Doubly Protected Estimation for Survival Outcomes Utilizing External Controls for Randomized Clinical Trials
Authors: Chenyin Gao
Shu Yang
Mingyang Shan
Wenyu Ye
Ilya Lipkovich
Douglas Faries
Venue: ICML 2025 poster (2025)
Keywords: Adaptive learning
Monotone coarsening
unmeasured confounding
data heterogeneity
Primary area: general_machine_learning->causality
TL;DR: We propose a doubly protected integrative estimator for survival outcomes that is more efficient than trial-only estimators and mitigates biases from external data.

Abstract: Censored survival data are common in clinical trials, but small control groups can pose challenges, particularly in rare diseases or where balanced randomization is impractical. Recent approaches leverage external controls from historical studies or real-world data to strengthen treatment evaluation for survival outcomes. However, using external controls directly may introduce biases due to data heterogeneity. We propose a doubly protected estimator for the treatment-specific restricted mean survival time difference that is more efficient than trial-only estimators and mitigates biases from external data. Our method adjusts for covariate shifts via doubly robust estimation and addresses outcome drift using the DR-Learner for selective borrowing. The approach can incorporate machine learning to approximate survival curves and detect outcome drifts without strict parametric assumptions, borrowing only comparable external controls. Extensive simulation studies and a real-data application evaluating the efficacy of Galcanezumab in mitigating migraine headaches have been conducted to illustrate the effectiveness of our proposed framework.

Doubly Protected Estimation for Survival Outcomes Utilizing External Controls for Randomized Clinical Trials
Chenyin Gao 12
Shu Yang' Mingyang Shan
Wenyu Wendy Ye² Ilya Lipkovich Douglas Faries
Abstract
event survival endpoints may not always be observed. As a complement to clinical trials, external controls offer a promising avenue to improve statistical inference when recruiting more patients is challenging. However, external controls can differ from clinical trials many aspects due to differences in the underlying data acquisition and generation mechanisms. Concerns regarding the plausibility of these assumptions have limited their broader deployment. Guid- ance documents from regulatory agencies, including the recent Food and Drug Administration (FDA) draft guidance on Considerations for the Design and Conduct of Externally Controlled Trials for Drug and Biological Products, note several potential issues with the use of external controls, including selection bias, lack of concurrency, differences in the definitions of covariates, treatments, or outcomes, and unmeasured confounding (FDA, 2001; 2019; 2023), Each of these concerns can result in biased treatment effect estimates if external controls are integrated with the trial without further serutiny.
Censored survival data common in clinical tri- als, but small control groups can pose challenges, particularly in rare diseases or where balanced randomization is impractical. Recent approaches leverage external controls from historical studies or real-world data to strengthen treatment eval- uation for survival outcomes. However, using external controls directly may introduce biases due to data heterogeneity, We propose a doubly protected estimator for the treatment-speciff re- stricted mean survival time difference that is more efficient than trial-only estimators and mitigates biases from external data. Our method adjusts for covariate shifts via doubly robust estimation and addresses outcome drift using the DR-Learner for selective borrowing. The approach can incor- porate machine learning to approximate survival curves and detect outcome drifts without strict parametric assumptions, borrowing only compa- rable external controls. Extensive simulation stud- ies and a real-data application evaluating the ef- ficacy of Galcanezumab in mitigating migraine headaches have been conducted to illustrate the effectiveness of our proposed framework.
2. Related Work
Data Integration with Non-survival Outcomes To re- liably leverage external data, it is crucial to address these potential issues with the use of external data. One primary concern is the distributional heterogeneity of the baseline disease characteristics between the two studies, leading to the issue of covariate shifts. Likelihood based frameworks have been explored mitigate covariate shifts arising from external datasets (Chatterjec et al., 2016; Huang al., 2016). Other propensity score weighting and matching methods have been proposed to construct new external data that have similar covariate distributions as the trial data (Li et al., 2018). However, these frameworks often rely on the invari- ance assumption, which posits that the conditional outcome distributions identical for the trial and external controls. This assumption can be problematie, as the distributions of outcomes may also vary across studies given the baseline covariates, leading to the problem of outcome drift.
1. Introduction
Understanding the risk of disease or death and how these risks evolve over time is critical for assisting clinicians in treatment assignment and disease diagnosis. In clinical tri- als biomedical studies, evaluating the effectiveness of drugs often suffers from limited sample sizes due to low disease prevalence or restrictive inclusion/exclusion criteria. This issue is exacerbated in survival analysis, as time-to-
'Notth Carolina State University, Department of Statistics, Raleigh, NC, USA Harvardd University, Department of Biostatis- tics, Boston, MA. USA 'Eli Lilly & Company, Indianapolis, IN, USA. Correspondence to: Shu Yang [0yang24@nnsu.cduu
In recent years, various adaptive learning methods have been proposed to address the study-specific generating mecha- nisms for outcomes, aiming to ensure robust estimation even when external controls differ significantly from the
Proceedings of the nnnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Doubly Protected Estimation for Time-to-event Borrowing
trial. The considered analytic frameworks include adaptive information borrowing from diverse populations in linear regression (Li et al., 2022; Yang et al., 2023; Gao & Yang, 2023), generalized linear models (Tian & Feng, 2023; Li et al., 2023a), and nonparametric classification (Cai & Wei, 2021).
tion equation (Gao et al., 2024a; Lee al., 2024) and the targeted maximum likelihood estimation (Rytgaard et al., 2022; 2023), have gained great popularity to draw inferences about the treatment effects for survival outcomes. There- fore, there is a pressing need for developing a flexible and data-adaptive integrative framework that accounts for out- come drift in time-to-event outcomes, coupled with valid inferential methods, enhance efficiency and reliability in survival model estimation.
Data Integration with Survival Outcomes However, the adaptive learning in survival analysis remains limited due to challenges in incorporating external time-to-event outcomes. Unlike regression settings, where models dynamically bor- row information for covariate effects, survival analyses need to address outcome drift in hazards as well. For instance, Liu et al. (2014) and Huang et al. (2016) proposed accommo- dating outcome heterogeneity by applying a constant factor to the cumulative hazard function. Additionally, Chen et al. (2022) and Wang et al. (2020) developed a propensity score- integrated Bayesian framework for the Kaplan- Meier (KM) estimator, which first selects external controls with similar hazard risks and then down-weights their impacts before incorporating them via the weighted KM estimator.
In this paper, we develop doubly protected estimation method for evaluating treatment effects for survival out- comes. To correct for covariate shifts, we utilize the density ratio of baseline covariates between two datasets in the con- struction of the doubly robust treatment estimator, motivated by the cemi-parametric EIF, Since our framework is devel- oped based the EIF, it offers an advantage over other pon-meraittric methods in the construction of confidence intervals.
Next, we recast this influence function into a selection-basee integrative framework to address outcome drift, identifying comparable subset of external data for borrowing. To ad- just time-varying hazards for survival outcomes, we propose to detect this subset based on differences in the subject-level restricted mean survival time (RMST) with DR-Learner By minimizing the biai-variance trade-off, our framework does not require stringent parametric assumptions on the survival curves and allows for the dynamic borrowing of external information with time-varying hazards.
Nonetheless, these approaches overlook the time-varying nature of the outcome drift for survival analysis. To ad- dress this, Chen et al. (2021b) developed an adaptive empir- ical likelihood estimation that incorporates constraints from external summary-level information to account for time- varying baseline hazard differences. Huang et al. (2023) proposed federated external control method to estimate hazard ratios in federated weighted Cox model for time-to- event outcomes. Due to privacy and logistical concerns with data-sharing, these frameworks are designed to utilize exter- nal aggregated survival information, which can be restrictive and less efficient.
Finally, we establish the asymptotic properties of our pro- posed data- adta-adeetive integrative estimator for survival out- comes with guaranteed consistency and efficiency improve- ment, even in the presence of external heterogeneity. Be- sides, we demonstrate the robustness and reasonable effi- ciency gains via extensive simulation studies and a real-data application. Our implementation codes will be made pub- licly available after the acceptance of this manuscript.
Furthermore, it is essential to control the information shar- ing between covariate effects and hazard risks simultane- ously. Li et al. (2023b) proposed transfer learning frame- work that allows for comparable information borrowing in both covariate effects and baseline hazards through penal- ized likelihood under Cox models. However, this frame- work lacks flexibility in modeling time-varying covariate ef- fects. Bellot& & van der Schaar (2019) proposed learning the shared representation between two populations via the flexi- ble nonparametric survival trees and correcting distribution mismatches with boosting, aiming to improve prediction performance without providing uncertainty quantification.
3. Methodology
3.1. Notation, Assumptions and Identifications
Following the potential outcomes framework, let T'(a) be the potential survival time if a subject received the binary treatment A = a. Let (t) and >,, (1) be the corresponding survival and hazard functions, defined as (t) = P(T(')
IV
t) and (t) lim h h < T(a) < t+h)/P(T(())
Our Contributions Existing integrative methods are lim- ited by the assumption of the Cox model, either on the cause-specific or subdistribution hazard scale, which re- quires to accurately model the survival curves. In recent years, semiparametric efficient and doubly robust estima- tors, which leverage the efficient influence function (EIF), including the methodology of solving the Els-b estima-
t). Under the consistency assumption, the observed survival time T is the realization of potential outcome under the actualized treatment, i.c., T = AT()) + (1 - -A)r()), In the presence of censoring, the survival time T is not always observable. Instead, we observe Y= min(T, C) and A =
1(T < C), where C' is the censoring time, and 1(.) is an indicator function. Let MC (dt X, r) dNC (t)
2

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Doubly Protected Estimation for Time-to-event Borrowing
1(Y 2 ( R be martingale with (1)
estimation into selective integrative procedure that first detects the biases and only retains a subset of comparable external data for integration (Section 3.3).
1(Y = 1, = 0, a), and ( X, R r) =
1 < C' <ttt
ax.A=a.R=r)
3.2. Efficient Integrative Estimation Assuming Population Homogeneity
Suppose we have two data sets: the trial data and the ex- ternal controls. Let be the baseline covariates and I 1 if the
be the indicator of the data source, where subject is from the trial data and R for the exter- nal controls. For the trial data, we observe R {V; (Y,, A., A, Xi, R = 1)}=r for the external controls, only = (V; = (Y; A. A, 0, x, R. = 0)) N are ob- served since no treatment is assigned. Denote the true dis- tribution for V; by P., and the empirical measure by PN as (N(f) = f(V)/N, where N = N₂ + Ne.
Under Assumptions 3.1 to 3.3, the average treatment effects 0r, or St R = 1) sufficiently, is identified based on the observed data. The following theorem provides the identification formulas.
Theorem 3.4. Under Assumptions 3.1 to 3.3, the follow- ing identification formulas hold for the treatment-specific survival curves Sa(t R [8=10 =1)
Let (X) = P(R- = 1 X), (X) = (x)(//11-
(a) Based on the trial data:
TR {x))) be the density ratio of the baseline covariates, (X) = P(A = X, R = 1) be the propensity score for the treatment, and (t R 1) P(T(a) 2 L R 1) be the ceatment-specifie survival curves for the trial population. The parameter of our interest 0, is the average treatment effect measured by the restricted mean survival time (RMST) difference up to т, defined by 0, = J (t R = 1) So = 1)]dt. To iden- tify the difference in RMST, the following assumptions are sufficient.
1
Sa(1 R R= 1) 1)
P(R P(R= E (RS [X, R 1))
(i(AAa)AA1p >t)
EF
P(R - P(A - X Xm (Y,
where of (1, P(C > X, 1) is the
censoring probability for the trial.
(b) Based on the external duta:
Assumption 3.1 (Internal validity for the trial data). (i) T(a) 1 A X, R 1 for 0,1; 1; and (ii) < A (X), (X) < 1 in the support of X.
Solt | R=1) 1)
(1 R)an(X)AI() >0
= P(R 1
E
Th (Y,
Assumption 3.2 (Informative censoring). T(a) 1 C X,A= =0=00. 0, R = r for a = 0, and r r 0,
where ~6 (1, = P(C 2 L X, 0) is the censoring probability for the external controls.
Assumption 3.3 (Comparabilityy for the external data).
So(t X)= (t X, R = 0) So | X, R = 1) for any < in the support of X, where St X, R)
Theorem 3.4 provides the identification formulas for S. (t R R=1). which is sufficient to identify the average treatment effect 6, among the trial population. In particular, Theorem 3.4(a) identifies St R = 1) based on the outcome impu- tation or the inverse probability censoring weighting with the trial data only; Theorem 3.4(b) uses the external controls to identify St R 1). As the covariate distribution of the external controls may not be representative of the trial population, the identification formula relies on the density ratio (x(X) to obtain the survival curves marginalized over the trial population. A detailed proof of Theorem 3.4 is provided in Appendix A.1.
((T(() 2 X, R):
Assumption 3.1 holds by the design of trial and is useful to detect the external heterogencity. Assumption 3.2 is a common censoring at random assumption for survival analysis, which is special case of the coarsening random (Tsiatis, 2006). Assumption 3.3 states the external data is comparable to the trial data if there is a rich set of covariates capturing all the outcome predictors that are correlated with the data source indicator R.
However, Assumption 3.3 is prone to violations in practice due to many blas-generating concerns, such as unmeasured confounding, lack of concurrency, and outcome validity. Our proposed framework is two-fold: 1) Under Assumption 3.3 where covariate shift can be present, we develop a semi- parametric efficient integrative estimator for the treatment effects evaluation using the combined data sets (Section 3.2);
However, the identification formulas provided in Theorem 3.4 could motivate infinitely many estimators for 0, under Assumptions 3.1 to 3.3. To construct more principled estimator, we derive the efficient influence function (EIF) of 0, in Theorem 3.5 based on the semiparametric theory (Tsiatis, 2006). The EIF, also known as the canonical gra- dient (Van der Laan et al., 2011), is a fundamental tool to achieve local semiparametric efficiency for estimation.
2) Considering the potential violation of Assumption 3.3 where the outcome drift is allowed, we adapt the efficient
3

