Orthogonal Survival Learners for Estimating Heterogeneous Treatment Effects from Time-to-Event Data
Authors: Dennis Frauen
Maresa Schröder
Konstantin Hess
Stefan Feuerriegel
Venue: NeurIPS 2025 poster ()
Keywords: Causal inference
treatment effect estimation
survival analysis
orthogonal learning
Primary area: probabilistic_methods
TL;DR: We propose a toolbox of orthogonal survival learners for HTE estimation from time-to-event data under censoring.

Abstract: Estimating heterogeneous treatment effects (HTEs) is crucial for personalized decision-making. However, this task is challenging in survival analysis, which includes time-to-event data with censored outcomes (e.g., due to study dropout). In this paper, we propose a toolbox of orthogonal survival learners to estimate HTEs from time-to-event data under censoring. Our learners have three main advantages: (i) we show that learners from our toolbox are guaranteed to be orthogonal and thus robust with respect to nuisance estimation errors; (ii) our toolbox allows for incorporating a custom weighting function, which can lead to robustness against different types of low overlap, and (iii) our learners are \emph{model-agnostic} (i.e., they can be combined with arbitrary machine learning models). We instantiate the learners from our toolbox using several weighting functions and, as a result, propose various neural orthogonal survival learners. Some of these coincide with existing survival learners (including survival versions of the DR- and R-learner), while others are novel and further robust w.r.t. low overlap regimes specific to the survival setting (i.e., survival overlap and censoring overlap). We then empirically verify the effectiveness of our learners for HTE estimation in different low-overlap regimes through numerical experiments. In sum, we provide practitioners with a large toolbox of learners that can be used for randomized and observational studies with censored time-to-event data.

Orthogonal Survival Learners for Estimating Heterogeneous Treatment Effects from Time-to-Event Data
Dennis Frauen, Maresa Schröder; Konstantin Hess, Stefan Feuerriegel
LMU Munich
Munich Center for Machine Learning
(frauen, maresa. schroeder, k.hess, feuerriegel/@lmu
Abstract
Estimating heterogeneous treatment effects (HTEs) is crucial for personalized decision-making. However, this task is challenging survival analysis, which includes time-to-event data with censored outcomes (e.g.. due to study dropout). this paper, we propose toolbox of orthogonal survival learners to estimate HTEs from time-to-event data under censoring. Our learners have three main advantages: (i) we show that learners from our toolbox are guaranteed to be orthogonal and thus robust with respect to nuisance estimation errors; (ii) our toolbox allows for incorporating a custom weighting function, which can lead to robustness against different types of low overlap, and (iii) learners model- agnostic (i.e., they can be combined with arbitrary machine learning models). We instantiate the learners from toolbox using several weighting functions and, as result, propose various neural orthogonal survival learners. Some of these coincide with existing survival learners (including survival versions of the DR- and R-learner), while others are novel and further robust w.r.. low overlap regimes specific to the survival setting (i.e., survival overlap and censoring overlap). We then empirically verify the effectiveness of our learners for HTE estimation in different low-overlap regimes through numerical experiments. sum, we provide practitioners with large toolbox of learners that can be used for randomized and observational studies with censored time-to-event data.
Introduction
Estimating heterogeneous treatment effects (HTEs) crucial in personalized medicine [T0] 15 HTES quantify the causal effect of a treatment on an outcome (e.g., survival), conditional on individual patient characteristics (e.g., age, gender, prior diseases). This enables clinicians make individualized treatment decisions aimed at improving patient outcomes. For example, knowing the HTE of an anticancer drug on patient survival can inform treatment decisions that tailored patient's unique medical history, thereby maximizing the probability of survival.
A common setting in medicine is survival analysis 26.5 51]. Survival analysis is aimed at modeling
medical outcomes, particularly in cancer care, where the dataset involves time-to-evell data [54). That is, the outcome of interest is time example, in cancer care, a treatment should maximize the time until death or tumor progression (44
event that are interested in maximizing. For
43
Hence, this requires tailored methods for HTE estimation from time-to-event data.
The fundamental problem that distinguishes survival analysis from standard causal inference is censoring 47). Censoring refers to the phenomenon that some individuals may not have experienced
"Equal contribution.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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the event (e.g., death, recovery) by the end of the study period (also called Tigh--censoring For example, older patients may be more likely to drop out of medical studies [35]. Censoring thus requires custom methods to allow for unbiased causal inference. For example, if we simply remove censored individuals from the data, the remaining population may be younger on average, which may lead to biased treatment effect estimates (if the treatment effect is, e.g., different for older patients).
are biased when used for As a result, standard methods for HTE estimation (e.g., [23 33 46]) censored time-to-event data.
In comparison to uncensored data, estimating HTEs from censored time-to-even data thus subject to three additional main challenges: 1 Complex confounding: Confounders might not only affect treatment and outcome, but also the event and censoring times. Thus, properly adjusting for these confounders is necessary to obtain valid treatment effect estimates. 2 Estimation complexity: Methods that adjust for confounding under censoring require estimating additional nuisance functions over multiple time steps, such as hazard functions. 3 Different types of overlap: A necessary requirement for estimating HTES sufficient treatment overlap, i.e., that every individual has a positive probability of receiving not receiving the treatment [e.g.. [ 32] However, under censoring. additional overlap conditions are required, which we refer to as censoring overlap and survival overlap. For example, every individual must have nonzero probability of being uncensored (i.e., experiencing the event).
Existing methods have limitations because of which they cannot deal with all the above challenges. For example, for uncensored data, state-of-the- methods for estimating HTEs are Leyman-orthogomm
meta-learners (such as the DR- and the R- R-learner) [e.g.. 23 33 46). - These have been extended to the survival setting to address challenges 1 and 2 from above, particularly via survival versions of the DR- and R-learners 7 52) However, these learners lack the ability to address challenge 2 of different types of overlap and, as a result, exhibit a large variance under low censoring or survival overlap (e.g., if certain individuals almost never experience the event).
In this paper, we address the above limitations by proposing novel, general toolbox with Neyman- orthogonal meta-learners for estimating HTEs from censored time-to-event data. Our proposed meta-learners address the challenges from above as follows: ! they use orthogonal censoring adjustments, which enable unbiased and robust estimation under both confounding and censoring; 2 they are model-agnostic in the sense that they can be instantiated with any machine learning method (e.g., neural networks) to effectively learn nuisance functions; In contrast to existing survival learners, they effectively overcome the difficulty of treatment effect estimation in the presence of lack of any of the three overlap types through targeted re-weighting of the orthogonal losses.
Our contribution arr: (1) We propose a novel toolbox with orthogonal learners for estimating HTES from time-to-event data. Our toolbox allows the specification of a custom weighting function for robust estimation under low treatment-, survival-, and/or censoring overlap. We also provide several extensions of our toolbox to different settings (continuous time, marginalized effects, different causal estimands, and unobserved ties) in our Appendix. (2) We provide theoretical guarantees that learners constructed within our framework are orthogonal and that our learners provide meaningful HTE estimates, regardless of the chosen weighting function. (3) We instantiate our toolbox for several weighting functions and obtain various novel orthogonal survival learners. These learners are agnel-agnostic and can be used in combination with arbitrary machine learning models (e.g., neural networks).
Related work
Below, we review key works aimed or- thogonal learning for HTEs, especially for censored data. We provide an extended literature review in Appendix A
Table 1: Overview of key orthogonal learners whether they can adjust for censoring / different types of overlap.
Comoring
Trise overlaps
evellpp
merlap
DR leumer
46
Orthogonal learning of HTEs from un-
learne:
censored data. Several meta-learners for
Survival
Sursival Leuner
HTE estimation have been introduced in
Ours
the literature, particularly for conditional
Code available https:: http:::///gttub httpt///ww.....oeoeoeeeeorrrrrre

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average treatment effects [e.g.. 9 Among them, the DR-learner [23 46 and the R-learner 33 are often regarded as state-of-the-arr because these are orthogonal, meaning they are based upon semiparametric efficiency theory [3 48 and robust with respect to nuisance estimation errors (5) Furthermore, orthogonality typically implies other favorable theoretical properties, such as quasi- oracle efficiency and doubly robust convergence rates [1T]. Recently, (32) showed that the R-learner can be interpreted as an overlap- version the DR-learner, thus addressing instabilities and high variance in low treatment-overlap scenarios.
Orthogonal learners have also been proposed for other causal quantities. such as the conditional average treatment effect on the treated (29), instrumental variable settings [I2 44 HTEs over time
[13] partial identification bounds [34] 41]. or uncertainty quantification of treatment effect estimates
[T. However, none of these learners are tailored to time-to-event data, and, hence, they are biased
under censoring.
Model-based learning of HTFS from censored data. There is some literature for estimating HTEs from time-to-event data that has focused on model-based learners, i.c., learners based on specific machine learning models [18]. Examples include tree-based learners [e.g.. 7 16 45
54] or
neural-network--bssee learners (8] 21 40 Note that these learners are neither -agnottg (i.e.,
cannot be used with arbitrary machine learning models) nor orthogonal. Furthermore, model-based
learners typically estimate the target HTE via a plug-in fashion and thus suffer from so-called plug-in bias [22]. In contrast, (model-agnostte orthogonal learners remove plug-in bias by fitting a second model based on a Seeyman--rthogggne second-stas stage loss. Nevertheless, model-based learners can be combined with model-agnostic orthogonal learners for the first stage (nuisance estimation).
Orthogonal learning of HTEs from censored data. Few works have proposed (orthogonal) meta- learners tailored to censored time-to-event data. Xu et al. (53) introduce an adaptation of existing learners time-to-event data based on inverse probability of censoring weighting (27 47). However, the proposed learner is only applicable to experimental data from randomized controlled trials and is sensitive overlap violations. Gao al. [14] propose orthogonal learners based on exponential family and Cox models, but not neural networks. Xu et al. (52) develop censoring unbiased transformations for survival outcomes; i.e., to convert time-to- mo-eetteeeent outcomes standard continuous outcomes, which can then be combined with existing orthogonal learners for estimating HTEs in the standard setting. However, the corresponding survival versions of the DR-learner and R-learner are not robust against survival or censoring overlap violations.
Research gap: So far, existing orthogonal survival learners fail account for different types of overlap violations (see Table []. To the best of our knowledge, we are thus the first to provide a general toolbox that includes custom weighting functions to ensure robustness against different types of overlap violations (such as survival or censoring overlap violations).
3 Problem setup
Data: We consider the standard setting for estimating HTEs from time-to-event data (8] [7.. That is, consider population A.A.T.C) ~ P. where E X' C R2 are observed covariates, AE (0, is binary treatment, T 'T event time of interest (e.g., death of the patient), and C' E T is the censoring time at which the patient drops out of the study. To ease notation, we assume throughout the main part
4
Causal graph for censored time-to-event data. Yellow variables are observed while blue variables unob- served. Intuitively, our goal is recover the red arrow from A to T based on the observed variables.
Figure 1:
discrete-time setting T == = "0,...,,mua of this paper. However, all our findings can be easily extended to continuous time, and we provide the corresponding results in Appendix
The challenge of time-to-event data is that we cannot collect data from the full population (X, A, T, C). Instead, we only observe dataset D- {x,, a;, F, 88 80 1mm } of size E N sampled i.i.d. from the population Z (X, T, A AG), where 7 = min mTn(T,C C),, the indicator A 1(T S C) equals one when the event of interest (e.g., death/recovery) is observed, and A 1(T C*) equals
3

