x)
(1 a.) Er ( (X)
(1 at)taa
where we used the consistency assumption Y Y(A), the randomization of A, and the density ratio property. Since the (Zi, St) for all i = 1,..., N are i.i.d., we can apply the multivariate central limit theorem to
N
N
1
Zi
S
N
N
-=1
1=1
to obtain
VN( ((
N N 1=
Zj
E - )11)
E E|2,]
3 S,(0.
N 1=
-Si
where
viZi] Cov((Sr) 7c)
Cov( Cov(Zr, Si) [[S.)
Σ
since 6a can be written as
N Σm 23
1
4 0,1
1 Σ 1
N
apply the Delta method to the map h : (I, v) + x/ y,, whose gradient evaluated at ((1-a) al)aa)
a)
1 (1)=1)1a)) 1-aa I-a 1-a
1
1b
Thus,
VN (Vā.c -ba) 4 NOO, Vila),
where
ViiG u Ev.
Tw.
Expanding, we obtain
V[Z + (1h4) )
2
V+
V[S.]
Cov(Z,, St).
a.sG
(1 - a)
(1 - a)²
(1 a)²
25

---

It remains to compute each term. Since S; ~ Bernoulliaa),
ViS.] a(1 a).
By direct computation,
viZi] = -1 (1- - a))aarr Br (K)) - ((1 a))aa).
Moreover, since (1 - Si) x Zi Zi we have
Cov Li, Si) = (1 a)ara.
Substituting into the expression for Va we find
1-a ET (x)) (wb)))
1
weighted G-formula. Define
Z: = St -((i)) (1.
so that
N Σラ i=1
Zi S:
Zn
Σ
SN
We first compute the expectations of Z; and S;. Using the definition of Z; and that S is binary,
E|Z,] = E S r(X) (x))
= o: Es [((x) 1/ (x)
a (x) ada.
where we used the importance sampling identity Es(r(X)S(X)) ET(S(X),, By the multivariate central limit theorem,
ZN
(av'a) )) 4 N(0,
VN
SN
where
ViZi| Σ = Cov (2, St)
(..
V[Si]
The gradient of the map h: (x, y) + r/y evaluated at (arta, a)) is
(1 a
3/a
u = Vh(ava, a))
By the Delta method,
V (Wa,wo= thas 4 NOO, Vi Va
where
V Va,wG u Eu
1
v[Zi
+ "
V(S)
2
Cov (z.
a²
a²
We compute each term:
vizl E (B|Z,))² aET r(x) (MS (x)))
222
ViS. a(1 -a).
2/. =a(11 a) Cov (2, E,Z... - B[Z]E[[.] = a ET [M]O(())]
26

---

Substituting into the expression for V e,wG we get:
+ 1/2
2
Vi
a,wG
aT
((x))u
(X))
u²
a a))
a(1 ax
a
a
a²
a (x))
Er
v2
v
-
2// (1 a)
a
a
v
Er (4x) (X))
a
Proposition 13. Grant Assumption to defining 3(a) := ,(a)], V := []. We rearrange the source Y; and V₁ so that the first observations of correspond to A 1. We then define Y1 = (Yi...., Yn,) and Yo = tIn........ Yn)", as well as V = ,.,....,, V,,) and Vo
(Vm,+1..... Vnn) Letting (2x S.N/N and for all a E {10,1].
[00
= and gla)
1
V.V.o
- V Ya.
1
-i=1 1s, ==1
n1a
Defining v Es [ and E Var Var(X|s 1), we have
VN - 000) 4 (0,E),
where
VSr
V(O) B(U)
V
0
(1-a)
0x0
B(0)
E
M
0
((1)
B(L)
o
x1-x)
a² M
1 VT Σー -I
with M-
Σ
Proof. Using M-estimation theory to prove asymptotic normality of the ON. we first define the following:
\z 8)
(1 - S) - 8b)
)
A(2,0)
(Z
5(1 A) (Velo) VVT (0, - B(0))) SA SA VVT (02 - -0(1)
Az(Z,6) 0)
where 0 = (8u, 01 02 0s). We still have that d is an M-estimator of type [see 42 since
XZn,0n)
Note that
E (2, (2,000) E[1 (111) (V - -Er(V))
=1--a)E, (V Er(Vi)
=0.
We also have
E (22(2,000)1 E S(1 - A)V (0)
aBs (1 ()
=a(1 Bs (0
a(1
Es
VIS (0)
0.
27

---

Similarly, we can show that E [2, [z))] 0. Since X(Z, 8) is a linear function of 8, 6ec is the only value of 0 such that E [A 8)] 0 Define
A (6xx
and B( E [(2
08 l mm 8
Next, we check the conditions of Theorem 7.2 in Stefanski and Boos and B (0 Since
42
First, compute A (0oo)
-(1 S)
0
0
ax
(2,8)
-S(1 V)
00
SAVVT
0
we obtain
-1 a1-a)
0
(0xx))
a
T)M
0
0
aU
where M = EEs [VV] which leads to
1
0
1-a
A (0x)
0
M
0
a
0
0
M-1
Regarding B00x),, since we have A(1 - A) = 0 and S(1 - 5) = 0, elementary calculations show that:
B(0x) B(0 2,
B(0x))s =0 and B(0
B(0x)1, = ((0x) )3.1 = O.
Besides
B(0x0)1,1 E. (1 5V Er (V)(V Er [V]
= a1 Var [VIS 0).
We can also note that:
B(0x)3,33 E ( S
=arss (cT)(( =
2
and similarly,
B(0 )222 = a(1 - x x)0² M.
Gathering all calculations, we have
(1 a) Var V 0]
0
B(0x=)
a - T)
0
OTO² M
Based on the previous calculations, we have:
A(2, 0) and its first two partial derivatives with respect to 0 exist for all and for all 0 in the neighborhood of e
For each 0 in the neighborhood 0zc, we have for all i, (1,3):
0²
Au(2, 0) 11.
00,00,
28

---

A(0x)) exists and is nonsingular.
x0) exists and is finite.
Since we have:
n X(T, 21, 0x) =0 and ON 2 0xx
Then, the conditions of Theorem 7.2 in Stefanski and Boos 42 satisfied, we have:
v (ŌN 000) -0x0 4 N
)
where:
Var[V
0
(1-a)
((0_)=0((0
0
M
0
0
0
o²
M
Corollary 1. For all {0, let a3G denote the transported G- G-formula estimator where linear regressions are used to estimate uaa. Then, under Assumption
and
FOLS 1/1 2015
VN
N No (0.))
v/0)
with
| g(1) + Er [V] a dM
Varlv|s=0| 1-a (B()) T Var[V]S-0 g(0)
Er(V
(B(1) T Var[V[S=0] g(o)
SOLS
1-a
(3(0)) T Var[VIS-0 g(0) ET[V] [V] a2M
a(1- ET [V!
I-a
1-a
Proof. Recall that for a E 1), we have:
Fa1s = (B(o))TT
From Proposition 13 we know that:
VN(ŌN 000) 4 N0,
with 0- (V (BO)), (T E R3++3 and where Σ is the block-diagonal covariance matrix.
By applying the delta method the map
((B)))TA V1O)
g( (V B00), (())
(B0)) VV
the asymptotic distribution of VN (F5 v.) is multivariate normal with covariance matrix:
OLS = JEJT
where J is the Jacobian of g evaluated at (Er (()))
J = (B()) (3(o)
ENIV1T)
ET[
we obtain:
(B(U)T [ar[V]S=0] ((1) E3 [V] [V] gfM [V 1-a
(g(1)) T Var Var[V]S==] 3(0)
POLS
1-a
(8(1)) T Var g(0)
(3(0)) T Var ((0) + ET [V] a(1 ET [V] (30)
M
1-a
29

