Doubly Robust Conformalized Survival Analysis with Right-Censored Data
Authors: Matteo Sesia
Vladimir Svetnik
Venue: ICML 2025 spotlightposter (2025)
Keywords: Conformal inference
Survival analysis
Uncertainty Estimation
Primary area: probabilistic_methods
TL;DR: This paper presents a conformal inference method for constructing lower prediction bounds for survival times from right-censored data.

Abstract: We present a conformal inference method for constructing lower prediction bounds for survival times from right-censored data, extending recent approaches designed for more restrictive type-I censoring scenarios. The proposed method imputes unobserved censoring times using a machine learning model, and then analyzes the imputed data using a survival model calibrated via weighted conformal inference. This approach is theoretically supported by an asymptotic double robustness property. Empirical studies on simulated and real data demonstrate that our method leads to relatively informative predictive inferences and is especially robust in challenging settings where the survival model may be inaccurate.

Doubly Robust Conformalized Survival Analysis with Right-Censonn Data
Matteo Sesia 12
Vladimir Svetnik
Abstract
approaches are gaining popularity (Ishwaran et al., 2008; Katzman et al., 2018), despite the difficulty of obtaining uncertainty estimates and statistical guarantees.
We present a conformal inference method for constructing lower prediction bounds for survival times from right-censored data, extending recent approaches designed for more restrictive type-I censoring scenarios. The proposed method im- putes unobserved censoring times using a ma- chine learning model, and then analyzes the im- puted data using a survival model calibrated via weighted conformal inference. This approach is theoretically supported by an asymptotic double robustness property. Empirical studies on simu- lated and real data demonstrate that our method leads to relatively informative predictive infer- ences and especially robust challenging set- tings where the survival model may be inaccurate.
A promising approach providing rigorous statistical in- ferences for ML models in survival analysis was recently introduced by Candès et al. (2023) and refined by Gui et al. (2024). Their conformal inference (Vovk et al., 2005; Lei & Wasserman, 2014) framework can use any survival model to compute a lower prediction bound (I.PB) for an individual's survival time, supported by rigorous statistical guarantees.
LPBs indicate the time beyond which a patient is expected to survive with at least (1 a)) probability, for some fixed level E 1). They can be useful in many applica- tions, including for priorizing treatments under resource constraints. When the data are limited or the model overfits, LPBs tend to be conservatively low, reflecting greater un- certainty. By quantifying uncertainty on an individual basis, LPBs are potentially able to distinguish between patients with confidently high survival expectations and those with greater uncertainty. This can lead to actionable insights for high-stakes applications, mitigating the risks associated with over-reliance on potentially inaccurate MI. predictions.
1. Introduction
1.1. Background and Motivation
Survival analysis focuses on time-to-event data, with appli- cations in many fields including clinical trials, engineering, and marketing. For example, in clinical trial, researchers may aim to predict how long cancer patient is likely to survive based on individual characteristics and treatments re- ceived. Two central goals are modeling the probability that an event will not occur before a given time and predicting the actual event time. These tasks are complicated by the fact that the data are censored-the the exact event time may be unknown due study limitations or participant withdrawal.
1.2. Main Challenges and Contributions
A limitation of the methods proposed by Candès et al. (2023) and Gui et al. (2024) is their focus on type-l censoring, a see- nario not representative of many practical cases. In type-I censoring. all censoring times need to be observed, includ- ing for individuals who experienced an event. Observations are represented as (T", C), where T min (I', C), with T as the event time and C' as the censoring time. For example, in a clinical trial with fixed end date, C' is the time from enrollment to the trial's end, and patients are either censored (T > C) or experience the event (T < <C).
While traditional methods, such the Kaplan- Meier esti- mator and parametric models like the Cox proportional haz- ards model (Cox, 1972), are valued for their interpretability, they struggle in nigh-dimensional settings or when their as- sumptions are violated. As a result, machine learning (ML)
These methods, however, do not extend situations where the censoring times unobserved for individuals who experience the event-a more common practical scenario known as right- Under right-censoring, we only observe (T, < CJ), where 1[T' < C) indicates whether the event occurred before censoring. If < C, the censor- ing time unknown. For example, in survival study, T is the time until death, and C is the time until withdrawal or the study's end. If patient dies (T < C), we observe
Depaarmeet of Data Sciences and Operations, University of Southern California, Los Angeles, CA. USA. "Thomas Lord De- partment of Computer Science, University of Southern Califomiia, Los Angeles, CA, USA. 'Mrrck&oo., Inc., Rahway, NJ, USA. Correspondence to: Mattco Sesia < marssalll.se..dus
Proceedings of the 4nnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Doubly Robust Conformalized Survival Analysis with Right-Censored Data
T T= T but not C. Conversely, for censored individuals we observe T C but not T. This incomplete information complicates conformal inference, requiring novel approach.
For each E [n], let X; X' C RP represent a vector of p features, T; 0 the survival time, and C. > 0 the censoring time. Define the event indicator E, = [(T < C.) E {0, 1]. The observed time for each individual is T; min C.).
We address this challenge by introducing a method for con- structing informative LPBs from right-censored data, appli- cable to any survival model. This extends the approaches of Candès et al. (2023) and Gui al. (2024) using a two- step process. First, we fit a censoring model to estimate the conditional probability of censoring and use it to impute un- observed censoring times, transforming the right-censored dataset into a semi-synthetic type-I censored dataset. Sec- ond, we fit a survival model to estimate survival probabilities and construct LPBs using the imputed dataset.
Candès et al. (2023) and Gui et al. (2024) study a type-I censoring scenario, where the available data are D := {(,, T2. C.) which includes both C2 and T, for all individuals, along with their features X,. In that setting, they construct LPB for the survival time -n of new random individual, with features Xn+1, from the same pop- ulation. Their methods are reviewed in Appendix Al.
In the right-censoring scenario considered in this paper, the data are D. == {(X,, T;, Ea) These data are gener- ally less informative than those found in a type-I censoring setting, because they only include the true censoring times for individuals who did not experience event. This makes existing methods not directly applicable.
This method works well in practice and is doubly robust (Bang & Robins, 2005) in theory, ensuring asymptotically valid LPBs if either the censoring model or the survival model is consistently estimated, even if the other is not.
Given a right-censored data set DrC our goal is to predict the survival time Ta+1 of a new test individual with covariates X sampled from the same distribution as the previous data points. Ideally, we would like to construct an LPB for T, denoted by L(X,+1; i; DF.), that provides (1 o))
1.3. Related Work
This is not first work extending conformal inference to right- censored data. Qi et al. (2024) tackled this challenge by using the Kaplan-Meier estimate to impute the latent sur- vival times, operating under the assumption that that-despite its lack of covariate adjustment-it can approximate the condi- tional survival function reasonably well; see Appendix A1.3. However, this assumption is not always easy to justify. Fur- ther, unlike ours, their approach is not doubly robust as it relies entirely on having good approximation of the condi- tional survival distribution-the very quantity we are trying to infer. As we shall see, our method performs similarly to that of Qi et al. (2024) in easier settings where the survival model already yields approximately valid LPBs, and is able to offer more robust coverage in harder scenarios.
marginal coverage at the desired level E (0, 1):
P P[Tnt1 2 L(x D'9] 21-a. a
(I)
However, since exact finite-sample guarantees for survival analysis are generally unachievable without strong assump- tions, we instead aim to construct LPBs that are approxi- mately valid in finite samples. These LPBs are designed to satisfy a relaxed, asymptotic version of (1), which can be understood as a form of double robustness: they are asymp- totically valid as long as at least one of two key population quantities is estimated consistently, even if the other is not.
This work contributes to growing literature on confor- mal inference beyond exchangeability (Barber et al., 2023), particularly for handling incomplete data. Other works fo- cused on unobserved counterfactuals (Lei & Candès, 2021), missing covariates (Zaffran et al., 2023), weak supervision (Cauchois et al., 2024), and label noise (Feldman et al., 2023; Sesia et al., 2024; Clarkson et al., 2024).
We will formally define this double robustness property in Section 3, but for now, we focus on presenting our method, starting with its main underlying assumption.
Assumption 2.1. The observed data set DrC {(,, T., E.) is generated by applying reght-censoring to a latent data set {(X,, Ti, J consisting of covari- ates, survival times, and censoring times for n + 1 individ- uals, sampled i.i.d. from some joint distribution Px,r,c = Px Prix PeIx. where T 1 C X. The features and true survival time of the test data point, (Xa+1, T241 J. are sampled independently from Px Prix-
Borrowing from Candès et al. (2023) and Gui et al. (2024), we use weighted conformal inference techniques (Tibshi- rani et al., 2019) to address the challenge that, under right- censoring, the missing data may not be missing at random.
This setup can be summarized as follows:
2. Methods
(X,, Ti, C.)
Lid Px Prix PaIx, ~
Vi [n],
2.1. Problem Setup and Assumptions
E, I(T, < C.). i. min(T, C.),
(2)
We consider sample of individuals, indexed by [n]] := [I.....n), drawn i..dd from some unknown population.
(Xn+1, Px Prix
ind.
2