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Doubly Protected Estimation for Time-to-event Borrowing
Theorem 3.5. Under Assumptions 3.1 to 3.3,
integrative framework can be generalized to any estimand that is function of the survival function (t) (e.g., the mean median of the survival time), with only a trivial algebraic extension, since the EIFs are derived for (t).
(a) the EIF for St | R = 1) is Us ott ott (V,V) = @s oft (tt V)- [t R= 1)/P(R = 1), where
We now give some intuitions behind the EIF Ss (t, V) for the combined data sets. First, we derive the full-data EIF efr (t, V) under Assumption 3.3, which involves a part being the weighted-averaged of the EIFs for the trial and external data with weights being inversely proportionately to the variances for the EIF's of the trial and external data, respectively. Next, we project the full-data EIF e to the coarsened data space induced by the censoring, and gives the observed- data EIF in Theorem 3.5(b); see Theorem 10.4, Tsiatis (2006). A detailed proof of Theorem 3.5 is provided in Appendix A.2.
RAI(Y > t)
eft, (t, V)=
P(R= (X-A-)))XX())) (1,
RA. (M (X) Si(l X, 1)
P(R = (X) (((r, (r, Si(r R-R=1)
RSI( (t) X,
A
P(R=1)
TA(X)
So. eff (f, V)
(b) the EIF for Solt | R = 1) is S eat (t, V) =
off (1, V)- - RSo(1 R= 1)/P(R = 1), where
Sy ell (t, V)= = R(1 A) (x =t)
However, constructing estimator based on the EIFs first requires approximating the unknown nuisance functions TR(X), т (X), St X), and SC X, It). By replac- ing the true distribution IP with its estimated counterparts and solving the empirical expectation IP, t eett (V)}, we have
IPRR = 1) of (t, Xt,
R1(1 - A) (R(x)dMM XX
1 P(R =
So(t X,R=11
So(r
X, R 1)
(1 - R) gR( 9x( |X, 0)
OpCw= N--N Σ 1 (( Vi)dt Ts (1, V.) di. VERUE
30 P(R=
(r(1;))((t,N)
So(t x=)
Under the conditions in Theorem 3.6, our proposed inte- grative estimator 0pcw is rate doubly robust, asymptotically normal, and locally efficient as established below.
So(r X
(x(X))(, z)
P(R=1 1)
T6. (1,(1,))((,
Theorem 3.6. Denote If = P(S(V)211/2 where P is the true distribution. Let xx()), Rx(X), Sa X, = 1), and #C X) be general remi-parametric models for T R π xX(), S, X, R = 1), and n. (t, respec- tively. Suppose Assumptions 3.1 to 3.3 and the regularity conditions A are satisfied, up to a multiplicative constant, the estimation error Oaew O, = err (Dach ,0,) is bounded by
Su(t| X,R
R. 1) {
( (AA-XX(AAAA(X)) xX(X))
P(R = 1)
X(t,
r(1, () (1 RX)
D(
X),
X)t, + {1 Ta(X)))n(X))
r(4, X) = VRI.AU/ Vio- VRI,A0
var (I(T > 1) R 1= = 0}, and Vno
err (0rC (,),,,,))))) = { (11o(t X, R 1) So(t X, R 1)1la, + (1 | X, R = 1) - (1 R= 1) }
var {1(T > 1) R=
(c) the EIF for 0, is:
of (V) = -f" (vs (t, - u.so.cn (t. jit
x | - =A + + - +x(X))l
|36 (r XX - 26 X, R))lZa }
+ max
rst
Theorem 3.)(a) shows the EIF for (t R = 1) with the trial data, which well-studied as the observed-data EIF un- der monotone coarsening in (Tsiatis, 2006); Theorem 3.5(b) shows the EIF for So(t R R 1). which is an extension of Gao et al. (2024a) with additional integral terms contributed by the censoring scores of concurrent controls and external data; Theorem 3.5(c) suggests that the EIF for 0, is an inte- gral of the EIFs for St(t) R = 1) as the average treatment effect 0, is an integral function of the treatment-specte specific survival curves (a(t) R = 1). Furthermore, the proposed
Thus, we have N1/2 (gacw - 0-) S N(0, V+), where N = N, + and V E E( oot (V)} is the semi-parametric lower bound for the variance.
Theorem 3.6 shows that the proposed estimator O, can incor- porate flexible nonparametric or machine learning methods for estimating the nuisances with the required convergence rates, while maintaining the parametric-rate consistency. The proof of Theorem 3.6 is deferred to Appendix A.3.
4