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one in the censoring case (study dropout) In other words, for every patient, we know only the time T and whether the patient experienced the main event censoring. A causal graph is shown in Fig.
Key definitions: We define the (conditional) survival functions (1 a) P(T t| X 2, A= a)
and Gz(z, a) = P(C > t X = z, A = a) of the main event and of the censoring event, respectively.
We further define the hazard functions Xf a) PP = t, A
1 T 2 x, A a)
and X (x( a) P(T t, A 1 T 2 t, x. = a) of the main event of interest and censoring, respectively. The survival hazard function X (z, a) denotes the probability that a patient with covariates x and treatment a experiences the main event (e.g., death) at time I, given survival up to time 1. Analogously, the censoring hazard function denotes the probability that the same patient drops out at time i, given no prior dropout. Finally, we define the propensity score as P = 1 X z), which represents the treatment assignment mechanism based on X = =
Causal estimand: We use the potential outcomes framework (39] formalize our causal inference problem. Let (a) E T denote the potential event time corresponding to a treatment intervention A = a. We are interested in the causal estimand
7y (x) P(T(1) > + -))
X x =)
for some fixed T. The estimand (x) is the difference in survival probability up to time t for a patient with covariates x. We also provide extensions of all our results to conditional means F(x) (1)) T(0) X
as well as treatment-specifie quantities such as 14(2, a) P(T(a) > 1 X = x) and paa, a) = (T)(a) X = x]] in Appendix
Identifiability: We impose the following assumptions to ensure the identifiability of (z)).
Assumption 3.1 (Standard causal inference assumptions). For all a E {0, and E X' it holds:
(i) consistency: T(a) T whenever
a: (ii) treatment overlap: 0 < ((x) < whenever
P(X - 2) > 0; and (iii) ignorability: A L ((1),T(0) X =
Assumption 3.2 (Survival-specifi assumptions). For all a E {0, and x E X' with P(X = A =
a) it holds: (i) censoring overlap: G (2, a) > O; (ii) survival overlap: S (2, a) > 0; and (iii) non-informative censoring: T T
A =
Assumption 3.1 is standard in causal inference 19 36 48) and ensures that there is (i) no interference
between individuals, (ii) we have sufficient observed treatments for all covariate values, and (iii) there are no unobserved confounders that can bias our estimation. Assumption 3.2 is commonly imposed for survival analysis [4] and ensures that we have sufficient non-censored and surviving individuals for each covariate value and that the censoring mechanism is independent of a patient's 's survival time (given covariates and treatments). Under Assumptions
3.1
and 3.2
we can identify ra(r)) via
Te T=(x) = 5x, 1) (z, 0) II (1- (x, i=0
II (x, i=0
(2)
We provide proof in Appendix
H Note that the hazard functions X (ar, a) only depend on the
observed population Z = (X, A, T, A AG) and can therefore be estimated from the data D.
Challenges in survival analysis: In classical causal inference, a major challenge is covariate shift, meaning a strong correlation of observed confounders with the treatment [42). For example, specific patients may almost always receive treatments, while others almost never receive treatment. This leads to the problem of low overlap, i.c., extreme propensity score π (I), and thus lack of data for specific patients with certain covariate values.
In survival analysis, the censoring mechanism adds two additional sources of data scarcity: (i) if (2,, a) is small, certain patients have a low probability of being uncensored beyond time I, implying a lack of uncensored observations (low censoring overlap). Similarly, if S.- i (æ, a) is small, most patients experience the main event before time I,, leaving few data to estimate the hazard function X (low survival overlap). Existing learners for estimating HTES from time-to-event data have not yet addressed challenges due to additional types of low overlap. In the following section, we provide remedy to these challenges by proposing general orthogonal learning framework that can incorporate custom weighting functions to address the different types of low overlap.
Related works often set A = 1 5, thus excluding ties (see Appendix
G

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Background on orthogonal learning
Why plug-in learners are problematic: A straightforward method obtain an estimator is the so-called plugin-learner Here. we first obtain estimates of the survival hazards X (a: a). We discuss methods for this in Appendix [
Then, we can obtain an estimator of our causal quantity of interest via (x) (zx, 1) S, (z. 0), where Sc a) II (1 X (æ, a)). That is, the approach is to "plug-in" the estimated hazards into the identification formula from Eq. (2). However, is well known in the literature that such plug-in approaches lead to so-called "plug-in in bias" and, thus, suboptimal estimation 22]. For details, we refer to Appendix B
Why we develop two-stage learners: As remedy to plug-in in bias, current ofate-of-the-ar methods for HTE estimation are built upon wo-stage estimation: First, so-called nuisance functions 77 are estimated, which are components of the data-generating process that we will define later. Then, second-stage learner trained via
fx(x) = arg min L(g, fi),, SEg
where L(g, fj.) is some recond-stage loss that depends the estimated nuisance functions ît. Two- stage learners come with two main advantages: (i) they allow to estimate the causal estimand directly, thus increasing statistical efficiency; and (ii) they allow to choose model class G of the causal estimand. For example, ix (x) can be directly regularized, or interpretable models such as decision trees be used.
The benefit of orthogonal loss functions: The current sfate-of-the-arr for designing second-stage loss functions are so-called Jorthooonot loss functions (5). Formally, sccond-stage loss C(g, m)) is orthogonal
DT DyDDyii DgG(g (gg m.)g 9, f mJ = 0,
for any 9 and ît. where Dr, and Dg denote directional derivatives [TT]. Informally, orthogonality implies the gradient of the loss w.r.t. is insensitive to small estimation errors in the nuisance functions. This robustness w.r.t. nuisance errors often enables favorable theoretical properties of orthogonal learners, such as quasi-oracle convergence rates
33
In the following, we carefully derive orthogonal losses for the survival setting, which are currently missing in the literature. As a result, our learners allows us to address not only a lack of data due to treatment overlap issues but also due to low censoring overlap and survival overlap.
A general toolbox for obtaining orthogonal survival learners
In this section, we provide our general recipe and theory for constructing orthogonal survival learners that can be used for estimating HTES from time-to-event data. We propose concrete learners that retarget for different overlap types in Sec. 6
Overview: Our toolbox for orthogo- nal survival learning proceeds in three steps: We fit nuisance models that estimate the nuisance functions тt- 2 We select a weighting function and corresponding weighted targel loss that addresses certain type of overlap violation. 3 We obtain an orthogo- nalized version of the weighted target loss that second-stai stage learner. An overview of our toolbox is shown in Fig. 2 In contrast to existing work, our step ? addresses overlap violations beyond treatment overlap within our orthogo- nal learning framework.
Nuisance estimation
Trzper
Sunfrll heand
Camnaring heuards
(c. (9,
Weighting function for retarged loss
Nos MeNyning
Teatrns count
Cansse gownt
[(x(X)) xX0 x(X)
((xxz) GXXX
to fit orthogonal
Orthogonal learning
argmin
((2t,h)yyta.))) dz.))²
S (ch))))
Figure 2: Overview of the three steps of our toolbox.
① Nuisance estimation. In step I, we estimate the relevant nuisance functions for the survival setting. These include the propensity score л (XX), well as the survival and censoring hazards
5

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X (2z a) and Xf a) for each time point up to t. Formally, we define:
m = ((X), (x, (X, (x,
(5)
Each of these nuisance functions can be estimated from the data using arbitrary machine learning methods. Details for estimating these functions are provided in Appendix
2 Weighted target loss. Once nuisance estimators (X) are obtained, the second step is to define a target loss that incorporates a weighting function designed to address overlap violations. Specifically, we consider positive weighting function f(] > 0 that depends on i)( (x()), (X,, 1), (X, XKK0) 0), G 1), GX, 0)) (i.c., the propensity score, the survival and censoring functions at time 1). The corresponding weighted target loss is defined
Cy no
1
E (
(6)
(x)))
The weighted target loss represents the population loss that we aim to minimize. Note that the minimizer of Ly 7/)) over function class G coincides with our target quantity Tz (X) whenever Te(X) E G, i.e., whenever our second-stage model class G is sufficiently complex. This holds no matter what weighting function we use as long as we ensure that (me(()) > 0.
Intuition for the weighting function. The weighting function (X)) ()>>>)>>)) > 0 allows us to retarget our loss towards more favorable population. For example, by choosing (X)( = x (X)), we downweight samples with low treatment overlap, and thus retarget to the population with large treatment overlap. This allows to prioritize estimation accuracy in regions with larger treatment overlap. Similar retargeting has been used in 220) for policy learning and in (32 for CATE estimation from uncensored data. Yet, to the best of our knowledge, we are the first to explicitly use retargeting for orthogonal HTE estimation from time-to-event data.
Weighting can be compared to clipping, i.e., removing observations with extreme weights from model training. Advantages of weighting over clipping include: (1) there is no need for choosing a cutoff value, which is otherwise often done heuristically, and (2) the weighting term is considered in the orthogonal objective we propose later, which makes the learner more robust to estimation errors in the weighting function. Clipping on the other hand can be insensitive to errors in, e.g., the propensity score.
How to choose the weighting function? functiion The different types of overlap can be estimated from the data at hand by estimating π, S,-1, and G.- Based on the estimated overlap types practitioners are able to choose a weighting function that addresses the specific overlap challenges of their setting at hand. We discuss the specific weighting functions and, thus, corresponding learners in Sec. 6
3 Orthogonal second-stage loss. In the final step 3, we obtain a Neyman -othhooorthooooall version of our weighted target loss that we use for the second-stage regression. Here, we follow similar approach as in (32. who derived weighted orthogonal learners for CATE estimation but from uncensored data.
How to choose such an orthogonal loss? First, we define corresponding averaged weighted estimand (X))] A natural candidate for an orthogonal loss is based on the so-called efficient influence function (EIF) dx,, f (2, 1T.) of the parameter 0.,f (see Appendix B for a more detailed background on ElFs). Further, a well-known result from semiparametric estimation theory states that the EIF satisfies the property D,, f me no ne]. i.e., its directional derivatives w.r.t. the nuisance functions are zero [5] 47). Hence, it remains to find a loss whose directional derivative w.r.t. g equals to the EIF. By deriving the EIF of fz, , we obtain the following main är as the partial derivative of the function f ) with respect to its (functional)
as or. = [ (X))
result. We denote af argument т, and use analogous notation for derivatives w.r.t. G,-1 and S.-1. Theorem 5.1. We define the (population) loss function
Ly(g,n) [x)) E. (((2))))(((
where
p(2, mc) = (%) + 0f (X
(8)
A
1
(88.1(.,4 SX-1 + (x)r
(x))
A)6o (G.-((,A) (x, (2))
7(X)
1 x(X)

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and
302, m)) Sx(X,1) -1(x,1)- (A- (X))()(ss(22 ())(7(()))
(9)
(())1-=(x))p))
with
Es(2, m))
Σ
1,
S2 (x..).
8a(2.
1(T - (XF(
S, (X.;-1(X.A))
(10)
and where we used the convention a= G-1(Tz,)) Then, Lylg, m²) is orthogonal with respect to the nuisance functions T₁-
Proof. See Appendix H
Therein, we calculate the functional derivatives according to Eq. (4
Theorem 5.1 shows that the loss (g. m)) is orthogonal for any choice of weighting function f and thus robust with respect to estimation errors in the nuisance functions. Of note, orthogonality implics beneficial convergence rates, as shown in (TT) for general orthogonal and convex losses. It remains to show that minimizing L g.g Me) actually leads to a meaningful estimator, i.e., we actually obtained an orthogonalized version of our weighted target loss.
Theorem 5.2. Let 9} arg mingeg L5 nt) ) be the minimizer of the orthogonal loss from Eq. over a class of functions g. Then, 9j also minimizes the weighted target loss
 arg min DEG B[((H((X)) (X) 9(x))²]
(11)
Hence, 93 = -o. any weighting function f as long as T EG.
Proof. See Appendix H
Theorem 5.2 implies that minimizing the orthogonal loss Cy ni)) indeed leads consistent estimator of The causal estimand of interest no matter what weighting function ff we choose (assuming the model class G is large enough to contain the ground-truth causal estimand and the nuisance functions are estimated sufficiently well). As result, Theorem 5.1 Theorem 5.22 together that the loss [g M))) is exactly what we wanted derive: orthogonalized verston of our weighted target loss. be readily used as a second-stage loss for obtaining the causal target parameter by minimizing its corresponding empirical version with estimated nuisance functions from step 1, i.c.,
Σ p(æ,, (e) (p(2z (e) g(x.))²
R (x) = arg min gEg D (G.().))
(12)
6 Orthogonal survival learners
We now explicitly instantiate our toolbox for specific weighting functions and write down the corresponding survival learners we obtain. We show that our toolbox both encompasses existing learners as special cases (Survival-DR-- and Survival-R-lcarner), but also leads to novel learners that address overlap types specific the survival setting. We the letters T/C/S in typewriter font to refer to survival learners addressing specific variants of overlap. We use @ to refer to a learner that does not address any type of overlap.
O-learner (no weighting; also known as Survival DR-learner 32 52 ): Here, no weighting is used in the target loss from Eq. (6), i.c., (X)) = 1. As a consequence, it holds that p(Z. m7. = 1, and the orthogonal loss becomes
LoR(g, m)) =
( (s.(X,))) (s((x,1) 5x x(X) 7 (X)) (Y -
(A- =(X)
S,(X, -)x))
(13)
using the transformed outcome Y(n) S, A)(1 Es (2,))))))) -1))). The drawback of the (Survival)-
DR-learner is that it is sensitive to all types of low overlap as it includes divisions by r(X), 1- 1-:11-x(]] (XX). S,(X, A), and G.(X, A). one of these quantities is small, the DR-loss becomes unstable, and the learner will exhibit high variance.
7