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Doubly Robust Conformalized Survival Analysis with Right-Censored Data
Above, Px, Prix, and Paix are arbitrary and unknown. The assumption T 1 C X, known "conditional inde- pendent censoring", states that I and C' are independent given X. This standard assumption in survival analysis is also made by Candès et al. (2023) and Gui et al. (2024).
tribution function, defined Fc x) - Jo Scix (c x:)de' Additionally, we assume the pre-trained censoring model M provides empirical estimates faxx of faxx and Faix of Faix, as it is typically the case in practice.
For example, if Mes is Cox proportional hazards model, it estimates the hazard function ho(c x) fcix (z x) / - (c x)] using a simple formula, from which fcix can be derived. Alternatively, if Meeas is random survival forest, standard implementations provide a non- parametric estimate of the conditional survival function 1 F Faix (c z). This estimate can be smoothly interpolated and differentiated to obtain fexx. For further details on computing fcix using standard survival analysis models, see the software repository accompanying this paper.
In addition to the calibration dataset Dre, our method as- sumes access to an independent training dataset unin+ con- sisting of analogous observations (X, T, F) from a separate group of individuals. These individuals are typically ex- pected to come from the same population, though this is not strictly required for the theoretical results in this paper.
Following a split-confomaal inference approach, we use Dis to train censoring and survival models, and D to transform their outputs into survival LPBs.
2.2. DR-COSARC
Next, leveraging our estimate fcix of fc|x, we will trans- form the right-censored dataset Die into a synthetic dataset Dit.e, designed to mimic the (unobserved) dataset Dil that would have been collected under a type-I censoring scenario.
2.2.1. 1. METHOD OVERVIEW
We name our method DR- COSARC, which stands for Dou- bly Robust COnformalized Survival Analysis under Right Censoring. This method uses two ML models: a survival model surl approximating Prix and a censoring model Moens approximating Palx- As detailed below, Maur guides the construction of a candidate LPB for Tn+1 as a function of X.+1. This candidate LPB can be adjusted either lower or higher via a scalar tuning parameter, cal- ibrated using D to (approximately) achieve the desired coverage level a. The censoring model Meses plays two key roles. First, it is used to impute the latent censor- ing times, simulating type-I censoring scenario. Second, it helps account for covariate shift in the calibration data, similar to Candès et al. (2023) and Gui et al. (2024).
For any i E [ri]. consider a right-censored random sample (x,, T. E,) from (2). Since C2 is latent, we replace it with a synthetic "imputed" time, C. computed as follows. If Fi- = 0, we know that C; < T, In this case, the true value of C is observed and equal Tū, so we can directly set C, T. Otherwise, if E, 1, we know that C, T, = Ti, but the true value of C, remains unknown. Fortunately, however, we have two key pieces of information that allow us to obtain a sensible "guess" C₁ of Cj: the property T IL C! X in Assumption 2.1 and the estimate fcix of fc
Concretely, if E; = 1, we sample C; from the distribution of C X | X,, C > T,, independent of everything else. Thanks the assumption that T JI C X, the probability density of C, as a function of the dummy variable e E R, can be written as:
2.2.2. TRAINING SURVIVAL AND CENSORING MODELS
The models Maury and Meens can be trained using any sur- vival analysis technique, like the standard Cox proportional hazards model, or more sophisticated ML approaches, in- cluding random survival forests (Ishwaran et al., 2008).
C₂ ~ fexx(el X1))
Ilc > T:).
(3)
AiX
where
Both models can be trained on the same right-censored training set - For M the event indicator is defined as usual, with a value of 1 indicating that T < C. For Moons the same techniques can be applied after flipping the event indicator: value of 1 now indicates that the event did not occur before the censoring time (C < T).
A(X,, T.) = - faillel X,)dc.
(4)
This procedure, outlined in Algorithm 1, provably leads to a synthetic sample (X,, T., C,) that shares the same distribu- tion as the ideal sample (X,, Tr, Ca) that would be obtained from (2) under type-I censoring, provided that Assump- tion 2.1 holds and faix is equal to feix.
2.2.3. IMPUTING THE MISSING CENSORING TIMES
To simplify the notation, and without much loss of gener- ality, we assume the conditional distribution of C given x = x has a continuous density, denoted by faxx (c x), with respect to the Lebesgue measure, for any x E X. Then, we let /f (c 2)) denote the corresponding cumulative dis-
Proposition 2.2. Under Assumption 2.1, let Px.ï.c denote the distribution of (X,, Tú, Ci), for any i E [n], obtained from (2) under type-l censoring. Then, Algorithm I applied with fcxx = fexx outputs independent triplets (T,, C.) whose distribution P Px.
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Doubly Robust Conformalized Survival Analysis with Right-Censored Data
Algorithm 1 Imputation of Latent Censoring Times
is filtered to include only samples where C2 2 co- The
input Pre-trained censoring model Mrrens right-censored calibration data {(x, Tú, F.)]==1-
filtered dataset is denoted by Ina = (i [n] : C; 2 co)- Third, a standard conformal prediction method for non- censored data is applied to calibrate an LPB for (Tn+1 Aco).
1: Using M compute an estimate fcix(e| (e| x) of the probability density fcix(e z) of C X = x.
In the third step above, the output LPB is obtained by cal- ibrating a candidate bound denoted as (X M"")), which depends on the survival model MM as well as on tunable parameter a. For example, leveraging conformal- ized quantile regression (CQR) (Romano et al., 2019), one can use [a; MM = qo (a;: M 1-a, a. for E R. where go. (2; M is an estimated a- a-quantile of the conditional distribution of T X, given by the model Masury
2: for = 1 to n do
3: if E; 0 then
4: Set C; = C.-
5: else
6: Randomly generate O₂ based on (3).
7: end if
8: end for
output Imputed censoring times (ÖI....,Ca,) Cn).
The main challenge is that the data (X,, (X,,T,A A.o) for E Teal are not exchangeable with the test point (Xn+1, Tu+1 A co) due the condition C' > co. which shifts their distribution. However, Candès al. (2023) noted that this difference is a covariate shift, enabling the use of weighted confor- mal inference (Tibshirani et al., 2019). This adjusts for the distribution shift by re-weighting the samples based on an estimate ((z) of the conditional censoring probabilities c(x) == P [C > co X = zr). obtained from Mcens
Implementing Algorithm 1 requires computing the normal- ization constant in (3), defined by the one-dimensional inte- gral in (4). This can be quickly evaluated either analytically or numerically, depending
the censoring model Meeess Then, sampling G. from (3) can be performed numerically using inverse transform sampling, which is also computa- tionally fast. Further implementation details are provided in the software repository accompanying this paper.
At first sight, it would seem that the method of Gui et al. (2024) can be directly applied to the imputed dataset Dimpuned output by Algorithm 1. Achieving double robust- ness, however, requires an additional step. Let L' ('(x,+1) denote the survival LPB computed by applying the method of Gui et al. (2024) to the imputed dataset Dimputed Instead of directly outputting L' (((n+1)), our method takes the mini- mum of i'(x,+1) 1) and uncalibrated estimate qa (1x+1) of the a-quantile of T' X Xn+1, provided by the sur- vival model M While this adjustment is important in theory to ensure the double robustness of our method, it often has a small impact in practice, as we will show empir- ically, because it is often true that i'(X,+1) S 9o (Xn+1).
After imputing (Ć) (Ci...., C,,) using Algorithm 1, either the method of Candès et al. (2023) or Gui et al. (2024) can be applied by substituting the imputed values C. for the unobserved C.. The specific steps for each approach are detailed in Sections 2.2.4 and 2.2.5, respectively.
We emphasize that, in practice, Algorithm I must be ap- plied using an estimate faxx of faxx. Nonetheless, as long as fcix is reasonably accurate, our two-step method is anticipated to yield approximately valid inferences. This parallels the expected behavior of the approaches proposed by Candès et al. (2023) and Gui et al. (2024), which achieve approximately valid survival LPBs using conformal weights derived from an estimated censoring model. We will formal- ize this intuition later in Section by establishing double robustness results for our method, which are qualitatively analogous to those derived by Candès et al. (2023) and Gui et al. (2024) under the simpler type-I censoring scenario.
Algorithm summarizes the main ideas of this implementa- tion of our method. See Appendix A2.1 and Algorithm A4 therein for further implementation details.
2.2.5. DR-COSARC WITH ADAPTIVE CUTOFFS
A limitation of the method of Candès et al. (2023), inherited by Algorithm 2, is its sensitivity to the choice of co. If co is too small or too large, the resulting LPBs tend to be too low to be informative, with the optimal choice often depending the data in a complex manner. While Candès et al. (2023) provide a heuristic for tuning co. it is not always guaranteed that a suitable value of co even exists, as noted by Gui et al. (2024) and confirmed by our numerical experiments.
2.2.4. CR-COSARC WITH FIXED CUTOFFS
We now describe how to implement our method by integrat- ing Algorithm with the approach of Candès et al. (2023), originally designed for data with type-I censoring, which apply with C, replaced by O for all i E [n].
The approach of Candès et al. (2023), outlined by Algo- rithm Al in Appendix AI.1, entails three main steps. First, the focus is shifted from constructing an LPB for T,+1 to constructing an LPB for (Tn+1 A co) = min[Ta+1, co). where 0o > 0 is a pre-defined cutoff constant. Second, Dille
To address this, Gui et al. (2024) extended the approach of Candès et al. (2023) by allowing Co to vary across individu- als based on their features X. Their approach can also use quantile regression (Romano et al., 2019) to compute can-
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Doubly Robust Conformalized Survival Analysis with Right-Censored Data
Algorithm 2 DR-COSARC with Fixed Cutoff'
method. Those bounds are valuable in theory to prove the asymptotic double robustness, but they too loose to be practically useful their own. Nonetheless, despite the difficulty of finite-sample analyses for this problem, DR- COSARC performs quite well empirically, especially in its adaptive-cutoff implementation, as shown in Section 4.
input Pre-trained censoring model Mr
pre-trained survival model M
right-censored data Dr = {(x, Tú B.)
significance level a E (0, 1), test covariates Xn+1. fixed threshold Co > 0.
1: Impute (C1...... C.) using Mgrens (Algorithm 1).
2: Assemble Dimputed: : {(x;, Tú, Ō.)
3.1. Double Robustness with Fixed Cutoffs
3: Compute an estimate ((x) of c(æ), using Means
We begin by studying Algorithm 2. For concreteness, we fo- cus on the implementation detailed by Algorithm A4, which uses quantile regression (Romano et al., 2019) to compute candidate LPBs in the form fa (z: M = qa (2; M 0, for R. where ga (1; MMry)) represents an estimate of go (z), the true conditional a-quantile of the distribution of T X, provided by the pre-trained survival model M
4: Apply Algorithm AI (Appendix AI.1) using Dimputed obtaining a preliminary LPB 1'(
5: Using M compute an estimate (x( of the a- quantile of the distribution of T |X= 2.
6: Compute L(X=+1) = min Í'(X ('(Xn+1): do (Xn+1)))
output A Al-a - a survival LPB ((x+1)).
Recall that Dirin denotes the independent training dataset, with size N = |Duninl, used to train M and M To establish double robustness, we assume that at least one of these models consistently estimates the relevant quantities. Assumption 3.1. The following two limits hold:
didate LPBs in the form [z; M) = qu (z; M - a, for E IR, where da (z; (2;MMurr)) represents an estimate of the true conditional a-quantile of the distribution Tf XX al- though this is not the only choice. Their method, described in Algorithm A2 in Appendix A1.2, often leads to much more informative LPBs.
1
lim E N-+00
= -0,
((x)
c(x)
Integrating Algorithm 1 with the approach of Gui et al. (2024) leads to the implementation of our method described by Algorithm 3, which often produces more informative LPBs compared to Algorithm 2. See Appendix A2.2 and Algorithm A5 therein for further implementation details.
fcix (c X)
fcxx (c X) A(X,
lim n-E N 71-000
de = 0.
A(X,T)
Assumption 3.1 requires that, with increasing training data, the censoring model Moens consistently estimates the two closely related relevant quantities: the conditional censor- ing probabilities c(x) used in the approach of Candès et al. (2023), and the conditional censoring density fcxx (c x) used in Algorithm 1. Importantly, fcix (c z) must con- verge to fcix (c æ>) at a rate faster than the growth of the calibration sample size n, suggesting that a larger portion of the available data should be allocated for training.
Algorithm 3 DR-COSARC with Adaptive Cutoff's
input Pre-trained censoring model Mo
pre-trained survival model M
right-censored data Dr = {(X,, 7, F,,E) F.)
significance level a E (0, 1), test covariates X
1: Impute (CI...., C,)) using Algorithm 1.
2: Assemble Dimpuredd = (X,, T,, C.)
3: Apply Algorithm A2 (Appendix A1.2) using Dimputed obtaining preliminary LPB 1'(Xn+1).
Assumption 3.2. The following two conditions hold:
4: Using M compute quantile of the distribution of T | X== = xz
estimate (x) of the a-
(i) There exists a constant >> 0 such that, for any > O, PP [ Z qa (x) + E X = x] 2 1-a a -a-be almost surely with respect to Px.
5: Compute ((xn+1) = min min(L'(Xxa) (''((++)) Ja (Xx+1))
output A 1 - a survival LPB ((xx+)).
(ii) lim E [l (X) yo (X) 0.
Assumption 3.2 requires that the survival model M consistently estimates (x),, the conditional a-quantile of T X, while also assuming that the true distribution of T X satisfies a relatively mild smoothness condition.
3. Double Robustness
Although it operates on right-censored data, DR-COSARC is asymptotically doubly robust the training and calibra- tion samples grow, akin to the methods of Candès et al. (2023) and Gui et al. (2024) under type-l censoring.
Theorem 3.3. Under Assumption 1. if either Assump- tion 3.1 Assumption 3.2 holds, the survival LPB i(xx+1) produced by Algorithm 2, implemented as detailed Ap- pendix 1, has asymptotically valid marginal coverage:
While we focus on the asymptotic regime here, Ap- pendix A5 also provides finite-sample coverage bounds for both the fixed- and adaptive-cutoff implementations of our
lim N --0
Tott 2 L(Xntz)) 2 1 a.
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Doubly Robust Conformalized Survival Analysis with Right-Censored Data
Further, under Assumption 3.2, it also has approximate conditional coverage, in the sense that, for any O,
(X1...., Xp) are generated independently, while T and C are sampled independendly conditional on X, from a log- normal distribution-log/ T' X ~ ((((X), (X))) (X))-or an exponential distribution C X ~ Exp(A(X)).
Jim P P [Tat 2 L(Xx+1) [x+1] > 1 - ---a0- cr- 1.
N-000
The three settings are ordered by decreasing difficulty. The first two simulate challenging scenarios where accurate sur- vival modeling is difficult, emphasizing the importance of conformal inference. In contrast, the third setting facili- tates easier survival model fitting, where raw LPBs from the model already provide approximately valid coverage.
3.2. Double Robustness with Adaptive Cutoff'
We study Algorithm 3, focusing for concreteness on the specific implementation detailed in Appendix A2.2.
Now, Assumption 3.1 is replaced by Assumption 3.4, which similar. Its first limit says that the function é should be an accurate estimate (up to a scaling constant) of ca. since in that case (Ca (z))) (ca(X)/(.(X)) R 1 for all x. Assumption 3.4.
Design and Performance Metries. We generate indepen- dent training, calibration, and test datasets, each with 1000 samples. Right-censoring is simulated by replacing the true T and C with T = min(T, C) and E = II( < C). The censored data are used to fit survival and censoring models, as specified below. Using these models and the calibration data, we compute 90% survival LPBs for the test set. Perfor- mance is evaluated by the average proportion of test points where the true survival time exceeds the LPB (targeting 90%) and the average LPB value, with larger values indi- cating more informative LPBs. To standardize comparisons across distributions, all LPBs are normalized by dividing by the average oracle lower bound in each setting. All experi- ments are repeated 100 times, and results are averaged.
The following two limits hold:
Co (o.Xn+11/Cc(Xntt
Jim sup E
-1 0,
N-000 aE(0,1]
E [xca(X)/Enll
fcix(e| x) fcix(e| lX) de = 0. T( A(X,T)
Jim n n-E -0000
Further, some mild technical conditions are needed.
Assumption 3.5. The function (z) used to compute can- didate bounds by the approach of Gui et al. (2024) in Algo- rithm 3 (see Algorithm A5 for details), continuous in a for Px-almost all x. Further, for any there exists constant % > O such that 1/ (x) S 9a for Px-almost all x.
Models. We consider four model families for Mrrens and Maurr ensuring consistent comparisons across different cal- ibration methods. The models are as follows: (1) grft a generalized random forest (R package gr (2) survreg, an accelerated failure time model (AFT) with a lognormal dis- tribution (R package surviva (3) rf. generalized ran- dom forest (R package randomF orest SRC)) (4) cox, the Cox proportional hazards model (R package survival).
Then, prove Algorithm 3 is also doubly robust.
Theorem 3.6. Under Assumptions 2.1 and 3.5, if either Assumption 3.4 or Assumption 3.2 holds, the LPB L(X.-1) produced by Algorithm 3, implemented as detailed in Ap- pendix A2,2, has asymptotically valid marginal coverage:
lim N-
PP P i(x,+1)) 2 a.
Calibration Methods. We compare six methods. Ora- cle is an idealized "method" that knows Prix and directly returns the lower 10% quantile, without using the data, Un- calibrated outputs the raw 90% LPB produced by Matrr without any calibration. Naive CQR applies CQR using
Further, under Assumption 3.2, it also has approximate conditional coverage, in the sense that, for any > O,
lim P N 400
P[p[Tn+ TT+1 2 i(Xn+1) [x+1] > 1 a--a--
Tat1 = Tu+1 A Cr+1 as the target of inference instead of Ta+1, typically leading to very small LPBs (Candès et al., 2023). KM Decensoring refers to the method of Qi et al. (2024), reviewed in Appendix 1..3. DR- CO-COSARC (fixed) and DR- DR-COSARC (adaptive) are our methods, implemented as detailed in Appendix A2.1 and A2.2, respectively.
Next, we will verify that the empirical behavior of our method mirrors this appealing theoretical property.
4. Numerical Experiments
To simplify comparisons, all calibration methods are applied using the same survival model. While the Uncalibrated ap- proach could, in principle, benefit from a survival model trained on a larger dataset, doing so would complicate com- parisons and does not affect our main conclusions. As shown in Appendix A3, increasing the training sample size does not resolve the reliability issues of the Uncalibrated
4.1. Setup
Synthetic data. We consider three data-generating distri- butions, summarized Table A1 (Appendix A3.1), which span a range of interesting settings, partly inspired by related previous works. In each setting, p = 100 covariates X =
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Doubly Robust Conformalized Survival Analysis with Right-Censored Data
method in the challenging settings where this approach fails.
with its double robustness property. When the censoring model is trained with all 1000 available samples, the cov- erage reaches the target level, and the experimental setup aligns with that of Figure 1.
Leveraging Prior Knowledge Pejx- To examine the effect of incorporating prior knowledge about the censor- ing distribution, we fit Moens using only the first P1 S P covariates, assuming C' is independent of (X.++1...., Xp) In the lata-generating distributions used in these experiments (Table A1), C X is inde- pendent of X1,,... X000 in all settings. Therefore, for 10 S P1 S p 100, this prior knowledge helps improve the censoring model by excluding irrelevant predictors and mitigating overfitting. We start with p1 = 10 and later evalu- ate the impact of larger pl. representing weaker prior knowl- edge. The case p1 = p corresponds to no prior knowledge, where all covariates are used fit the censoring model.
4.4. Additional Experiments
given (X Xp,).
Appendix A3.2 presents the results of additional experi- ments examining the impact of varying the number of train- ing samples for the censoring model, similar to Figure 2 but under the relatively easier settings 2 and 3. In those settings, our method consistently achieves valid coverage, and the performance of its predictions appears largely unaffected by the number of training samples for the censoring model.
Appendix A3.3 summarizes the results of experiments sim- ilar to those in Figure 2. but with both the survival and censoring models fitted using a varying number of train- ing samples. These results show that our method tends to achieve valid coverage as the sample size increases, even when other approaches either continue to underperform in terms of coverage or produce overly conservative inferences.
4.2. Results
Figure 1 compares the performance of the six methods in settings 1 1-3, based on the grf models.
In the first setting, the Uncalibrated method leads to under- coverage. KM Decensoring provides no improvement in this case, as the Kaplan-Meier survival curve it uses to impute T T > C' fails to reasonably approximate the true distribution of T T > C, X. In contrast, DR-COSARC achieves coverage close to the desired 90% level. However, its coverage is still slightly below the target, and the average value of its lower bounds is noticeably lower than that of the oracle, reflecting the high intrinsic difficulty of this setting.
Appendix A3.4 investigates the effect of varying numbers of covariates used to fit the censoring model. The results show that our method performs better when the number of covariates used in the censoring model not too large, highlighting the advantage of leveraging accurate prior in- formation to prevent overfitting.
Appendix A3.5 examines the effect of the calibration sample size. The results indicate that the average performance of all methods is generally not heavily influenced by the number of calibration samples, although larger calibration sizes do tend to reduce variability.
In the second setting, the Uncalibrated method continues to be invalid, as does KM Decensoring. However, DR- COSARC performs well, achieving 90% coverage and pro- viding relatively high (more informative) LPBs, approach- ing the oracle's performance. This success is due its ability to model the censoring distribution accurately.
Appendix A3.6 shows empirically that the double robustness adjustment in Algorithms! 2 and 3 often has a small effect in practice.
In the third setting, all methods except Naive CQR perform similarly. In this simpler scenario, Mysure is highly accurate, making conformal calibration less necessary.
Appendix A3.7 studies the algorithmic stability of our method with respect to the randomness in the imputation of latent censoring times. The results show the adaptive version of our method is more stable than the fixed- cetoff variant, with stability improving as the calibration set grows.
4.3. Impact of the Censoring Model Quality
Figure 2 presents results from experiments similar to those in Figure 1, using only subset of the available 1000 train- ing samples to fit the censoring model. The aim is to ex- amine how the quality of the censoring model affects the performance of our method, specifically in the challenging setting 1. The results indicate that when the censoring model is trained using fewer samples- leading to lower-quality imputation-our method fails to provide valid coverage, performing comparably to the approach of Qi et al. (2024). However, as the number of training samples increases and the quality of the censoring model improves, the coverage of our method approaches the desired 90% level, consistent
Appendix A3.8 examines the effect of using different sur- vival and censoring models. These results show our method performs robustly across different survival and censoring models, except for the Cox model, which struggles with complex censoring patterns in the most challenging cases.
Appendix A3.9 presents the results of additional experi- ments conducted under different synthetic data settings, bor- rowed from Candès et al. (2023) and Gui et al. (2024). These experiments yield similar findings. A complete list of all 10 settings considered is provided in Table A2.
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Doubly Robust Conformalized Survival Analysis with Right-Censored Data
Coverage
Lower Bound
raacle
Uncalibrated
[|-
Naive CQR
KM Decensoring
DR-COSAR C (fixed)
DR-COSARC (adaptive)
racle
Method
Uncalibrated
Naive CQR
KM Decensoring
DR-COSARC (fixed)
DR -ROSCACC (adaptive)
Oracle
Uncalibrated
Naive CQR
KM Decensoring
DR-COS AR (fixed)
DR-COSARC (adaptive)
0.00
0.25
0.50
0.75
1.000.00
0.25
0.50
0.75
1.00
Figure 1. Performance synthetic data under three different settings of our method for constructing lower confidence bounds on the true survival time a new individual based on right-censored data, compared to existing benchmark approaches. Performance measured by empirical coverage, aiming for 90% nominal coverage (dashed red line), and the average value of the lower bound (higher is better, provided coverage valid). The number of training samples available to fit survival censoring models is 1000. three settings correspond to situations in which fitting accurate survival model is increasingly easy, with rigorous conformal calibration being most essential in setting and less crucial in setting 3.
designs and sizes; Table A3 in Appendix A4 provides details on the number of observations, covariates, and data sources.
Coverage
Lower Bound
1.0
6
0.8
We apply standard preprocessing to each dataset to handle outliers, missing values, and ensure compatibility with all learning algorithms. Zero survival times are replaced with half the smallest non-zero time in the dataset, missing values are imputed using the median for numeric variables and the mode for categorical variables, and rare factor levels are merged into an "other" category or removed for binary fac- tors. Features with high pairwise correlations are iteratively filtered, and linearly redundant variables are removed. See Appendix A4 for additional details about preprocessing.
0.6
0.4
0-
100
300
1000
100
300
1000
Number of training samples for censoring model
Oracle
KM Decensoring
Method
Uncalibrated
-COSARC (fixed)
Naive CQR
DR-COSARC (adaptive)
We compare method against the same three benchmark approaches considered in Section 4: Uncalibrated, Naive CQR, and KM Decensoring. Because the ground truth data distribution is unknown for these data, the Oracle method cannot included. Additionally, as the experiments Sec- tion demonstrate that the adaptive-cutoff implementation of our method consistently outperforms the fixed-cutoff im- plementation, focus solely the adaptive version here, referring to it simply as DR-COSARC for clarity.
Figure Performance on synthetic data of our method as a fune- tion of the number of training samples used to fit the censoring model, compared to other benchmark approaches. The number of training samples available to fit the survival model is fixed equal to 1000. These experiments are conducted under setting 1. where fitting accurate survival model most difficult. The results demonstrate the double robustness property of our method, which requires only of the survival censoring models be accurate in order to achieve valid coverage. Other details are as in Figure 1.
All methods use the same four types of model as in Section 4 to estimate the survival distribution (grf. survreg, if. and cox), with the censoring distribution always estimated using grf. The datasets are split into 60% for training, 20% for calibration, and 20% for testing, and each experiment is repeated 100 times using independent random splits.
5. Application to Real Data
We apply our method to seven publicly available datasets: VALCT. PBC, GBSG, METABRIC, COLON, HEART, and RETINOPATHY. These datasets cover range of study
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Doubly Robust Conformalized Survival Analysis with Right-Censored Data
We evaluate the performance of the survival LPBs pro- duced by each method the test set in terms of esti- mated average coverage (targeting the nominal 1 - level) and average LPB value (higher is better). Since the test data are censored, the true survival times for censored individuals are unobserved, making exact coverage eval- uation infeasible. Following the approach of Gui et al. (2024), we estimate empirical lower and upper bounds for the average coverage: Bww ((1(X I Ta+1] and Supp ([
6. Discussion
This paper introduces a novel conformal inference method for constructing lower prediction bounds (LPBs) for survival times from right-censored data, extending recent methods designed for type-I censoring. The proposed approach is asymptotically doubly robust in theory and demonstrates strong empirical performance, producing LPBs that are both informative and robust compared to alternative methods.
Tn+1 S Catil. These bounds satisty Slow S ([L((x-1) I Ta+1]= I Supp- To simplify comparisons, we also report a point estimate of the coverage, defined as the midpoint Baid = (fow + Supp)
Our numerical experiments revealed two key insights. First, the adaptive implementation of our method, inspired by Gui et al. (2024), significantly outperforms the fixed-cutoff ver- sion (Candès et al., 2023), and we recommend its use in prac- tice. Second, real data experiments showed that uncalibrated survival models often produce reasonably well-calibrated raw LPBs, though they may fail in more complex scenarios. Our method performs relatively well in these challenging cases, where conformal inference is most critical.
0.95
Estimated coverage
0.90
A limitation of our method is its focus on lower prediction bounds, similar to Candès et al. (2023) and Gui et al. (2024), However, Holmes & Marandon (2024) very recently pro- posed a method for constructing also corresponding upper bounds, suggesting opportunities for combining these ap- proaches. Another promising direction for future work is to extend our method handle possible data errors, such as in- accuracies in observed times mislabeled events, building ideas from Sesia et al. (2024).
0.85
0.80
0.75
Uncalibrated
Naive CQR
KM Decensoring
DR DR-COSARC
Method
Figure 3. Distribution of average (estimated) coverage of survival LPBs computed by different methods, across seven real datasets and four survival models. The nominal coverage level is 90%.
Finally, future work could explore strategies to reduce the algorithmic variability of our method, which arises from both random data splitting and stochastic imputation of latent censoring times. Potential directions include the use of e-values (Vovk& & Wang, 2021; Bashari et al., 2023) or adopting full-conformal approach (Vovk et al., 2005).
Figure summarizes the distribution of average estimated coverage across the seven datasets and four models, at level = 0.1, with Tables A4-A7 in Appendix A4 reporting the detailed results obtained in each setting. Figure A18 in Appendix A4 summarizes similar results obtained using different values of a. Overall, the results indicate that Uncal- ibrated and KM Decensoring tend to achieve slightly lower- than-expected coverage, while Naive CQR is overly con- servative. In contrast, DR-COSARC consistently achieves average coverage closer to the desired level.
Software Availability
A software implementation of the methods described in this paper is available online at https: / / /github.... com/ msesia/ cot.i//aooooooaaaaccmm sawooooooomlaauuvvaaalaaaaaa survival.
Acknowledgements
These findings align with the results on synthetic data pre- sented in Section 4. The relatively modest undercoverage observed with Uncalibrated and KM Decensoring in Fig- ure 3 reflects the comparatively simpler nature of these datasets, where the survival model seems reasonably well calibrated even without conformal inference, in contrast to the more challenging synthetic scenarios discussed earlier. Of course, in practice one would typically not know whether the fitted survival model is sufficiently accurate, and our con- formal inference method is precisely designed to offer an additional layer of protection in those cases.
The authors thank the anonymous referees for their helpful comments on an earlier version of this manuscript.
Impact Statement
This paper presents work whose goal is to advance the field of Machine Learning- There many potential societal consequences of our work, none which we feel must be specifically highlighted here.
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Doubly Robust Conformalized Survival Analysis with Right-Censored Data
References
Kalbfleisch, J. D. and Prentice, R. L. The statistical analysis failure fime data. John Wiley & Sons, 2002.
Bang, H. and Robins, J. M. Doubly robust estimation in missing data and causal inference models. Biometrics, 61 (0):962-973,, 109997000000 2005.
Katzman, J. L., Shaham, U., Cloninger, A., Bates, J., Jiang, T., and Kluger, Y. Deepsurv: personalized treatment recommender system using Cox proportional hazards deep neural network. BMC Med. Res. Methodol., 18: 1-12, 2018.
Barber, R. F., Candès, E. J., Ramdas, A., and Tibshirani, R. J. Conformal prediction beyond exchangeability. The Annals of Statistics, 51(2):116-845 2023.
Lei, and Wasserman, Distribution-fres prediction bands for non-parametric regression. R. Stat. Soc. Ser. B, 76 (1):71 1-96, 2014.
Bashari, M., Epstein, A., Romano, Y., and Sesia, M. De- randomized novelty detection with FDR control via con- formal e-values. In Advances in Neural Information Pro- cexsing Systems, volume 36, pp.
2023.
Lei, L. and Candès, E. J. Conformal inference of counter- factuals and individual treatment effects. J. Stat. Soc. Ser. B, (355):9119938 2021.
Blair, A., Hadden, D., Weaver, J., Archer, D., Johnston, P., and Maguire, C. The 5-year progmosis for vision in diabetes. The Ulster Medical Journal, 49(2):139, 1980.
Moertel, C. G., Fleming. T. R., Macdonald, J. S., Haller, D. G., Laurie, J. A., Goodman, P. J., Ungerleider, J. S., Emerson, W. A., Tormey, D. C., Glick, H., et al. Lev- amisole and fluorouracil for adjuvant therapy of resected colon careinoma. New England Journal of Medicine, 322 (6):352-33) -58, 1990.
Candès, E., Lei, L., and Ren, Z. Conformalized survival analysis. J. R. Stat. Soc. Ser. B, 85(1):24-45, 2023.
Cauchois, M., Gupta, S., Ali, A., and Duchi, J. C. Predictive inference with weak supervision. Journal of Machine Learning Research, 25(118):1 45, 2024.
Qi. S.-A., Yu, Y., and Greiner, R. Conformalized survival distributions: A generic post-process to increase calibra- tion. In Proceedings of the 41st International Conference Machine Learning, volume 235, pp. 41303 41339. PMLR, 2024.
Clarkson, J., Xu, W., Cucuringu, M., and Reinert, G. Split conformal prediction under data contamination. In Con- formal and Probabilistic Prediction with Applications, volume 230, pp. 5-27. PMLR, 2024.
Cox, D. Regression models and life-tables. R. Stat. Soc. Ser. B, 34(2):187-202, 1972.
Romano, Y., Patterson, E., and Candès, E. Conformalized quantile regression. Advances neural information pro- cessing systems, 32, 2019.
Crowley, J. and Hu, M. Covariance analysis of heart trans- plant survival data. Journal of the American Statistical Association, 72(357):27-36, 1977.
Sesia, M., Wang, Y. R., and Tong, X. Adaptive conformal classification with noisy labels. J. R. Stat. Soc. Ser. B, pp. qkae 14, 2024.
Curtis, C., Shah, S. P., Chin, S.-F., Turashvili, G., Rueda, O. M., Dunning, M. J., Speed, D., Lynch, A. G., Samara- jiwa, S., Yuan, Y. al. The genomic and transcriptomic architecture of 2,000 breast tumours reveals novel sub- groups. Nature, (86674033)334--3323 2012.
Therneau, T. M., Grambsch, P. M., Therneau, T. M., and Grambsch, P. M. The Cox model. Springer, 2000.
Tibshirani, J., Foygel Barber, R., Candès, E., and Ramdas, A. Conformal prediction under covariate shift. Advances in neural information processing systems, 32, 2019.
Feldman, S., Einbinder, B.-S., Bates, S., Angelopoulos, A. N., Gendler, A., and Romano, Y. Conformal prediction is robust to dispersive label noise. In Conformal and Probabilistic Prediction with Applications, pp. 624-626. PMLR, 2023.
Vovk, V. and Wang, R. E-values: Calibration, combination and applications. The Annals of Statistics, 1736- 1754, 2021.
Gui, Y., Hore, R., Ren, Z., and Barber, R. F. Conformalized survival analysis with adaptive cut-offs. Biometrika, 111 (2):459-477, 2024.
Vovk, V., Gammerman, A., and Shafer, G. Algorithmic learning in a random world, volume 29, Springer, 2005.
Zaffran, M., Dieuleveut, A., Josse, J., and Romano, Y. Con- formal prediction with missing values. In Proceedings of the 40th International Conference on Machine Learn- ing, volume 202 of Proceedings of Machine Learning Research, pp. 40578 40604. PMLR, 23-29 Jul 2023.
Holmes, C. and Marandon, A. Two-sided conformalized survival analysis. arXiv preprint arXiv:2410.241336 2024.
Ishwaran, H., Kogalur, U. B., Blackstone, E. H., and Lauer, M. Random survival forests. The Annals of Applied Statistics, 2(3):841 860, 2008.
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Doubly Robust Conformalized Survival Analysis with Right-Censored Data
A1. Review of Existing Conformal Inference Methods
A1.1. Conformalized Survival Analysis for Type-I Censored Data (fixed)
Algorithm Al outlines the conformalized survival analysis method proposed by Candès et al. (2023), designed for data subject to type-I censoring. The method requires several key inputs in addition to the censored calibration data: (1) a pre-trained survival model that approximates the conditional distribution of T X: (2) sequence of functions used to compute candidate survival lower bounds; (3) pre-trained censoring model that approximates the conditional distribution of C' X and is utilized to compute the necessary weights to account for covariate shift (Tibshirani et al., 2019); and (4) fixed threshold cu > 0 for the censoring times. These inputs are typically trained on a separate dataset, independent of the calibration data (Candès et al., 2023).
Algorithm A1 Conformalized Survival Analysis with Fixed Cutoff (Candès et al., 2023)
Require: Pre-trained survival model M sequence of functions (fa (2; M
to compute candidate survival lower bounds, pre-trained censoring model Moens fixed cutoff co > 0, calibration data with type-I censoring {(X,, T,. C-) significance level a E (0, 1), test covariates Xn+1-
Jaejo.1] M
1: Define the filtered calibration subset T.ul = {i E (1,...,n) C 2 co)-
2: for each é Teal do
3: Compute the conformity score:
V, inf E A: fa(,)) S 7, not
4: end for
5: Define the function vi(x), estimating 1/c(x) = 1/PPCC 2 co x = x)) using Moonn Va.
6: For each E Teal compute the weight W; = (Xii) E oo).
7: Compute the weights for Xn+1:
W (X = Ejer W, + ((((++))
(i(Nn+ (Xnt2)))) = ΣjEr, W3 + (Xn+1)
8: Compute:
((nn+1) = Quantile 11 az Σ P:(X+t1)6n + t (Xo
*ET
9: Output the calibrated lower prediction bound:
i(X.+1) (n(Xn)))(Xat) A co.
Implementation Details. While Algorithm A1 is quite flexible, allowing the candidate survival lower bounds to be computed using any set of functions (x; MM""M) indexed by a real-valued calibration parameter E A C R, in practice, we adopt one of the simpler implementations proposed by Candès et al. (2023), This approach is inspired by the conformalized quantile regression method of Romano et al. (2019) and defines fa (z; Maure) = 4a M a, for A = RR where (a; M represents the estimated a-quantile of the conditional distribution of T X, given by the survival model M In this case, the conformity scores are simply given by V42 (X,) (T, A co). and the output prediction lower bound is i(x.+1)
m)(x++))) co-
Data-Driven Tuning of co. Candès et al. (2023) proposed algorithm for tuning the cutoff parameter co adaptively, using the training data set. In this paper, we apply their method using simpler approach for tuning co, which we always set equal to the median of the observed censoring times. We found this approach to be relatively more stable in practice.