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Doubly Protected Estimation for Time-to-event Borrowing
3.3. Robust Selective Borrowing with DR-Learner for Outcome Drift Detection
Lemma 3.7. Let E(V) be the pseudo-outcome with un- known nuisance functions replaced by their estimated coun- terparts and E (V) ) be its probability limit, the conditional expectation [[E - -bo XX HH is bounded by
In practice, Assumption 3.3 is often violated, that is, (t X, R 1) * So X = 0) for some & [0, r]. and d, may be biased. Suppose there exists a comparable sub- set A C (1, (1,....,.. N of the external controls such that So X,, R = 1)= So X,, R = 0) for any time t and subject i E A, and we aim to selectively borrow this com- parable subset. However, the subset A is often unknown a priori. In this section, we propose a robust selective borrow- ing framework to incorporate comparable external controls in estimating the average treatment effect. We introduce vector of bias parameter by (b1, .0: .N..o)) where bi.0 = (t X, R, = 1)at- - S" So X; R. = O)dt. To prevent bias in f,. our goal is to identify the zero-valued subset of the bias parameter and leverage only this subset for the integrative estimation.
(x) - (x)))l1
x [| X, K.R=r) - - X, = x-)l
Σ | () (t,
x [|Su(t |X, -) - Solt
X, R = r)1|kz
+ [|a(()) - aX)
x ||So(t =x So XX 1)11
up to a multiplicatiie constant.
One simple estimator detect the bias is the "plug-in" es- timation, defined as by = b(V.) = fo So(l Xi, R, = 1)dt- So St X,, R. = O)dt. However, the "plug- in" estimator might have large finite-sample biases if one uses flexible models for the conditional survival functions. Heuristically, bias detection is equivalent to estimating the conditional differences in expected control means over two datasets, which parallels the conditional average treat- ment estimation in the causal inference literature when the study source indicator is perceived as the treatment indicator. To more accurately approximate the biases, the DR-Learner approach (Kennedy Edward, 2020; Kallus & Oprescu, 2023) is utilized to construct the initial pseudo- outcome & for the bias b,, by & = {(V)) = So ko(t, Vi R = 1)dt- - So (t, V R- = O)dt, where
The results in Lemma 3.7 show that nuisance errors will have smaller impacts on the pscudo-outcomes & as its esti- mation error is bounded by quadratic form of the nuisance errors. The proof is provided in Appendix A.4.
We now present our adaptive integrative framework in Al- gorithm 1. The cutoff value r for computing the RMST is crucial in practice, as the distribution of the tail beyond this point is neglected. Typically, the event rates at this cutoff value should exceed 10% to ensure sufficient data for model development.
In Algorithm 1, Step is a typical strategy to use machine learning techniques for estimating the nuisance functions; The penalty term in Step 2 is chosen to guarantee the se- lection consistency such that P(Á A) + 1, where = b, 0} is the estimated comparable set. For ex- ample, the penalty term p(-) can be the adaptive lasso (Zou, 2006), the smoothly clipped absolute deviation (SCAD) (Fan & Li, 2001). and the minimax concave penalty (MCP) (Zhang, 2010); Finally, Step 3 outputs the adaptive integra- tive estimator Gelups with the comparable set A by
(t, V, R 1)= = So(t X,, R= =)
R1 - (r R= Solt| X1, R= 1)
TR(X,)(1 - ( (x,)) (f Solr X1, R=
R1 - A,)
<x(X,((1 - (X,)]
1(Y =t) -So(t| X,, R= 1)
(1,
Garlep N-1 Σ
L' Ss. (t, Vi)dt
iERUE
and
N-1
Saf (t, Vi)at,
Ro V/ R 0) 0)= = St X,, 0)
VERUE
1 (1 - R.)dMM (r X,, 0) So
X,, R 0)
Jo {1 [1-Tn(X,) (X,), 1 Xt, So(r X,, R R= 0)
where the difference between & $o. (t, V) in Step 3 and @s (4, V) in Theorem 3.5 lies in the focus on leveraging the comparable set A instead of the whole external controls E for the integrative analysis.
(1 - R₂)
> 1)
(t X,, R X, - })
(X,) X.)
Lemma 3.7 provides bounds for estimation error for the pseudo-outcomes, which is the key step for bias detection.
Theorem 3.8. Let Vaiqu be the variance of the trial-only estimator. Under the same conditions Theorem 3.6, we
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Doubly Protected Estimation for Time-to-event Borrowing
Algorithm 1 Doubly Protected Adaptive Integrative Analy- sis for Survival Outcomes
X = (X,,, .xp)) (0,, 1xxp) with p = 3 for each subject i. Next, generate the data source indicator R by Bernoulli sampling in Table I, where R is chosen such that the average of R is around N where R = 1, the treatment A is generated by Bernoulli sampling:
Input: trial data R = (V = (Y, A, A, X;, R = 1) the external controls & = {V; = (Y,, Ai, A; =
For the trial data,
0, O,X, R, = > penalty function p(-), tuning
parameter AN, and the cutoff value T.
A X, R =1 ~ Bernoulli { exp(ax.. + 1IX) }.
4 Preparation
I exp((a + 11X)
Randomly split the data R E into two folds I and Σz.
where a, is chosen such that the average of A is around Ni/N, and N1 is the size of treatment group. We consider the following conditional hazard function Xa X, R 1) = cxp(=.50 - 1 0.2), and X X, 1) exp( 0.1 + Bc) for the trial. The time-to-event out- comes T and the censoring times C' are generated by invert- ing the survival functions induced by the hazard function (t X) and X X), respectively. The parameter Bc = 1 controls the expected censoring time for the trial data where the censoring rates P(C < '') is around 40%; additional simulations where the data is subject to a differ- ent intensities of censoring with various values of Be are available in Appendix B.
4 Step 1
Fit the conditional survival curves (t X, R r) and SC(t
on In for r 0, 1.
Fit the propensity scores (X)) and x (X) on I
Compute the pseudo-outcomes & by ko(1, V/ R 1) and (t, V2 R 0) on Iz-
4 Step2 2
Refine the ostudo-sutcomes by b, = arg minging,CC CG 6,)2 + Xxp((b+]).
Obtain the comparable set A = {i: b, = 0}.
Next, to mimic the bias-generating concerns raised by the FDA, namely, selection bias, unmeasured confounding, lack of concurrency, and different covariate effects and time- varying baseline hazards, we consider five simulation set- tings for the external controls, as summarized in Table 1. Under Setting One, the survival and censoring times for the external controls are generated using the exact same pa- rameters as Aa X) with fixed 0 (i.e., treatment). Under Setting Two, the hazard functions for both studies are confounded by an unobserved factor U' to maintain the same level of hazard variability across the two datasets. In particular, U ~ (O, 1) (zero mean) is included in the haz- ard function for the trial data, whereas + 1 ~ (1, 1) (non-zero mean) is used for the external controls, which is expected introduce greater outcome drift. Under Setting Three, the external controls are subject inconcurrency bias, where 8; takes values from {0, 5) with equal proba- bility 1/2. Lack of concurrency could occur when the trial data and external control data are collected during different time periods or under varying healthcare settings. Under Setting Four, the external controls have different covariate effects, whereas under Setting Five, they exhibit different baseline time- vnre-varying hazards. Such discrepancies may arise if the two populations respond differently to the treat- ment (or placebo), even when they share the same baseline covariates.
4 Step 3
Output the adaptive integrative estimator Oettupe
have /112 (gudapp
0.) S. (O, Vindupr), where
Tn (o(X)r(t,X)P(C) = =0 0 0) P(R 1)3 (1
Vradon
Proips
Di (t x(t,X)
Doo X), rr (t(t VRI.AO.
where Di Xt, - (t, X) = x) (1- T xX(X)]9 and r- X) = AR. /io: AR are defined in Theorem 3.5, and the modified variance terms VRI,AO and Vio are defined in Appendix A.5.
Theorem 3.8 highlights the benefit of selective incorporating external controls, where the asymptotic variance of Gelaps is strictly smaller than the variance Viniw of the trial-only estimator unless the external study is in extremely poor quality (i.e., rt., X)= = 0) or the comparable external subset is empty (i.e., P(bg = 0 | X, R = 0) = 0). A proof is provided in Appendix A.5.
4. Simulation
In our evaluation, we compare the proposed selective bor- rowing estimator Oidlpp with the trial-only estimator Ozipw (Tsiatis, 2006), the naive full borrowing estimator Peew the propensity score-integrated estimator (jeawee (Chen al., 2020), and the transfer learning Cox regression @aanscox (Li
In this section, we conduct several simulation studies to eval- uate the finite-sample performance of the proposed selective integrative estimator. The sample size for the external con- trols are fixed at Ne = 500, and the parameter of our interest the difference in RMST with 7 2. First, we generate
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Doubly Protected Estimation for Time-to-event Borrowing
Table 1. Summary of considered three simulation settings
Bias
Setting
Details
Covariate shift
R | X ~ Bernoulli
explorgro
Selection bias
exp( x1
An(t X, R exp(-11 -0.2)
Outcome drift
R X ~ Bernoulli
Unmeasured confounder
1+exp(aR. o +,X++ -
(a X,U, = exp -0.5a 17x 0.2 0.2 (()
R X ~ Bernoulli 1 expp(R 1,,)
explerg.+11XX
Lack concurrnncy
(t X, R) expl-550- 1,, -02 35
0.2
(a
X, 1) = exp( -50 11,
Different covariate effect
Aa(1
X, R = 0) = exp( - 1- X 0.5)
Different baseline hazard
Aa(t
X, R 1) expl 5a 1 11x-0.2 0.2)
do(t
X, R = 0) = 2t exp -1 0.2)
et al., 2023b). The penalty term p(.) is chosen to be the adaptive lasso (Zou, 2006). The conditional survival curves (a | X) and S (t X) are modeled by the Cox model, and the propensity scores (X) and (X) are modeled by the SuperLeermerr with the logistic regression and random forest as the base learners (Van Der Laan & Rubin, 2006).
highest powers for detecting treatment effects over a varying size of concurrent controls. However, it has inflated type-I errors and deteriorated coverage probabilities under Setting Two and Three when the outcome drift is present. In con- trast, our proposed borrowing estimator effectively controls the type-I error, maintains the satisfactory coverage rates, and achieves improved or comparable power levels com- pared to the benchmark @rip", .irrespective of the presence of the outcome drift.
Figure ((Top) presents the bias, standard eгror (SE), and the square root of the mean squared error (Root-MSE) for each method across three simulation settings, with the size of the concurrent control No ranging from 50 to 400. As expected, the trial-only benchmark estimator @.ip" exhibits small biases across these three settings by design. Under Setting One, where the external controls do not present any outcome heterogeneity, all integrative estimators demon- strate improved Root-MSE relative to the benchmark ":in" Our proposal Gohupt is less efficient compared to other inte- grative estimators
5. Real-data Application
This section presents an application of the proposed selec- tive borrowing methodology to evaluate the effectiveness of Galcanezumab (120mg) versus placebo in patients with episodic migraine. The primary trial study is EVOLVE-1, a phase 3 double-blinded trial for patients with episodic migraines that randomized patients 1:1:2 to receive monthly Galcanezumab 120mg, 240mg, or placebo for up to months (Stauffer et al., 2018). addition, placebo subjects from the REGAIN study were used as external controls to augment the control arm from the EVOLVE-1 study. The REGAIN study is phase-3 double blinded trial for patients with chronic migraine headaches that randomized patients 1:1:2 to receive monthly galcanezulab 120mg. 240mg, or placebo for up to months, with a subsequent 9-month open label extension follow-up period (Detke et al., 2018).
induces extra variability due to bias detection for selective borrowing. However, peom Opowe and @TransCox can be substantially biased under Settings Two and Three, where the unmeasured confounder or time inconeur- rency are present. In particular, our data-adaptive integrative estimator can detect the incompatibility of external controls and selectively borrows the comparable subset, resulting in similar level of biases, but improved standard errors com- pared to the benchmark p The results under Settings Four and Five are presented in Appendix B.
To evaluate the asymptotic properties of our proposed esti- mators, Figure ((Botoom) presents the type-l error, the cov- erage probability, and the power for detecting 0, > -0.3 for the estimators Oripp Oaew and O.llpi The bootstrap- based variance estimation with bootstrap size 50 is used to construct the 95% Wald confidence intervals for evaluation. Under Setting One, geew successfully controls the type-I errors, maintain the nominal coverage rates and achieve the
The primary objective is to assess whether galcanezumab 120mg is superior to placebo in helping patients with episodic migraine achieve meaningful improvement in migraine headache days (MHD), defined as a 50% reduc- tion in mean MHD per month from baseline. To estimate the difference in time to first meaningful MHD reduction up to 6 months post-baseline, the galcanezumab 120mg and placebo arms from EVOLVE-1 1 are augmented with
7

---

Doubly Protected Estimation for Time-to-event Borrowing
the placebo from the REGAIN study. The treatment
effect 0, is defined as the RMST difference of the time first occurrence of 50% MHD reduction up to time 6 months.
oTransCor
0
6
Bias
SE
Root-ASSE
(A)
RECAN LANC
EVOUE JANE
86
Seeting One
1.00
RMST.
0.75
3-
Scetro Two
Ng
100
-
150
200
0.0
0.6
0.3
P/SST
0.9
Sctting Theee
2
0.0
0.9
0.8
200
300
400
200
400
230
300
400
S Size Concurrent Control No.
Allow
ASCW
dadapt
Figure 2. (A) Kaplan-Meier survival for the EVOLVE- study and the placebo group of the REGAIN study; (B) The estimated treatment effect in RMST for EVOLVE-1 study; (C) Probability of study success of detecting 0, < -0.05, -0.1, -0.15 using Hune and Gettpt over range of re- stricted times under varying sizes of sub-samples from the placebo group the REGAIN study.
Typo error
C Coverage Probabilty
Power
200r0 One
0.50
0.25
0.00
1.00
Figure 2(A) presents the unconditional Kaplan-Meier curves for the placebo group of these two studies. The log-rank test indicates that the time to meaningful MHD improvement different between the placebo arms the EVOLVE-1 study and the REGAIN study, suggesting the need to account for outcome drift. Figure 2(B) provides the estimated treatment effect of Galcanezumab (120mg) in terms of RMST as function of the restricted month т. All estimators exhibit a trend of decreasing RMST and show significant improve- ments in mitigating migraines severity, implying a shortened response time for least 50% reduction MHD after being treated with Galcanezumab (120mg). Our proposed estimator @.dapt a RMST that closer to the trial-only estimator @uipw highlighting its capability to control the bias arising from external heterogeneity.
0.75
5 1NO
0.50
1.00
0.75
Settro TTV8
0.50
0.25
0.00
100
700
300
400
160
200
300
400
100
200
300
400
Size Concurrent Control No
Figure 1. (Top) estimattin results for RMST over 500 Monte Carlo experiments; (Bottom) Asymptotic results for RMST over 500 Monte Carlo experiments under Settings 1) selection bias only; 2) unmeasured confounding: 3) lack of concurrency.
Next, we benchmark the performances of @udapt with the trial-only estimator @rpp to emphasize the efficiency gain of our selective borrowing framework. To accomplish this, we retain the treatment group of the primary EVOLVE-1 study with a size of N₁ 212, and create 50 sub-samples
8

---

Doubly Protected Estimation for Time-to-event Borrowing
by randomly selecting N6 patients from its placebo group with No 100, 150, 200. 'he placebo group of the RE- GAIN study is then augmented to each selected sub-samples. Figure 2(C) presents the empirical probability of study suc- cess (PrSS) over a range of restricted times T under dif- ferent sizes of concurrent controls. The empirical PrSs computed by the proportion of successfully detecting 0, < -0.05, -0.1,- -0.15 over the repeated sub-samples. When combined with the placebo group of the REGAIN study, 0: enhances the time-to-event analyses and yields a higher PrSS compared to 0r"h across all the sizes of sam- pled concurrent controls.
valuable external controls that are partially comparable to the trial data. Future research should explore approaches that select comparable information rather than entire com- parable subjects. One promising direction involves jointly estimating the average treatment effect and bias functions, as suggested in recent studies (Yang et al., 2020; Wu & Yang, 2023).
Software and Data
Our R codes with illustrative examples are avail- able at Seleetiveintegratiee
For example, suppose that we aim reach PrSS 2 0.6 of detecting 0- < 0.1 at month r = 6, @ only need to recruit 100 patients for the placebo group (solid red line at r 6 of the middle panel in Figure 2(C)), however, the trial-only estimator ":: needs at least 150 patients for the placebo group (dash green line at r 6 of the middle panel in Figure 2(C)). Therefore, our approach could attain similar levels of PrSS with fewer patients and a shortened patient enrollment period by appropriately leveraging the external controls, which could eventually accelerate the drug development for rare diseases where the event rates are typically low and imbalanced trials are often considered.
Acknowledgements
We would like to thank the anonymous (meta-)reviewwe of ICML 2025 for their helpful comments. This project is supported by the Food and Drug Administration (FDA) of the U.S. Department of Health and Human Services (HHS) as part of a financial assistance award U01FD007934 total- ing 1, 674, 013 over two years funded by FDA/HHS. is also supported by the National Science Foundation under Award Number SES 2242776, totaling 225, 000 over three years. The contents are those of the authors and do not nec- essarily represent the official views of, nor an endorsement by, FDA/HHS, the National Science Foundation, or the U.S. Government.
6. Discussion
In this paper, we introduce a doubly protected borrowing framework that utilizes external controls to enhance treat- ment estimation for survival outcomes. Unlike most existing approaches, our method built on semi-parametric efficient estimation coupled with the DR-Learner detect outcome drift for selective borrowing. This approach effectively in- corporates external controls without introducing biases into the integrative treatment evaluation. Moreover, the pro- posed approach offers new perspective on the integrative analysis for survival outcomes with a proper method for inference, which could be a valuable contribution to many survival analyses in the machine learning community, such as those involving eustomer churn (Larivière & Van den Poel, 2004; Gao et al., 2024b), and multi-source domain adaption (Mansour et al., 2008; Shaker & Lawrence, 2023).
Impact Statement
Our proposed method for analyzing censored survival data enhances the evaluation of treatment efficacy in clinical trials with small control groups by incorporating external controls while addressing data heterogeneity to reduce bias. This approach promotes ethical research practices by im- proving statistical efficiency and minimizing patient expo- sure ineffective treatments.
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Doubly Protected Estimation for Time-to-event Borrowing
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Doubly Protected Estimation for Time-to-event Borrowing
A. Proofs
A.I. Proof of Theorem 3.4
We only prove the identification formulas for So(t R 1), and similar proofs follow for Si(t R = 1). Based on the trial data, we have
So R= = 1E( X, R 1) 1}
1
R (t X, R. 11)
(1)
P(R 1)
1
P(R = B(RP(T (0) > X, 1)}
1
P(R I) B(RP(T (0) > X, 0,0' m)}
1
R1-A)A113 >
E
P(R 1)
[[1 - xA(X))SC X, A R
where the fourth equality holds under Assumptions 3.1 and 3.2. Based on the external data, we have
So = 1) = (t X, R = 1) R 1)
1
E BR X, R 1))
P(R 1)
1
E ((TR(())((( X, R = 0))
P(R
1
[(TR(X))((10 > 0,
P(R
=
1
(1-
>1]
E [1
1[
SCCY X, A, R=
where the third equality holds under Assumptions 3.3.
A.2. Proof of Theorem 3.5
1. PREIIMINARIES
We first derive the full-data efficient influence function (EIF) for the treatment-specific survival curves S. R 1) without censoring, i.e., the full data W, (T, X,, R, r). We next employ the semi- comipparaeetric theory in Bickel et al. (1993) to derive the EIFs. In particular, we consider a one-dimensional parametric submodel fo(W), which contains the true model f(W) at 0, i.e., fWW lØ=0= f(W). We use dot to denote the partial derivative with respect to 0, and (.(.) to denote the score function the submodel. For example, we have
Eo (xo( / (x(X) Ofo(X) dx ax
jAn 20
| (o(
fo(X) FolX)
Fo(x) (X)xx E(po(XXsso
where (X) Olog /000 Further, we factorize the full-data likelihood function
S(W) -1 1 I-R
f X
X, 1) R(1-A)
fT X, R = 1, - 0) ((1-4)
f(T R)
12