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T-learned (treatment overlap; also known as Survival R-learner) [52]: To address treatment overlap, choose (7) x (1 (X)). In other words, individuals with small treatment
overlap will be weighte in the weighted target loss. By noting that p m)) = (A - π (x))?, the orthogonal loss becomes
Cx Tp) = BB (v(m) A(mDs(x))
(14)
with transformed variables A(m) A (X) and (me) = (m²) (X) with (x) -
& X) = T SXX), (1- (X))),(((( as proposed in [52]. Compared to the DR-leamer, the R-learner does not divide by (XX) propensities (low treatment overlap). However, the R-learner still divides by Sr(X, A), and Gx(X, A), and is thus sensitive w.r.t. low censoring or survival overlap.
(X) and is thus less sensitive w.r.t. small or large
C-learner (censoring overlap): To address low censoring overlap, we can choose the weighting (X,1) 0), which down-weights patients who have large treated or untreated censoring probability before time t. In other words, if, for patient with covariates X, either the treated untreated probability to remain in the study until the time t of interest is small, the patient will be dein-weighted in the target loss. This type of weighting results in
function ( ff (X))
p(Z, = GG-1
(1 (1-
A
+ 1 x
1-A
EG Ea(Z, (e12.
(15)
((X)
(A - x (x)))ss (., m. mX),((X.) A)
((Z, =
- -) 5x(,0) 7 xXX) (X)) (1 (1- 1- (xA)) - Eo (1))
(Z, M)t 1) below downweight possible extreme loss values induced by low censoring overlap via division by G3 (X,, A) in (Z, 17--1). However, in contrast to the R-learner, divisions by propensities π (X) still occur in the loss (sensitivity to treatment overlap).
Both the multiplication by G,-1 (X, () in Eq. (15) and the division by &a Cc(Z,
s-learner (survival overlap): Analogously to censoring overlap, we can also weight for survival overlap via (x)( ( (XX,,-(X(00)))) 0). That is, we down-weight patients who have a low treated or untreated survival probability beyond time - 1. This results in weighting results in
m( = (X-
(1
+ 1- (X)
1
Es(2,
(16)
(()
(A - {X))sss (Z, IX, A)
(( = - S,(X,0)
T (X)(1- xX)) (1
1 =(x) -
) Es (Z, m1-1)
which is less sensitive to divisions by S,( as compared the DR-learner loss.
Combined overlap types: is also possible to arbitrarily combine weighting to accom- modate different overlap types simultaneously. This results in the following learners:
T+C-learner (treatment-ccnsorrngg overlap): [x)( =(X) (X)
T+S-learner (treatment-survivall overlap): fX)) = - (xx
T+C+S-learner (all): /tg../a.........................
C+S-learner (censoring-survival overlap): - = 55..0
= ==g__p_|__e00..............0001-
Choice of learner/ weighting function. The choice of weighting function depends on the type(s) of overlap we would like our learner to be robust for (similar to choosing between DR- and Relearer in standard causal inference). In practice, we recommend using the estimated nuisance functions to inspect overlap (e.g., by visualizing the propensity score, censoring, and survival functions).
7 Experiments
We follow pest-practice in causal inference literature (e.g., 88 [3] and perform experiments using synthetic and real-world data to demonstrate the effectiveness of our toolbox to different types
WWe denote the Survival-B R- Rearnrr as T-learner to make the connection to the different weighting schemes explicit.
8

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No viotation
Propensity
Censoring
Survival
No violation
Propensity
Censocing
Survicl
0
1.36
+ 0.25
4.56 2:75
3.41 + 2.54
2.70 1.53
8
1.64
0.19
1.11 0.31
0.61 0.15
4.54 0.36
T
1.86 + 0.72
194 + 1:90
3.37 2.52
2,66 = 1.59
1.12 1: 0.41
0.55
:: 0.15
0.75 4: 0.2
6.77 +: 1.8
0.82
0.64
347
1.91 1: 0.46
0.98 0.27
0.60 a: 0.144
4.82 1: 0.46
8
199 0:85
5.11
2.92
2:72
0.87
T4C
1.95
+ 0.82
4.19 + 2.20
1.87 = 0.50
5.02 = 1.67
7+0
1.40
0.29
0.65 0.18
0.74 1: 0.23
4.52 0.60
T+8
202
+ 0:85
4.30 + 2.28
2.97 2,40
2:74 4: 1.56
748
3.55
1: 1.13
0.56 +: 0.14
0.71 0.19
9:23 1.27
2,10
0,90
5.91
0.76
2.83
0-8
0.70
0.86 0.31
4:
5.38 0.7
T+0-8
200 0.88
1.86 0.58
2.73
T+0+8
0.32
Table 2: PEHE in Scenario 1: Mean and stan- dard deviation of PEHE averaged over the first time steps across 10 runs. Targeted learners per setting (column) in gray background. = Overall, targeted weighting improves performance.
Table 3: PEHE in Scenario 2: Mean and stan- dard deviation over all assessed time steps across 10 runs. Targeted learners per setting in gray background. = Again, targeted weighting gener- ally improves performance.
of overlap violations. We instantiate all models with the same neural network architectures and hyperparameters. This allows us assess the effect our proposed weighting scheme, as differences in performance can be merely attributed to the different orthogonal loss functions for training the second-stage model. Implementation details are in Appendix !I
Synthetic data. Data generation: We consider two different data generation mechanisms:
Scenario I considers one-dimensional confounder and sigmoid propensity and hazard func- tions across five time steps. Scenario follows [8] by generating 10-dimensional multivariate normal confounders with correlations across 30 time steps. From each scenario, we generate multiple different datasets, in which we introduce propensity, censoring, or survival overlap violations or combination of them. All datasets consist of 30,000 samples. For details, see Appendix
30
CiD
S/D
Tea (T4
IT+SVT
(T+SUT
(T+
FEHE Ratio
20
(T+C-SUT--5
2.0
(T-C+SUC+5)
26
(T+C-SVIC-S)
10-
0.5
0.5
Time Stop
Time Stop
Time Stop
Figure 3: Benefit of targeted weighting over time. Ratios of PEHE of the targeted learner wrt. the learner without the correct target (data scenario I across 10 runs). Blue: Low censoring overlap scenario. Green: Low survival overlap scenario. Red: Low treatment overlap scenario.
Results: We evaluate the performance of our various orthogonal learners based on the precision in the estimation of heterogeneous effects (PEHE) with regard to the true CATE [17]. We compare the PEHE for scenario 1 (Table 2 and scenario (Table 3) across different types of overlap violations (i.c., no. treatment overlap, censoring overlap, and survival overlap violation, respectively). For better comparability, report the PEHE x10-1 Across both scenarios, observe that the learners targeted for the low-overlap scenario achieve the lowest PEHE. Furthermore, targeted weighting reduces the estimation variance but unsuitable weighting
harm performance.
In Figure 3 we further show the benefit our targeted weighted learners terms of PEHE ratios at different time steps. For data scenarios with low censoring survival overlap, we observe decreasing benefit of our learners over time after an initially equal performance of all learners. This in line with our expectations: (i) At timestep 0, no censoring or time-dependent survival hazards present. Thus, the respective survival- and mansoring-overlas weighting does not affect the prediction performance. (ii) The benefit of the targeted weighting reduces with increasing timesteps due to increasing hazards, decreasing sample size with 7 > t, increasing hazards, and thus decreasing effect of (f) (X). (iii) Treatment overlap is independent of t. Therefore, the benefit of learners targeted to low treatment overlap is constant over time.
Medical data. Data: We perform a case study on the Twins dataset as [3] to showcase the applicability of our learners to high-dimensional medical data. The dataset considers the birth weight of I 19844 pairs of twins born the USA between 1989 and 1991 with respect to mortality in the first year of life. Treatment 1 corresponds to being born the heavier twin. The dataset contains 46

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confounders, We provide more information on the data and preproeessing in Supplement detailed description of the dataset, see
|| For
CA510
Tec30
2003
Figure 4: Twins: 10- and 30-day Figure Validation loss for Figure 6: Validation loss for 30- effects across 10 runs Esti- 10-day survival: Fastest con- day survival: As in Fig the mated survival effects align with vergence for the C- and T+C- C and T+C learners converge the literature. Variance decreases learners, indicating the presence fastest, reproducing the former for weighted learners.
of censoring and treatment over- finding. lap violations.
Results: We analyze the effect survival after 10 days and one month. We observe generally negative effect on mortality, i.c. a positive effect on survival, for being born heavier (see Fig. 4 This is in line with the literature and medical results [e.g.. We also observe increasing effect over time, i.e. from 10 to 30 days, which again is in line with medical domain knowledge. Overall, we observe lower estimation variance for the T+C (for ceatment-censoriing overlap) and the C+S-learner learners (for cansoring-survivee overlap). This suggests the presence of multiple forms overlap violations, especially censoring overlap, in the data, and underlines the necessity of our survival learners targeted for specific overlap types.
The benefits of our proposed orthogonal survival learners show when inspecting the validation loss during training (Fig. [): The C- and C+T-leamers show by far the fastest convergence, whereas survival-overlap weighting in the S-learner hinders fast convergence. This affirms that suitably weighted learners are close to the data-generating process, enabling fast convergence, which brings important benefits for estimation in sample regimes common in medical applications. We note that high learning rates are likely to result in oscillating behavior on the loss, as the reweighting can be initially unstable to optimize. Therefore,
follow best practice and employ low learning rate together with the optimizer [25), which adapts learning rates over time and may switch from small to larger gradients, which can thus result in larger drops in the validation loss.
8 Discussion
Limitations. We observed that appropriately targeted weighted learners achieve better estimation performance in terms of PEHE and improve convergence speed. However, in practice, we recommend carefully assessing the necessary weighting before applying our toolbox, as inappropriate weighting. i.e., overlap weighting even if there sufficient overlap, can significantly slow down convergence. We recommended plotting the estimated weighting function (e.g., propensity or censoring overlap) as a measure "trustworthiness" (or uncertainty) of the model predictions. Predictions in low-overlap regions may then be discarded or deferred to domain experts, while the model predictions in large- overlap regions benefit from weighting and may be leveraged. Finally, we note that complete overlap violations can still lead to unstable training and result in a high variance of the estimate, is common with weighted orthogonal learners.
Broader impact. Our toolbox has a crucial impact on HTE estimation in personalized medicine. Time-to-event data is common in medical settings, but frequently suffers from cansoring-induces censoring and survival overlap violations. Our toolbox offers way to ensure reliable and stable treatment effect estimation in such settings.
Conclusion. We proposed a toolbox for constructing custom-weighted orthogonal survival learners estimate HTES from time-to-event data. Our learners can be constructed in a model-agnostic way, are semi-parametrically efficient, and ensure stable training in the presence of treatment, censoring, or survival overlap violations. As result, our work makes an important step towards reliable estimation of heterogeneous treatment effects in survival settings.
10

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Acknowledgements
We thank Lars van der Laan and Jonas Schweisthal for helpful discussions.
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20

---

A Extended related work
Semiparametrie inference and orthogonal learning: The concept of Neyman orthogonality is deeply rooted in semiparametric efficiency theory (22) 49 Neyman-ortin noymannorthooonll and efficient- influence function- baseed estimators have a long tradition causal inference, primarily for the estimation of average treatment effects. Examples include the AIPTW estimator [37], TMLE [48). the doubleML framework [5). and doubly robust policy value estimation (2). Recently, the concept of Neyman orthogonality has been extended to HTEs (1T), which allowed the construction of various orthogonal learners, including the DR- and R-learner for conditional average treatment effects [2
46 32 33
Efficient average treatment effeet estimation for time-to-event data: Most work on semiparametric
inference for time-to-event data has focused on average treatment effects (ATEs). For example, (38)
proposed so-called doubly robust censoring unbiased transformations for semiparametric efficient inference on the ATE under censoring. Furthermore, various works proposed one-step Targeted Maximum Likelihood Estimators (TMLE) for estimating average causal quantities in survival settings
50 56
21

---

B Background on influence functions and orthogonal learning
In the following, we provide short background on efficient influence functions and orthogonal learning. We mostly follow Kennedy [22].
Efficient influence function (EIF). In statistics, estimation is formalized via statistical model
{IP E P). where P is family of probability distributions. We are interested in estimating a functional :: P + IR. For example, (P) E[S((X,1) S,(X, 0)] (X), If yb is sufficiently smooth, it admits the so-called von Mises or distributional Taylor expansion
(bP / ((t. P)(() (P,
(17)
where (I, P) is a second-order remainder term and @(t, P) is the so-called efficient influence function of 1b, satisfying J @tt, P) and f (tt, dTP)( < oc.
Plug-in bias and debiased inference. Let now P be estimator of IP and (FP) the so-called plug-in -ing-in estimator of (P). The von Mises expansion from Eq. (17 implies that ((P) yields first-order plug-in bias because
( +(P)
= (PP
ott, (tt, PP) dP(() + (2(
(18)
due to that f ott, IP) dP(t) = 0. In other words, simply plugging the estimated nuisance functions into the identification formula can result in a biased estimator.
simple way to correct for the plug-in bias is to estimate the bias term from the right-hand side of Eq. (18) and add it to the plug-in estimator via
A-II ((P) Po(o(T,
(19)
this estimator is called (one-step) bias-corrected estimator.
Debiased target loss. One-step bias correction generally only works for sitee-dimensional target quantities (e.g., average causal effects such as [X)]). In this paper. however, we are interested in the HTE Te (X), which an mnfnnte-dimensionll target quantity. Hence, direct one-step bias correction is not applicable. The EIF can nevertheless be used for obtaining "good" estimator of the HTE. The idea that, instead of de- biasiig the HTE TE (X), we can de-bias the target loss that aim at minimizing. This leads to orthogonal versions of the target losses, which is precisely what we derive our paper.
22