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Doubly Robust Conformalized Survival Analysis with Right-Censored Data
A1.2. Conformalized Survival Analysis for Type-I Censored Data (adaptive)
Algorithm A2 presents the conformalized survival analysis method proposed by Gui et al. (2024), a follow-up to the work of Candès et al. (2023). The goal of this more recent approach is to enhance the adaptability of conformal survival analysis for data subject to type-I censoring by incorporating more flexible, rovariate-dependent threshold for the censoring times. As demonstrated in Gui et al. (2024) and corroborated by our numerical experiments, the adaptive strategy of Algorithm A2 often leads to more informative lower prediction bounds compared to the original method proposed by Candès et al. (2023).
Similar to Algorithm AI, this method requires the following inputs in addition to the censored calibration data: (1) a pre-trained survival model that estimates the conditional distribution of T X; (2) sequence of functions used compute candidate survival lower bounds; and (3) a pre-trained censoring model that approximates the conditional distribution of C X, which is used compute weights for adjusting covariate shift (Tibshirani et al., 2019). These models are typically trained on separate dataset, independent of the calibration data (Candès et al., 2023). Unlike Algorithm A1, however, Algorithm A2 does not require the specification of a fixed censoring threshold co > 0, and this is its main advantage.
Algorithm A2 Conformalized Survival Analysis with Adaptive Cutoffs (Gui et al., 2024)
Require: Pre-trained survival model M sequence of functions { (z; M
Jaejo.11 to compute candidate survival lower bounds, pre-trained censoring model Myeens calibration data with type-I censoring ((x,, Tú C,)) significance level a E (0, 1), test covariates Xn+1-
Define the function uh (z)), estimating (z) == 1/P(C 2 fa()) X = x) using Moems Va, x.
2: Determine AA using the computational shorteut from Gui al. (2024), as detailed in Equation (A6).
3: for each E A do
4: Compute the estimated miscoverage rate:
a(a) = ET th (XX C)
Ei (X.) SC)
5: end for
6: Compute the threshold:
à = supfa E A: sup a(d') sa). of sa
(A5)
7: Output the calibrated 1 lower prediction bound:
i(Xn+1)= = fx(n11)).
Computational shorteut. In general, the threshold à in (A5) can be computed efficiently using the following shortcut, originally described in Gui et al. (2024), which can also be utilized to implement Algorithm A5 efficiently. Note that su á(a') is non-decreasing piecewise constant function in with no more than 2n knots values of at which the indicators I[T, < (X,) S C.}. or (X,,) S C.} change signs. Denote a = sup.c(o.1] [x,,)(X,) s T.) and à su (X,, C.), and let A {ā, ; = 1,...,, ......... n) and A = (ā. : .. n). Then, by definition, the breakpoints of the piecewise constant map + a(a) must all lie in A₁ U Ag. Therefore, to compute à, we only need to search through the finite grids
A= U, (0).
(A6)
Implementation Details. Similar to Algorithm AI, Algorithm A2 is quite flexible, allowing the candidate survival lower bounds to be computed using any set of functions fx( Manr) indexed by a continous calibration parameter E [0, 1]. []] In this paper, we adopt one of the simpler implementations proposed by Gui et al. (2024). This approach defines
fz;;; Ma: M.sur aa (z; Moury,, for A = [0, 1]. where ÿa (z; M represents the estimated a-quantile of the conditional distribution of T X. given by the survival model M
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Doubly Robust Conformalized Survival Analysis with Right-Censored Data
A1.3. Conformalized Survival Analysis via KMM Decensoring
Algorithm A3 presents the conformalized survival analysis method proposed by Qi al. (2024), which, similar to our paper. focuses the analysis of data subject to right censoring.
This approach uses the Kaplan-Meier estimate to impute the latent survival times, and is thus very different from our method, which imputes the latent censoring times.
Algorithm A3 Conformalized Survival Analysis via KM Decensoring (Qi et al., 2024)
Require: Level a, calibration data Deat = {(X,, Th C)) test features æ, functions (zz, yi D) (conformity score), (x;;; D) (weight function), C(D) (threshold selector), right-censored calibration data ((X,, T, E,))
Phase 1: Imputation of Latent Event Times via KM Sampling
2: Compute the Kaplan-Meier (KM) survival function:
dg
(t)
II S
1
based on {(T), E.)_
3: for 1 n do
4: if E, then
5: Define the conditional KM survival function:
(t 1t> [;) min { SKM(()
(kM((A)) (i))
6: Set Ti equal a random sample generated with density:
Sku(l 1 C.)[(I C.).
7: else
8: Set T T,
9: end if
10: end for
11: Phase 2: Conformal Prediction with De-Censoree Data
12: for i = to n do
13:
Compute the conformity score V; = V(X,,Ty).
14: end for
15: Compute:
1
1
(πx) Quantile
1
Σ
5v. +
800
n +
+
i=1
16: Compute the lower prediction bound:
Úx, Dea) = inf V(x,)) v) S n(x))
17: Output the calibrated lower prediction bound:
i(xn+1;Daa)
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Doubly Robust Conformalized Survival Analysis with Right-Censored Data
A2. Additional Methodological Details
A2.1. DR-COSARC with Fixed Cutoffs
Algorithm A4 provides further details on the implementation of our method sketched by Algorithm 2, which integrates Algorithm 1 with Algorithm Al in Appendix AI.1, the approach of Candès et al. (2023) for conformalized survival analysis under type-I censoring.
Algorithm A4 DR-COSARC with Fixed Cutoffis (detailed implementation)
Require:
Pre-trained survival model Masii sequence of functions (ffa (a: M JaE to compute candidate survival lower bounds, pre-trained censoring model Myses, right-censored calibration data {(X,, T,, E,) fixed threshold co > 0, significance level a: E (0, 1). test covariates Xu+1-
1: Phase 1: Imputation of Latent Censoring Times
2: Generate (C,...., C,,) using Algorithm I, and define Dew = ((X,, C.
3: Phase 2: Conformal Calibration
4: Define the filtered calibration subset Ien ii E [.. 2 cn..
5: for each i Tcal do
6: Compute the conformity score V; inffa E A: fx,,)) <T, Ao).
7: Compute the weight W, ),), where vi()) estimates 1/c(x) = 1/P((C 2 x== using Moons
8: end for
9: For each E Teat, compute the weights:
W (Xxx+1))= Ejere W, + ((x+1))
(x())+1) Poo(Xn+1)= Ejetal W3 + u (((x+1))
10: Compute:
((a+1)) = Quantile 11 CK; Σ
(x(()1)) + p:o (Xn+1))))
iEIca
Compute ('(xn+1: = fx(x,+))) (xne++(((++))))))) (Xn+1) cp-
12: Phase 3: Adjustment for Double Robustness
13: Extract the function J. (z) from M uncalibrated estimate of the a-quantile T' X
14: Adjust the lower prediction bound:
= mxi()))=nn((( Dai), 2n (Xn+1))
15: Output: A calibrated 1 - lower prediction bound L(x,-1).
Implementation details for Algorithm A4. While Algorithm A4 is quite flexible, allowing the candidate survival lower bounds be computed using any set of functions (2; M indexed by real-valued calibration parameter a E A IR, in practice, we adopt one of the simpler implementations proposed by Candès et al. (2023). This approach is inspired by the conformalized quantile regression method of Romano al. (2019) and defines fa Mrs")) qa M
for A = R. where a (x; Masurr represents the estimated a-quantile of the conditional distribution of T X, given by the survival model Maury, In this case, the conformity scores are simply given by V = (X,) - (T, A co), and the output prediction lower bound is L(xa+1) = (â. (Xa+1) ) (X(++1)) A Cp-
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Doubly Robust Conformalized Survival Analysis with Right-Censored Data
A2.2. DR-COSARC with Adaptive Cutoffs
Algorithm A5 provides further details on the implementation of our method sketched by Algorithm which integrates Algorithm with Algorithm A2 in Appendix A1.2, the approach of Gui et al. (2024) for conformalized survival analysis under type-I censoring.
Algorithm AS POR-COAARC with Adaptive Cutoffs (detailed implementation)
input Pre-trained survival model M sequence of functions (2; M ]eE[..1] to compute candidate survival lower bounds, pre-trained censoring model Myrens right-censored calibration data {(X,, Tú, E.)) significance level o' E (0, 1). test covariates X,+1-
1: Phase I: Imputation of Latent Censoring Times
2: Generate (Ci...., ,,,,,,,,)))))))))))) using Algorithm I, and define Deat ((,;,T;, CD
3: Phase 2: Conformal Calibration
4: Define the function via (2), estimating (z) == /PC(C fa(s) x= x) using Moens Va, x.
5: Determine A using the computational shortcut from Gui al. (2024), as detailed in Appendix Al.
6: for each a E A do Compute the estimated miscoverage rate:
á(a) = D th (X, f((,)) CD
R We wa (x) s C
8: end for
9: Compute the threshold:
à supfa A: sup a(a') s a). of d'
10: Phase 3: Adjustment for Double Robustness
11: Extract the function (z) from M an uncalibrated estimate of the a-quantile of Tf X æ.
12: Adjust the lower prediction bound:
((X,+1) = fa (X
13: Output the calibrated 1 lower prediction bound L(Xn+1).
Implementation details for Algorithm A5. In this paper, we implement Algorithm A5 using candidate LPBs in the form [a; M 9. (z; M for E R.
The threshold à in (4) can be computed efficiently as follows. We note that sa á(a') is non-decreasing piecewise constant function in a, with no more than 2n values a at which the indicators 1(T; < fX, S C.} or (X., < C.} change signs.
Denote ā; = suP.c(o. (X,, < } and a; = supaejo. (x,, S Ci), and let A = (ā. = 1,..., n) and A2 (ā; i = 1,...,7, n). Then, by definition, the breakpoints of the piecewise constant map a + a(a) must all lie in A U A2. In the implementation, in order to obtain à, we only need to search through the finite grids
A = A U A2 U (0).
15