---

Doubly Protected Estimation for Time-to-event Borrowing
and the associated score function under the submodel can be decomposed
R - P(R X)
(W) = (x( P(R X) P(R 1 XX] Po(R - -
R(A - xA))
T (X) TA (X)) Po(A=1) = [
+ RAss (T' XX = 1 R( A)sa(T X, = 1,
+ (1 - RT |X = 0),
where (x( = θ log [o(X) (T X, 1,A d So(T X, = = = for a 0, 1. and (T | X, R 0) log fo
X, 0)/00. Analogous our definition fo( 10=0= f(W), we have so(:) l==0= s()), which is the true score function evaluated at the true parameter under the one-dimensional submodel.
A.2.2. FULL-DATA EFFICIENT INFLUENCE FUNCTION
From the semiparametric theory, the orthogonal complement of the full-data nuisance tangent space A equals to
A = H1 110H 1 H3 + H4.
(2)
where
Hi = (T(X): -(T())))
H₂ = {{R- - 1 [[R=1|X))(XX)} H3 (A[A
H4 H4i 042 = -T RR, : (,, =3
n (r(.. E
[1-R) R(- 1) 0(R=0|X) P(R=1,A=01 X)
r(T X,T, x - 0,1<7]
for any two arbitrary qquare-integraa measurable functions a(X) and b(X). The tangent space H42 is induced by the conditional mean exchangeability in Assumption 3.3, where So(1 X, R 1) S X, R = 0) for any < The EIF for So R = 1), denoted by v.S. (t, W) E Ap should satisfy
0t.(t R = 1)/00 10-0= B(V en (t, (W)).
X, R 1))/P)) 1), which is ratio for with numerator N (() St X, R 1)) and denominator D TP(R 1). Therefore, strategy is to first derive the EIF for the numerator and denominator, and then combine them to have the final full-data EIF & W).
Based our identification formula (1), So ===))=((=((())))
Let No and Do denote N and D being evaluated at the submodel fo(W). For the numerator, pathwise derivative is evaluated by the chain rule:
NN,
10=0
E(P(R = X)P > R= 1, 0,X)s(X)}
00
OIp OR ( P(T / == 1, - 0, x)
E
10-0
(3)
00
(P R = 1,4 0.
E
1P(R=111 1
10=0
(4)
a0
Next, we show the second part of the pathwise derivative (3) is
O. - l
E[(R - PPR = 1| ()) R X)).
80
However, the pathwise derivative (T > 1 X, = 1)/00 in the third part can be derived in different ways under
13

---

Doubly Protected Estimation for Time-to-event Borrowing
Assumption 3.3 with the trial data by (5) and the external controls by (6):
Po
R 1, A= X)
00
0
Eo {1(T) t) R 1,A 0,
00
1(T t) 00 fo - 0)
1(T > (T R = 0, R = 0xX
R(1 - (T (1T) St
X, R = -, So(T R, X)TT
- X)
R(1 - A) (1(T -)((( X, R= 1)} (T A, X
(5)
P(R = 1,A = 0 x)
and
Po(T > >1 --
00
0
Po(T -
00
0
Eo [1(T) > R 0.4 0,X}
00
1(T > t) 00 fo R= 0. A = 0, X)
1(T > (T R 0, 0, x)fo(T R 0, A X)dT
-1- R) X) P(R x)
(1(T > 1) - So(t X, R 0)) (T X, R)
dT
[1- R) (1(T 1) So(t X, R- 0)} 8p (7'
X) X
E
(6)
P(R
X)
To obtain the efficient influence function of N, we need find the proper functions C₁ and C2 of (X, R, A) such that the third part (4) belongs to the tangent space H4, which satisfies:
C1(1 H)g(() t) |o 1)3 CC A) 1(T 1(T t) Solt Xx, 1) R(1 - 1)1(7 1)
E
(
- (X))
(1 - [
P(R=0|X) 0 11) 1xx= =0.
By simple algebra, can show that
G E R -0110=00-56(1XX==))0000--
C2
[1- (1--(())((==111=00(
-1
E (1
0
x(t,X)
(1 - (X())RaR(X)
where x(t,X) = var (1(T>t) R =- /var (I( > 1) R = 03. Plugging the above formulas, we obtain the
14

---

Doubly Protected Estimation for Time-to-event Borrowing
EIF for the numerator N as:
of (t) Rt X, R =)
+ (1 (1- R)r(t, X)qn(X) {1(T >t) -) R- 1))
D)t,X)
R(1 - A)gn( X) (1(T > 1) - -So(L] X,R= = 1)}
D(t,X)
where D(1, X) = xl1, + (1-28 (x)))(( For the denominator Da, we have 0Do/00 l0-0= RRR@PO 1)/00} l=--- By Lemma SI in Jiang et al. (2022), the full-data EIF of N/ D is obtained by:
N off (4, W) - RS R=1)
J. Sy (t, ()
P(R= 1)
R(1 A) (x( (1(T 1) So X,R R 1)} P(R
D11,
1 R r(1,X)u((X) (1(T) > 1) - Soll X, R=
P(R 1)
(t,
R
(So(t) X, = = 1) - So(t R R=
P(R [ 1)
R(1 A) (X(X) {1(T' - So | R= = =1}} D)t,X)
P(R = 1)
(1 R)qR(X) r(t, X) {1(T)) > >t) -So(t R= = 1))
P(R 1)
X)
Solt X, 1) So R 1) {
R(A xA(X))9R(X)
r(t,x)(R (1 (-)))R(()) D(1,X)
+
PPR 11
D)1,))
which belongs to the tangent space A
A.2.3. OBSERVED- DATA EFFICIENT INFLUENCE FUNCTION
In the presence of censoring, i.e., special form of monotone coarsening, we observe the data set V; (Y, A, X,, R. r) instead of W. Define the many-to-one linear mapping K. A A to be KC(h) (E((V) W) for any A where A the orthogonal complement the observed-data nuisance tangent space by Lemma 7.3, Tsiatis (2006). Let bF (t, W) be typical element of A by Theorem 7.2 from Tsiatis (2006), the space A,, consists of all elements that can be written as
A = - A (1, W) W) +KC
(8)
P(A =1 P(A=1
where KC-1 the inverse operator, and (C) consists any arbitrary functions L(t,, V) such that B(I(L, W) 0. The first part of (8) is motivated by the inverse censoring weighting of the complete-case estimator where the event time is observed, indicated by A = 1. The second part of (8) is referred to as the augmentation space due to censoring. The optimal element (8) with the greatest efficiency improvement is obtained by projecting A&" (t. W)/P(A = 1 | W) to the tangent space K. (O) By Theorems 9.2 and 10.4, we derive the observed-date EIF for So 1) under our monotone
15

---

Doubly Protected Estimation for Time-to-event Borrowing
coarsened data:
R(1 A)A
(R(X) (1 (t)--S((t R= 1)}
U et, (t, V) P(R - (Y
(9)
1)
(),,
(1 (X)
x),, (1(Y > So(t R 1=
(10)
P(R = (Y X, R = 0)
D(t,X)
So(t X, = 1) - So(t R [R=1)
( R(A +x(X)))R(X)) + x(t, - (1 RX)3(X}3} }
(11)
P(R = 1)
D(t,
X)
R(1 - A) dMf (r) X)
(n(()) (1(T > 1))- R = =1]} |T> r,
P(R- - 1) ( 1) E
(12)
X,
D)
(1 - RX) dMD (r x)
x) (I(T > - Sult | R R= 1)}
E
|T> 1,
(13)
P(R=1) (r(r) X, R 0)
x(t,
where d.M. (r IX) = d.o 1(Y 21 -r X). The last two terms (12) and (13) belong the augmentation space
K-1(()). (0). Next, we can simplify it to be the EIF in Theorem 3.5. Note that (12) can be expressed by
P(R = 1) ( X, R 11 E qn(X)) {1(T > t) So R R= 1)}
R(1 A) dMf (r X)
T |T>,X]
X),,
R(1 - A) dMg' (r X)qn(X)
1(r < t) { Sutt X, R 1)
Solt R R =1}
P(R = 1) ( X, R = ())
So((
X, R 1)
R(1 A) dMg (r (X)
+
1(r 2 f) So(t| R = 1)}
0
P(R = (r X, R = I)D((,
R(1 A) dM (r) x))((x) Solt X,
(14)
P(R- = 1) SC X R ()) Solr
X, R 1)
R(1 A)
dMg (r| (X)
+
(15)
P(R = 1) ( X, = 1))((x))
So(l | R 1= /.
R(1 -A) A) dMg (r XX)
(16)
P(R = 1) (r
Xx R = 1)D(1,
The second term (15) equals to
R(1 A)
dMo (r| X))
P(R = 1) SO X, R = 1)D X)
R(1 (X)
000
aMg (r |X)
P(R- = X)
SC(r X, R = 1)
R(1 (X)
dN - 1(Y 2 (r IX)
P(R- X)
So(r X,R=I]
R(1 - A)) 2t)
1-A
(C X, R R- }
A6 (r X)dr
P(R = X))
1)
R(1 - AA)xx(X1((Y 2))
A
P(R = 1)D(t,X)
SC X, R 1)
S X, R = 1)
where the last equality holds as we know that (r (r Xx -oggg SCi(r
X, R 1)/Or, and A x)/s((r)
x, R
| = 1). For the third term (16), we have
So(t R 1) /0-0000
R(1 - A)
d X))
P(R 1) S X, R= X))
=Sp(t R 1) R(1 A))R(X)
dNC - 1(Y 2 rr X)
P(R = L)D(t, X)
5 R=
R(1 A))
A
-t R= 1) P(R = ()D X) {
- X,
16