---

C Estimation of nuisance functions
Here, we discuss the estimation of the nuisance functions, that is, the propensity score (x), and the hazard functions X5 (ze a) and X a).
Propensity score. The propensity score P(A = 1 X = x) defines a standard binary classification
problem. Hence, any standard classification algorithm (such feed-forward neural network with softmax activation and cross-entropy loss) can be used for propensity estimation.
Hazard fuucction. We can estimate the hazard functions via maximum likelihood. We can write the (population) likelihood as
P(T = T 55 = 5 1= x, =
(20)
7-8
II (1- a)) (Z a) gr(2, 0r 00 (2-090)00--09) ==
(21)
where (3 a) denotes the (conditional) probability mass function of C. Hence, parametric models 2 (xt, a,, 0) parametrized by (e.g., neural networks) minimize the resulting the resulting log-likelihood loss
(0) =
log (1 - X3 (2i 0)) + 0, log (V (Ii, (i:i:::;0)) 0))
(22)
Analogously, 29 (x;, ag, 8) can trained to minimize
T 6
(0)
log (1 X4 (ri ai 0)) +000
log (Ag 589log
(x(()
(23)
23

---

D Extensions to marginalized effects
We now discuss an extension to marginalized effects, i.e., the case where interested in the causal estimand
(1e)=P(T()) > t / V u) P( > VV==)
(24)
conditioned subset of confounders V C X. This can be relevant in many applications where certain confounders are not available during inference time (called runtime confounding [6] or
should not be used (due to, e.g., fairness or privacy constraints). Under Assumptions 3.1 and 3.2 identification holds via
T(() = [[+
(25)
For our framework in the marginalized case, we consider weighting function f (V)). where the weighting only depends on the marginalized propensity score (V) = (=(V) = P(4 = V)).
The orthogonal loss is given by
(g m²)
E (p(2m)
(26)
Of (V))
where p(Z, m)) is the same as in Eq. 7 and
plz m.) = ((T((V)) + Of (V))(A-T(VV)
(27)
Ôn The following theorem states that this actually targets a meaningful weighted loss. Theorem D.1. Let 9 = arg mingeg Lg (i) be the minimizer of the orthogonal loss from Eq. over a class of functions G. Then, 9ÿ also minimizes the oracle loss
266
gEg (V(I B. [IGN(V) (TVV) (vx(V)
1
9j arg min
(28)
Hence, 9e Tt. for any weighting function J as long as EG.
Proof. We write the orthogonal loss as
1
()(g.m)
E
(Z (6(2, - -
(29)
E[f(m((V) (V))
1
(((, m)) - (v( + - +v((V)
(30)
(V)))
(V)( (E (B[p(2.n) (p(2, m) ((()))
(31)
+1E (2(Z, me) (z(ZZ))) ((V))
(32)
(T. (-2B me) (10(2, m.) - (V))) oV))
(33)
+ [p(2,n)) -(()))]])
(34)
where only the terms in the last equation depend on g. The first term we can rewrite as
IE [p( mt) (v((Z, mt) - g(v))
(35)
- 0; (A (4-=00)))51222255544444401 -
-E
p(Z, m²) 1,((x1)-ss(x,,)
g(V)
(36)
-E
E |,
(37)
[AA [ (2)))))(( g ((V)) X A|
E
EE
(38)
x[X) xX))
plz, m )V)
(E (x,(X,1) - (.()))s((x,))) 0) - (A (X)) (Z. m))
(39)
X, (X) )(V)
E
(40)
. xX)((1 (X))
0,
(41)
24

---

where (+) follows from Lemma
H.2 For the second term, note that
E (z, no) V]
(42)
of
E
(((((V)) + θn
((())(A-
(43)
-|v]
(44)
=((i(V)) + of (V)) -VV))
(45)
2 (()(V))
(46)
where (**) follows from Lemma H.2
Hence,
(() (T.(V) - vl]
IB (TV)
= p n))
(47)
((C - -((()) r]]
= E
(48)
= (T(V) -
(49)
Putting everything together, we obtain
Ly(g, = (V))
(+( To (V))
(50)
+ [p( Z, me) (cp(Z, -
(51)
(V))) E [(m (T((V) -gV))
,(v))²]
(52)
which proves the claim because the first two summands do not depend on g and do not affect the minimization.
As consequence, we obtain the following two learners (which are, to the best of our knowledge, novel).
Marginalized survival DR-learner (no weighting; in our taxonomy: marginalized (-learner). Here, we set f (Vi) = 1, which implies p(Z, 17.) = and
Lpulg, mp) = (10 (()(Z,))) - -
(53)
with
(( -x (A (x= A)
(54)
x(X) - (XX)
Equivalently, this can be written as standard DR-learner
(A - +())
(( = (.( - x(X)(1 - x(X)
(Y())) - S,(,, ((XX)))) A))
(55)
with the transformed outcome Y Y(n) - S,XX A1 - (Z(,,)))
Marginalized survival R-learner (marginalized treatment overlap; in our taxonomy: marginal-
ized T-learner). Here, we set (X)) = (V) which implies p(Z, m)) (A- (V))² The orthogonal loss becomes
[[(((V((1 [[V)]] (v((, (c(2,n)-9(V) -
(A (A- (V))2
CR(9, n.) = E
(56)
where
((Z, = 1, - -S,x) - w
(57)
with
w(X) = ((V(( - क (V)) (A- x(X)
(58)
*(X) - -10)(1-((1)) (X)) (A-
25

---

This is equivalent to minimizing the R-learner loss
Lnlg.m) E (o((m)-AArile )
(59)
where A(m) = A =VV and
(nn) (()) + A[(1 1111-wa + - 1),:(X(0)]]
(60)
- (x,) + (w(X)- - (()))((X)))
(61)
Note that this coincides with the standard survival R-learner for V XX as this implies w(X) 1.
26

---

E Extension to the continuous-time setting
Our survival learners also extend to the continuous-ting setting by making two key changes: (i) we have to estimate the hazard function in a different way, and (ii) we have to add a small modification to our weighted orthogonal loss.
(i) Hazard functions in continuous time. In continuous time, we can write
(z, = P( = A = 2 2,, =1,
(62)
X = =
(63)
2t x
P(T = A
= X = 2, 2, =
(64)
Σxz - a) =
(65)
(66)
and analogously
(z, = P( =t =1 AC = =1 = =
(67)
Σxzr = x x
Hence, we can estimate the hazards by estimating the conditional probabilities PPT A 1, = = = a). PAAS = X = A a), and P(T"= = X = I, A a) for all t. This can be done by using standard classification algorithms such feed-forward neural networks with softwax activation and cross-entropy loss. As an alternative, one can impose parametric assumptions such as the Cox model, done in (21).
(ii) Orthogonal luss. For the second stage, we can use the same loss as in Eq. but where we define
Es( 1( st,
57(X, L' 1( 1) A) A)G, A)
di
(68)
as well
1 i) A) di. J. 5, A)
Ea n.)
(69)
Here, the integrals can be approximated using a numerical integration method.
27

---

Extensions to further causal estimands
F.1 Treatment-specific survival probability
We aim to construct (weighted) orthogonal learners to estimate the survival probability
Pa (2) = ((T(a) > X =
(70)
specific for a fixed treatment A = a. We consider a weighting function f (x)( that depends on the treatment-specffie nuisance functions 7)e ( a), Gr-1(X, a)), where Ta (z) (PA = a X = z). Following the same derivation as in our main paper, we first define the corresponding weighted average treatment effect via
0 f.ttf (T))) (x))
(71)
One can show that the efficient influence function of Ou,es is given by
Po(Z, ma)
(2.1,
(Pa(2, m1) 0o,,))
(72)
(X))
where
Pa(Z, f()(()) Of
(T) a) =a)-m((X})
(73)
OTo
1(A = a) (0)(7((X))
St (X_1(X, (X,a) 11
(74)
(X)
005--1(;,0)
(X))
G. (X, a Eo Za. m-
+ (G.-1(;a)
(75)
and
La (Z, m1) a) 1(A = 0)5. (Zu. Ta (a(X)polZ,
(76)
and we used the notation Zu = (X,a, a, T, A A C In particular,
Es n1) I( i, A 1) -
(77)
S(XX,)))_1(X.aa)
The orthogonal loss is given by
Cialg, nh) = E[(((())
Pa (P. -(())²
(78)
F.2 (Restricted) mean survival times
For some h 5 maax. we consider
max
(x) = E(T(a) A X= 2
Σ
(79)
t=
1=0
and
h
F (σ) = Ah -h- -1) Σ St 1) Se(x,, Σ (x)
(80)
Our weighting function now depends on h and is defined via ( (X)). To derive the orthogonal loss, we define the averaged causal quantity
Els(i (A))
fo Elf (X)) (X))
o
(81)
ElS (X))
E (X))
28

---

Using additivity, the efficient influence function is given by
(r.)(2 = p(Z, m))
(x))) (m(2, - O).1):
(82)
where
Th) (X)m((x) + 0f Ôπ
- (m (X)
(83)
((X) + 1 T (x) )
(as(m((X)
(8s--1(,,A) (1,,s(X,))s(,,
(84)
(f (X)) Gh-1 (X, (2, (2,
(85)
8G -1: A)
and
h Σ S, S) (A 11-
mhZ
(86)
-())(1=(
1=0
The orthogonal loss is thus given by
m))
(X)) E
(87)
29

---

G Extensions to unobserved ties
We assume now that A is unobserved and we only observe A A In this case, it holds that
2 (z. =) P(T - = -2 = P = t. - | 2
(88)
(89)
and thus the censoring hazard is not identified from observational data as it depends on the unobserved T and C' through P(T C T x. z, A a). In the following, we proposed two methods that still allow us to apply our toolbox in this setting.
Method
The term -
quantifies the conditional probability of having tie at time I. Given the independent censoring assumption, we may assume this probability will be small if T is sufficiently large, i.e., if we observe fine-grained time steps. Then, we can approximate
(x, P(
A 0 2 a=
(90)
and we can apply our orthogonal loss from the main paper with A = A. In the extreme case where T continuous, equality holds and we can ignore ties altogether.
Method 2
Here, we reparametrize our orthogonal loss using identifiable nuisance functions even if A is not observed. First, we define the survival function of the observed time
Pe(z,a) IPT X I, A A= a)
(91)
and note that Pt( a) = S1( a)G,(z), a) because T = min (T,C) C} (independent censoring assump- tion). Then, define nuisance functions
nh (X) = (X) U (A 1), (x, p. (X, (X 0))
(92)
Note that the correction term Es( L, nh) from Eq. 110 can now be written as
2 Σ
1(T = 1) 1(π 2 i)
Es Mt))
(93)
P A) - X3 A))
We consider weighting function [(i where (X) (iT (), P-P-1 (X, xX11) 1),p:-((X,0)) The efficient influence function of the weighted average treatment effect 0z.s is given by
p(Z. 77)
(x)( (p(2,
- 0,,))
(94)
where
p(Z, m)) (X)( + 0f
(7e (m(
(95)
-(A
A
1 1 )
(01((((1)
(Xx
+ 1 क (x)
(0n-1(;4)
(96)
(X))
and
= S)( -)(x)))
(97)
(
The orthogonal loss becomes
Lg nh.) (Tr
p(zz n.) (1 -
(98)
The orthogonal loss in Eq. (98) requires us to estimate (a a) instead of the censoring hazards (f( a). Estimating Pr( a) for all t is standard problem of estimating discrete conditional c.d.f. For example, one could fit a multi-output MLP with softmax activation and cross-entropy loss to
estimate PTT = X = x, A = a) and then (I, a) = - Ei P(T = X x, A a)-
In the reparametrized nuisance setting, we obtain the following orthogonal survival learners:
30

---

(i lllaarner (Survival DR-learner) corresponds to setting f()((X)) = 1;
(ii) T-learner (Survival R-learner) corresponds to setting (()(()) - (1-x(11 (())),
- TT
(iii S+C-learner (survival-censoring weighting) corresponds to setting f(ii(
Pt- P (X,1)pt-1 (XX,t-XX,000000 0); and
(iv) T+C+S-learner (full weighting) corresponds to setting f(ī((X)) = ax)(1
x ((X)) (X,1)xx-1(X,0))) 0).
31