---

Proposition 14. For all E {0, 1), let Fow denote the weighted G-formula estimators where the density ratio is estimated using logistic regression and linear regressions are used estimate piaa Then, under Assumption 7 Ioo
VOLS VI,
VN
// 1/1
4 N(0, IWG).
Proof. Let Z X, A,S x Y) and define 0 (01, 02 03. B. 04, 0s). Consider the estimating
function X(Z, 0) defined as:
1-0(X,0) 50 V 01,
x(x,9) T-8s
(xx SO, V 02,
(xx3)
I-8s
S- 0g.
V(S - o(X, 8)),
S(1 A) (Ve(0) - VV (04 - -(0)), SA
VV (0; - -B(1))
Define dN = (Vo io1s
Fl, â, 19,, (()), where By is the MLE from the logistic regression
of S X, and r( r(x,, BN) 1_o(X.,,N)
I-a Then, the estimators ROLS take the form:
a(Xi,N)
N Σ S. 1 1X,
o(X,, Bx) (B(a))
The estimator y solves the estimating equation:
Σ ~x(2,0w)=0
This setup is structurally identical to that proposition and 13 and the M-estimation theory in Stefanski and Boos [42 applies directly. Specifically, the regularity conditions (e.g., smoothness of X, identifiability, uniqueness of root) are satisfied.
As before, the asymptotic distribution of the M-estimator is:
v VĀx - 0z)) 4 (N, -
where A and B are the Jacobian of the estimating function and its variance, respectively, evaluated 0x (No, vi, a, So0, ((0), 3(2)). Focusing the top-left 2 2 block of the sandwich covariance matrix coreesoiing (JOLs Go.s -wee we denote this block by 2o15, and conclude:
VN
Po.wG v/o L
4 AO 2OL) where >
0,wG COLS
COLS
2/1
V.,wG/
where using Lemmal I we simplify the expression to:
(B Ag(aa)
2/2
+ (3(0)) ET(VV (3ta) - T (V
V-OLS
a
1
1-a
(31)
ET MITTIMMM M M-
+
(32)
aTa
For the covariance terms get:
CoLs (B(0)) TA
( Er [VT (V [VV
1-a +2
1,00
1 1
(33)
where A = Erb((X)VVT]
30

---

Lemma 2. Grant Assumption 7 to Linear mode ! then:
the matrix DoL
Lg is semi-definite positive,
the matrix EWHTT I is semi-definite positive.
Consequently, VoGG S VONG S Va,wHT-
Proof. First statement We first start with EWHT E015. Under Linear model
y(a) (Bla))TV + ela), where
E(e(a)
| 0,
Var(e(w)
XX= = 0².
we can express the variances VIITII and the covariance term Cwir defined in 19 of the weighted Horvitz-thompson in terms of the model parameters and the distribution of X. Starting with WIT we expand each expectation by substituting the linear form of ( The first term of V Va, becomes
(r(r(X)(x)1))) Er [x(x) (((B))) + ) vv + (c(o))))
Taking expectations and applying the assumptions E(ela) X 0 and Var(el*) X) = o², this simplifies to
Er(r(X)((st))) (v((())
T V> + orEr-(())]-
The second term, becomes (ET[(((a)) T x])², since the error has mean zero. Moving to the third term, we use linearity to write
E ( (B(a)) T LVV
and thus the quadratic form becomes
vo ((8la)) " TB(o).
For the fourth term, we compute
(x) = =x)
which leads to Noting that Q - a) (xx BVVT] this further simplifies to:
2 ET
= 2 (3(0)) T T VVTT
2E ( QQ (X 2 (3(a))
ET VV (zaa)
1 1-a
Combining all components, using linearity and the definition of A, we have
((8(0))TAB') - (Er(((()))T11)
Va,
aTa
1-a
+2 (B(a)) Ey IVV 13(a)
Turning to the cross-covariance term OWHIs we proceed similarly. Noting that
[y(a)) = Er((((()) v1. Er(y (3(0) [V[]V
Ey (x) = (x
we substitute these into the original definition to obtain
CWHT = Ey (8()) T v] Ey T
(3(1)) Er Er(VV T (3(0)
+
- 1-a
1
31

---

Now, we can compute the following quantities:
VH,HH VOLS 1 Te (B(6))
Ag(aa)
+ a²
(Erlr(X) Ex]]] M B.
a
aTa
and
CWITT - COLS (310)) A,3(1)
Since < na < 1, we immediately have that
1 Na ( 2 0.
Q.T
Now, turning to the second term in the expression for VH, VOLs observe that
Er(r(X)]
Er - M
can be rewritten using the fact that the target distribution T is defined via reweighting from the source distribution S, with r(X). This yields:
Br(r(X) [v] M Mr = - [
To interpret this expression, we recognize it as a variance-type quantity. In particular, we can rewrite it as:
Es[r(X)9 - 2Es(r(X)) (
+ (X Es EE -
which simplifies to:
Es (f(x) -,
This is the expected squared residual from projecting r (X) onto the linear span of V, and is therefore nonnegative. Hence,
Ex|((x) M- 2 0.
Combining this result with the earlier inequality, we conclude that
V Va,HTT VOLS 0.
Since this holds for both 0 and 1, and noting that
(.(00) (3300)
CWHI
wG
we now analyze the overall matrix difference. The trace satisfies:
tr( tr(EwHT E (V1,wHT
VOLS + (Vo,HHT VOLU 2 0,
and the determinant becomes:
det(2wHT 201) (V1,wHT VOLS (Vo, WHT VOLL
(CWHT CO
1
((B")T A3 (5(0)) T A ((3())TA30))) A, 0,
a²
where the final inequality follows from Cauchy-Schwarz. Since both the trace and determinant of
EwHr SOLS are nonnegative, we conclude that the matrix EwHI EOLS is positive semi-defnitt.
Second statement Now, want to prove that E LG is semiidefinite positive. Observe that using the expression defined in 19 and 30 we have
VOLS - VOLS (8(0)) T H13l))
and COLS
COLS = (8(1))T3800
where the matrix is defined as
3X VVT + -B "
BrIVVT)
H =
ET
1 a
(X,3)
32

---

This can be rewritten more compactly as:
H (C - AB-
where we define:
C=Er VVT
A= A-Er B = (x
o(X,3)
First, define the vector:
Vo(X,3)
ER?4.
Vo(X,3)
(Note that here we changed the definition of V to simplify later steps.)
Then the outer product VVT is:
VVVT VVT
VVT VV
VVT =
Taking expectation, we define the matrix:
B AT
A C
M = R. [
=
We now show that M is PSD. For any 2 E Rd2d we have:
M Ey VV Er (z T [)² 2 0,
since each term in the expectation is a square. Therefore, M 2 0. By the Schur Complement Lemma for block matrices, provided that B is invertible and square and B and M ž O, get that
H 1
(C AB A)
Now, by definition:
[(B(0)] H3
H (8(1) (
OLS WG
OLS GG
(B(W)) H33(0) (
H3(())
Since this matrix is Gram matrix induced by H 2 0, it follows that EQLS
2G ž 0.
Proposition 15. Grant Assumption 7 to and let +015 denote the transported G-formula estimators where linear regressions are used to estimate wlay Then, 2OLS is asymptotically normal:
T $ N (0, volg) VN (fa,GG
where
0Φ
VoIS 4.1G
((1)
+
((0)
1
Ovhy
Dvh
Er [V] M [V
1
OΦ
2
0Φ Oui
27
a
Delo
π
Proof. We begin by analyzing the transported OLS estimator, defined as
OLS 4.1G
((3N)) Voo (3(0)) ViO))
where Vio is the empirical mean of covariates in the target population. Under Linear model | the corresponding population estimand is
T
[x],) (B(0) BB))))
33