---

Doubly Protected Estimation for Time-to-event Borrowing
Plugging these formulas back with (9) and (12), we obtain
R(1 - A)A
(R(x))1(()) > - Solt R= 1)}
P(R 1)50 X, R 1)
XI1,
+ /'' R(1 A) dMg (r X))((X)
So(t X, R=
0 P(R= 1) (r()) X, = X) So(r
1)
+ R(1 - (X) 2 2t)
1
A
P(R = (t)
(C
(([1X SC I-
Solt R 1) P(R = 11-)(((--)--)
R(1 A)gn(()) (X)
{1-
A
(1) X)
- I)
R811 A)gn(X)1(x 2t)
R(1 A))n(X)So(l R= 1)
IP(R- = (1, x) X, 1)
P(R 1)D((, X)
% R(1 ( A) dMp't (r (X) Solt X, 1)
F(R= = 1) ( X, R = 1), X) So(r I)
Analogous simplification applies to the combination of (10) and (13), which leads to
(1 - ((,, > t)
(1 - (x),(()) R R= 1)
P(R = 1))(, X)50 0)
P(R= (), X)
+ %-- (1 R) dMf (r) (x), Solt 1=
IIR=1) 5 X, R = - X) Solr X)
So the final observed-data EIF now becomes:
R(1 - ())n(X)1(( 21) P(R (), X)Sc(t (t X, 1)
ef (t, V)
+ 1." R(1 A) dMf (r XX) So X = 1)
1)) SC | X, R
L)D(t, So(r X,R= 1)
(1 (1- ()t,(()((, >t)
P(R = ())(( 0))
" (1 () dMp(rr X))n(X)r(t, X) So =
Jo P(R 1) 50(r
X, R= (t, X) So(r ==
So(t R,R=11
R[A- ((()))) + (R- (1-R))(1(()X)) R))
P(R- = 1)
X)
D(t,))
RSo(t R= 1)
P(R = 1)
where
RŞA TA (X)))n(X)+ A))n(x)
RM1 R (x)
((R=1)D(14
P(R - P(R=1)D((,))
rl,, X1, - (1- (1-)())(111))x + (1 ()))))
Xr
P(R = xt)
P(R= = 1), X)
A.3. Proof of Theorem 3.6
Assumption A.1. Assume (a) [x)( (x(()), = (P)), (X))xx)AAA -
= (X)))) X, R)ll,, or(1): 0 < a S Tx(X),,a(X)) (t X, R)
1)
X, R = 1)]l,, (1,, and [| X, R) - SC
X, R) < 02 < 1 and their estimated counterparts are bounded away from 0 and 1 for some constants and oz; (c)
|| X,R 1) Sa(t | X, R 1) ||aa(X) TX(iil (N 11.. | X, 1) S.
X, R =2=1]] = 1)]l,, || - = = (N and (t R = 1) - St X, = 1)7l,, max,<1 [7
x, R) 2 R,]l,,,, = (1121 for a E 10.1).
17

---

Doubly Protected Estimation for Time-to-event Borrowing
Assumption A. is analogous those for double machine learning estimation for average treatment effects (Kennedy, 2016).
To investigate the asymptotic properties of Ör, we need to understand the components that constitute 0, - 0r. Let PN be the empirical measure, we have
/ 30, (V)dP -
0- - 0, =
- (n.
/ en (V
(17)
+ / (ōo, (V) - 00,, ef,, (V)
(18)
/ (0o, (V) Φe, .at (V) - P),
(19)
where the first term (17) is asymptotically normal by the central limit theorem. The third term (19) is the empirical process which is negligible if Φ0,,eff (V) belongs Donsker classes the cross-fitting technique is employed. Under the assumptions in Theorem 3.6, and the regularity conditions A. we have
0 +
Σ
v0, offf (V) |Rem [P, IP]l + oP(NN1/2),
N
VERUE
where PP is the estimated counterpart of the true distribution P, and
|ee (P, -f(oo..a ee (V)))
is second-order remainder term (18). By the definition that oft of So (0s, (t, V) @s (t, V)) will first characterize the remainder term induced by of the remainder term induced by ds (V) follows in similar techniques. Combining parts (9), (10), (12), and (13) in efl (V), we can show
m1(t, V) + (Y X, R 1) - 1 mma((, (t(t V) + m)(t. + (C X, R == =) 0)
- - -1]matt, V)
dMg (r X)
dMG (r X)
E V) T
+
E (m2((1, TT> r}
Jo ( X, R 1)
50 X, R 0)
-m V)- ())
dMg (r XX)
(m,(1, V) - E (mi (t, V) T> r|]
5C (r X, R 1)
dM (r XX)
(m2(1, V) - E (m2(1, V) T r,,,)]
5 X, R = 0)
where
m1(t, V) = R(1 A)qn( (1(1)))))))) R=1))
PRR = 1)D(t,X)
(1 - R)qa(X)r(( (t, (1(T)t)) Solt R R=
mz(t, V)
P(R = X)
dMG (r X)
5O X, R r)
S X, R 1)
by Lemma 10.4 from Tsiatis (2006); Zeng Lin (2007). Combine them with (11), the second-order remainder term
18

---

Doubly Protected Estimation for Time-to-event Borrowing
((sooee
oso. esl (V) becomes
So(t x, R 1) - So(t -
- (x))))/((( ()(()) - =n((1]))
P(R -1)-)))((,))()r((A
(r x)
(m. -m, |2,,)
P
(20)
(r R
dM (r X)
P
most, V) B [
(21)
( X, R = 0)
since P.mi (t, v)} = (mm2(t, V)} 0, where
(1, V) R(1 A) (x) So(t| R 1=
PRR = 1)D(t,)
m2(1, (1- -
t) Sult R=1))
1)(1))
Next, compute the expectations of d X) and E (1(T> t) T r, X) under the true distribution PP by condi- tioning on {T' r, r, X):
(g [x)]dr,
[I > |
where the first equality holds since [2 > > = = 0, 1(Y > rr X), and the second equality holds under Assumption 3.2. By conditioning on [T>r,>>r,]] for < L, we show that the expectation (20) is equal to
1(Y > 1
P(R=1)D(,x)3311 X, (20 (r - (6 R= 1))dr
E (1(T ( x] - (1(T> (1 t)
T>r, x>r,,)|)
Similar iterated expectation can be applied to (21). Under the regularity conditions A.I, we collect all the terms above and use the Cauchy- -aechy-Schhwrz inequality:
10sc (V) off (V) VdP dP
(5 X, (8 X, = 1)1l,a) }
$ (xa( - *x( + (X) - (X)]] + max
= --
ret
Sott| x, So X, R = 1) L22
where X indicates that the inequality holds up multiplicative constant. Thus, we can show the second-order remainder term is bounded:
Rem P,)), = /f 00 off 2V)
$ || x,R= - -So(t) X, R 1) + 11 X, = S | = =1)lle+ }
x | - (X)II + (X) - [x(x) + max 1 (r X, R) - Xg 1 X. R2) 12) )
ret
which is negligible under the regularity conditions A.1. Thus, we have 0, = 0, + EiERUE de, eV)) + 1122) which achieves semiparametric efficiency V- = [V (V)).
19

---

Doubly Protected Estimation for Time-to-event Borrowing
A.4. Proof of Lemma 3.7
Follow the similar technique in the proof of Theorem 3.5, the observed-data ElF-motivated estimator for So(t X, R = 1) with the trial data only
ko V R = 1) = Soll I
+ R(1 - - 1)}
( (X X,R=1)
R(1-A)dMo = B(1(T |X, -)- Sol( X, R = 1) T>r))
X; TR ( (=X,
By simple algebra, we obtain the simplified formula in the main paper
kott, V R= 1) = So(t =
R(1 - (X >1
X))
(x (X X, =
R(1 - () X, R =) So X, R = 1)
(1 (r X, So(r X, =
R(1- - A)1(Y
1
A
TR(X)L - (X)]
(c((t X, 1)
SC(Y X, 1)
R(1 - () X)
{
A
1-
TR(X){1- TA((X)
{ 5C (x
=So(t X R = 1)
R(1 A)
(5 X,, R 1= So(1 X, R }
1(Y t)
xx())(1 - [x)
R(1 - () X,
=) 1))
So(t X, R = 1)
( X, X, R R=
where
(M X, = 1( >1)
{
1
SC(r X, [ 1)
(S X-
5 X, R
dM (r X, =
A
5C(T R- =1- SC X, R-
by our arguments in Theorem 3.6. Following another representation of ko V R = 1). we can show that
Ro(t, V R= = 1) = St X,R 1)
R(1 A)(1(T > 1) - So X)}
+
[1 (X}}
A
R(1 A){1(T > t) -11-))17>=))-6((XX------
SC X, R 1) 1
TR (X)(L-TA((M)
R(1 A)dME X)R)
[R(X)(1 - A X- B(1[T >) So(t -6
-So(1| X, R =1)+ R(1 - () X,R= 1)}
(n(X)(1-T1(X)
R(1 11-A)AMM (r X,R)
+
1=), 1)).
X-R
20

---

Doubly Protected Estimation for Time-to-event Borrowing
where 1 (Y r) Jo (M ( Let Let V R 1) be the probability limit of
R,
(1, V R = 1), we can show that
B( V R= So X, -
S (t X, R 1) So(t X, 11} [1- m
-x(X)(1-Ra(x}
x E (I(T (t) | - E(I((T])) X,
5
Thus, we establish the bound for kā(t, V R = by
|E V R= So X, R 1
S [TT((X) - xR(X)]| |15 X, - S X, R 1))1
+ I (|5)(() X, -1)=1))So(( X, =1)ll
+ ( (t(1| X,R 11)11111-11664 X 1) -So(t X, R 1)
Similarly, we can establish the bound for (t, V R 0) as
(t V R= 0) (t X, R 0) |
S (X) x | X 0) - St X, = 0)
- (18 X, R = 0) - - (1 X, R =220) - X, R 0)]
Putting the bounds for nttt, V R 1) and (t, V R 0) together, we obtain the desired result:
[B(E(V) - bo! 13111
(x) xx(X) -100)) [ R- -) So((| X, R = =r)ll
||sCa(t (t
5t( X, R = r)) (t X, R = r) -So(t X, R r)
+ (X) T * (t X R = 1) St X, R zz
which completes the proof of Lemma 3.7.
A.5. Proof of Theorem 3.8
Under the condition that there exists a subset Aof the external controls such that S(t X,, R= =)= any time t and subject E A, the tangent space Ha in (2) is modified for the updated restricted moment condition
= S(t XX,, R = 0) for
F = XTT,,X,R, BAT(T, X, X, A) R, 0}
NI(Y :E E
(1 O(R=0,b-0|))
R(1 A) ) ((Y,X, 1 X 0, T
= T[R=1,A=0|X)
21

---

Doubly Protected Estimation for Time-to-event Borrowing
Similarly, we find the proper functions Ci and C in (7) obtain the full-data EIF for So(l R 1):
So.eff (t, W) = R(1 A) qn( x) {1(T > 1) Soll x, R 1)}
P.A.A
P(R 1)
Day X)
(1 - (bo = 0) r(t, ((, () {1(T) > t) - Sult x, 1)) Dbo (1,
IP(R= 1)
R
P(R = 1) (50(1) X, R
X, = 1)- So(1
- R [[1}}]
R(1 A) ( () (1(T > 1) Soll| R == 1)}
F(R = 1)
Xt)
(1 (bo (X) X) (1(T > t) - Solt R 1)} Dby X)
P(R=1)
R(A TA (x()))
{So(t X, R,R=11 1) So(t R ===a 1))
P(R b) (t)
r(t, (RFF(b 0
- X, R 0) - (1 RR)1(bo = (x)R( (So(t X, = 1) So R= 1)}.
P(R D) (1,
where Dbp Xt. = r(t, Pb = 0 X,R 0) + {1 A (X)))/ By finding the optimal element of the
augmentation space with some algebra similar to Section A.2.3, we obtain the modified observed-data EIF under the updated restricted moment condition
R(1 - A)
(R(X)1(Y > t)
S..p (t,
P(R 1) SC X,R = I)
(1- - R)o 0)
> t)
P(R 1) SC(t
0)D, X, | R
R(1 A) gR(X)dMg X) So(t X)
+
P(R 1) R
So(r X)
(1 (1- [b) = 0) @R(()) (x(X)r(, (r X)
+
P(R= (C(r) X, = Do (t, X) So(r IX)
0
X)
+
P(R=1)Dh, X)
+ (1, X, R = 0) - (1 (il-))ii ()g((X)]So((] RSo(t R 1)
P(R = 1) Xt.
P(R = 1)
Rt R R= 1)
Sof (t. V)
P(R=1) 1)
which belongs to the orthogonal complement of the updated observed-data nuisance tangent space A From our previous arguments, we know that the trial-only efficient influence function is vir (t, V):
R(1 A) t)
St (t. V)
P(R = 1) - n. A (t X,
So(1 | X) P(R = 1)(1 - (r [r|X,, X, R 12=1] 1) So X) R(A T |X,]] 1) RS R 1) 1)
R(1 - A)AMS (r X)
P(R= 1)(1 - (x))
which leads to the EIF of 0, by vis ell (V) = So (Wss,en (t, V) - So.ee (t, V) with the asymptotic variance waipw
Compare the asymptotic variance yadaps to vaipa we have
graipw
vradaps E[vag seff! (v))= E(vr .aft (V)}2
= EE(up off (V) - ( -io (V)}2
2TE( off (t, (2 off (V) eot (V))
off
(22)
22