---

H Proofs
H.1 Identifiability of the target estimand
Lemma H.1. Under Assumptions 3.1 and
3.2
the causal estimand Ty Tx))) is identified from the
observational data Z via
(3)
II (a, ())
II (z,
(99)
1=
Proof. We show w.l.o.g. identifiability for pa a) = P(T((a) > t X =x). The result for (x)
follows by taking the difference. It holds that
P(T(a) > + x P(T(() X =
(100)
P(T x ==
(101)
II - P(T = 7,
(102)
(88)
P( = i,
A=
(103)
i=0
(2,a),
(104)
==
where (i) follows from ignorability and treatment overlap, (ii) from consistency, and (iii) from non-informative censoring, censoring, and survival overlap.
H.2 Proof of Theorem 5.1
We make use of the following lemma.
Lemma H.2. Let (s(7,, n.) and Ea( 1-- be defined as in the main paper. Then,
E LEss n)) X, A] (Eo (- A] =0.
(105)
Proof. We have
1(T 1, = 1 1) () AA
[[ss(Z,)) X.
Σ E
X,A
(106)
S, A
1=0
P(T i, = 1 X(A) 2 i X, (X)
(107)
S,(X, (X,
=1 - - P(T i, X, A)
= (108)
S (X)
The result for (Z T)-1) follows analogously.
Now we turn to the actual proof of Theorem 5.1
Proof of Theorem 5.1 By taking the first-order directional derivative
II
we obtain
DyCr(g, 1g 9]
(109)
E (x)) dt [
(o()) -(x))))]]
(110)
-2
E [p([ Z, m)) (p(Z. -(X))
(111)
(X))]
32

---

-2
(S, -00 x(x))
(112)
(X)]
(A- (X))))((XXX)) H
(113)
x(x)(1 - ((x))
-2
[((( (()) - - (X() 1-4
(114)
(X)]
((x) 11(X, ()) 1-x(X) S,(x,0)ss(2,n)
(115)
A (0f(((x)) St (151(111)552m
0f(
(A x(x)) - *(X) (0St-1(-1)
G (x)1(())
8G--(511)
(116)
(1)) ) (0,
(0 S (xs,) (X))
8G.-1(-00) G G (1)6
(117)
-xx(,))) ((x))}
(118)
To show orthogonality, have show that the second-order directional derivatives w.r.t. to the
nuisance functions are zero. We start by computing the derivative w.r.t. the propensity score, resulting in
mi) - 9. - n]
(119)
d |DCf(g
"e.li x f( - =)>(g- - glle=o
(120)
di
-2
E (0(X) x(X))((XX) x(x)) { (1G((T) (m(A) 1.(X,1)6s(24
(121)
B[ (X))
1 1-
(122)
(1 (1-x (X))²
(f()((X))
()(X, - -S((X - -(X)- - A
1-A
1)) +
S((X, ())
O
*(X)
1- x(X)
(123)
0 (X)
(A - x(X)) (x))
(124)
pT
A
(x)
a8,-1(;1) S - (x, 12) + (x))
aG.-1(;1) Gx-1(X,1)6o(2
(125)
T(²²²
A
8 (7)
82 f
S._
(2.
0r0G G G (X,
(126)
+X)
P7081-1(.1.
1 1-4 A
(f(m
(X)) Gi- (X, 1)6
(aS--(.,1) (,,1 S._ (X, (Z.v))
(127)
(1 - (X))²
1G,-((,1)
1-4 1-4
)
( (()(() (Z) + 0TōG,-1(-.0)
0 (x)(
1- (
())00 n--1)
@T0SS-1(50)
(128)
(S.(X, - (x x(X)))
(129)
2D,C, n)) 19 9] (x)))
(130)
(x))2
dx
-2
E (0(X) g(X))(((X) x(X)) {
(x)) (S.(X, S,(,0) 9(X))
(131)
(X))
δT
(X)) (5.(X, 0((X,0) (00x000-((300 [((1))]
+
(132)
On
0,
(133)
where in (x) we applied the law of total expectation and Lemma H.2 to remove all terms including any of Es(( ),) Ea (Z, 1) or 4-m +(X) and the same argument to show that DyCy(g. me) - g) This shows orthogonality w.r.t to the propensity score.
33

---

To show orthogonality w.r.t. survival and censoring hazards, note that
k
Daj Dx8(,a)s((X,0)))) - = 11 s -
- II11- - a))
xx( a))
#Fj
(134)
- 15(X. ()) -x5(X.) 1 k
= -
1 1- (X,
(135)
and also that
21 -2 (-,a)
(136)
1( (;
-x(((x) a)) Σ
(137)
1=0
+ [( = - 1 1)9 11)
(138)
(X)
(1f = 1 102 -1 (X,
1(
(139)
1(T2))
Es( ne)
= -x)((x)))- ax)
(140)
() 1- xx(
Here, it holds that BlEs ZZ. m/) X, A = 0 following the same arguments as Lemma
H.2
For the cross-term, obtain that
D Mt- (. - (3)
(141)
1(T = i, i, =1) -,)==,,)-)- ()) 1 1 1
=())(x,a)- -xx(x.)))((xxx))
(142)
1=0
(Eo(Ze: 1) 1- (X, a)
=(}5(Xxa)) - -xx(xx)))2$(Xx.1) a))
(143)
with (Z, 7/- 1)- x, A| 0.
Using these calculation we show orthogonality w.r.t survival hazard functions via
D DeL 9. - (. d - (00
(144)
99] 1=
(145)
-2 (X)]
(aS(T(()))
E (0(x)
-
x(X)
(x.((.1)
(146)
A
+ (x) s((x,0)ss(2.m)) + +
1-4
- 1)
(147)
T(X)
1-
A.
-S, 1))
-1 + Ez(Zn.
+ 1X),)
(148)
*(X)
1x),)
(X,
1- (x,
8 (X))
(A- x(X)) A
0 (()(
(x-1 12,s mc)
(149)
(x5(1 1)āT
T(X)
D2S 1)a5,-((11))
(X))
SS 1) (Z, 1-
- [Za 1)
+ (x1,
(51)
1 - (x.1)
1 (X, 1)
(150)
34

---

a. 1);G,1((,1) (), m-))+ (x))
8 fl
EalZa,
0G,-1(:11) Gx-1(X,1)
(x,
(151)
-5((x,))-((x,0)) 5((X,0) - -(()))
-
(152)
2D 16 9 [)((((()))
(153)
(Ag(-.1)
-2
(6(X) - -xx
(s(
(S((X,1)- - -(((,,))-s(x,,))--- o()))
(X))
(,
(154)
1( A
-1( 2 1)
f(F(()))
1((X1))
(155)
1 1-X,(X11) (XX)
1G,,
(x))
21)
)
}
(x,1)
-xx
=(X) (8,-1(;1)
(((x11)6,_((xx111
(156)
-2
[(0(X) - (((x))(())((())) (xx))(2 1x)
(157)
(X))
(x))
(x.- (S. 56(X,0) 2(x)
(158)
(17(,1) (5.-1(.1)
1) + (x
(T-x((X.)) )
(()(()))
(159)
1 1 1X),
(X)) OS
- (6(xx,))-s(x,0)) - -x(x,,)) o(x3) }
(160)
(5,-1(11) (1)
= 0,
(161)
to remove all terms including any of n-1). or A - + (X) and the same argument to show that DyCy(g, g] We can apply an analogous argument for (.
where (+) we applied the law total expectation and Lemma H.2 (Z, nc). Es
n.)) (2,
(-,0) 0) and obtain
3 g. (3.0) -((())))((.))) D Dafc.o) PgC (g
(162)
DgCylo .. x5( + -f - 9) = di
d
(163)
(164)
which proves orthogonality w.r.t. survival hazard functions.
To show orthogonality w.r.t. censoring hazard functions. note that
GxX. -1),)))))))))))))))))))(((())()) - ((99 - 25(x,a)) 1(j 1 ()Gr(X,a))
(165)
1- (x)
and also that
Dxgraaflllan n- -9((,
(166)
1(T25) =x) -x9(xx))))())) (-o, [_][(X,)))__((x,_) (1
Ec( 11-1)
(167)
1 19 (X)
with EiEa(Za,77i1) X, A] 0 following the same arguments as Lemma
H.2 Furthermore,
( Es(Za, n1)
Dzg(.a)Es(Za, [3 (- 2(,))) (,, 19(x,)))
(168)
1-
with E(Es(Z...))) X = 0.
35

---

Hence, for the second recond-order directional derivative w.r.t. censoring survival functions, we obtain
D 16 g. -.
(169)
d ...
s1 1=
(170)
di
(O((((()) (.9(.1)
(0(X) - s x(x) (x.)
x(x)
(171)
A
(15
+ 1- ((x) S. (0)))))((..)))
1-1
(172)
+(X)
A
(s((x.1) ( (1- (-
Es(Za: m/))
+ ((m(())
(173)
+(X)
(X))
(A (A- xx-x(x))--- A
8 [(T)(())
+ (.
S. (X, n-11)
(174)
=(X)
(.
(x)
S 1) Es 1/-1))
8 ((T(())
+
G1-1(X,1)6o (175)
08.-1(-1)
1 19(X,1)
(.9(11)000--1
((()(())
C (2, (2,
-1(T 2
Ez 1/-1)
+ GX,1(X,11)
8G:-1(;1)
1- 1x(
(((X.1)5)-1XX11)
1- 19 1)
(176)
- -0(()) }|
2D 2D.Lr(g.Tb)e 9l B
(x))
(S) - -)
(177)
(x))
(,
2 (x)(
(0(X) (G0X)- ((00))(xX)((xXx111 -x1
9(X))
(19(111
(178)
A (X))
1
]
Gz-1(X.1)
T(X) 0G,-1(:,1)
(179)
2 (x))
(G(X) -xx))
(s(T(()) (x.-)(X.11
(S((X, - St(( 0) - -((X))))((X))
[GG-1(;1) (.9(:11
(180)
(X) 1x11
(s.(X. (6(x,1)-s(xx00)) -x(x)) -
}
(181)
0Gr-1(,11) 019(,1)
(182)
where in (+) we applied the law of total expectation and Lemma H.2 to remove all terms includ- ing any of (Z( no). (Z, nt). mn(2, A (X) and the same argument to show that D Lg) m) gl = O. Finally, we can apply the same line of arguments to Xy (,,0) to show
oo DyCr melo - d
(183)
dt (.cc(o)...,,,(6,,0))- -1)0))) g) 1=
(184)
-0,
(185)
which completes the proof.
H.3 Proof of Theorem 5.2
Proof.
We write the orthogonal loss as
(g = 1
E [p(Z. mx ( o(x))²
(186)
(X))
36

---

(X)] TR ( -
(187)
(o(2.))) ((())) (x)]))
(X)]]
(188)
(X)] (-2B m/) ( (T. 9(x))²])
(189)
where only the terms in the last equation depend on y. The first term we can rewrite as
E (p( m)) (p( x())
(190)
=B p(
15(x,1)-5:(C)
(A - (20 (1X) - (X)) m.)
Tx(x)) s(x)
(191)
E
(A - n (
x(x) x, A
(192)
x(x)(1 - (x))
(A n (()) m.) X, ( ()F(((()) 5(x)
E
(193)
x (x][))
0,
(194)
where (+) follows from Lemma H.2
For the second term, note that
E [p(ZZ ne) X
(195)
EE ((m(1)) + of Ôt
(M(X))(A (100((--(x)) x(X))
(196)
(X) 1xx(Xx) )
A
+ 1- A
(a
(aS ()
(197)
(x)) G GI (x_1(X, (L_ |
(198)
(G,-1(,A)
(X)) + 0s (7x (x)) xx())
(199)
0π
x(X) (X) S (X IX, A 1]
(200)
=(X)
a8,_1(,11
(x))
G.-1(X, (1-
(201)
0G,-1(;1)
1 (Xx)
(0)(m(())
( X, A
(202)
7(X)
( (.0)
(x))
G.-1 (X, (x- |x, 01
(203)
0G,-1(;00)
(X)(,
(204)
where (x) follows from Lemma H.2
Hence,
E p2z nd) (T((X) - s(x))]
= E E ((Z (T(()) - 9(x))² | |x]
(205)
= E E [((T(X)) (Tx()) - -((x)) [x]
(206)
= E ( -((x))
(207)
Putting everything together, we obtain
(208)
+2E [p(Z, m)) (v((Z, (()))
(209)
37

---

Blf (X))) E (X) (Tx( 9(x))
(210)
which proves the claim because the first two summands do not depend on and do not affect the minimization.
L
38