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Doubly Robust Conformalized Survival Analysis with Right-Censored Data
A3. Additional Numerical Results
A3.1. Additional Details on the Experiments of Section 4
Table AI. Summary three synthetic data generation settings considered Scction 4, listed in decreasing order of intrinsic difficulty.
Setting
p
Ref.
Covariate, Survival, and Censoring Distributions
X : Uni 1J2)
1
100
T X: LogNormal, (x(X) = (X2 > 2) + (Xa < 2 + (1 x1)0
(X() = 1-X1
C | X: LogNormal, (x (X2 > 2)) + (X3 < 2)) + (1 X1) + 10.
(x( X2
X : Unif([0,1}")
2
100
T X: LogNormal, (x(X) X (x(X)
C X: LogNormal, (e(X) Xi (x 0.1
X : Unif[[-1,
3
100
(Candès et al., 2023)
T X: LogNormal, (x) = log(2) + 1 + 0.55 XgX5), a(X) =
X +1
C X: Exponential, Xx 0.4
Setting: 1 (hard)
S Setting: (medium)
Setting: 3 (easy)
4
1e402
3-
10
Time
Time
5-
2-
1e400
Censoring
Event
3-
1e-01+
1e-02
0.50
0.75
1.00
0.00
0.25
0.50
0.75
1.00
-0.5
0.0
0.5
1.0
Figure AI. Visualization of the synthetic data distribution under the three experimental settings outlined in Table A1. The censoring and event times (in different colors) are plotted against the first covariate, X1.
16