---

Doubly Protected Estimation for Time-to-event Borrowing
By directional derivative, we have
0, --eEer (V)s(V) EAvss (V)s,))
where s(V) is the score function for the observed data, and therefore ( eff (V) vig (V)) = O. implying Vo eef (V) - v .eff (V) belongs to the updated observed nuisance tangent space A, Note that v/p .e (t, V) E A
we have (22) equals to zero, and
y/aipw
Tradapt
[V. (V) - 1/ 1/0 (V))
where 1/tt
-Vor,e (V) So SS (t, V)dt So ei (t, V)dt. Next, we can show
Sp (t, V) - 2 So V)
R(1 A)
9R(X)
1(Y > t)
P(R 1) X, R ( TAlx)(
Dou (1,
R(1 A)dM5 (r x) So(t X)
gr((X)
1
}
Jo P(R = (t X, So(r X)
Dan X))
1 (A(X)
Rbu 0) qx(x)r(t,x)1
P(R 1)
Xt
R)1(bu 0) (x())r(t, (r X) So | X)
P(R=1) 50(r X,R= Do (1, X) So(r X)
R(A- (x(
(x)(X){ (X) T (X)) XX,
= 1( (0(11=xaX))
P(R = 1) (1,, X) 0) (1 11) ())
r(1, (R)P(b 0 X,
11)
P(R = 1)Dbn (t, X)
By some algebra, we have
RíA RIA-
R
1 1-4
}
1
Solt X, R 111). 1),
-(R=1)(1-7xXX)
P(R=1)
1 (x)
and
(X)) X, R=
PPR = Dr. Xt
((tx))RP((( 0 X,R R 0) - (1 Rb) = 0)qR X, R =1) 1)
X,
P(R = 1)Dro (1, X)
R- R(1 ()) ((t,X)(1 1b) 0)
So(l X, R = 1).
P(R =) 1)
P(R = D0 X1)
P(R = Do. (t,
Plugging these terms back to th (t, V)
1/s (t, V), we have
23

---

Doubly Protected Estimation for Time-to-event Borrowing
So,ooff (t, V)- - 1/ So, (t, V)
R(1- A)
(x(X))
1( t)
(23)
F(R 1)8C(1 (t X, - 1) 1 11-T((X)
Dos (),
R(1 A)dM (r IX) So(t X) {
(R(X)
1
J0 P(R = (t ( So(r XX)
Doo (t,
1 1-1 H xX()) $
(24)
(1 (bu =0 9x(X)r(t,x)11( > t)
(25)
P(R- =1) 1) SC(( X, R= 0) Xt
(1 1bo =0 (tt, (r X) So(t X)
(26)
P(R 1) ( X, 0) (t, X) So(r X)
R(1 A)
1
(x(X)
} So(( | (X)
(27)
P(R = 1) 1 TA
Doo (t, x)
r(t, X) - (bu ()) Solt| (x).
(28)
P(R= = Dio (1, X)
Combining (23), (24), and (27) gives us
R(1 A)
(R(X)
1(8 1({>1)
P(R 1) X, R R- 1) 11 (X))
(t, X)
J/ R(1 A)dM8 (r (x] Solt (x)
(x(X)
1
P(R = 1)5c (1 X, R= = So(r (X)
Doo (t, x)
1 - (X()
R(1 A)
(x(X)
Solt lx)
P(R- = 1) (1- (t,
R(1 - A)
[r(t, = 0 R,
mi((, V):
P(R= 1)
(1- (X())
where
mi(tvV)= = S X, R = + L'' dMg X) So(t X)
1(Y > 1)
- So(t X)
Sc X, R- 1) Su(r) X)
SC X, R R- 10/ (r X) Su(t X)
Y > t)
x) X) -So(t] |X).
Similarly, (25). (26), and (28) together gives us
11 - Rb1 = 0) (n(X)((,x)) > t)
P(R 1) 1)
SC(t X, R 0) Xt
(1 - Rb) = 0) qn( XYdMS (r X) Solt X)
= IPR =1 (0(r XxR= Dn (1, X) So(r X)
(11 - 1b) = (X) So(( X)
PP(R=1))) 1) (1, X)
(1 (b, = 0) (x)(X)((,,X))) X)
mz(t, V),
P(R - 1) (t,
where
mi([V) S0 X, R= L'' dMG (r X) Solt X)
1(Y > >t)
So(l |X)
5 X, R = 0) So(r X)
- R,R=0) + 1. 8o
AI(Y > 1)
dMg (r x) So(1 X)
-oot X).
s X, 0) Sp(r X)
24

---

Doubly Protected Estimation for Time-to-event Borrowing
Then, can show that [ So (1, V)
Sof (1, 2))² X] equals
TA (x( (X))
- x,
{1- 11-*A(h (t,) )
2 var (mi R == 111,1
P(R= = 1)
{1 -1- [I-TR(X))P((( = 0 X, R =0) [g [g((X)r(1x))1 (m2(1, V) R 0, = 03
P(R=1)2
(1,
(x(X)r((,X)P(() x 0) DE (1, X) x),,X) V
VRI,AO:
= 1) (t, X)
[X(()} Dop (t, X) r (t, X)
where
V $1.00
r" (1, (1, X)-
Dr (t, = (1, x, ( (x)(a))
Vo
RI var {mi (1, V) R = 1, A = 0},
Vio = var {m² (4, V) || = 0, = 0}.
Thus, the proof of Theorem 3.8 is completed.
B. Additional Simulations
Additional Bias-gene Settings Figure 3(Left) presents the simulation results under Settings Four and Five. Both Oglupt and @TrnssCxx account for heterogeneity covariate effects and the risk associated with varying baseline times in these settings. However, @.ramsCux is only valid under the Cox model. For example, when the conditional survival curve (a X) does not follow the Cox model, as in Settings Two and Three of the main paper where the marginalized curves (a X) over U (or 8) result in model that no longer satisfies the Cox proportional hazards assumption, @Taansox may exhibit substantial bias, whereas the proposed estimator Oulupi continues to control for bias due its double robustness and demonstrates improved performance.
Asymptotic Properties of the Proposed Estimator Figure 3(Right) provides more details of our proposed selective integrative estimator, specifically focusing on its average borrowing proportion of the external controls and its relative efficiency. The relative efficiency is measured by the ratio of the width of its confidence intervals to the trial-only estimator. In Setting One, it is reasonable to observe that the selective integrative estimator O:clppt is always more efficient compared to the benchmark as every external control is comparable, and the proportion of borrowing approaches 1 as No increases. Under Setting Two, the proportion of borrowing diminishes to zero as it detects that nearly all the external controls are not comparable when more concurrent controls become available. Under Setting Three the borrowing proportion approaches 50%, aligning well with the true proportion of comparable external subset our data-generation process. One side note that our proposal might be subject slight relative efficiency inferiority compared the benchmark in some cases, which is also observed in other literature (Chen et al., 2021a).
Varying Censoring Intensity We conduct additional simulation studies to evaluate our selective borrowing estimator @adape under varying censoring rates for the trial. In particular, we vary the value of Bc in the hazard functions X X, R= 1) to represent different censoring levels, with 3e = 0 indicating high censoring (the censoring rate is around 60%) and 3c - indicating low censoring (the censoring rate is around 20%). The results, presented in Figure 4, demonstrate that our proposed estimator effectively controls external biases across all settings and achieves improved estimation, as shown by smaller Root-MSEs compared the trial-only estimator.
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Doubly Protected Estimation for Time-to-event Borrowing
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Figure 3. (Left) Point estimation results for RMST over 500 Monte Carlo experiments under Settings 4) different covariate effects, and 5) different baseline time-varying hazards; (Right) Average borrowing proportion of external controls and relative efficiency of the selective integrative estimator G:ang over 500 Monte Carlo experiments.
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Figure Point estimation results for RMST over 500 Monte Carlo experiments under Settings One, Two Three when (Left) 3c -2 (low censoring rate) and (Right) Bc (high censoring rate).
26