---

Implementation details
L1 Data generation
Synthetic data generation: We generated two different scenarios for our experiments on synthetic data, from which we each generated four different datasets with full overlap, low treatment overlap, low censoring overlap, and low survival overlap, respectively. For Scenario 1, we sample a one-
dimensional confounder from a standard normal distribution and set T' = 5. Then generate the propensities as well as the censoring and survival hazard for the full overlap setting as
<xx 0.50(z) + (-20r-x))
(211)
X ( a) 0.50 +
(212)
(2, a) (z5 t);
(213)
where denotes the sigmoid function. For the low overlap settings, we then replace (I)) X (1, a) and 25 (a, a) by
π(x) = (2x)
(214)
X (x,a) = o(1.6(x + t))
(215)
(x, = a-aa// (t 1)),
(216)
respectively.
For the more difficult Scenario 2. we follow 8 and sample a ten-dimensional standard normal confounder. In this scenario, the full overlap setting with T = 30 is generated
T (x)
Σ (x)
(217)
2 (xx a) = 0
(218)
0.10 -
0.5 (Ex)))
1 i 10,
(x = 0.10 10 Σx ))
(219)
t> 10.
To generate the different low-overlap settings, we then update the functions by xxx = a (3xx)
(220)
2 (x(x))) = 0.1
10
Σx x
+ al
(221)
0.
- -0.5
Σx
a + 1 xx x2))00)))) 6 10,
(9 a)
(222)
0.10
10
Σx
100.5+11022000000 01)). 1> 10,
respectively. From all datasets, generate 30000 train samples with train- train-validann validation split of 0.6 and 3000 test samples. For Scenario I, we evaluate our model on time steps to 5, and for Scenario on time steps 0,5,10, and 15.
For our medical case study, we employ the Twins dataset as in [31]. The dataset considers the birth weight of 11984 pairs of twins born in the USA between 1989 and 1991
Twins data:
Censoring
Survival
Probabliity
with respect to mortality in the first year life.
Treatment = corresponds to being born
the heavier twin. The dataset further contains 46 confounders associated with the parents, the pregnancy, and the birth. For a detailed descrip- tion of the dataset, see [3T]. We follow the pre- processing steps and censoring mechanism in g to create the time-to-event outcome.
50
120
150
200
250
700
350
Time (days)
Figure 7: Survival and censoring curves over time averaged across all samples.
In Figure 7 we assess the survival and censoring overlap averaged across all samples. Overall, there no sign of systematic survival or censoring overlap on average, i.e., the mean-aggregated curves across all covariate realizations are bounded away from zero.
39

---

1.2 Implementation:
All experiments were run in Python and run on an AMD Ryzen PRO 6850U 2.70 GHz CPU with eight cores and 32GB RAM. Our experiments can be easily computed on standard computing resources. We provide our code at https://gittub. cwm/m /tt.:///ooorrrrrrooorrrrraareroe
Model architecture and parameters: Throughout our experiments, we instantiate all models with the same four-layer neural network architectures and hyperparameters. This allows assess the effect of our proposed weighting scheme, as differences in performance can be merely attributed to the different orthogonal loss functions for training the second-stage model.
For the synthetic datasets, our propensitiy networks consist trained over 10 epochs with batch size 64, learning rate 0.0001 dropout factor 0.1. Our hazard networks were trained without droput across 40 epochs and with batch size 256. Finally, the second- stage models, consisting of 64 neurons per layer, were trained without droput across 30 epochs with batch size 64 and learning rate 0.00001.
hidden neurons per layer and
For the real-world dataset, we adapted the training of the propensity network to 20 epochs and the training of the hazard networks 120 epochs. We increased the hidden dimension of the second stage model to 32 neurons per layer and wich was trained over 300 epochs with learning rate 0.000001. All other parameter specifications remained as for the synthetic datasets.
40

---

Additional experiment
J.1
Ablation study
Here, we provide additional empirical results by evaluating our method across varying sample sizes using synthetic Dataset with low censoring overlap (see Tab. +.. We report the results for the C and the T+C learner. PEHE results are averaged over the time steps to the power of 10 As expected, the variance increases with smaller sample sizes, but overall, the proposed learners remain robust to changes in dataset size.
Table 4: Results by learner and dataset size (PEHE + standard deviation).
Learner
50% dataset
100% % dataset
150% dataset
C
0.2541 + 1.0848e-3
0.0836
8.1418e
0.0571 + 3.0890e
S
0.0722 + 3.1591e-6
0.0677
1.0696e 5
0.0535 + 1.0176c -5
T+C
0.2227 + 1.2506e-3
0.1100
2.2593e
0.0637 + 4. 4.6489c -5
T+S
0.0720 + 3.0045e-5
0.1602 1.0559e
0.0533 + 1.0353e
J.2 ADJUVANT dataset
The ADJUVANT trial 30 55 enrolled 171 patients with EGFR-mutant stage II-IIla non-small cell lung cancer (NSCLC). Baseline characteristics such as age, sex, and smoking history were recorded, and comprehensive sequencing of 422 cancer-related genes (including CDK4 and MYC) was performed. The trial's primary objective was to compare the efficacy of adjuvant gefitinib versus chemotherapy.
While previous analyses of the ADJUVANT trial relied on linear survival models, we demonstrate how nonlinear survival learners provide new insights. Specifically, we observed more pronounced effects on discase-free survival (DFS) time in patient subgroups with co-alterations TP53 and SMAD4 (average subgroup effect of 0.2087 vs. total ATE of 0. 081882), co-alterartions TP53 and NKX2-1 (average subgroup effect of 0. 0.1997) or who are of 65 years and older (average subgroup effect of 0.2052) across all learners. Additionally, patient subgroups with co-amplification of TP53 and SMAD4 (average subgroup effect of 0.2075 vs. total ATE of O. 1870) or of old age (average subgroup effect of 0.2040) showed a benefit in overall survival (OS) time when treated with gefitinib.
41