---

where Er(V] is the expectation of covariates under the target distribution.
To study the asymptotic behavior of the estimator, we perform first-order Taylor expansion of Ab around the point (Vi, 1/6) = ((3()) (TrT)) (B(O)) Er(V]). This yields:
OLS 0GG
V(0): T = + (B (8(0)) (1801) T Vio: - (3(1) (88)) EErlVi)
R(00) (B - (( ErI (B101) ( Brlvi)
04
Jwh
(191)))
0C
(13:0))T Vo()) - (B(01) TErTTBII (1)
avio (09.06)
(1/ (1)0))
V(( - (B (((0) ( 1)
+ 00
To further decompose the linear terms, note that for each a E {0, 1}, we can write:
(B(1) ()( [V = (8(%) - g(o)) ( [V] + (B())T (Vo) Er(V))
Combining these expressions, we obtain:
T
(
- 3(2)) Er [V] (BOI)T (Vo) Er(vi]
4.1G
Onhy
(3(00) - ) TE [V] + (B T (Vo) Ex( + (NV-113
+ Orio (g):vg)
By the multivariate Central Limit Theorem, the Law of Large Numbers, and Slutsky's theorem-along
with 13 we conclude:
N COSS
- $ N (0, 0 Eax))
where E is defined
30 and the influence vector aeo is given by
O 8xh
11,00 + Oyho (29.05)
00,000
(w9,05)
and each oo RR is defined as
02 = (181a))T, ET (a==), Er[V] [(a=1)) e 30+3
Combining this with the block-diagonal form of E, we obtain the explicit asymptotic variance:
0 Eax =
1
OΦ
(())
+
B(0)
Oth
arvo
Var[V]|=0]
2 [T M M- [
2
27
+
a
1 n Erbo
π
Ou'y
Proposition 16. Let POLS denote the weighted G-formula estimators where the density ratio is estimated using logistic regression and linear regressions are used estimate play. Then, under G is asymptotically normal:
Assumption
to
OLS
-OOS VN N (T&.WG
4 N (0, VOLS -T)
where
OΦ 2
OΦ 2
Op
0Φ Ox'o
VOLS
VILL +
VOLS +
COLLS
Ouh
Ovioo
Ouhn
34

---

Proof. Using 14 asymptotically normal vector. By the delta method:
we apply the delta method to *.wG
2018
( vo, of the two-dimensional
VN (fo.GG D(11, vo)) S N (0. V fo)
( vo)).
where ( vh)
88 0ei
u 060
the gradient of & evaluated at the population means. Therefore
we get:
OΦ
a
aΦ O40
a Ovo
VOLS f,wG
VI.WG
+
VO.GG
+2
COLS
Oih
Orio
B.5 Semiparametrie Efficient Estimators under Exchangeability of conditional outcome
Proof of Proposition t In this proof, we use the usual machinery of influence function computation, as described for instance in [26]. In particular, we will for the sake of the computation, assume that X is a categorical variable taking value in a countable space. We first recall that if
(Pbaa == Er.. [AA
(34)
for some measurable function h and some event of positive mass A, then the influence function of at Pa Pobs is given by
1{A}
[Z) == Paba (A) (h(Z) - 6).
(35)
We now rewrite th using functional of the form (34
Ve (0)S 0]
EPau [r X, S 0J{S 0]
Epor [E (a) X, 5 1JʃS 0]
Epasa EP.... Y Y|X,S 1,A a] 0]
Σ Ep [1[X = xJ|S 0] EP.. Y x 1, A a).
FEX
Using 35
and usual properties of the influence functions 26 Sec 3.4.3], we find
Σ S
-1(x 1 (1{x =} - Eo [1 z)|S 0])
(Z)
TE
1 x, 1,
EP.n 1-XX = =3|5= =
(Y EP |Y| x, S= 1, 1, a]).
Pobs (X x,
The first sum simply rewrites
1- S
1 1-a (Maa) (X) (P)),,,
and for the second, notice that, using that and S are independent:
P. [1 x315 0] = aP(A=a))
r(X)
so that the sum rewrites
= 1,5= ((X(Y Moa
aP(A=aa
In the end, we indeed find
IF())((Z) = 1-a (M() - v2) + S1(A = a}
1 1-5
aP(A = a) ((X)(Y (x)x(
35

---

Proof of Proposition و these functions as deterministic. By using the law of large number, we see that m/N goes to and that (1)) converges towards
By conditioning with respect to the randomness of P(aa and f, we can treat
E[(1 ()) (X)] + a
(x)
[SF(X)) -Ma
[x)
If (a)) = M(a). then the second term in the above formula cancels and we are left with
[[( () (X) (l = 0] = tha
If = r, then the second term yields
[x(x) - P(a) (X) = 1] = (a) [S = 0] = =v. - IE (X))S =0],
which also yields results. Using continuity of allows to conclude.
B.6
Computations of one-step estimators and estimating equation estimators for the OR.
Note that Ve is solution to Σ (Z,, f. (EE) = 0. Then, given the expression of 9a in Proposi-
tion 3 holds that for any other estimator v/a:
N
Σ
Pa (Z 17 = m
EE
We will use this observation in the subsequent computations.
(RD) For the risk difference, the estimating equation approach yields:
EE
EE
ZEE
RDD
The one step approach, based on initial estimators Vi and th yields an estimate of the form:
OS
RDD
th +
N(1
m
o.)
(v1
EE
11)
N(1 a)
m
(vo
EB
711
EE RD
+
N(1 a) FRD
m
N(1 - a)
In particular, we see that starting from estimators E. of the form Es, -0 ((a)(X))) (X,) yields a
final estimator that has the same structure the estimating equation estimator, up scaling factors depending on a, N and m that are asymptotically close to 1.
(RR) For the risk ratio, the estimating equation approach yields:
EE
EE
In contrast, the one step approach, based initial estimators en and vin yields estimate of the form:
1 m
TRR
-0
N(1
EE
11
V
N
(vo a)
EE
In particular, we see that in general, - apply the one-step correction.
+
SE FOR
unless of course we initially picked TEE
(OR) For the odds ratio, the estimating equation approach yields:
EE
EE
BBB
FOR
BE
EE
36