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Doubly Robust Conformalized Survival Analysis with Right-Censored Data
A3.2. Impact of the Censoring Model Quality
Setting: 2
Setting:
Coverage
Lower Bound
1.0
2.0
0.8
1.5 -.5-8
0.6
1.0
Method
0.5
0.4
r
0.0
100
300
1000
100
300
1000
Uncalibrated
Setting: 3
Setting: 3
Naive CQR
Coverage
Lower Bound
KM Decensoring
1.0
1.00
DR-COSARC (fixed)
0.8
0.75
DR C (adaptive)
0.6
0.50
0.25
0.4
0.00
100
300
1000
100
300
1000
Number training samples for censoring model
Figure A2. Performance synthetic data of our method function of the number of training samples used to fit the censoring model, compared to other approaches. The number of training samples available to fit the survival model fixed equal to 1000. These experiments conducted under settings 2 and 3 (see Table Al), where fitting accurate survival model is easier. Other details as in Figure
A3.3. Impact of the Total Number of Training Samples
Figure A3 summarizes the results of experiments similar to those in Figure 2, but with both the survival and censoring models fitted using varying number of training samples. These results show that method tends to achieve valid coverage as the sample size increases, even when other approaches either continue to underperform in terms of coverage or produce overly conservative inferences. Figure A4 reports analogous experiments conducted under settings 2 and 3, where our method consistently achieves valid coverage, with its performance remaining largely unaffected by the total number of training samples.
Coverage
Lower Bound
Method
1.0
Oracle
0.8
Uncalibrated
Naive CQR
0.6
2
KM Decensoring
0.4
DR OR-COSARC (fixed)
100
300
1000
100
300
1000
DR-COSARC (adaptive)
Number of training samples
Figure A3. Performance our method (DR- COSARC) as a function of the total number of training samples used to fit the survival and censoring models, in experiments based on synthetic data under setting 1, similar to Figure 2.
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Doubly Robust Conformalized Survival Analysis with Right-Censored Data
Setting:
Setting:
Coverage
Lower Bound
1.0
2.0
0.8
1.5
0.6
Method
0.5
0.4
rracle
0.0
100
300
1000
100
300
1000
Uncalibrated
Setting: 3
S Setting:
Naive CQR
Coverage
Lower Bound
KM Decensoring
1.0
DR-COSARC (fixed)
0.9
0.8
DR CCOSARC (adaptive)
0.6
0.6
0.3
0.4
100
300
1000
100
300
1000
Number of training samples
Figure A4. Performance of method function of the number of training samples used fit both the survival and censoring models, in experiments based on synthetic data similar to those of Figure A3. These experiments are conducted under settings and 3 (see Table A1) where fitting an accurate survival model easier. Other details are Figure A3.
A3.4. Impact the Number of Covariates in the Censoring Model
Figure A5 presents the results experiments similar to those Figure 2, but with varying numbers covariates used to fit the censoring model, while keeping the number of training and calibration samples fixed at 1000. The results show that method performs better when the number of covariates used in the censoring model is not too large, highlighting the advantage of leveraging accurate prior information prevent overfitting. Figure A6 reports similar experiments conducted under the easier settings 2 and 3. In these scenarios, our method consistently achieves valid coverage, and its performance remains largely unaffected by the number of covariates used to fit the censoring model.
Coverage
Lower Bound
Method
1.0
Oracle
0.8
Uncalibrated
Naive CQR
0.6
KM Decensoring
0.4
DR -COSARC (fixed)
10
30
100
10
30
100
DR C (adaptive)
Number of covarariates in the censoring model
Figure A5. Performance on synthetic data of our method and henchmark approaches as a function of the number of covariates used to train censoring model, with the number training samples fixed at 1000. The results demonstrate larger numbers of covariates may increase overfitting, resulting in less accurate censoring models and lower coverage for our method. Other details are as in Figure 2.
18

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Doubly Robust Conformalized Survival Analysis with Right-Censored Data
Setting: 2
Setting:
Coverage
Lower Bound
1.0
0.8
1.5
0.6
1.0
Method
0.5
0.4
raalee
0.0
10
30
100
10
30
100
Uncalibrated
Setting: 3
Setting: 3
Naive CQR
Coverage
Lower Bound
KM Decensoring
1.0
1.00
DR-COSARC (fixed)
0.8
0.75
DR C (adaptive)
0.6
0.50
0.25
0.4
0.00
10
30
100
10
30
100
Number of covarariates in the censoring model
Figure A6. Performance on synthetic data of method and other approaches function of the number of covariates used train the censoring model, with the number of training samples fixed at 1000. These experiments are conducted under settings 2 and 3 (see Table A1) where fitting an accurate survival model easier. Other details are in Figure A5.
A3.5. Robustness to Small Calibration Samples
Figure A7 presents the results of experiments similar to those in Figure 2, but with varying numbers calibration samples, while keeping the number of training samples fixed at 1000. The results indicate that the average performance of all methods is generally not heavily influenced by the number of calibration samples, although larger calibration sizes do tend to reduce variability across independent repetitions of the experiment. Figure A8 shows similar results for settings 2 and 3.
Coverage
Lower Bound
Method
1.0
Oracle
0.8
Uncalibrated
Naive CQR
0.6
KM Decensoring
0.4
DR-COSARC (fixed)
10
100
1000
100
1000
DR C (adaptive)
Number of calibration samples
Figure A7. Performance synthetic data of method (DR-COSA COR-COSACC) and benchmark approaches function of the number of calibration samples, with the number training samples fixed at 1000. The results demonstrate notable robustness to small calibration samples. Other details in Figure 2.
19