Reviews:
[{"review_id": "hNEPqmgMeq", "reviewer": "Reviewer_id4e", "summary": "The authors propose a \"doubly protected\" estimator for treatment-specific restricted mean survival time difference in RCTs, focusing on alleviating biases commonly encountered when employing additional (i.e., non-trial-derived) external control data. Their estimator accounts for both covariate shift and outcome drift, addressing some of the most common issues preventing the usage of external control data for such problems.\n\nMethodologically, the paper makes two main contributions:\n1. Development of an integrative estimator under an assumption of comparability\n2. Extension of the estimator from (1) to settings in which comparability may be violated\n\nIn terms of theory, the authors prove two (main) theorems (Theorem 3.6 and Theorem 3.8) establishing the estimation error of the estimator from (1) and the variance of the estimator from (2).\n\nEmpirically, the authors consider three simulation scenarios, each focused on a potential bias that might occur with external control data, including selection bias, unmeasured confounders, and lack of concurrency. The authors show that their adaptive estimator (2) performs well and can choose adaptively to what extent and which external control samples should be included for estimation, resulting in comparable bias to the trial-only estimator while achieving moderately reduced MSE and SE.\n\nLastly, the authors consider an exemplary real-data application of their method on an RCT of a drug for episodic migraines.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for the careful review and nice words. We hereby provide one-to-one responses to your concerns.\n\n**Methods And Evaluation Criteria**\n1. The current three settings represent three typical scenarios we often encounter in practice: Setting 1, where all the ECs are comparable after adjusting for covariate shift and should be included; Setting 2, where the unmeasured confounding is present, none ECs are comparable and the external data should not be used; Setting 3, where there is lack of conccurency, only 1/3 of ECs are comparable and that portion of external data should be borrowed.\n\n2. As suggested, we consider two additional simulation settings, and the results are presented here (https://anonymous.4open.science/r/ICML2025-7977/fig1.png).\n* (Setting 4) **different covariate effects**: for RCT, $\\lambda_0(t\\mid X)=\\exp(-0.2X_1 -0.2X_2 -0.2X_3)$; for EC, $\\lambda_0(t\\mid X)=\\exp(-0.5X_1-0.5X_2-0)$\n* (Setting 5) **different time-varying hazards**: for RCT, $\\lambda_0(t\\mid X) = t \\exp(-0.2X_1-0.2X_2 -0.2X_3)$; for EC, $\\lambda_0(t\\mid X) = 2t \\exp(-0.2X_1-0.2X_2 -0.2X_3)$.\n\n3. Both the proposed estimator and TransCox (Li et al. (2023b)) could handle differences in covariate effects and time-varying hazards. **However, TransCox is only valid under the Cox model.** If the conditional survival curve $S_a(t\\mid X)$ is not a Cox model (e.g., under Settings Two and Three in the paper, if we integrate out $U$ (or $\\delta$) out in the hazards, the model generation will no longer be Cox model), TransCox will have large biases whereas our proposed estimator still controls the bias due to its double robustness and achieves improved performance.\n\n**Experimental Designs Or Analyses**\n\n1. The penalty term is chosen to be the **adaptive lasso** (Zou, 2006). The conditional survival curves $S_a(t\\mid X)$ and $S^C(t\\mid X)$ for the event and censoring are modeled by the **Cox PH model**, and the propensities $\\pi_R(X)$ and $\\pi_A(X)$ are modeled by **SuperLearner** from the `SuperLearner` R package, which is an ensemble model of the Logistic Regression and Random Forest. These details will be added to the Simulation section.\n\n\n**Other Comments Or Suggestions**\n\n1. The comments on the tables and figures are well-received. We have updated the colors for figures by the *Nature Platte*. Also, the shapes of points are changed to be different for estimator and scenario. The updated figures are provided here (https://anonymous.4open.science/r/ICML2025-7977/fig2/fig2_1.PNG; https://anonymous.4open.science/r/ICML2025-7977/fig2/fig2_2.PNG; https://anonymous.4open.science/r/ICML2025-7977/fig2/fig2_3.PNG).\n\n**Questions For Authors**\n\n1. The conditional hazard function **should be indexed with a**. The original table only includes the hazard functions that have been modified under each considered setting. To avoid confusion, we explicitly list the hazard function as $\\lambda_a(t \\mid X, R)$ under each setting.\n* (Setting 1) $\\lambda_a(t \\mid X, R) = \\exp(-0.5a - 1_p^T X \\cdot 0.2)$;\n* (Setting 2) $\\lambda_a(t \\mid X, R) = \\exp[-0.5a - 1_p^T X \\cdot 0.2 + 3\\{U + \\mathbf{1}(R=0)\\}]$;\n* (Setting 3) $\\lambda_a(t \\mid X, R) = \\exp(-0.5a - 1_p^T X \\cdot 0.2 + 3\\delta \\mathbf{1}(R=0))$.\n\n2. Under Setting 2, we include $U$ in the hazards for RCT as well to keep the variability of hazards the same level across two datasets. In particular, for $R =1$, $U \\sim N(0,1)$ with zero-mean is included in the hazard model, whereas for $R=0$, $U+1 \\sim N(1,1)$ with non-zero mean is included, which is expected to **introduce more outcome drift for the external controls**.\n\n3. **Lack of concurrency** will be discussed further in the section of Introduction, where FDA drafted the Guidance documents on the use of external controls: \"Lack of concurrency could occur when RCTs and ECs are collected in **different time periods** or under **varying healthcare settings**. Therefore, directly integrating ECs with RCTs without any adjustment could introduce biases into the treatment estimation\". The guidance reference from FDA will be added to the paper\n\n---\n1. Li, Z., Shen, Y., and Ning, J. Accommodating time-varying heterogeneity in risk estimation under the cox model: A transfer learning approach. Journal of the American Statistical Association, 118(544):2276–2287, 2023b.\n2. H. Zou. The adaptive lasso and its oracle properties. Journal of the American statistical\n association, 101(476):1418–1429, 2006.", "claims_and_evidence": "The authors are relatively modest in their claims and do not (IMO) oversell, focusing primarily on the point that \"This approach effectively incorporates external controls without introducing biases into the integrative treatment evaluation.\", which I agree with based on the presented evidence.", "methods_and_evaluation_criteria": "The general simulation setup makes sense. The real dataset also seems appropriate. I have several comments regarding the simulation design:\n\n1. The authors seem to assume time-constant hazards (i.e., data is effectively simulated from an exponential distribution) throughout - coming from a survival perspective, this seems quite restrictive, especially given that some competing methods consider e.g., Weibull distributions in their simulation [1]. Just to be clear, I think this is fine for the censoring hazard function but not necessarily the event hazard function. I don't think the current ablations on $\\beta_C$ are enough and would like to see simulations including non-time-constant hazards.\n2. Similarly, given the fact that lack of accounting for time-varying outcome drift and time-varying covariate effects are presented as drawbacks of current methods in related work, I was surprised to not see the simulations directly addressing these.\n3. The simulations in [1] also investigate changes in covariate effects separately and jointly with time-varying baseline hazard differences (see 1 + 2). Unless the authors have strong reasons for not investigating these, I think they would strengthen the simulations and thus the paper.\n\n[1] Li et al. (2023b)", "theoretical_claims": "I only skimmed the proofs and have no concerns.", "experimental_designs_or_analyses": "4. Several experimental details regarding the empirical results are either missing or unclear: (i) which penalty term is used for step 2 in 3.3 in the experiments and how is the $\\lambda$ tuned? (ii) How are the nuisance functions estimated throughout the experiments?", "relation_to_broader_scientific_literature": "The first part of the proposed estimator (3.2) seems like a relatively straightforward (novelty-wise) extension of [2] to the survival outcome setting. Despite this, I think the adaptive proposed method (i.e., 3.3) which stems from a combination of 3.2 and the DR-Learner framework [3, 4], is interesting, given its low bias and low-moderate improvements in terms of MSE, relative to the trial-only estimator.\n\n[2] Gao et al. (2024)\n\n[3] Kennedy Edward (2020)\n\n[4] Kallus & Oprescu (2023)", "essential_references_not_discussed": "None that I am aware of.", "other_strengths_and_weaknesses": "Overall, I think this paper handles an interesting and timely problem, especially in the context of survival outcomes. While novelty could be higher, the paper itself is interesting, especially given the proposed estimators' basically non-existent cost in terms of bias.\n\nDespite this, I think there are several points (especially in the simulations) that need to be extended and or better explained (see e.g., also 4).", "other_comments_or_suggestions": "- Table 3, simulation scenario 2 is missing a right closed parentheses before the final curly brace.\n- AFAIK, the default ggplot2 color scheme is not particularly color-blind friendly, so I would suggest the authors switch their figures to a different palette.\n- Some colors are not matched between figures (e.g., Fig 1 top has acw in olive, bot. in green) and some figures use colors very close to another figure for something very different (e.g., Figure 2 has blue for sim. scenario two, while Fig. 1 bot. uses it for one of the estimators).\n- Some Figures use panels (Fig. 3) and some top bottom - visually, I think it's easier for readers to have it consistent.\n- The y-axis of Figure 2 is very (too, IMO) tight for \"Relative Efficiency\", presumably due to being forced to share it with the other facet - I would suggest relaxing this and likely making it symmetric for the relative efficiency. It may also make sense to flip the y-axis, to keep with the high -> good direction of the other facet (\"Proportion of Borrowing\").", "questions_for_authors": "5. The second simulation scenario was a bit unclear to me, primarily due to two things:\n  - Is there a particular reason why the cond. hazard function is not indexed with a? If it is since there is no $−.5a$ term, what is the reason that term was left out?\n  - What is the indicator on $R = 0$ doing? AFAIU, $U$ is sampled independently for $R=1$ and $R=0$, so why is the additional indicator needed? (as, also according to the text, $U$ reflects differences in the baseline hazards).\n6. Some terms are never really defined, for things like [lack of] concurrency, having at least an informal definition and/or citation to somewhere would make it easier to read for people coming from adjacent fields (e.g., survival).", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "CwJUKOygUD", "reviewer": "Reviewer_q7gt", "summary": "The paper introduces a new way to estimate treatment effects in survival analysis using external controls, which is especially helpful when clinical trials have small control groups, like in rare diseases. It introduces a doubly protected estimator for the restricted mean survival time (RMST) difference, combining doubly robust estimation to adjust for covariate shifts and a DR-Learner to mitigate outcome drift. By leveraging machine learning, the method flexibly models survival curves and selectively borrows external data while ensuring robustness.\nEmpirical validation through simulations and real-data application demonstrates its practical utility.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the comments. Here is a detailed response to your concerns.\n\n**Other Comments Or Suggestions**\n\n1. We will restate each theorem and lemma in the supplementary material. In the main text, we will **cross-reference the proof** of each theorem and lemma in the Appendix. \"Theorem 3.4 is proved in Appendix A.1, Theorem 3.5 is proved in Appendix A.2, Theorem 3.6 is proved in Appendix A.3, Lemma 3.7 is proved in Appendix A.4, and Theorem 3.8 is proved in Appendix A.5\".\n\n2. We have already prepared the codes along with one implementation example for the proposed method in the supplementary material.\n\n**Questions For Authors**\n\n1. One short paragraph of **double robust estimators** will be included for the related work section: ”However, existing integrative methods are limited by the assumption of the Cox model, either on the cause-specific or subdistribution hazard scale, which requires to accurately model the survival probability. In recent years, semiparametric efficient and doubly robust estimators, which leverage the efficient influence function (Bickel et al., 1993; Tsiatis, 2006; van der Vaart, 2000; van der Laan & Robins, 2003), including estimation equation methodology (Hubbard et al., 2000; Robins & Rotnitzky, 1992; van der Laan & Robins, 2003) and targeted maximum likelihood estimation (van der Laan & Rubin, 2006; Rytgaard et al., 2022), have gained great popularity in many fields and are increasingly used to draw inference about treatment effects.”\n\n---\n1. Bickel, P. J., Klaassen, C. A. J., Ritov, Y., & Wellner, J. A. (1993). Efficient and adaptive inference in semiparametric models, Forthcoming monograph.\n2. Tsiatis, A. A. (2006). Semiparametric theory and missing data (Vol. 4). New York: Springer.\n3. van der Vaart AW (2000) Asymptotic statistics, vol 3. Cambridge University Press, Cambridge.\n4. van der Laan MJ, Robins JM (2003) Unified methods for censored longitudinal data and causality. Springer, Berlin.\n5. Hubbard AE, van der L MJ, Robins JM (2000) Nonparametric locally efficient estimation of the treatment specific survival distribution with right censored data and covariates in observational studies. In: Statistical models in epidemiology, the environment, and clinical trials, Springer, Berlin, pp 135–177.\n6. Robins JM, Rotnitzky A (1992) Recovery of information and adjustment for dependent censoring using surrogate markers. In: AIDS epidemiology, Springer, Berlin pp 297–331\n7. van Der Laan, Mark J., and Daniel Rubin. \"Targeted maximum likelihood learning.\" The international journal of biostatistics 2.1 (2006).\n8. Rytgaard, Helene C., Thomas A. Gerds, and Mark J. van der Laan. Continuous-time targeted minimum loss-based estimation of intervention-specific mean outcomes. The Annals of Statistics 50.5 (2022): 2469-2491.", "claims_and_evidence": "-   **Doubly Protected Estimator for RMST Difference**:\n    -   Theorem 3.5 and Theorem 3.6 provides a theoretical foundation for constructing valid confidence intervals and ensuring asymptotic properties, which strengthens the claim.\n        \n-   **Handling Covariate Shifts and Outcome Drifts**:\n    -   The use of the  density ratio  to adjust for covariate shifts and the  DR-Learner  to address outcome drift is conceptually sound. These approaches are grounded in semi-parametric theory and machine learning.\n        \n-   **Asymptotic Properties**:\n    \n    -   The authors claim to establish asymptotic consistency and efficiency improvements  for their estimator. They use efficient influence function, which is known to provide such guarantees under regularity conditions.\n        \n-   **Empirical Validation**:\n    \n    -   The claim that the method performs well in simulations and a real-data application is supported by sunthetic and real-data analysis of Galcanezumab for migraine headaches in section 4.", "methods_and_evaluation_criteria": "The claim that the method does not require stringent parametric assumptions is plausible, as the framework incorporates flexible machine learning techniques for survival curve approximation.", "theoretical_claims": "I have briefly reviewed the correctness of Theorems 3.4, 3.5, and 3.6, and they appear to be correct.  