---

In contrast, the one step approach, based on initial estimators 인 and Do yields an estimate of the form:
OS OO
1
1
400
1 V0 (V1 F)
1/1 4
(1
1/))²
400
N(1 (K)
11
1
m
(40
1
N
06)
In particular, we see that in general,
20S FOR
*
=EE OO
unless of course we initially picked EE
to apply the one-step correction.
C
Transporing/Reeviighting a causal effect under exchangeability of CATE
C.1
Identification under exchangeability of conditional outcome
Risk Difference (RD)
T RD
= ET (X)]
Risk Ratio (RR)
ET
TRn (X) 1 (x)
TRR
ET [Y(0]
Odds Ratio (OR)
OOR ( (X (X)
Er [0]
ET
1+T (X), (X) - (X)
T FOR
1 Er [(0)]
( (0) (X)
1 Er
14TOn (x) (x) x)-1 (X)
C.2 Semiparametrie efficient estimators under exchangeability of treatment effect
Proof of Proposition
6
We use the same tricks as in the proof of Proposition 3 We first notice that
Σ
PX 2|S = (o) (x)).
ZEX
so that, with a slight abuse of notation
(F+ Σ - S
(1(X- x) Pohs ())
Potns (X = x/$ = (T)
#Ex
Using that (z) = (E|Y 1,5 - 1,X x)), we further find that
1|A=1,5=1. (Y (1,
IF(T(()))
-,A=
+ 0, (Y - (1)
Again, we know that
P(A = a, - -
an =
x 0).
Furthermore, since (( (a, (( b). b)) + ( = and 0 (( b), Patching all of this together yields the result.
for all a, b. differentiating with respect to or yields
D
37

---

C.3 exchangeability of CATE
Computations of one-step estimators and estimating equation estimators under
Example 3 (Application to the usual causal measures.). We give the expression of & for the most usual causal measures.
(RD) For the risk difference, we find
EE
T ((o (X,) + (X.))
5,=0
P(X.) (
Ai (Y; - -x,().) Pio) (Xi))
1 Ai (Y; 1 (Xi)) 1 π
am
(RR) For the risk ratio, we find:
EE
1
(o (X,,)
S:=0
1
[ A (Yi (X)7,,
1- A (Y, 100 (x.)
f(Xi)
Om
π
π
(OR) For the odds ratio, we find:
1 (X))
EE
1
$1
(1
(Xi)
(1) (xx)
5
1 f(X,)
Aj
m (x,
Y; 11, (X:) + 110 (Xi)
am
π
Ai (Y: - pio0 (Xi))
P (X,)
1
(1 (0) (Xi) 1 (X;) (x
D
Simulation
For the simulations we have implemented all estimators in Python using Scikit-Learn for our regres- sion and classification models. All our experiments were run on a 8GB M1 Mac.
Linear setting under Assumption 3: we evaluate estimators under a linear response surface:
((aa) (V) 3. with B1 (0.5, 1.2, 1.1, 3.3, -0.6) and 3o (-0.2, -066, 0.6, 1.7, 0.3).
Since 30 and 31 remain unchanged across the source and target domains, Assumption is satisfied, results are depicted Figure As expected from the linear generative process, all estimators perform well across all measures, with the transported and weighted G-formula exhibiting particularly low variance in this setting oottefo-iuurrromminggg the influence function-based estimators.
38