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Doubly Robust Conformalized Survival Analysis with Right-Censored Data
Setting: 2
Setting:
Coverage
Lower Bound
1.0
2.0
0.8
1.5
0.6
1.0
Method
0.5
0.4
raalee
0.0
10
100
1000
10
100
1000
Uncalibrated
Setting: 3
Setting: 3
Naive CQR
Coverage
Lower Bound
KM Decensoring
1.0
1.00
DR-COSARC (fixed)
0.8
0.75
DR C (adaptive)
0.6
0.50
0.25
0.4
0.00
10
100
1000
10
100
1000
Number calibration samples
Figure A8. Performance synthetic data of method and other approaches function of the number of calibration samples, with the number of training samples fixed at 1000. These experiments are conducted under settings 2 and 3 (see Table Al), where fitting an accurate survival model is easier. Other details are in Figure A7.
A3.6 Impact the Double Robustness Adjustment
Coverage
Lower Bound
Oracle
Uncalibrated
.I-
Naive CQR
- Compe
KM Decensonin
DR COSAR C (fxed]
PR-COSARC (fxee, not DR DR -COS-COAARR ARC (adaplive)
DR DR-COSARC (adaptive, not DR)
Oracle Uncalibrated
Method
Naive COR
KM Decensoring CO-SCAAAC (fixe] DR-COSARC (fied, not DR -COSAR (adaptive)
&
DR-COSAR (adaptive not DR
Oracle
Uncalibrated
Naive COR
KM Decensoring DR-COS -CSAAR (fiee] DR-COSARC (fixed, not DR DR CRCARC (adaptive
DR DR-COSARC (adaptive, not DR
0.25
0.50
0.75
1.000.00
0.25 0.50 0.75
1.00
Figure A9. Performance synthetic data under threc different settings of our method for constructing lower confidence bounds the true survival time of a new individual based on right- data, compared to existing benchmark approaches. Our method is applied with (default) and without the double robustness adjustment [(Xx+11 min(L'(X. 11), d. (X.+1)) explained in Algorithms 2 and 3. Other details are as in Figure 1.
20

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Doubly Robust Conformalized Survival Analysis with Right-Censored Data
A3.7. Algorithmic Stability
We evaluate the algorithmic stability of our method and the KM Decensoring approach, both of which rely on random imputation of unobserved variables. Our goal is to assess how this randomness impacts the stability of the predicted lower function of the calibration sample size.
bounds for fixed dataset and test instance,
To this end, we perform experiments similar to those in Figures A7-A8, varying the calibration set size as a control parameter. For each configuration of the experiment, we independently apply our method (with both fixed and adaptive cutoff strategies) and the KM Decensoring method 10 times per data realization, using different random seeds for the imputation step. Each of these 10 runs uses the same training, calibration, and test sets, and the same trained models. The test set contains 100 points instead of the usual 1000 reduce the computational cost.
We measure stability by computing the empirical coefficient of variation of the predicted lower bounds across the 10 runs. This coefficient of variation is then averaged across the 100 test points and over 20 independent data realizations.
Figure A10 presents the resulting average coefficient of variations for each method as a function of the calibration sample size. Error bars represent two standard errors of the estimated average coefficient of variation, reflecting the uncertainty in estimating the ideal population coefficient of determination that would be obtained with an infinite number of data realizations (instead of 20).
The results show that the fixed-cutoff implementation of our method tends to have the highest algorithmic variability, while the adaotive-cutoff implementation of our method and the KM Decensoring approach are relatively stable and increasingly become more stable as the calibration sample size grows.
Coefficient of Variation
S Setting: 1
S Setting:2 2
Se Setting: 3
0.6
0.05
0.6
Method
0.04
0.4
0.03
0.4
KM Decensoring
0.2
0.02
DR DR-COSARC (fixed)
0.2
0.01
DR DR-COSARC (adaptive)
0.0
100
1000
100
1000
10
100
1000
Number of calibration samples
Figure A A10. Average coefficient of variation of predictive lower bounds ohtained with different methods across 10 independent imputations, plotted as a function of the calibration sample size. Results are averaged over 20 independent data realizations. Error bars represent two standard errors. Lower coefficient of variation indicates greater algorithmic stability. Other details in Figures A7-A8
A3.8. Impact of Different Survival and Censoring Models
We examine here the effect of using different survival and censoring models the performance of our method and the benchmark approaches. Figures A11 and A12 present results from experiments similar to those in Figure 1, but with different survival and censoring models applied to synthetic data generated under setting 1. Similarly, Figures A13 and A14 report results from setting 2, while Figures A15 and A16 show results from setting 3, where fitting an accurate survival model is relatively easy. Overall, the results are consistent with those presented earlier, empirically demonstrating the double robustness of our method. Notably, Figure A12 reveals that when using a Cox model for the censoring distribution, our method fails achieve the target 90% coverage, even with a training sample size of 1000, underscoring the limitations of the Cox model in capturing the complex censoring patterns simulated in the particularly challenging setting 1.
21

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Doubly Robust Conformalized Survival Analysis with Right-Censored Data
Coverage
Lower Bound
Oracle
napoo JunS
Uncalibrated
Naive CQR
KM Decensoring
DR CR-COSARC (fixed)
10
DR OR-COSARC (adaptive)
raclle
Uncalibrated
MI
1T5 neeell IunS
Naive CQR
KM Decensoring
DR DR-COSARC (fixed)
Method
DR-COSARC (adaptive)
raalle
Uncalibrated
If japou Juns
Naive CQR
KM Decensoring
DR DR-COSARC (fixed)
DR DR-COSARC (adaptive)
Oracle
japoa Juns
Uncalibrated
Naive CQR
KM Decensoring
CC-COSACC (fixe)
DR -COCSARC (adaptive)
1
|||
0.00
0.25
0.50
0.75
1.00
0.0
2.5
5.0
7.5
10.0
Figure A11. Performance synthetic data of method for constructing lower confidence bounds the true survival time of new individual based on right-censored data, compared to existing approaches. These experiments are conducted under setting (see Table Al), where fitting accurate survival model most difficult. Each calibration method applied with different type of survival model (fitted using 1000 training samples) and the grf censoring model (also fitted using 1000 samples). Other details are as Figure 1.
22

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Doubly Robust Conformalized Survival Analysis with Right-Censored Data
Coverage
Lower Bound
Oracle
japoo Juns
Uncalibrated
Naive CQR
KM Decensoring
DR OR-COSARC (fixed)
DR -C-COSARC (adaptive)
Oracle
Uncalibrated
1T maeell Iuns
Naive CQR
KM Decensoring
DR DR-COSARC (fixed)
Method
DR-COSARC (adaptive)
raalle
Uncalibrated
If japoa Juns
Naive CQR
KM Decensoring
DR-COSARC (fixed)
DR DR-COSARC (adaptive)
||
Oracle
japoa Juns
Uncalibrated
Naive CQR
KM Decensoring
CC-CCAACC (fixe)
10
DR -C-CSARCC (adaptive)
0.00
0.25
0.50
0.75
1.00
0.0
2.5
5.0
7.5
10.0
Figure A12. Performance synthetic data of method for constructing lower confidence bounds the true survival time of new individual based on right-censored data, compared to existing approaches. These experiments are conducted under setting (see Table Al), where fitting accurate survival model most difficult. Each calibration method applied with different type of survival model (fitted using 1000 training samples) and the cox censoring model (also fitted using 1000 samples). Other details are as in Figure A11.
23

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Doubly Robust Conformalized Survival Analysis with Right-Censored Data
Coverage
Lower Bound
Oracle
jeeoll Juns
Uncalibrated
Naive CQR
KM Decensoring
DR OR-COSARC (fixed)
DR OR-COSARC (adaptive)
raclle
napeo Suns
Uncalibrated
Naive CQR
KM Decensoring
DR DR-COSARC (fixed)
Method
DR-COSARC (adaptive)
raalee
Uncalibrated
IF japow AULS
Naive CQR
KM Decensoring
DR DR-COSARC (fixed)
DR DR-COSARC (adaptive)
Oracle
iapoa 'uJS
Uncalibrated
Naive CQR
KM Decensoring
CC-COSACC (fixed)
+
DR -C-CSARC (adaptive)
1
0.00
0.25
0.50
0.75
1.00
0.0
0.5
1.0
1.5
2.0
Figure A13. Performance synthetic data of method for constructing lower confidence bounds the true survival time of new individual based on right-censored data, compared to existing approaches. These experiments are conducted under setting 2 (see Table Al), where fitting an accurate survival model moderately difficult. Each calibration method is applied with different type of survival model (fitted using 1000 training samples) and the grf censoring model (also fitted using 1000 samples). Other details are as in Figure 1.
24

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Doubly Robust Conformalized Survival Analysis with Right-Censored Data
Coverage
Lower Bound
Oracle
meeoll Juns
Uncalibrated
Naive CQR
KM Decensoring
DR OR-COSARC (fixed)
DR OR-COSARC (adaptive)
raclle
nepeo SNn
Uncalibrated
Naive CQR
KM Decensoring
DR DR-COSARC (fixed)
Method
DR-COSARC (adaptive)
raalee
Uncalibrated
Jur
Naive CQR
IF japou
KM Decensoring
DR DR-COSARC (fixed)
DR DR-COSARC (adaptive)
Oracle
japoa JuJS
Uncalibrated
Naive CQR
KM Decensoring
CC-COSACC (fixe)
DR -C-CSARC (adaptive)
+
0.00
0.25
0.50
0.75
1.00
0.0
0.5
1.0
1.5
2.0
Figure A14. Performance synthetic data of method for constructing lower confidence bounds the true survival time of new individual based on right-censored data, compared to existing approaches. These experiments are conducted under setting 2 (see Table Al), where fitting an accurate survival model moderately difficult. Each calibration method is applied with different type of survival model (fitted using 1000 training samples) and the cax censoring model (also fitted using 1000 samples). Other details are as in Figure A11.
25

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Doubly Robust Conformalized Survival Analysis with Right-Censored Data
Coverage
Lower Bound
Oracle
Uncalibrated
Naive CQR
KM Decensoring
DR-COSARC (fixed)
DR O (adaptive)
Oracle
Uncalibrated
SuN. modelt
Naive CQR
KM Decensoring
DR-COSARC (fixed)
Method
DR DR-COSARC (adaptive)
Oracle
Uncalibrated
Sun Seppo 11
Naive CQR
KM Decensoring
DR DR-COSARC (fixed)
DR-COSARC (adaptive)
Oracle
neepl yuns
Uncalibrated
Nalve CQR
KM Decensoring
DR DR-COSARC (fixed)
DR C (adaptive)
0.75
1.00
Figure A15. Performance on synthetic data of method for constructing lower confidence bounds the survival time of new individual based on right-censored data, compared to existing approaches. These experiments conducted under setting 3 (see Table Al), where fitting an accurate survival model relatively easy and conformal calibration is not crucial. Each calibration method applied with a different type of survival model (fitted using 1000 training samples) and the grf censoring model (also fitted using 1000 samples). Other details are as Figure A11.
26

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Doubly Robust Conformalized Survival Analysis with Right-Censored Data
Coverage
Lower Bound
Oracle
Sun. m (nodel COX
Uncalibrated
Naive CQR
KM Decensoring
DR-COSARC (fixed)
DR O (adaptive)
Oracle
Uncalibrated
SuN. modelt
Naive CQR
KM Decensoring
DR-COSARC (fixed)
Method
DR DR-COSARC (adaptive)
Oracle
Uncalibrated
Sun. POOOL
Naive CQR
KM Decensoring
DR DR-COSARC (fixed)
DR-COSARC (adaptive)
Oracle
neepl yuns
Uncalibrated
Nalve CQR
KM Decensoring
DR DR-COSARC (fixed)
DR C (adaptive)
0.75
1.00
Figure A16. Performance on synthetic data of method for constructing lower confidence bounds the survival time of new individual based on right-censored data, compared to existing approaches. These experiments conducted under setting 3 (see Table Al), where fitting an accurate survival model relatively easy and conformal calibration is crucial. Each calibration method applied with a different type of survival model (fitted using 1000 training samples) and the cox censoring model (also fitted using 1000 samples). Other details are as Figure A15.
27