Reviews:
[{"review_id": "y7Uz7OOu96", "reviewer": "Reviewer_27xr", "summary": "This work proposes a toolkit of orthogonal causal learners for estimating heterogeneous treatment effects (HTEs) from censored time-to-event data in survival analysis. The authors generalize existing survival learners by introducing a flexible weighting function that recovers known estimators (e.g., survival DR- and R-learners) under specific settings, and yields novel learners that improve robustness in low survival and censorship overlap regimes. The proposed learners are orthogonal, model-agnostic, and perform well empirically across different censoring and overlap scenarios.", "questions": "(From Strengths and Weaknesses section)\n\n* **Reweighting justification:** How does the proposed reweighting help with HTE estimation in regions of low overlap, where variance is high and information is scarce? Why not just restrict estimation to covariate regions with sufficient overlap?\n* **Overlap tradeoffs:** Does downweighting low-overlap samples lead to biased estimates in those regions? How should practitioners interpret the resulting HTEs?\n* **Estimand relevance:** Is estimating $\\tau_t(x)$ for times $t$ with low survival or censoring probabilities meaningful in practice? Why focus effort on these regimes?\n* **Plot survival/censoring curves:** Can you plot the survival and censoring curves in the twins study, as you recommend practitioners do, to help contextualize the overlap?\n* **Figure 4 formatting:** The caption for Figure 4 is unreadable. Could you fix the formatting?", "limitations": "The authors have satisfactorily addressed the limitations of their work in Appendix J.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n\n* The paper is technically correct and well-executed.\n* The real-world application (twins study) is somewhat compelling and helps ground the proposed methods.\n\nWeaknesses:\n\n* The work is quite derivative. It largely builds on [50] by adapting their approach to survival settings and generalizes [30] through a flexible weighting scheme. The theoretical contributions regarding these general orthogonal learners are fairly limited besides identification (i.e. do the orthogonality benefits extend to these more complex estimators as expected?)\n* The reweighting strategy aims to reduce overall estimation error by downweighting regions with low survival or censoring overlap, but this comes at the cost of discarding information in those regions. Estimation there becomes essentially extrapolation, and the paper does not clearly justify why one shouldn’t simply restrict estimation to covariate regions with sufficient overlap and acknowledge limitations elsewhere.\n* It’s unclear whether estimating $\\tau_t(x)$ for times $t$ where survival or censoring probabilities are low is even meaningful. These functions are monotonically decreasing in $t$, so the problem of poor overlap worsens over time. In contrast, treatment assignment overlap is fixed over time and more interpretable. It seems like a lot of work to enable estimation in regions that may not matter in practice, especially since [50] already handles treatment-overlap and [30] generalizes this to non-survival outcomes.\n* The authors recommend plotting survival and censoring probabilities to assess overlap and guide estimator choice, but do not do so themselves in the twins study. This would be helpful to support the empirical claims and demonstrate overlap issues in practice.\n* **[Minor]** The caption for Figure 4 is unreadable due to a formatting issue and should be corrected.\n\n**Conclusion:**\n\nOn one hand, the paper is technically solid but the contribution feels incremental, i.e. it is another application of the orthogonality hammer. On the other hand, I can see how the proposed learners, with their flexible weighting schemes, may be practically useful for researchers dealing with survival data in low survival and censorship overlap regimes, if such settings are as common as purported. Since I lack domain expertise in survival analysis, I’d like to defer to other reviewers regarding the practical significance of the contribution. For now, I lean toward weak reject, but I’m open to revising that score in the rebuttal period.", "was_revised": true, "final_justification": "After reading the rebuttal, I am satisfied that the authors have addressed the main concerns raised in the initial review. Below is a summary of key points that informed my updated recommendation:\n\n* **Novelty of contribution**: The authors clarified how their overlap-weighted orthogonal learners generalize ideas from existing literature to the survival setting and potentially to other causal inference settings. While orthogonal learners themselves are not novel, adapting orthogonality concepts to this setting with multiple sources of low-overlap is sufficiently novel and practially useful. \n\n*  **Reweighting rationale and interpretation:** The authors provided a convincing explanation of why downweighting low-overlap regions is appropriate, along with the advantages over hard clipping and standard plug-in approaches. In particular, my worry was that these additional overlap issues worsen with increasing $t$ due to drop-out, so I was wondering whether these methods are useful over just cutting off the population that drops off early. The authors have addressed this point and I am looking forward to seeing this discussion in the camera-ready version.\n\n* **Clinical and empirical relevance:** The rebuttal included compelling examples of clinical relevance and real-world scenarios where survival or censoring overlap is an issue (the twin study). I was unable to review the overlap-over-time curves in the twins study, but I trust that the authors' description is accurate and that these curves will be included in the camera ready. \n\n* **Commitment to improvements:** The authors have promised several concrete revisions in the camera-ready version, including plotting survival and censoring curves, fixing formatting issues, and clarifying theoretical assumptions and practical guidance. I have revised my score under the expectations that these will be executed as promised.", "rebuttal": "Thank you for the constructive feedback. We will take all your comments and suggestions to heart, and we will make improvements to our manuscript accordingly. Below, we provide answers to all the questions and concerns you raised. \n\n**Changes to our paper.** In the camera-ready version of our paper, we will make the following key changes: (1) We will report plots of the survival and censoring curves in our real-world study to help contextualize the overlap for the reader. Unfortunately, the rebuttal guidelines prevent us from directly showing our revised plots during the rebuttal period. Where possible, we provide tables below, which will then convert to figures and be added to our paper. (2) We will fix the formatting issues in Fig. 4 (thank you for pointing this out!). \n\n**Novelty.** Thank you for giving us the opportunity to elaborate on our contribution. While it is true that we leverage tools from existing literature to derive our learners (e.g., orthogonal learning), the following contributions are novel and non-trivial:\n\n1. To the best of our knowledge, **we are the first to extend the idea of overlap weighted learners beyond propensity scores.** The survival setting is special as it allows for different sources of overlap beyond treatment overlap, but we suspect that such overlap-weighted learners could be beneficial in various other settings such as IV or causal mediation settings that also yield different sources of overlap. Hence, our paper potentially paves the way for substantial method innovation in causal inference.\n2. The mathematical derivations of our learners are non-trivial as they required the construction of the efficient influence function for the overlap weighted averaged target quantity (see derivations in Appendix H).\n3. We provide a **complete toolbox of learners as well as various extensions** (e.g., to continuous time, marginalized effects, ties, different estimands) in the appendix. Our results are readily applicable for practitioners in e.g., medicine, where accounting for censoring is crucial.\n\nAdditionally, we would like to emphasize that a **large part of the literature in causal inference proposes influence-function based estimators** in different settings and thus leverage existing tools to derive novel estimands [e.g., 2, 3, 4, 5]. Our novelty is thus in line with established works in that we leverage existing mathematical tools to derive novel results.\n\n**Reweighting justification:**\n\nThank you. We would like to clarify that **overlap-weighting does precisely align with your intuition:** in low-overlap regions, estimation becomes indeed extrapolation and hence predictions are downweighted to not disturb the overall estimation. You may think about this as an adaptive version of your idea to “restrict estimation to regions with large overlap” via e.g., clipping. **Advantages of our learners over clipping include:** (1) there is no need for choosing a cutoff value, which is otherwise often done heuristically, and (2) the weighting term is considered in the orthogonal objective, which makes the learner more robust to estimation errors in the weighting function. Clipping on the other hand can be insensitive to errors in e.g., the propensity score. (3) In contrast to clipping, weighting is a smooth operation which allows for deriving an influence function and thus enables orthogonal learning in the first place (pathwise differentiability).\n\nIn standard causal inference, the R-learner can be seen as a propensity-overlap weighted learner, and is widely applied in practice. Our work enables applying such learners in survival analysis.\n\n**Action:** We will include the discussion above in the camera-ready version of our manuscript.\n\n**Overlap tradeoffs:**\n\nThank you for this important question. First, note that downweighting low-overlap samples **does not introduce additional population bias** for the weighted estimand if the model is correctly specified (see Theorem 5.2). In finite samples, overlap weighting might increase MSE in low-overlap regions as these regions are essentially \"sacrificed\" to reduce overall estimation variance. However, **this is deliberate and similar in behaviour to established causal inference methods**. For example, the same applies to fitting an R-learner in the standard (non-survival) setting, or fitting a CATE model using propensity clipping. In practice, it is always recommended to plot the weighting function (e.g., propensity or censoring overlap) as a measure of “trustworthiness” (or uncertainty) of the model predictions. Predictions in low-overlap regions may then be discarded or deferred to domain experts, while the model predictions in large-overlap regions benefit from weighting and may be leveraged.  \n\n**Action:** We will include the discussion and practical guidelines on how to interpret the results in low-overlap regions in the camera-ready version of our paper.\n\n**Estimand relevance:**\n\nThank you for this thoughtful question. First, we would like to clarify that **the benefit of our method is not restricted to settings with *overall* low survival or censoring overlap.** Instead, overlap weighting can be **beneficial even in settings in which a small subgroup of the population is affected by low overlap.** For example, in a medical study, older participants may have a large probability of dropping out within a short time after the start of the study (low censoring overlap). Even if the majority of the population is younger, the small subpopulation of older people may yield a large variance when applying the standard DR-learner due to divisions by extremely small censoring functions (see e.g., Eq. 13). Overlap weighting focusses estimation accuracy on the younger population, as this is the population for which we have available outcome data and for which we can obtain causal insights.\n\nAdditionally, based on our collaborations in medicine, we frequently encounter settings where estimating τₜ(x) with low survival or censoring overlap is both common and clinically relevant.\n\n**(1) Clinical relevance:**\n\nEstimating τₜ(x) despite low survival overlap is important in many oncology settings. For example, the rapid adoption of new targeted therapies in advanced non-small cell lung cancer recently led to immediate uptake in some subgroups, resulting in limited data for others. Similarly, low censoring overlap may arise when treatments are discontinued due to toxicity, often linked to very heterogeneous reasons such as co-medications for some patients and genetic biomarkers for others. In both cases, the oncologists from our collaborations seek to generate clinically meaningful evidence about the treatment effectiveness while carefully addressing low survival or censoring overlap.\n\n**(2) Empirical relevance:**\n\nLow treatment overlap can reflect clear clinical decision rules and may be addressed through changes in the research design (e.g., excluding patients before estimation). In contrast, low survival or censoring overlap often stems from structural limitations, such as small high-risk subgroups or informative censoring. Here, simple fixes to the research design are typically not feasible for clinical researchers because ignoring these regions risks overlooking treatment effects for certain subpopulations. Therefore, developing methods that remain robust under limited survival or censoring overlap is crucial for valid inference in such real-world settings.\n\nImportantly, our orthogonal learners aim to improve treatment effect estimation in survival settings (regardless of whether there is low overlap or not). It is well known in the literature that simple plug-in learners lead to so-called “plug-in bias” [1] and, thus, suboptimal estimation. A remedy is offered by our orthogonal two-stage learners, which are designed to prevent such plug-in bias. (ii) Existing learners lack the ability to address different types of low overlap and, as a result, exhibit a **large variance under low censoring or survival overlap**. As a remedy, our toolbox allows the specification of a custom weighting function for robust estimation under low treatment-, survival-, and/or censoring overlap. \n\nFinally, we emphasize that we also provide several **extensions of our toolbox to different settings and effects** (continuous time, marginalized effects, different causal estimands, and unobserved ties) in the **supplementary material**.\n\n**Survival & censoring overlap:**\n\nWe are happy to report the survival and censoring curves in the twins study to help contextualize the overlap. While we are unfortunately not permitted to include new figures showing the curves during the rebuttal phase (as per NeurIPS guidelines), we now report a summary of the curves. Overall, there is no sign of *systematic* low survival or censoring overlap on average, i.e., the mean-aggregated curves across all covariate realizations are bounded away from zero at all assessed times $t&lt;=30$. Across the complete study time (of one year), we observe low overlap towards the end of the study time amplified through various covariates such as tobacco usage throughout the pregnancy for the survival curve. Rare risk factors such as cardiac disease are associated with low censoring-overlap. \n\n[1] Edward H. Kennedy. “Semiparametric doubly robust targeted double machine learning: A review”. arXiv preprint (2022).\n\n[2] Nathan Kallus. \"What's the Harm? Sharp Bounds on the Fraction Negatively Affected by Treatment\". NeurIPS 2022.\n\n[3] Coston et al. \"Counterfactual predictions under runtime confounding\". NeurIPS 2020.\n\n[4] Syrgkanis et al. \"Machine learning estimation of heterogeneous treatment effects with instruments\". NeurIPS 2019.\n\n[5] Lan et al. \"A Meta-learner for Heterogeneous Effects in Difference-in-Differences\". ICML 2025."}, {"review_id": "TEGiiU9mwW", "reviewer": "Reviewer_QWg4", "summary": "The paper presents a toolbox of orthogonal learners for estimating heterogeneous treatment effects (HTE) from time-to-event data which supports custom weighting and compatibility with machine learning models. Theoretical and empirical results are provided.", "questions": "1. When referring to censoring, clarify this is for right censoring instead of left censoring.\n\n2. For main theorems, please provide a proof sketch in the main text.\n\n3. Section 5 mentions that the paper is the first work to explicitly use retargeting for HTE estimation in survival analysis. I suggest reconsidering this claim because the paper provides a unified perspective rather than the formulation of retargeting (i.e. weighted) estimation.\n\n4. How to construct targeted weighting for real world data that potentially violates overlap? \n\n5. L242: missing parenthesis.", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 2, "strengths_and_weaknesses": "The paper addresses the key challenge in time-to-event analysis with censored outcome. It is overall well-written and easy to understand. The authors use informative subtitles, which is helpful. The paper’s method is well-grounded in clearly outlined related work, with natural motivations. See the questions for weakness.", "was_revised": true, "final_justification": "I have carefully read the reviews and responses. I decided maintain my score (5: accept). During the rebuttal period, my concerns are well-addressed. I think this paper is technically solid and novel--estimating HTE in survival is highly nontrivial due to right censoring, and the paper provides a toolbox to address this challenge. I think this would be of interest and bring insights to the broader causal inference community.", "rebuttal": "Thank you for your positive evaluation of our manuscript and the helpful comments. We will take all your comments and suggestions to heart and make improvements to our manuscript accordingly. Below, we provide answers to all the questions and concerns you raised. Will will incorporate all points marked with **Action** into the camera-ready version of our paper.\n\n\n\n1. Thank you for pointing out the ambiguity related to the term *censoring*. Upon reading your comment, we realized that we should be more explicit that we deal with the problem of *right censoring*, which is very common in survival analysis settings. We thus assume that events have not happened before time t=0. \n\n    **Action:** We will update the wording to emphasize that we are focusing on right censoring in the camera-ready version of our manuscript.\n\n2. We are happy to follow your advice and also provide proof sketches in the main text, thereby fostering the reader's understanding.\n\n    **Action:** We will include brief sketches of all proofs in the main section of the camera-ready version of our manuscript.\n\n3. Thank you for this suggestion!\n\n\t**Action:** We will rephrase the respective parts in the text to emphasize that we are the first to provide a unified perspective on orthogonal HTE estimation in survival settings.\n\n\n\n4. This is a very relevant question. For practitioners analyzing real-world data, we recommend to estimate the propensity score as well as the survival and censoring functions to inspect potential overlap violations. Then, depending on the type of overlap violation found (e.g., low treatment and censoring overlap), an appropriate weighting combination can be selected, and an appropriate orthogonal survival learner from our toolbox can be trained that accounts for said overlap violations. Finally, in the prediction step we recommend plotting the estimated overlap weights as a measure of “uncertainty” to inspect whether the predictions of the model can be trusted. This overall procedure is consistent with best-practices in standard causal inference using the propensity score.\n\n**Action:** We will add a note stating the difficulties for HTE estimation on real-world data with complete overlap violations to the camera-ready version of our paper.\n\n\n\n5. Thank you. We will fix the typo!"