---

Horvitz: Thocpsson
W Iormula
G-formala
Esimaing--qutiies
Ono-dup
-0.10
Risk
Figure 4: Comparison of estimators across different causal measures under linear outcome model with sample size of N = 50,000 and 3,000 repetitions.
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Reviews:
[{"review_id": "3y9d5SoedL", "reviewer": "Reviewer_LpQj", "summary": "This paper studies the problem of causal meta analysis, and proposes a unified framework for transporting a board class of first moment causal effect measures. It aims to generalize treatment effect estimates from RCTs to target populations under covariate shifts. The authors extend existing methods for generalizing causal effects by deriving identification results under conditional exchangeability assumptions. The framework incorporates both classical estimators and semiparametric efficiency theory to provide doubly-robust estimators for various causal measures. The method is evaluated through synthetic simulations and real-world applications, showcasing its utility in generalizing both absolute and relative causal effects to broader populations.", "questions": "Could the authors please clarify the justification for my questions on lemma2 proof? As proposition 2 seems crucially assumes correctness of lemma 2.", "limitations": "See above.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strength:\n- In general I think formulating meta analysis under a causal framework is an important direction. One way of approaching this is to reframe the problem as generalizing treatment effects from controlled settings to some target populations that may differ in their distribution of covariates. Then, the paper's focus on a diverse set of causal measures beyond the risk difference, which is a major strength. \n- The work is quite thorough. The authors systematically derived identification formulas and a complete suite of estimators (weighting, G-formula, one-step, EE) under two clearly articulated transportability assumptions. This is very organized and I like it a lot. \n\nWeakness:\nThere are a few minor weaknesses especially on discussions, evaluation and potential mistakes in proofs.\n\n-  The paper presents two parallel frameworks based on exchangeability in mean vs effect measure. The latter is weaker but imposes the strong practical requirement of having access to control outcomes for the target. The paper would be strengthened by a more detailed discussion of applied scenarios where this condition is met and more explicit guidance on how a practitioner should choose between these two sets of assumptions and corresponding estimators\n\n-  I am not sure if lemma 2 proof is correct. Its correctness depends on the component matrix\n$$\nE_{T}[VV^{\\top}] - E_{T}[VV^{\\top}] E_{T}[\\sigma(X,\\beta)VV^{\\top}])^{-1}E_{T}[VV^{\\top}]\n$$\nbeing PSD. As the authors suggests, this matrix has the form $A-AB^{-1}A$. But I think the authors might have missed an additional condition for $A-AB^{-1}A$ to be PSD which is $B \\succeq A$ should hold.\n\nHowever, given the definitions $A=E_{T}[VV^{\\top}]$ and $B=E_{T}[\\sigma(X,\\beta)VV^{\\top}]$, and the fact that $\\sigma(X, \\beta) \\in (0,1)$ (if I understand correctly, the authors can elaborate), it would seem the reverse inequality, $A \\succeq B$, holds true instead.\n\n- on eval:  In the real-world application (sec 5.2), the authors conclude that the treatment appears to have \"little to no impact on the target population\". However, the confidence intervals presented in Figure 3 are quite wide. The claim of \"little to no impact\" seems stronger than what can be concluded from the point estimates and their associated uncertainty.\n\n\n\n\n\n**Please note: I do find I enjoy this paper, I am giving borderline reject simply due to those concerns. If the authors can address my concerns, especially the one on lemma 2, I will consider increasing the score.**", "was_revised": true, "final_justification": "Most of my concerns has been addressed", "rebuttal": "We are grateful for the reviewer’s positive remarks. In particular, we appreciate the recognition of our effort to present a unified causal framework that accommodates different effect measures and transportability assumptions. We aimed to make the methodology both general and practically accessible. We are encouraged that this structure was found to be clear and well-organized.\n\n---\n## Response to Comment on Exchangeability Assumptions and Applied Guidance\n\nWe thank the reviewer for this thoughtful and constructive comment. We fully agree that providing concrete examples and clearer practical guidance on when to prefer each assumption significantly improves the applied relevance of the work.\n\n### Applied Scenarios Where Exchangeability in Effect Measure Is Plausible\n\nIn pre-treatment settings, the intervention has not yet been introduced, so outcomes under usual care can be observed for all individuals. We are currently collaborating with an Inserm team to generalize the effect of paracetamol (acetaminophen) for preterm infants to prevent patent ductus arteriosus  from the RCT **TREOCAPA trial** to 23 cohorts from the **RECAP preterm** project. In this setting, outcome data under the control condition are available for the cohorts, as the infants in these cohorts did not receive paracetamol. The aim is to predict the treatment effect across these different populations. These untreated cohorts provide high-quality data on clinically important outcomes—including survival without severe morbidity, defined as the absence of major complications such as bronchopulmonary dysplasia (BPD), necrotizing enterocolitis (NEC), severe intraventricular hemorrhage (IVH), and cystic periventricular leukomalacia (cPVL). \n \nWe have revised the discussion section to highlight a real-world setting in which control outcomes are available in the target population.\n\n\n### Understanding the Assumptions and Their Plausibility\n\nWe also clarify the structural differences between the two assumptions:\n\n- Exchangeability in effect measure assumes that treatment effects are portable but requires access to control outcomes in the target population and access to shifted treatment effect modifiers.\n\n- Exchangeability in mean, is stronger and assumes that conditional outcome models are stable across populations, but relies on  shifted prognostic covariates.\n\nTo assess which assumption is plausible, it is possible to perform some statistical tests. To help practitioners we now include the following guidance:\n\n- **Model-based diagnostics**: To test exchangeability in mean, one can regress the outcome on covariates and a study indicator. If the study coefficient is significant, it suggests the conditional mean function differs across populations (Robertson et al., 2021).\n\n- **Illustrative example**: Suppose the potential outcomes in the source population follow  \n  $$\n  \\mathbb{E}_S[Y^{(a)} | X] = b(X) + a \\cdot m(X),\n  $$  \n  while in the target population they follow  \n  $$\n  \\mathbb{E}_T[Y^{(a)} | X] = \\gamma + b(X) + a \\cdot m(X).\n  $$  \n\n  where $m(.)$ and  $b(.)$ respectively correspond to the treatment effect and the baseline. The constant $\\gamma$ modifies the baseline of the target population and may be due to the fact that the target population is more likely than the source population to undergo undesirable events, due to exogeneous variables whose information is not fully contained in the covariates $X$. In this case, the local effect of the Risk Difference can be generalized but the distribution of the potential outcomes cannot. \n\n- **Identifying effect modifiers** is an ongoing research challenge, but recent methods (e.g., Hines et al., 2022; Benard et al., 2023; Paillard et al., 2025) provide tools for detecting covariates that drive treatment effect heterogeneity.\n\n\nTo conclude, the choice among different approaches often depends on data availability—particularly whether outcomes in the target population are accessible. When outcome data are available, both identification assumptions can be empirically tested. Close collaboration with clinicians is essential to develop a DAG and to identify key variables: those likely to act as shifted prognostic factors, and potential treatment effect modifiers. In the latter case, specific statistical tests can also be conducted to support the analysis.\n\n\n\nWe believe these additions help clarify when each assumption is likely to be valid and how practitioners can evaluate them in applied settings.\n\n### Estimator Recommendation\n\nRegardless of which identification assumption is used, we recommend the corresponding estimating equation (EE) estimator (and not the one-step estimators): either the standard version (under exchangeability in mean) or the Γ-formula-based version (under exchangeability in effect measure). These estimators are doubly robust, providing consistent estimates if either the outcome model or the density ratio model is correctly specified. Our theoretical and empirical results show that EE estimators offer favorable bias–variance trade-offs under both assumptions.\n\n---\n\n## Clarification on Lemma 2: Positive Semi-Definiteness Argument\n\nYou're absolutely right that the expression\n\n$$\n\\mathbb{E}_T[VV^\\top] - \\mathbb{E}_T[VV^\\top] \\left( \\mathbb{E}_T[\\sigma(X,\\beta) VV^\\top] \\right)^{-1} \\mathbb{E}_T[VV^\\top]\n$$\n\nis **not guaranteed to be PSD** unless additional structural assumptions (e.g., $B \\succeq A$) are imposed which does not hold in our setting. However, this can be easily solved. The second statement holds if the matrix H is symmetric and positive semidefinite. We can rewrite H up to a positive scaling factor as the regrouped form:\n\n$$\nH := \\mathbb{E}_T\\left[\\frac{1}{\\sigma(X,\\beta)} VV^\\top\\right] - \\mathbb{E}_T[VV^\\top] \\left( \\mathbb{E}_T[\\sigma(X,\\beta) VV^\\top] \\right)^{-1} \\mathbb{E}_T[VV^\\top].