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Doubly Robust Conformalized Survival Analysis with Right-Censored Data
A3.9. Numerical Experiments under Additional Settings
Table A2, Summary of synthetic data generation settings considered in addition those outlined Table A1.
Setting
p
Ref.
Covariate, Survival, and Censoring Distributions
X
Unif([-1,1]] 1j))
4
100
(Candès al., 2023)
T X: LogNormal, (x(X) log(2) + 0.55 (3 (x()
C X: Exponential, (x( 0.4
X : Unif[[0,4]P)
T X: LogNormal, (X() = 0.632 x X1, xsX) =
5
(Gui et al., 2024)
C X: Exponential, de (X() 0.1
X
Unif([o,
6
1
(Gui et al., 2024)
T X: LogNormal, (x(X) = 3(X, > 2) + X1 < 2x)) =
C X: Exponential, (x 0.1
X Unif([0044]
7
(Gui et al., 2024)
T X: LogNormal, (x( 2(X1 > 2) + x <2)),
C X: Exponential, (x() 0.25 + X1+6
X : Unif([O, 4JP)
8
(Gui al., 2024)
T X: LogNormal, (x 3(X, 2) + 1, (X) = =
C X: LogNormal, (x(X) 2 2-X,
ox(X)= =
50
X : Unif([0,,4]
9
10
(Gui et al., 2024)
T X: LogNormal, (X() = 0.126(X, + Vxxx,) + +1,0s(X) =
C X: Exponential, de (x(X) Xu + 26
X Unif([O, 4)P)
10
10
(Gui al., 2024)
T X: LogNormal, (x() 0.126(X, vX3X3) 11, o,() Xx+2
C X: Exponential, \e(X) 10
X1u +
28

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Doubly Robust Conformalized Survival Analysis with Right-Censored Data
Coverage
Lower Bound
Oracle
Uncalibrated
Naive CQR
Saniie 4
KM Decensoring
DR-COSA (fixed)
DR-COSARC (adaptive)
Oracle
Uncalibrated
Setting:3
Naive CQR
KM Decensoring
DR-COSAR (fixed)
DR-COSARC (adaptive)
Oracle
Uncalibrated
Setting:
Naive CQR
KM Decensoring
DR-COSAR (fixed)
DR DR-COSARC (adaptive)
Oracle
Method
Uncalibrated
Seiting:
Naive CQR
KM Decensoring
DR-COSARC (fixed)
DR-COSARC (adaptive)
Oracle
Uncalibrated
Seating
Naive CQR
KM Decensoring
DR-COSAR AR C (fixed)
DR DR-COSARC (adaptive)
racle
Uncalibrated
Naive CQR
KM Decensoring
DR-COSAR (fixed)
DR DR-COSARC (adaptive)
Oracle
Uncallbrated
Setting 10
Naive COR
KM Decensoring
DR-COSAR C (fixed)
DR DR-COSARC (adaptive)
0.00
0.25
0.50
0.75
1.000.00 0.25 0.50 0.75 1.00
Figure A17. Performance synthetic data under additional settings of method for constructing lower confidence hounds the true survival time of new individual based on right- censored data, compared to existing benchmark approaches. These seven settings are borrowed from Candès al. (2023) Gui al. (2024), outlined in Table A2, and correspond to situations which fitting an accurate survival model is relatively easy, with rigorous conformal calibration not being as crucial in settings and of Figure 1. Other details as Figure
29

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Doubly Robust Conformalized Survival Analysis with Right-Censored Data
A4. Additional Results from Real Data Applications
A4.1. Data and Pre-Processing
We apply our method to seven datasets: the Veterans' Administration Lung Cancer Trial (VALCT); the Primary Biliary Cirrhosis (PBC) dataset; the German Breast Cancer Study Group (GBSG) dataset; the Molecular Taxonomy of Breast Cancer International Consortium (METABRIC) dataset; the Colon Cancer Chemotherapy (COLON) dataset; the Stanford Heart Transplant Study (HEART); and the Diabetic Retinopathy Study (RETINOPATHУ), Table A3 provides details the number of observations, covariates, and data sources.
The datasets were obtained from various publicly available sources. VALCT. PBC, COLON, HEART, and RETINOPA- THY are included in the survival package. GBSG was sourced from GitHub: wwwp:.cciooottal.oo///ooooooo barecdleekttmaaa/Dessssxz. summaryyldd-bra__etaaaaaa
comm
METABRIC was accessed via https: htops://wwww.. www.:/cccoppaa..oooooooo
Each dataset underwent a pre- pre-processing pipeline ensure consistency and prepare the data for analysis. Survival times equal to zero were replaced with half the smallest observed non-zero time. Missing values were imputed using the median for numeric variables and the mode for categorical variables. Factor variables were processed to merge rare levels (frequency below 2%) into an "Other" category, while binary factors with one rare level were removed entirely. Dummy variables were created for all factors, and redundant features were identified and removed using an alias check. Additionally, highly correlated features (correlation above 0.75) were iteratively filtered.
Dataset
Observations (n)
Variables (p)
Source
Citation
VALCT
137
survival
(Kalbfleisch Prentice, 2002)
PBC
418
17
survival
(Therneau et al., 2000)
GBSG
2232
6
github.com
(Katzman et al., 2018)
METABRIC
1981
41
cbioportal.org
(Curtis et al., 2012)
COLON
1858
11
survival
(Moertel et al., 1990)
HEART
172
4
survival
(Crowley & Hu, 1977)
RETINOPATHY
394
5
survival
(Blair et al., 1980)
Table A3. Summary of the publicly available survival analysis datascts used in Section 5 of this paper.
A4.2. Additional Results
Target: 0.80
Target: 0.90
Target: 0.95
Uncalibrated
Naive CQR-
KM Decensoring
DR -R-AAS
0.85
0.80
0.85
0.90
0.95
Estimated coverage
Figure A18. Distribution of average (estimated) coverage of survival LPBs computed by different methods, across seven real datasets and four survival models, using different nominal levels Other details in Figure 3.
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Doubly Robust Conformalized Survival Analysis with Right-Censored Data
publicly available data of different methods for constructing survival LPBs. All methods utilize the same survival model (grf) and aim for 90% coverage. Since the performance is evaluated on a censored test set, the coverage cannot be evaluated exactly. Instead, report empirical lower and upper bounds for the truc coverage. Values in parentheses represent twice the standard error. Coverage point estimates that are more than two standard errors below the nominal level are highlighted in red.
Table A4. Performance
Estimated Coverage
Method
Point
Lower bound
Upper bound
LPB
COLON
Uncalibrated
0.91 (0.00)
0.90 (0.00)
0.91 (0.01)
299.15 (3.89)
Naive CQR
0.90 (0.01)
0.90 (0.01)
0.91 (0.01)
296.97 (8.86)
KM Decensoring
0.90 (0.01)
0.90 (0.01)
0.90 (0.01)
306.27 (9.12)
DR-COSARC
0.91 (0.00)
0.91 (0.01)
0.91 (0.00)
286.03 (5.06)
GBSG
Uncalibrated
0.90 (0.00)
0.90 (0.00)
0.91 (0.00)
13.10 (0.11)
Naive CQR
0.91 (0.00)
0.90 (0.01)
0.92 (0.01)
12.80 (0.25)
KM Decensoring
0.89 (0.00)
0.89 (0.01)
0.90 (0.01)
13.70 (0.23)
DR-COSARE
0.91 (0.00)
0.90 (0.00)
0.91 (0.00)
12.98 (0.13)
HEART
Uncalibrated
0.84 (0.02)
0.78 (0.03)
0.90 (0.02)
14.27 (1.40)
Naive CQR
0.94 (0.01)
0.92 (0.02)
0.97 (0.01)
3.58 (0.81)
KM Decensoring
0.84 (0.02)
0.77 (0.04)
0.91 (0.02)
15.06 (2.28)
DR-COSARC
0.88 (0.02)
0.83 (0.03)
0.92 (0.02)
9.74 (1.43)
METABRIC
Uncalibrated
0.90 (0.00)
0.89 (0.01)
0.91 (0.01)
40.86 (0.53)
Naive CQR
0.92 (0.00)
0.91 (0.01)
0.93 (0.01)
37.84 (0.69)
KM Decensoring
0.89 (0.00)
0.88 (0.01)
0.90 (0.01)
42.55 (0.83)
DR-COSARC
0.90 (0.00)
0.89 (0.01)
0.91 (0.01)
40.10 (0.56)
PBC
Uncalibrated
0.92 (0.01)
0.89 (0.01)
0.95 (0.01)
853.89 (24.37)
Naive CQR
0.93 (0.01)
0.91 (0.01)
0.95 (0.01)
784.62 (38.83)
KM Decensoring
0.85 (0.01)
0.81 (0.02)
0.90 (0.01)
1035.49 (32.76)
DR-COSARC
0.92 (0.01)
0.89 (0.01)
0.95 (0.01)
852.27 (24.16)
RETINOPATHY
Uncalibrated
0.88 (0.01)
0.87 (0.01)
0.89 (0.01)
7.58 (0.38)
Naive CQR
0.90 (0.01)
0.89 (0.01)
0.91 (0.01)
6.66 (0.56)
KM Decensoring
0.88 (0.01)
0.86 (0.02)
0.89 (0.01)
8.03 (0.68)
DR-COSARC
0.90 (0.01)
0.89 (0.01)
0.91 (0.01)
6.19 (0.56)
VALCT
Uncalibrated
0.93 (0.01)
0.93 (0.01)
0.93 (0.01)
12.35 (0.69)
Naive CQR
0.94 (0.01)
0.94(0.03)
0.94 (0.02)
10.66 (1.50)
KM Decensoring
0.90 (0.02)
0.90 (0.02)
0.90 (0.02)
14.47 (1.78)
DR-COSARC
0.94 (0.01)
0.94 (0.01)
0.94(0.01)
10.17 (0.80)
31

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Doubly Robust Conformalized Survival Analysis with Right-Censored Data
Table A5. Performance seven publicly available data of different methods for constructing survival LPBs. All methods utilize the same survival model (cox) and aim for 90% coverage. Since the performance is evaluated on a censored test set, the coverage cannot he evaluated exactly. Instead, report empirical lower and upper bounds for the coverage. Values in parentheses represent twice the standard error. Coverage point estimates that are more than two standard errors below the nominal level are highlighted in red.
Estimated Coverage
Method
Point
Lower bound
Upper bound
LPB
COLON
Uncalibrated
0.89 (0.00)
0.89 (0.01)
0.89 (0.01)
320.96 (4.88)
Naive CQR
0.90 (0.01)
0.90 (0.01)
0.91 (0.01)
299.68 (9.27)
KM Decensoring
0.90 (0.01)
0.90 (0.01)
0.90 (0.01)
308.78 (9.56)
DR-COSARC
0.91 (0.00)
0.90 (0.01)
0.91 (0.01)
288.30 (9.46)
GBSG
Uncalibrated
0.89 (0.00)
0.89 (0.00)
0.90 (0.00)
12.36 (0.09)
Naive CQR
0.91 (0.00)
0.90 (0.01)
0.92 (0.01)
11.38 (0.20)
KM Decensoring
0.90 (0.00)
0.89 (0.01)
0.90 (0.01)
12.23 (0.18)
DR-COSARE
0.90 (0.00)
0.89 (0.01)
0.91 (0.00)
11.87 (0.17)
HEART
Uncalibrated
0.74 (0.02)
0.64 (0.03)
0.84 (0.02)
28.52 (1.81)
Naive CQR
0.94 (0.01)
0.91 (0.02)
0.97 (0.01)
6.97 (1.17)
KM Decensoring
0.84 (0.02)
0.77 (0.04)
0.91 (0.02)
17.65 (2.54)
DR-COSARC
0.84 (0.02)
0.78 (0.03)
0.90 (0.02)
13.19 (2.40)
METABRIC
Uncalibrated
0.89 (0.00)
0.87 (0.01)
0.90 (0.01)
42.41 (0.50)
Naive CQR
0.91 (0.00)
0.90 (0.01)
0.92 (0.01)
35.02 (1.02)
KM Decensoring
0.89 (0.00)
0.88 (0.01)
0.91 (0.01)
40.55 (1.01)
DR-COSARC
0.90 (0.00)
0.88 (0.01)
0.91 (0.01)
38.79 (1.08)
PBC
Uncalibrated
0.85 (0.01)
0.79 (0.02)
0.90 (0.01)
1198.29 (37.23)
Naive CQR
0.94 (0.01)
0.91 (0.02)
0.96 (0.01)
768.98 (56.89)
KM Decensoring
0.87 (0.01)
0.82 (0.02)
0.91 (0.02)
1110.57 (51.64)
DR-COSARC
0.88 (0.01)
0.85 (0.02)
0.92 (0.01)
1020.75 (66.27)
RETINOPATHY
Uncalibrated
0.87 (0.01)
0.85 (0.01)
0.88 (0.01)
8.77 (0.47)
Naive CQR
0.91 (0.01)
0.90 (0.02)
0.92 (0.01)
6.55 (0.62)
KM Decensoring
0.88 (0.01)
0.86 (0.02)
0.90 (0.02)
8.24 (0.80)
DR-COSARC
0.89 (0.01)
0.88 (0.01)
0.91 (0.01)
6.57 (0.54)
VALCT
Uncalibrated
0.88 (0.01)
0.87 (0.02)
0.88 (0.02)
24.64 (5.55)
Naive CQR
0.94 (0.01)
0.94 (0.02)
0.94 (0.02)
16.36 (5.63)
KM Decensoring
0.89 (0.02)
0.89 (0.03)
0.90 (0.02)
21.93 (5.45)
DR-COSARC
0.91 (0.01)
0.90 (0.02)
0.91 (0.02)
18.05 (4.41)
32