}, {"review_id": "bs6v9n39U7", "reviewer": "Reviewer_Wcbe", "summary": "The authors propose a new framework for estimation of Heterogeneous Treatment Effects, specifically for Time-To-Event data. The key contribution is the framework's ability to better handle low overlap in all combinations of treatment, censoring, and survival via custom weighting functions. The framework is model-agnostic. The authors motivate their framework and situate it in the existing literature, thoroughly introduce the method, and evaluate it on synthetic data and one real-world dataset.\n\nNote: My expertise lies mostly in survival analysis and methods for time-to-event data. I am less qualified to rigorously assess the theoretical contributions related to causal inference and HTE.", "questions": "* Could you elaborate on the training dynamics in Figures 5 & 6? Is the long plateau followed by a sudden drop expected, and is it tied to the very low learning rate? Does this pose a practical challenge?\n\n* What are the specific disadvantages of using the combined T+C+S learner by default? Why was this learner excluded from the real-world experiment?\n\n* Can you comment on the expected performance on other real-world datasets to strengthen the empirical claims beyond the single dataset used?\n\n* As the estimated survival effects on the Twins dataset align with existing literature, what is the primary additional insight gained from using this framework?", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n* Clarity and Presentation:\n\n   * The paper is exceptionally well-written, organized, and presented.\n   * It clearly introduces and motivates the main challenges of HTE estimation with censored data.\n\n* Originality and Significance:\n\n   * To the best of my knowledge and understanding, the work is novel and addresses a clear and important research gap.\n\n* Theoretical Soundness:\n\n    * The theoretical derivations appear solid and well-motivated.\n\n\nWeaknesses\n\n* Empirical Validation:\n\n   * Compared to the very extensive theoretical contributions, the empirical validation feels lacking. For the synthetic datasets, only two settings are evaluated. It would e.g. be interesting to see how the performance of the method changes for datasets of different sizes. Furthermore, only a single real world dataset is evaluated, with the main result apparently being that \"Estimated survival effects align with the literature.\" with reduced variance. How does this compare to prior works based on this data? Why was this dataset specificaly and only this dataset evaluated?\n\n   * The training dynamics shown in Figures 5 & 6 appear problematic, with a long, flat loss plateau followed by a sudden drop.\n\n   * Likely related: The use of an extremely low learning rate (1e-6) for the real-world experiment seems impractical and requires justification.\n\n    * A more in-depth discussion of the trade-offs when selecting a weighting function and the drawbacks of always opting for the (T+C+S) learner would be helpful for practitioners: Surely almost all real world datasets have some amount of overlap of each case?\n\nMinor Points:\n\n* A typo (\"retarged\" instead of \"retargeted\") in Figure 2.\n\n* Some mathematical notations appear to have missing closing braces.\n\n* The discussion was moved into the appendix, likely due to the page limit. Even though it will probably be moved to the main text for the final version, the page limit exists for a reason and should not be disregarded by moving parts of the paper (same as for the extended related work section) into the appendix.", "was_revised": true, "final_justification": "The authors have thoroughly addressed all my concerns and provided compelling new results using real-world data. Following the discussions with the other reviewers, I am confident that no major unresolved issues remain in the sections of the manuscript where I had initially expressed less certainty.\n\nWhile I remain somewhat concerned about the training dynamics presented in Figures 5 and 6, the authors have adequately addressed this point in their rebuttal. Given that none of the other reviewers, who possess greater expertise with these models, have identified this as problematic, I no longer consider it a significant concern.", "rebuttal": "Thank you for your positive evaluation of our manuscript and your extensive comments and suggestions. We will take everything to heart and make improvements to our manuscript accordingly. Below, we provide answers to all questions and comments you raised.\n\n**Answers to weaknesses:**\n\n* **Empirical validation:**\n\n    Thank you for this valuable suggestion. In general, our choice of datasets was motivated to **show primarily the theoretical properties of our proposed learners** and by a practical limitation: many survival datasets do not include treatment information, and, as a result, only a small number of publicly available datasets are commonly used in the literature => our selection reflects the availability of such established benchmarks. Nevertheless, to address your comment, **we have strengthened our experimental evaluation as follows**.\n\n    First, we added new results evaluating our method across varying sample sizes using synthetic Dataset 1  with low censoring overlap (see below). Due to space constraints, we report the results for the C and the T+C learner, though we observe similar patterns for other learners as well. PEHE results are to the power $10^2$. As expected, the variance increases with smaller sample sizes, but overall, the proposed learners remain robust to changes in dataset size. We include a condensed version of the results here; we will provide a more comprehensive version (along with visualizations) in the camera-ready version.\n\n    | Learner     | 75% dataset         | 100% dataset          | 200% dataset           |\n    |--------------|--------------------|--------------------|--------------------|\n    | C             | 0.0974 $\\pm$ 0.0799   | 0.0677 $\\pm$ 0.0506   | 0.0330 $\\pm$ 0.0219   |\n    |  T+C        |0.1015 $\\pm$ 0.0819  | 0.0732 $\\pm$ 0.0535  | 0.0324 $\\pm$ 0.0218 |\n\n    Second, we chose established datasets from the causal inference literature, such as the Twins dataset (e.g., [1]), which has the rare property of allowing proxy access to both potential outcomes. This makes it ideal for validating our method. Importantly, we show that our method successfully recovers known effects from this dataset, which we regard as a strength to demonstrate the reliability of our survival learners.\n\n    Third, to highlight the practical utility of our learners, we repeated our analysis using the ADJUVANT trial from oncology [2,3]. We provide details below.\n\n* **Minor weaknesses:**\n\n    Thank you for noting the typos in the figure and the notation. We will address these issues in the camera-ready version of our manuscript. We apologize for not including an in-depth discussion in our main paper and will include all relevant parts of the paper in the main section of the camera-ready version.\n\n**Answers to questions:**\n\n\n* **Figures 5 & 6**: The training dynamics in Figures 5 & 6 are as expected for our weighted learners. High learning rates are likely to result in oscillating behaviour on the loss, as the reweighting can be initially unstable to optimize. Therefore, we follow best practice and employ a low learning rate, as noted by the reviewer. To counteract the low learning rate, we employ the Adam optimizer, which adapts learning rates over time and may switch from small to larger gradients, which can thus result in larger drops in the validation loss. **\n\n**Action:** We will include a discussion on the training dynamics in the updated version of our paper.\n\n\n\n* **Choice of learner:** Thank you for this interesting question. Always opting for the (T+C+S) learner is not optimal, as ‘unnecessary’ reweighting can decrease the convergence of the learners (as seen in Fig. 5 & 6). In the extreme case, the training can become very unstable. Thus, it is important to consider the overlap present in the data when choosing/designing the targeted learner. In practice, we recommend using the estimated nuisance functions to inspect overlap (e.g., by visualizing the propensity score, censoring, and survival functions) to aid in deciding on the optimal learner for the data at hand.\n\n    **Action:** We will discuss the benefits and disadvantages of the different learners and highlight a guideline on how to choose a good option for the data at hand in the camera-ready version of our paper.\n\n\n* **Expected performance on other real-world datasets:**\n\n    We initially chose the Twins dataset due to its widespread use in the literature (e.g., [1]), which allows for benchmarking against established findings. This choice was deliberate: by demonstrating that our survival learners recover known effects, we highlight the reliability of our framework, which we consider a key strength. A major benefit of this dataset is the ability to directly compare our results with prior work, and indeed, we observe that our method successfully reproduces these known treatment effects.\n\n\n    Building on your suggestion, we realized that adding a second dataset from a medical domain could further show the practical value of our survival learners in clinical applications. However, many clinical datasets for survival analysis are not publicly available, which hampers reproducibility. To address this, we specifically sought out a public dataset and chose the ADJUVANT clinical trial [2,3]. This trial, conducted in patients with EGFR-mutant non-small cell lung cancer (NSCLC), compares the efficacy of gefitinib versus chemotherapy (vinorelbine plus cisplatin).\n\n\n    While previous analyses of the ADJUVANT trial relied on linear survival models [3], we demonstrate how nonlinear survival learners provide new insights. Specifically, we observed more pronounced effects on disease-free survival (DFS) time in patient subgroups with co-alterations in TP53 and CDK4 or in NKX2-1 and CDK4 across all learners. Additionally, patient subgroups with co-amplification of NKX2-1 and SMAD4 showed a benefit in overall survival (OS) time when treated with gefitinib. Overall, we observe notable differences compared to the conventional linear models reported in [3], which can be attributed to the greater flexibility of our toolbox in handling nonlinear relationships. This improved accuracy may provide a more nuanced understanding of the relative benefit of cetuximab versus bevacizumab in metastatic lung cancer and thus may help in new insights for personalized cancer care. \n\n    **Action:** In the revised version, we will explicitly explain our motivation for using established benchmark datasets to demonstrate reliability. While we are unfortunately not permitted to include new figures for the ADJUVANT dataset during the rebuttal phase (as per NeurIPS guidelines), we will incorporate Kaplan-Meier plots and related visualizations in the final version to illustrate survival differences captured by our learners.   \n\n[1] Christos Louizos et al. “Causal effect inference with deep latent-variable models”. NeurIPS. 2017.\n\n[2] Zhong, W.-Z. et al. Gefitinib versus vinorelbine plus cisplatin as adjuvant treatment for stage II-IIIA (N1-N2) EGFR-mutant NSCLC (ADJUVANT/CTONG1104): a randomised, open-label, phase 3 study. Lancet Oncol. 19, 139–148 (2018)\n\n[3] Data access details: [https://www.nature.com/articles/s41467-023-41011-4](https://www.nature.com/articles/s41467-023-41011-4)"}, {"review_id": "dYQmTCyyQT", "reviewer": "Reviewer_TJGC", "summary": "This paper introduces a novel set of orthogonal survival learners designed to estimate heterogeneous treatment effects (HTEs) from time-to-event data, particularly in the presence of censoring.", "questions": "1.\tWhat is the domain of $T$? Does it only take discrete values? Can it be extended to continuous values, which are more common in practice?\n2.\tHow do you estimate $\\phi(z, \\eta)$ in (7)? What is the precise definition of the derivative in (8)? Does this involve estimating the derivative? If so, this might lead to a larger variance in the estimator compared to existing methods such as DR-learner or T-learner.\n3.\tYou claim in Theorem 5.2 that $g^*_f = \\tau_t$. However, since $\\tau_t$ is unknown and you estimate it by minimizing equation (12), can you provide consistency results for your final estimator $\\tau_t$? At least at the population level? \n4.   Moreover, the benefits of orthogonality remain unclear. It would be helpful to provide theoretical results on estimators (i.e.,  convergence rate) to demonstrate and support its advantages. \n5. After you get estimator $\\hat{\\tau}_t(x),$ which is a function of $t$ and $x,$ how do you determine the treatment effect?", "limitations": "No.", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "The authors derive orthogonal losses tailored to address treatment effect estimation for survival data. However, it will involve many complicate nuisance when using these orthogonal losses.", "was_revised": true, "final_justification": "I will raise the score.", "rebuttal": "Thank you for the in-depth evaluation of our manuscript. We will take all your comments and questions to heart and make improvements to our manuscript accordingly. Below, we provide answers to all the questions and concerns you raised. \n\n\n\n1. **Domain of T:**\n\n    Our framework applies to both **discrete and continuous** time domains. In the main paper, we focus on discrete time for ease of notation. **In Appendix E, we extend all derivations for the continuous time domain**.\n\n2. **Estimation in Theorem 5.1:**\n\n    Thank you for giving us the chance to explain the notation-heavy theorem in more detail. The function $\\phi(Z, \\eta_t)$ in Eq. 7 denotes the (unweighted) pseudo-outcome. It is estimated following Eq. 9 based on the nuisance functions $\\eta_t(X) = \\left( \\pi(X), \\left( \\lambda^S_i(X, 1), \\lambda^S_i(X, 0), \\lambda^G_i(X, 1), \\lambda^G_i(X, 0) \\right)_{i=0}^t \\right)$. \n\nThe derivatives $\\frac{\\partial f(\\widetilde{\\eta}_t(X))}{\\partial m}$\n\nfor $m \\in (\\pi, S_{t-1}(\\cdot, A), G_{t-1}(\\cdot, A) )$ in Eq. 8 denote the (standard finite-dimensional) partial derivatives of the function $f(\\pi, S_{t-1}, G_{t-1})$ w.r.t. one of its components. **These formulas can be computed analytically and yield closed-form solutions, without the need to numerically estimate derivatives.** As an example, we take a look at theT-Learner in Eq. 14: \n\nTo address the treatment overlap, we have $f(\\widetilde{\\eta}_t(X)) = \\pi(X) (1 - \\pi(X))$. In this case we have $\\frac{\\partial f(\\widetilde{\\eta}_t(X))}{\\partial \\pi} = 1-2\\pi(X)$, resulting in the simplified $\\rho(Z, \\eta_t) = (A - \\pi(X))^2$ according to Eq. 8. Note that the T-Learner (=treatment overlap learner) equals the survival R-Learner. As shown in Eq. 13, the survival DR-Learner is *also* an instantiation of our framework in Theorem 5.1. In conclusion, **the derivative in Eq. 8 does not lead to higher variance compared to e.g., the DR-learner, and is easily computable in analytic form.**\n\nThe (standard) T-Learner consists of two outcome regressions, each trading off bias for variance in its regression problem, which does not guarantee a joint optimal bias-variance trade-off (see, e.g., [1]). Our framework, however, provides statistically efficient learners and thus guarantees that *our learners have the lowest variance among all unbiased estimators* (see, e.g., [2]). Overall, the derivative in Eq. 8 does not lead to high variance.\n\n**Action:** We will explain the definition of the derivative more clearly in the camera-ready version of our manuscript.\n\n3. **Consistency result:**\n\n    Our theorem 5.2. states that the orthogonal loss from Eq. 7 has the same minimizer as the oracle loss from Eq. 11, namely $\\mathcal{L}_{f}(g, \\eta_t) = \\frac{1}{\\mathbb{E}[f(\\widetilde{\\eta}_t(X))]} \\mathbb{E}\\left[f(\\widetilde{\\eta}_t(X))\\left(\\tau(X)- g(X) \\right)^2\\right]$.\n\n\n    Hence, Theorem 5.2. immediately implies consistency of our learners: if the nuisance functions are consistent (i.e., well specified), the empirical loss in Eq. 12 converges in probability to the orthogonal loss in Eq. 7 by applying the standard law of large numbers. If $\\mathcal{G}$ is well-specified, this implies $g^{\\ast}_f = \\tau_t$ for the minimizer of Eq. 12 asymptotically.\n\n\n    **Action:**  We will state consistency more clearly in the camera-ready version of our paper.\n\n\n\n4. **Benefits of orthogonality and rates:**\n\n    Thank you for raising this important point. Theoretical aspects and particularly rates for orthogonal learners are well-studied, we refer to the foundational work “Orthogonal statistical learning” [4] by Foster and Syrgkanis for details. In particular, their work provides slow and fast rates for general Neyman orthogonal losses under a convexity assumption. In Theorem 5.1. we proved Neyman orthogonality of our loss, which implies that their rates are readily applicable to our setting. The additional key assumption of convexity (Assumption 4 in their paper) is also satisfied as we consider a weighted squared loss. Hence, their Theorem 1 shows that nuisance estimation errors impact the estimation accuracy of our second-stage learners only in second order, as it is common for orthogonal learners.\n\n\n    **Action:** We will add a discussion on rates to the camera-ready version of our paper.\n\n\n\n5. **Final effect estimation:**\n\n    Recall our definition of $\\tau_t(x) = \\mathbb{P}(T(1) > t \\mid X = x) - \\mathbb{P}(T(0) > t \\mid X = x)$ for some fixed $t \\in \\mathcal{T}$ (see Section 3). Thus, $\\tau_t(x)$ represents the difference in survival probability up to time $t$ for a unit with covariates $x$, which defines the survival treatment effect. Note that, in survival analysis, the HTE, which is our main effect of interest, does depend on both the covariates $x$ and the time $t$.\n\n\n    In practice, one might also be interested in related but different treatment effects. In **Appendix F**, we thus also provide **extensions of all our results to other causal quantities**, such as the conditional means $\\mathbb{E}[T(1) - T(0) \\mid X = x]$ as well as the treatment-specific quantities $\\mathbb{P}(T(a) > t \\mid X = x)$ and $\\mathbb{E}[T(a) \\mid X = x]$.\n\n[1] Morzywolek, P. et. al.. “On a general class of orthogonal learners for the estimation of heterogeneous treatment effects”. arXiv preprint, 2023\n\n[2] Kennedy, E. H. Semiparametric doubly robust targeted double machine learning: A review. arXiv preprint, 2022. \n\n[3] Victor Chernozhukov et al. “Double/debiased machine learning for treatment and structural 387 parameters”. The Econometrics Journal 21.1 (2018), pp. C1–C68\n\n[4] Foster & Syrgkanis. \"Orthogonal statistical learning\". The Annals of Statistics (2023)."}]

Author rebuttal:

Meta-review: This paper proposes a toolbox of orthogonal survival learners for estimating heterogeneous treatment effects (HTEs) from censored time-to-event data, featuring orthogonality guarantees, custom weighting functions to address low-overlap regimes, and model agnosticism, with both theoretical foundations and empirical validation.

During the rebuttal process, authors thoroughly addressed reviewers’ key concerns, including clarifying the novelty of extending overlap weighting to survival settings and non-trivial mathematical derivations, justifying the rationale and tradeoffs of reweighting versus hard clipping, supplementing empirical evidence with additional sample size experiments and a new real-world dataset (ADJUVANT trial), and elaborating on theoretical details such as consistency, convergence rates, and derivative definitions. All reviewers acknowledged that their concerns were adequately resolved, with all of them give final ratings positively.

The paper nominated for spotlight status presents a truly novel approach in the domain of causal inference. It not only fills a significant gap in the existing literature but also demonstrates remarkable potential for real-world applications. And the novelty, scientific rigor, excellent presentation and technical quality of the paper is highly recognized by the reviewers as well. It has the potential to attract significant attention and inspire further research in the field, thus the recommendation for this paper is acceptance.
Decision: Accept (poster)