\n$$\n\nLet us define:\n\n* $A := \\mathbb{E}_T[VV^\\top]$,\n* $B := \\mathbb{E}_T[\\sigma(X,\\beta) VV^\\top]$,\n* $C := \\mathbb{E}_T\\left[\\frac{1}{\\sigma(X,\\beta)} VV^\\top\\right]$.\n\nThen we can write:\n\n$$\nH = C - A B^{-1} A.\n$$\n\nThis expression is PSD **if and only if** the following block matrix is PSD (by the Schur complement condition):\n\n$$\nM := \\begin{bmatrix}\nC & A \\\\\\\\\nA & B\n\\end{bmatrix}.\n$$\n\nNow observe that this matrix can be written as an expectation over rank-one PSD matrices:\n\n$$\nM = \\mathbb{E}_T[\\tilde{V} \\tilde{V}^\\top],\n$$\n\nwhere\n\n$$\n\\tilde{V}^T := \\begin{bmatrix}\n\\frac{1}{\\sqrt{\\sigma(X,\\beta)}} V^T  \\\\\n\\sqrt{\\sigma(X,\\beta)} V^T\n\\end{bmatrix} \\in \\mathbb{R}^{2d}.\n$$\n\nSince each outer product $\\tilde{V} \\tilde{V}^\\top$ is PSD and expectations preserve positive semi-definiteness, it follows that $M \\succeq 0$. By the Schur complement identity, this implies that $H = C - A B^{-1} A \\succeq 0$ as well.\n\nWe have updated the manuscript to reflect this corrected structure and clarify the PSD argument. This correction ensures the validity of the key step used in Proposition 2.\n\n---\n\n## Clarification on Interpretation of Real-World Results\n\nWe agree that the phrase “little to no impact” was unclear. We have revised the sentence as follows: \n>*All estimators (except one) indicate a positive treatment effect; however, the confidence intervals are wide and include the null value (zero or one), preventing any definitive conclusions about the treatment’s effectiveness.*\n\nThis revised phrasing better captures the uncertainty in the estimates and avoids overinterpreting the results. We thank the reviewer for pointing this out.\n\n---\n\nWe thank the reviewer again for their feedback. We have addressed all raised concerns, including the clarification of Lemma 2, improved interpretation of the real-world results, and the addition of practical guidance and examples to support assumption selection. We believe these revisions substantially strengthen both the theoretical clarity and applied usability of the paper.\n\n\n---\n\n### References\n\n* Hines, O., Diaz-Ordaz, K., & Vansteelandt, S. (2022). *Variable importance measures for heterogeneous causal effects*. arXiv preprint arXiv:2204.06030.\n* Benard, C., & Josse, J. (2023). *Variable importance for causal forests: breaking down the heterogeneity of treatment effects*. arXiv preprint arXiv:2308.03369.\n* Paillard, J., Reyero Lobo, A. D., Kolodyazhniy, V., Thirion, B., & Engemann, D. A. (2025). *Measuring variable importance in heterogeneous treatment effects with confidence*. ICML 2025.\n* Robertson, S. E., Steingrimsson, J. A., Joyce, N. R., Stuart, E. A., & Dahabreh, I. J. (2021). *Center-specific causal inference with multicenter trials: Reinterpreting trial evidence in the context of each participating center*. arXiv preprint arXiv:2104.05905."}, {"review_id": "DqiliATgRU", "reviewer": "Reviewer_naME", "summary": "This paper studies the generalizability of several nonlinear causal estimands under distribution shift, proposing a family of estimators under mean exchangeability and effect measure exchangeability conditions. When mean exchangeability holds, the authors propose weighted, regression, and doubly robust strategies for transporting an estimand that is a known mapping of the mean of two potential outcomes. In addition, the authors discuss generalization strategies for such estimands based on the inverse mapping. Theoretical analysis shows the inferential guarantees of the proposed estimators, and the performance of them is demonstrated via simulations and real data analysis.", "questions": "1. How would you interpret the discrepancy in the messages in the real experiments? Since the truth is not available, it would be helpful to investigate why there is a discrepancy to decide whether it makes sense.\n\n2. I think connections to existing results should be more explicitly discussed to more clearly position the contributions. In particular, [1] discusses quite a few identification strategies for nonlinear causal estimands. How does this work compare to them? If all identification strategies have been studied and this work just focuses on estimators, then the estimators would look not very novel/surprising given the existing results, and this work might be better suited to a (bio)stats journal with more in-depth simulations and real data analysis to show the performance of various estimators in diverse conditions. \n\n\n[1] Colnet, B., J. Josse, G. Varoquaux, and E. Scornet (2023). Risk ratio, odds ratio, risk difference... which causal measure is easier to generalize? arXiv preprint arXiv:2303.16008.", "limitations": "Yes.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strength:\n\n1. This paper studies an important problem of generalizing causal estimands.\n2. The results are solid and rich, covering identification, estimation, and inference. \n3. Numerical results show promising performance.\n\nWeakness:\n\n1. The contributions need to be more clearly stated by comparing to relevant literature (see Questions).\n2. The real data analysis has vague message.", "was_revised": true, "final_justification": "The author's response clarified the contributions which I had concerns about.", "rebuttal": "## Clarification on discrepancy in the messages in the real experiments\n\nThank you for this comment. We have clarified the real data analysis in the revised manuscript. While the RCT analysis (not included in the paper; see Colnet et al., 2024) suggested a slightly positive effect of the treatment, the estimators applied to the target population suggest a slightly deleterious—but non-significant—effect. This discrepancy is expected due to differences in the underlying populations: the RCT population differs from the target population used in generalization.\n\nAll causal estimators applied to the target population—whether based on generalizing from the RCT or using purely observational data—converge on the same conclusion: a non-significant effect in the Traumabase cohort. As you rightly point out, the true effect is unknown. However, because we observe outcomes in the target population, it is possible to directly estimate treatment effects from the observational data (Mayer et al., 2020). While this relies on different assumptions—particularly, the absence of unmeasured confounding—it is reassuring that both approaches (generalization from RCT and causal inference from observational data) lead to the same conclusion. This is true for the risk difference as well as relative measures (Boughdiri et al., 2025).\n\nIn addition, if one suspects that some identifiability assumptions are not met, it is possible to resort to sensitivity analysis.\n\n\n## Position with Colnet et al. (2023)\n\nWe thank the reviewer for this comment and appreciate the opportunity to clarify the novelty and scope of our contributions relative to prior work, particularly to Colnet et al. (2023).\n\nWhile Colnet et al. (2023) explores identification strategies for causal estimands—most notably the risk ratio and the risk difference—in the context of generalization, **our work extends the identification and estimation landscape in several important directions**:\n\n---\n\n### **1. Unified Framework for Transportability**\n\n**We introduce a general identification and estimation framework for transporting a broad class of first-moment causal estimands**—including more than a dozen measures, both **collapsible** (e.g., RD, RR) and **non-collapsible** (e.g., OR)—across covariate shifts. Specifically:\n\n* Under exchangeability in mean, we **recover and extend existing identification results** and **derive asymptotic properties** for all proposed estimators. For instance, under a logistic selection and linear outcome model, we show that the **transported G-formula is asymptotically more efficient than the Horvitz–Thompson estimator**.\n\n* Under exchangeability in effect measure, a **strictly weaker assumption**, we provide **a different result from Colnet, which is more general**  to cover **any first-moment causal estimand**, including **non-collapsible ones like the odds ratio (OR)**. To our knowledge, these are **novel identification results** in the generalization context and are especially relevant given the widespread use of the OR in applied research.\n\n---\n\n### **2. New Estimators for Transported Estimands**\n\nWhereas Colnet et al. focused on identification **we propose new estimators designed for generalization from randomized trials**, including:\n\n* Semiparametric efficient estimators (one-step and estimating equation), and we **demonstrate that their behavior diverges for nonlinear estimands** such as RR and OR—**a key theoretical and empirical contribution**. Notably, we show that the **estimating equation (EE) estimator is doubly robust** for any causal measure (unlike the one-step estimator), and **consistently outperforms it empirically**. **This difference had not been previously emphasized** in the literature.