The derivations and proofs seem consistent with the theoretical framework and align with established principles in the field.", "experimental_designs_or_analyses": "For synthetic data,  extensive simulations show robustness and efficiency gains compared to trial-only estimators and other methods. For real data: The method is applied to evaluate the efficacy of Galcanezumab in mitigating migraine headaches, illustrating its practical utility.", "relation_to_broader_scientific_literature": "Doubly robust estimators are well-established in causal inference and missing data literature. These estimators are robust to misspecification of either the outcome model or the propensity score model, making them attractive for handling covariate shifts (e.g., Bang & Robins, 2005; Van der Laan & Rose, 2011). The authors extend doubly robust estimation to the context of survival outcomes with external controls. They use the density ratio of baseline covariates to adjust for covariate shifts and derive the  efficient influence function  for the restricted mean survival time (RMST) difference. This builds on semi-parametric theory (Tsiatis, 2006) and provides a principled framework for integrating external data while maintaining efficiency and robustness.", "essential_references_not_discussed": "I think it's well discussed.", "other_strengths_and_weaknesses": "The theoretical contributions, such as the derivation of the efficient influence function for the restricted mean survival time difference and the establishment of asymptotic properties, provide a rigorous foundation for integrating external data into survival analysis. The application of the method to evaluate the efficacy of Galcanezumab for migraine headaches demonstrates its practical utility.", "other_comments_or_suggestions": "- In the supplementary material, it would be easier to read if the author restate the theorem and lemma before the proof, and in the main text, use one or two sentence to mention the proof is at which section in the appendix. \n\n-  It would be better to include the synthetic experiment implementation code as supplementary material, rather than stating, \"Our implementation codes will be made publicly available after the acceptance of this manuscript.\" (Additionally, \"codes\" should be revised to \"code", "questions_for_authors": "- It would be great if the author can add a small paragraph of double robust estimators in the related work section.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "5L9gQ32JEf", "reviewer": "Reviewer_wZ8p", "summary": "Authors study the estimation of restricted mean survival time in a randomized controlled trial where external controls are leveraged to increase statistical power. Since there may be a conditional shift (outcome drift) between trial controls and external controls, it is well understood that doing this is not trivial and requires adjustment for it (e.g., reweighting external controls to mimic the trial controls' covariate distribution). Authors derive the efficient influence function for that estimand which motivates a doubly-robust estimator. They also propose an estimator  (based on selective borrowing) that can operate when there are unmeasured confounders that drive the outcome drift between the trial and external controls.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank reviewer for the through assessment. We here provide the detailed response to each of these points.\n\n**Experimental Designs Or Analyses:**\n\n1. We have **refined and reorganized the experiment sections** to align with the objectives of the simulation, including how to design the data generation mechanisms accordingly, competitors, evaluation metrics, results, and detailed discussions. \n\n2. The names of real-world dataset have been changed to **EVOLVE-1 study** and **REGAIN study** and made **consistent** throughout. The **EVOLVE-1 study** (Evaluation of Galcanezumab v.s. Placebo in the Prevention of Episodic Migraine) serves as the *randomized clinical trials* (**CGAG** is its protocol name). **REGAIN study** (Evaluation of Galcanezumab v.s. Placebo in Patients with Chronic Migraine) serves as the *external controls* (**CGAI** is its protocol name). The references will be included in the paper.\n\n3. The threshold $-0.1$ for the real-data is chosen by **domain knowledge**, that is, reducing the RMST at month $\\tau=6$ by $0.1$ is considered clinically meaningful. Further, the PrSS could be computed under **various other thresholds** and similar conclusions could be drawn; see https://anonymous.4open.science/r/ICML2025-7977/fig3.png. \n\n4. To interpret the results in Panel (C) of Figure 3, we could use an example to illustrate. Suppose that we aim to **reach PrSS at most $0.6$ at month $6$**, our method “adapt” only need to recruit **$100$ patients** for the placebo group (solid red line at month $6$), however, the benchmark method “aipw” needs **at least $150$ patients** for the placebo group (dash green line at month $6$). Therefore, our approach could attain similar levels of PrSS with **fewer patients** by leveraging the external controls and thus **shorten the patient enrollment period**, which could eventually accelerate the drug development for rare diseases.\n\n**Relation To Broader Scientific Literature**\n1. First, the proposed method can be generalized to **any estimand that is a function of the survival function $S_a(t)$** (e.g., mean or median of the survival time), not necessarily limited to one particular estimand as the EIFs are derived for $S_a(t)$. Let the estimand of interest be $\\theta_\\tau(t)=\\Phi_\\tau(S_a(t))$, the associated EIF for the estimand of interest can be obtained as $\\psi_\\theta(t) = d\\Phi_\\tau(q)/dq \\cdot \\psi_{S_a}(t)$ by Taylor expansion, where $\\psi_{S_a}(t)$ is the EIF for $S_a(t)$, and the DR-learner is directly applicable to detect the outcome drifts in the estimand of interest. Thus, our proposed method should be **useful to any integrative (causal or not) analysis for survival outcomes**. We will emphasize this point in the paper.\n\n2. **Survival problems should be important in the machine learning (ML) community, such as dropout and customer churn.** In many ML problems, there are heterogeneous data sources that can be integrated for the same task or domain adaptation; the critical issue is to handle data heterogeneity. Our proposed selective borrowing method offers a new perspective of the integrative analysis for the survival outcomes with a proper way for inference, which could be a valuable contribution to the general ML community. We will include such discussions in the paper as well.", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "Synthetic experiments seem reasonable and authors methods perform better under cases they are expected to. There a few things I could not quite follow about the real-world experiments which I elaborated on below (Experimental Designs Or Analyses)", "theoretical_claims": "I did not check the proofs in detail, as they are rather lengthy. The parts I skimmed through seemed correct, and the resulting expressions make sense. As the authors mention, most of the techniques for deriving EIFs are borrowed from the literature, so I would not expect any mistakes in the results.", "experimental_designs_or_analyses": "Experiments seem exhaustive but this section can benefit from polishing & organization. For real-world experiments, what are CGAI and CGAG trials? Creating a table for abbrevations can help.\n\nI could not really understand the evaluation criteria for real-world experiments through PrSS. Where does that ground-truth threshold (-0.1) come from? Even if we take that as given, I had a hard time understanding what were you looking at. The real-world experiments are extremely important to justify the complicated theory in the paper. I think you have a nice dataset & experimental setup, but just need to be much more clear about what you are doing & how you are evaluating.", "relation_to_broader_scientific_literature": "One of the main drawbacks of this paper for me is its relevance/contributions to the broader machine learning community. It focuses on a very particular estimand that is relevant for causal inference from censored data, and it develops an efficient estimator which can leverage external controls using recipes from the efficient influence functions literature. I do see how this could be useful in practice, but have a feeling that this work would be a great fit for a statistics conference/journal, but it does not contribute much on the methodology/technical side that could be generalized to/used in other ML problems.", "essential_references_not_discussed": "NA", "other_strengths_and_weaknesses": "see above (Relation To Broader Scientific Literature)", "other_comments_or_suggestions": "NA", "questions_for_authors": "NA", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "YjE5fqdckL", "reviewer": "Reviewer_KmLa", "summary": "Estimating the average treatment effect (ATE) from both trial and external control datasets is challenging due to data heterogeneity, specifically covariate shift and outcome shift. This paper proposes a doubly protected estimation framework to address these challenges.\n1. When the external control dataset is comparable to the trial dataset, the authors propose a doubly protected estimator, which corrects for covariate shift using density ratio weighting of baseline covariates.\n2. When comparability is violated (i.e., outcome shift occurs), the method selectively \"borrows\" only a subset of the external control dataset that remains comparable to the trial data, improving robustness.\n\nThe effectiveness of the proposed method is demonstrated via extensive simulation studies and illustrated through a real-world application in migraine treatment evaluation.\n\n## update after rebuttal\nThank you for the response and the additional experiments. I think the new results strengthen the paper and make the overall argument more convincing. I would like to keep my score as Accept.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for the careful reviews. Here are our detailed responses to your questions.\n\n**Methods And Evaluation Criteria**\n\n1. The selection of the cutoff value $\\tau$ for computing RMST is crucial in practice since the tail distribution after $\\tau$ is neglected. Typically, the event rates at this cutoff value should exceed 10% to **ensure sufficient data for model development.** A common rule of thumb is to set $\\tau$ to be around the 75%-80% quantile of observed event times. However, this choice varies case by case and should be informed by domain expertise with real-world data. A brief discussion on this will be included in the paper.\n\n**Experimental Designs Or Analyses**\n\n1. We have included additional set of simulation, where for RCT, the hazard is $\\lambda_a(t\\mid X) = \\exp(-0.5a-0.2X_1-0.5X_2 -0.1X_3)$; for EC, $\\lambda_0(t\\mid X) = \\exp(-0.1X_1-0.5X_2 -0.2X_3)$ (i.e., **the coefficients are not the same anymore**), and the covariates are generated by the multivariate normal distribution with pair-wise correlation 0.5 (i.e., **non-independent features**). The results are presented here (https://anonymous.4open.science/r/ICML2025-7977/fig4.png), and main findings stay the same as before.\n\n2. Moreover, **transformation on the covariates** (e.g., polynomial transformation) can be considered as data pre-processing to capture the non-linear relationship between the covariates and the time-to-event outcomes. Furthermore, our method allows to use flexible survival models (e.g., survival random forest) for estimating the survival curves (e.g., $S_a(t\\mid X)$), which could also capture the non-linear relationship.\n\n**Other Comments Or Suggestions**: We have changed Assumption 3.2 to *Informative censoring* as $T^{(a)}\\perp C\\mid X, A=a, R=r$ for $a=0,1$ and $r=0,1$.\n\n**Questions For Authors**: Thanks for catching this typo. We have changed Assumption 3.1 (ii) to \"$0<\\pi_A(X),\\pi_R(X)<1$ in the support of $X$\", where $\\pi_A(X)=P(A=1\\mid X, R=1)$ and $\\pi_R(X)=P(R=1\\mid X)$.", "claims_and_evidence": "The theoretical and empirical claims made in the paper are well-supported by rigorous derivations, asymptotic properties, and empirical validation through simulations.", "methods_and_evaluation_criteria": "The proposed methodology and evaluation criteria are appropriate for the problem at hand:\n1. The doubly protected estimator improves efficiency by integrating external controls while mitigating biases using density ratio weighting and DR-Learner.\n2. The paper extends semiparametric efficiency theory to survival analysis, allowing dynamic selection of comparable external controls.\n3. The approach is flexible, accommodating machine learning models to estimate survival curves without strong parametric assumptions.\n\n\nThe ATE estimation is based on restricted mean survival time (RMST). However, when the survival curve remains at a high probability at $\\tau$ (as seen in Figure 3A), RMST may underestimate survival differences by neglecting the tail distribution, which can significantly contribute to the total effect. The authors should discuss this limitation in the future work.", "theoretical_claims": "I briefly reviewed the correctness of the proofs and did not identify any errors. The derivations are rigorous and well-structured.", "experimental_designs_or_analyses": "The experimental design is sound and well-structured:\n1. The benchmarking against multiple baselines provides a comprehensive comparison.\n2. The simulation study covers various settings, including different types of bias-generating mechanisms (selection bias, unmeasured confounding, and lack of concurrency).\n\n\nThe only concern I have is that, the feature generation process in simulations assumes independent features and linear relationships (moreover, the coefficients are all 1s) between the covariates, treatment assignment, and hazard function.", "relation_to_broader_scientific_literature": "This paper develops a flexible and data-adaptive (for covariates shift, and outcome shift) framework that accounts for survival analysis datasets. The consistency and", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "Other strengths\n1. The paper clearly articulates the problem, the proposed solution, and its impact on clinical trials.\n2. The derivations are mathematically sound and insightful, providing clear intuition alongside proofs.\n3. The simulations and real-world application convincingly support the claims.", "other_comments_or_suggestions": "I believe the correct term for Assumption 3.2 should be *informative censoring*, rather than *non-informative censoring*, as stated in the paper. This assumption acknowledges that event and censoring times are dependent and only become independent when conditioned on covariates. In other words, knowing the censoring time provides information about the event time. However, this is merely a terminological distinction and does not affect the validity of the paper.", "questions_for_authors": "Assumption 3.1 states $q_R(X)<1$, which implies $P(R=1\\mid X) < P(R=0 \\mid X)$. Should this instead be $P(R=1\\mid X) < 1$, allowing for the trial population to be dominant in some covariate regions?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This work introduces a new way to estimate treatment effects in survival analysis using external controls, which is especially helpful when clinical trials have small control groups, as is the case in rare diseases. The work introduces a doubly protected estimator for the restricted mean survival time (RMST) difference, combining doubly robust estimation to adjust for covariate shifts and a DR-Learner to mitigate outcome drift. 


Most reviewers agree that this paper handles an interesting and timely problem, especially in the context of survival outcomes. While novelty could be higher, the paper itself is interesting, especially given the proposed estimators' basically non-existent cost in terms of bias. I think the paper will be of interest to many in the ICML community.
Decision: Accept (poster)