\n\n* **Γ-formula estimators under exchangeability in effect measure**, which are, to our knowledge, novel contributions, especially in the context of **generalizing non-collapsible estimands**.\n\n---\n\n### **3. Complementary Goals to Prior Work**\n\nWhile Colnet et al. emphasize which causal estimands are easier to identify and generalize, our work offers a unified framework that supports the generalization of all first-moment causal measures, with a strong focus on estimation strategies and efficiency (which allows practical deployment of such methods with theoretical guarantees in contrats to their work). Their focus is on separating treatment effect from baseline; our focus reveals how many estimands share a common structure, enabling unified treatment in both identification and estimation. We see these as complementary contributions but different that together helps the understanding of causal transportability. \n\n---\n\n### **4. Relevance to NeurIPS**\n\nAlthough we agree that real-world validation is valuable, our primary goal is to offer a principled, broadly applicable framework for researchers and practitioners at the interface of machine learning, statistics, and application. Specifically, our use of influence functions, robustness analysis, and complex simulation designs (e.g., under model misspecification and nonlinear responses) directly targets challenges common in machine learning applications that require both generalizability and estimator efficiency.\n\n---\n\nWe hope to have convinced you that our contribution is both theoretical and methodological, encompassing a unified transportability framework and novel results at both the identification and estimation levels, especially for **nonlinear estimands**. We believe this goes well beyond a routine application or minor extension and is a strong fit for a generalist ML venue like NeurIPS. Furthermore, **our framework is broadly applicable across domains**—biostatistics, econometrics, and public policy—underscoring its potential impact.\n\n\n---\n\n### References\n\n* Colnet, B., Mayer, I., Chen, G., Dieng, A., Li, R., Varoquaux, G., Vert, J.-P., Josse, J., & Yang, S. (2024). *Causal Inference Methods for Combining Randomized Trials and Observational Studies: A Review*. Statistical Science, 39(1), 165–191. \n\n* Mayer, I., Sverdrup, E., Gauss, T., Moyer, J.-D., Wager, S., & Josse, J. (2020). *Doubly robust treatment effect estimation with missing attributes*. arXiv preprint \n\n* Boughdiri, A., Josse, J., & Scornet, E. (2025). *Quantifying Treatment Effects: Estimating Risk Ratios in Causal Inference*. ICML2025"}, {"review_id": "oEZyZ6uSG7", "reviewer": "Reviewer_247y", "summary": "This paper aims to generalize transportability theory from average treatment effects to a broader class of first-moment population-level causal measures. The authors derive identification results under two regimes—exchangeability in mean and in effect measure—and propose several estimators (classical, regression-based, semiparametric). They show empirically that estimator performance diverges sharply across nonlinear estimands and assumptions.", "questions": "- What if the observed covariates are not fully overlapping among training and target experiment?  What if some covariates are changing distribution on target but they are not directly observed? (see PPCI formulation [1] for reference).\n- Are the assumptions and estimators scaling to an higher number of observations? The real world experiments only consider 6 covariates.\n\n[1] Causal Lifting of Neural Representations: Zero-Shot Generalization for Causal Inferences, arxiv, Cadei et al., 2025", "limitations": "See questions.", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 1, "significance": 3, "originality": 2, "strengths_and_weaknesses": "*Strengths*\n- Motivation: The paper tackles an under-addressed problem (generalising nonlinear causal estimands across populations) motivated by several real-world application, far beyond medicine. \n- Method: The paper is methodologically rich, providing numerous instantiations of classical and doubly robust estimators. \n\n*Weaknesses*:\n- The introduction is difficult to follow. Several concepts are not properly introduced, e.g.:\n   - l27 the risk difference is introduced with no definition or assumptions on the setting (binary outcome?).\n   - l54-55 'collapsible and not collapsible measures' are not defined.\n\n- Imprecisions/Error (?):\n  - Risk ratio assumes continuous outcome for interpretation, but I cannot find where/if assumed in Section 2. Similarly Odds ratio etc.\n  - l110 m and n are DEPENDENT binomials (use multimodal distribution instead).\n  - Finishing the intro was still unclear if some transportability assumptions are required. I find them crucial and I would suggest to better stress them in the introduction too.\n- Experiments\n  - Synthetic experiments are under-defined and no code is provided. How many covariates are considered (d)? What are the source values in Figure 1 caption?\n  - I find difficult to interpret the objective of the experiments. Additionally I find unclear how to evaluate the real world experiment results in Figure 3.\n\nOverall I appreciate the attempt of this work in formalising transportability of population-level causal measures but I find the writing still immature, containing imprecisions with unclear story conclusions.", "was_revised": true, "final_justification": "The rebuttal addresses some theoretical points and acknowledges important limitations. However, the empirical evidence remains very limited and underspecified, with no demonstration on realistic or robust settings. Additionally, while the authors mention providing code, no link has been made available, which hinders reproducibility and transparency. Given these unresolved issues, especially the weak empirical support, I maintain my recommendation to reject.", "rebuttal": "We thank you for your carefull review. We will fix the several typos you pointed out. \n\n**Covariate Non-Overlap Between Source and Target Populations**\n\nThe overlap assumption is critical for identifying causal effects in a target population. It requires that for any covariate pattern present in the target population, there is sufficient representation in the trial population ($P(S = 1 | X = x) > 0$). When this assumption is violated—i.e., when the observed covariates are not fully overlapping between training (RCT) and target populations—standard identification results no longer hold, and estimators proposed in the paper may become biased or unstable due to extreme or poorly estimated weights.\n\nThere are two main strategies to address this:\n\n1. **Restricting the target population**: One approach is to redefine the generalization target to individuals in the target population who are within the support of the RCT sample—that is, those who could plausibly have been enrolled in the RCT. This amounts to generalizing the effect only to the subpopulation with overlapping covariates, thereby restoring identifiability at the cost of external validity (the target population has changed).\n\n2. **Performing sensitivity analyses**: An alternative is to formally account for violations of overlap through sensitivity analyses, which quantify the robustness of conclusions under deviations from the assumption. See for instance Huang (2025) for recent developments in this direction.\n\n\n**Unobserved Shifting Covariates**\n\nIf unobserved covariates drive differences in outcome distributions between the source and target populations, the assumption of exchangeability in mean is violated, undermining identification based on outcome modeling. In such cases, a weaker but potentially more realistic assumption—exchangeability in the effect measure—can still hold. Indeed, if the unobserved covariates influence only the outcome level but not the treatment effect heterogeneity, transportability of the effect measure is still possible. However, identification under this assumption requires access to control outcomes in the target population, which may not be feasible in many  scenarios. If shifted treatment effect modifiers are not observed, then, this boils down to an identifiability issue and consequently, one has to resort to traditional methods to give bounds on treatment effects, for instance using sensitivity analysis. We added a paragraph in the discussion section on limitations, and the verification of assumptions.\n\n\n**Scalability of Estimators with Increasing Covariate Dimensions**\n\nIt’s true that the example uses only six covariates, but this reflects the limited data collected in the clinical trial. In randomized trials, covariates are often less critical, since treatment effects can be estimated directly by comparing average outcomes between treated and control groups. The six covariates used are those shared between the trial and observational data, and which show distributional differences across the two datasets.\n\nThat said, there’s no theoretical limitation on the number of covariates. The estimators rely on the estimation of nuisance functions—such as regressions of $Y$ on $X$—which can be modeled using parametric (e.g., linear regression) or nonparametric methods (e.g., random forests). When using g-formula-type estimators, more data is required as the number of covariates increases, especially with nonparametric methods due to slower convergence rates. However, doubly robust estimators can recover parametric convergence rates even in high-dimensional settings.\n\n\n### Weaknesses\n\n\n**1. \"The introduction is difficult to follow. Several concepts are not properly introduced, e.g.: l27 the risk difference is introduced with no definition or assumptions on the setting (binary outcome?). l54-55 'collapsible and not collapsible measures' are not defined.\"**\n\nWe thank the reviewer for this comment. We have revised the introduction for clarity and now explicitly define key concepts such as the *risk difference* and *collapsible vs. non-collapsible measures*, including references for further reading.\n\n**2. \"Risk ratio assumes continuous outcome for interpretation, but I cannot find where/if assumed in Section 2. Similarly Odds ratio etc.\"**\n\nThe *risk ratio* can be interpreted for both binary and continuous outcomes (as long as the outcome remains positive), while the *odds ratio* applies only to binary outcomes. Our results hold in the continuous and binary setting, which we have now clarified in Section 2.\n\n**3. \"l110 m and n are DEPENDENT binomials (use multimodal distribution instead).\"**\n\nWe infer you meant multinomial and not multimodal. You are right , $(n, m)$ follow a **multinomial distribution** with parameter $N$ and probabilities $(\\alpha, 1 - \\alpha)$.\n\n**5. \"Synthetic experiments are under-defined and no code is provided. How many covariates are considered (d)? What are the source values in Figure 1 caption?\"**\n\nThank you for the feedback. We now also provide a link to the experimental code for reproducibility. We have updated the experiments section to clarify that:\n\n* The number of covariates (*d*) is **5**, consistent with real-world RCTs that often involve few covariates.\n* The *source value* in Figure 1 denotes the **true treatment effect in the source population**, shown to illustrate the shift in effect when applied to the target population."}, {"review_id": "7N2O5fcbnh", "reviewer": "Reviewer_nTva", "summary": "This paper introduces a unified framework for transporting a wide range of population causal effect measures from randomized controlled trials (RCTs) to target populations under covariate shift. It provides identification results under two key assumptions (exchangeability in mean and in effect measure), and derives classical, semiparametric, and doubly-robust estimators, revealing that estimators often differ across causal measures due to nonlinearity.", "questions": "Do you expect significant performance differences between your estimators in small or moderate sample sizes, especially when nuisance functions are estimated with limited data?", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**:\n- The paper generalizes causal transportability to a wide class of first-moment estimands, which is a notable expansion beyond existing methods\n- The theoretical claims are convincingly supported \n- The paper is well written and easy to follow\n\n\n\n**Weaknesses**:\n\n- All the simulations use large sample sizes ($ n=50,000$), masking potential pathologies under finite or moderate samples, such as weight instability, overfitting of nuisance models, or violation of positivity. Arguably, most trials have significantly smaller sample sizes and therefore it would be valuable to evaluate the proposed estimators in a small sample setting", "was_revised": true, "final_justification": "I stand by my original recommendation to accept the paper. The authors have addressed my concerns about small-sample behavior with additional simulations. Assuming the correctness of Lemma 2 (which I reviewed based on their rebuttal and found reasonable) the contributions of this paper remain solid.", "rebuttal": "Thank you for your thoughtful and constructive review. We are glad that you found the contributions of the paper valuable and the exposition clear.\n\nRegarding your comment on small or moderate sample sizes: while our theoretical results are asymptotic, the methodology itself is not restricted to large samples. In practice, when RCT data is limited and nuisance functions (e.g., outcome or propensity models) must be estimated from this same limited data, a pragmatic approach would be to use parametric models to reduce variance and mitigate overfitting.\n\nTo assess finite-sample behaviour empirically, we conducted simulations with total sample size \\$N = 5000\\$ and \\$500\\$, of which respectively 1,500 and 150 for the RCT group. The results in Figure 4 (first table) correspond to the setting under exchangeability in mean, and those in Figure 2 (second table) to the setting under exchangeability in effect measure. Across both settings, the **Estimating-equation** estimator—whether standard or Γ-adjusted—shows the best empirical performance, achieving the lowest combination of bias and variance across RD, RR, and OR. While in some specific cases it may be marginally outperformed by a G-formula or a One-step estimator, the **Estimating-equation remains doubly robust**, offering consistent performance even under model misspecification. For this reason, we strongly recommend its use in practice, particularly when nuisance functions are estimated from limited data.\n\nThat said, we agree that further exploration of finite-sample properties would be valuable. In particular, understanding how the sizes of both the source (RCT) and target populations affect the performance of different estimators—especially those based on efficient influence functions—is a promising direction for future research. We have added a sentence to the conclusion to reflect this.\n\n**N=5000**\n\n| Estimator           | RD                  | RR                  | OR                  |\n| :------------------ | :------------------ | :------------------ | :------------------ |\n| Estimating-equation | **0.0033 ± 0.0867** | 0.0098 ± 0.1916     | **0.0021 ± 0.2762** |\n| One-step            | 0.0049 ± 0.0868     | **0.0027 ± 0.2003** | 0.0024 ± 0.2771     |\n| T G-formula         | 0.0115 ± 0.0798     | 0.0198 ± 0.1780     | 0.0208 ± 0.2606     |\n| W G-formula         | 0.0604 ± 0.0559     | 0.0991 ± 0.1555     | 0.1449 ± 0.2172     |\n| W Horvitz-Thompson  | 0.0669 ± 0.0581     | 0.0001 ± 0.2894     | 0.0718 ± 0.3586     |\n| W Neyman        | 0.0759 ± 0.0557     | 0.0438 ± 0.2698     | 0.1261 ± 0.3320     |\n\n\n| Estimator             | RD                  | RR                  | OR                   |\n| :-------------------- | :------------------ | :------------------ | :------------------- |\n| Estimating-equation   | **0.0015 ± 0.0383** | 0.1293 ± 0.0648     | 0.0536 ± 0.1317      |\n| W Horvitz-Thompson    | 0.0435 ± 0.0349     | -0.0288 ± 0.0555    | -0.0885 ± 0.1501     |\n| W Neyman              | 0.0448 ± 0.0296     | -0.0314 ± 0.0493    | -0.0963 ± 0.1315     |\n| W Γ-formula           | 0.1573 ± 0.0278     | -0.1327 ± 0.0328    | -0.1343 ± 0.0884     |\n| Γ Estimating-equation | **0.0015 ± 0.0383** | **0.0113 ± 0.0678** | **-0.0119 ± 0.1109**     |\n| Γ One-step            | **0.0015 ± 0.0383** | **0.0113 ± 0.0678** | -0.0124 ± 0.1108 |\n\n\n**N=500**\n\n| Estimator           | RD                   | RR                   | OR                  |\n| :------------------ | :------------------- | :------------------- | :------------------ |\n| Estimating-equation | **-0.0153 ± 0.1354** | **-0.0664 ± 0.5310**     | 0.1662 ± 1.7489     |\n| One-step            | -0.0157 ± 0.1356     | -0.0956 ± 0.4791     | 0.1307 ± 1.7063 |\n| T G-formula         | -0.0210 ± 0.1233     | -0.1130 ± 0.4482 | **-0.0072 ± 1.4014**    |\n| W G-formula         | -0.2270 ± 0.0876     | -0.7003 ± 0.1600     | -1.5368 ± 0.3619    |\n| W Horvitz-Thompson  | -0.2100 ± 0.0811     | -0.6106 ± 0.4439     | -1.4052 ± 0.6020    |\n| W Neyman            | -0.2073 ± 0.0707     | -0.6106 ± 0.4124     | -1.4104 ± 0.5589    |\n\n\n\n| Estimator             | RD                  | RR                  | OR                   |\n| :-------------------- | :------------------ | :------------------ | :------------------- |\n| Estimating-equation   | **0.0062 ± 0.0999** | 0.0154 ± 0.1457     | 0.1330 ± 0.5173      |\n| W Horvitz-Thompson    | 0.0647 ± 0.0879     | -0.0835 ± 0.1988    | 0.0416 ± 0.5925      |\n| W Neyman              | 0.0569 ± 0.0793     | -0.0968 ± 0.1759    | -0.0207 ± 0.4897     |\n| W Γ-formula           | 0.1586 ± 0.0754     | 0.0688 ± 0.2283     | -0.1641 ± 0.2978     |\n| T Γ-formula           | 0.0097 ± 0.0892     | 0.0971 ± 0.2766     | -0.0157 ± 0.3539 |\n| Γ Estimating-equation | **0.0063 ± 0.0998**     | **0.0871 ± 0.3054** | **-0.0044 ± 0.3654** |\n| Γ One-step            | **0.0062 ± 0.0998**     | **0.0870 ± 0.3053** | -0.0157 ± 0.3603 |"}]

Author rebuttal:

Meta-review: This paper develops a unified framework for transporting a broad range of first-moment population causal effect measures under covariate shift. Unlike most prior work that focuses on the risk difference, the authors address both absolute and relative effect measures, which are commonly reported in clinical and policy research. The paper presents identification results, develops classical and semiparametric estimators, and supports the framework with simulations and empirical applications.

While the reviewer team did not reach full agreement, the concern regarding Lemma 2 appears to have been adequately resolved during the discussion. The remaining issues (specifically, the need to acknowledge the limitations of the empirical analysis and to release code) can be addressed in the camera-ready version.

Given the strength of the contribution and the practical relevance of the framework, I recommend acceptance.
Decision: Accept (poster)