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Doubly Robust Conformalized Survival Analysis with Right-Censored Data
Table A6. Performance
publicly available data sets of different methods for constructing survival LPBs. All methods utilize the same survival model (survreg) and aim for 90% coverage. Since the performance evaluated on a censored test set, the coverage cannot he evaluated exactly. Instead, we report empirical lower and upper bounds for the true coverage. Values in parentheses represent twice the standard error. Coverage point estimates that are more than two standard errors below the nominal level are highlighted in red.
Estimated Coverage
Method
Point
Lower bound
Upper bound
LPB
COLON
Uncalibrated
0.90 (0.00)
0.90 (0.01)
0.90 (0.01)
369.05 (7.39)
Naive CQR
0.91 (0.00)
0.91 (0.01)
0.91 (0.01)
351.31 (10.46)
KM Decensoring
0.90 (0.01)
0.90 (0.01)
0.90 (0.01)
361.87 (10.75)
DR-COSARC
0.91 (0.00)
0.91 (0.01)
0.91 (0.01)
339.93 (10.70)
GBSG
Uncalibrated
0.90 (0.00)
0.89 (0.00)
0.91 (0.00)
13.43 (0.16)
Naive CQR
0.91 (0.00)
0.90 (0.01)
0.92 (0.00)
12.69 (0.21)
KM Decensoring
0.89 (0.00)
0.88 (0.01)
0.90
13.80 (0.24)
DR-COSARE
0.90 (0.00)
0.89 (0.00)
0.91 (0.00)
13.17 (0.14)
HEART
Uncalibrated
0.83 (0.02)
0.75 (0.03)
0.90 (0.02)
22.72 (2.74)
Naive CQR
0.94 (0.01)
0.91 (0.02)
0.97 (0.01)
8.91 (1.69)
KM Decensoring
0.84 (0.02)
0.77 (0.03)
0.91 (0.02)
20.75 (2.53)
DR-COSARC
0.86 (0.02)
0.81 (0.03)
0.92 (0.02)
14.22 (2.32)
METABRIC
Uncalibrated
0.88 (0.00)
0.87 (0.01)
0.90 (0.01)
46.04 (0.61)
Naive CQR
0.92 (0.00)
0.91 (0.01)
0.93 (0.00)
38.98 (0.86)
KM Decensoring
0.89 (0.00)
0.87 (0.01)
0.91 (0.01)
44.98 (0.94)
DR-COSARC
0.89 (0.00)
0.88 (0.01)
0.91 (0.01)
43.66 (0.80)
PBC
Uncalibrated
0.85 (0.01)
0.79 (0.02)
0.92 (0.01)
1197.47 47 (41.55)
Naive CQR
0.95 (0.01)
0.92 (0.01)
0.98 (0.01)
738.45 (54.18)
KM Decensoring
0.86 (0.01)
0.79 (0.02)
0.92 (0.01)
1167.58 (40.41)
DR-COSARC
0.87 (0.01)
0.82 (0.02)
0.93 (0.01)
7698.76 (43.67)
RETINOPATHY
Uncalibrated
0.87 (0.01)
0.85 (0.01)
0.88 (0.01)
9.22 (0.49)
Naive CQR
0.91 (0.01)
0.89 (0.02)
0.92 (0.01)
6.93 (0.65)
KM Decensoring
0.87 (0.01)
0.86 (0.02)
0.89 (0.02)
8.79 (0.78)
DR-COSARC
0.89 (0.01)
0.88 (0.02)
0.91 (0.01)
7.13 (0.59)
VALCT
Uncalibrated
0.89 (0.01)
0.89 (0.02)
0.90 (0.02)
21.47 (0.97)
Naive CQR
0.94 (0.01)
0.93 (0.02)
0.94 (0.02)
16,92 (1.92)
KM Decensoring
0.89 (0.02)
0.88 (0.02)
0.89 (0.02)
22.24 (2.15)
DR-COSARC
0.92 (0.01)
0.91 (0.02)
0.92 (0.02)
17.27 (1.67)
33

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Doubly Robust Conformalized Survival Analysis with Right-Censored Data
Table A7. Performance seven publicly available data of different methods for constructing survival LPBs. All methods utilize the same survival model (rf) and aim for 90% coverage. Since the performance is evaluated on censored test set, the coverage cannot be evaluated exactly. Instead, report empirical lower and upper bounds for the truc coverage. Values in parentheses represent twice the standard error. Coverage point estimates that are more than two standard errors below the nominal level are highlighted in red.
Estimated Coverage
Method
Point
Lower bound
Upper bound
LPB
COLON
Uncalibrated
0.90 (0.00)
0.89 (0.01)
0.91 (0.01)
403.71 (9.60)
Naive CQR
0.91 (0.01)
0.90 (0.01)
0.91 (0.01)
379.64 (9.90)
KM Decensoring
0.89 (0.01)
0.89 (0.01)
0.90 (0.01)
406.86 (10.83)
DR-COSARC
0.91 (0.00)
0.90 (0.01)
0.91 (0.01)
379.60 (12.36)
GBSG
Uncalibrated
0.89 (0.00)
0.88 (0.01)
0.90 (0.00)
14.48 (0.13)
Naive CQR
0.91 (0.00)
0.90 (0.01)
0.92 (0.01)
13.43 (0.26)
KM Decensoring
0.89 (0.00)
0.88 (0.01)
0.90 (0.01)
14.55 (0.25)
DR-COSARE
0.90 (0.00)
0.89 (0.01)
0.91 (0.00)
14.05 (0.19)
HEART
Uncalibrated
0.83 (0.02)
0.76 (0.03)
0.90 (0.02)
18.26 (1.93)
Naive CQR
0.95 (0.01)
0.93 (0.02)
0.97 (0.01)
4.55 (1.08)
KM Decensoring
0.83 (0.02)
0.77 (0.04)
0.90 (0.02)
17.58 (2.45)
DR-COSARC
0.87 (0.02)
0.82 (0.03)
0.93 (0.02)
11.45 (1.69)
METABRIC
Uncalibrated
0.89 (0.00)
0.88 (0.01)
0.91 (0.00)
43.21 (0.59)
Naive CQR
0.91 (0.00)
0.90 (0.01)
0.93 (0.00)
39.45 (0.75)
KM Decensoring
0.89 (0.00)
0.87 (0.01)
0.90 (0.01)
44.67 (0.81)
DR-COSARC
0.90 (0.00)
0.88 (0.01)
0.91 (0.01)
42.22 (0.70)
PBC
Uncalibrated
0.88 (0.01)
0.82 (0.01)
0.94 (0.01)
1057.31 (31.32)
Naive CQR
0.95 (0.01)
0.91 (0.01)
0.98 (0.01)
662.76 (54.8I)
KM Decensoring
0.86 (0.01)
0.79 (0.02)
0.92 (0.01)
1117.82 (42.77)
DR-COSARC
0.88 (0.01)
0.83 (0.02)
0.94 (0.01)
1012.21 (49.52)
RETINOPATHY
Uncalibrated
0.88 (0.01)
0.86 (0.02)
0.89 (0.01)
9.75 (0.62)
Naive CQR
0.91 (0.01)
0.89 (0.01)
0..2
7.90 (0.63)
KM Decensoring
0.88 (0.01)
0.86 (0.02)
0.90 (0.01)
9.78 (0.67)
DR-COSARC
0.89 (0.01)
0.88 (0.01)
0.91 (0.01)
8.19 (0.61)
VALCT
Uncalibrated
0.90 (0.01)
0.90 (0.02)
0.90 (0.02)
33.90 (9.67)
Naive CQR
0.94 (0.01)
0.9400.08
0.94 (0.02)
19.12 (6.61)
KM Decensoring
0.91 (0.02)
0.91 (0.02)
0.91 (0.02)
29.50 (8.73)
DR-COSARC
0.93 (0.01)
0.93 (0.01)
0.93 (0.01)
21.33 (6.13)
34

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Doubly Robust Conformalized Survival Analysis with Right-Censored Data
A5. Mathematical Proofs
AS.1. Proof of Proposition 2.2
Proof of Proposition 2.2. Since Px,F.c = Px.=- Poxx,7 it suffices to demonstrate that è is a random sample from P This immediately true when π C, as those cases, we have Ć C. Therefore, remains to show that Ô is random sample from Pcx,c>>:: when Č P.
To this, apply the definition of conditional probability along with the assumption of conditionally independent censoring, T 1 C X. The distribution of C given (X x, < C,T = t) for any and t can be written
Pexx,cst:6e a,t) p(C X x, < <CT==))
p(C X z, =x,(>(T==
x P(C c, <, |=r,
x c/C==c,>> x
(since T T. C |X)
fcxx(c (z) I(c I(c>t) t),
A(z,t)
where A(x,t) S, fc|x(c r) is normalizing constant.
If fcIx fc|x, this matches the procedure used in Algorithm A4 sample Ö' when C' > T, completing the proof. For clarity, note that the subscripts for probability density functions p have been omitted where possible, to simplify the notation without introducing ambiguity.
AS.2. Auxiliary Theoretical Results
Lemma A1. Ler {(X;, T;, C:))==1 be i..dd random samples from some distribution Px Px P Paix, under Assumption 2.1. For each E [n], define T; T, C, and E; I[T; Ct). Let CI....,C, C, denote the set of imputed censoring times output by Algorithm 1, based on the data {(X,, Tú, Ei) and any fixed censoring model Meens, Then, the total variation distance between the distributions of ((X,, Tú C.) and {(X,; Tú, ()) denoted as d Pxx.t.oosPPP.FF...... satisfies
fc|x(e| XX) fax(e|x) de
drv(P Poxtorr) (x.
f
A(X,T) A(X,T)
where Px denote the joint distribution of (X,T), for (X,T,C) Px..Oc TTAC
Proof of Lemma Al. Note that
div(P (e.Pxxx.cpppPxx.tcee n drv x. (Px.fx. Px.I,)))
n n.E.x (x Px
dTv(Pox..i) Po
n ...TxxTPPP
drv drv(Pax, PoTx)esst)
fcix(e| X) fcxx(e|x) A(X,T) A(X,T)
= 2 E
where the first equality follows directly from the definition the total variation distance, and the second equality follows from the fact that C" almost surely if < T.
Theorem A2. Consider a random sample (X, T) from some distribution Px, = Px PT For any x. E X, let (x) denote an estimate of the a quantile of the conditional distribution of T X I. Assume the function qa X' + R, depends on training data set independent of (X,T). Assume also that
35

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Doubly Robust Conformalized Survival Analysis with Right-Censored Data
(A.A2.1) There exist a constant b 0, and fincciin ya: : X R4, such that, for any >> O,
P T 2 qa (z) + X = =]
be, almost surely with respect to Px.
For any x. E X, define (xx) = 10 (x) qa (-)) Then, for any constant > O.
P [ 2 La (X)) 2 a (6 (E+ 1/0+0)
(A7)
and, for any 3 E (0, 1), with probability at least 1 - 13
E [E{( BI
PT Q. X] a
(A8)
Proof of Theorem A2. Define E(x) (x) qa (2) Note that
PP T > 0 (x) X x]
2 P [ 2 (z) + E + da (z) (x) e X =]
2 [T' 2 (z) + (4 (a))))a((()) - e| x== = x]
(A9)
2 PP 2 (x) X = x) [((()> >
+
21 a be [() > c),
where the last inequality follows from Assumption A.A2.1. Therefore, by Markov's inequality, for any 1> O.
P P 2 dn (X)) 2 be (X))
Consider now the choice
= (E (x)())) 1/(2+))
which leads to:
IP T 2 (x; Dai)) 2 1 (b (B
a
This completes the proof of the unconditional result (A7).
We will now prove the conditional result (A8). For any 3 E (0, 1), consider
(B (x))) 2/e
c=
Bi/e
which, through Markov's inequality, leads to
E [x) 3.
P (x) 1s
e
Combined with Equation (A9), this implies that, with probability at least :3.
E 1 (x)))
PT 2 (x)
X 2 1
B1/E
36

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Doubly Robust Conformalized Survival Analysis with Right-Censored Data
A5.3. Double Robustness of Algorithm A4
A5.3.1. NON-ASYMPTOTIC THEORY
We begin by establishing non-asymptotic theory for the double robustness Algorithm A4. To achieve this, we present two key results: Theorem A3 and Theorem A4, each addressing one side of double robustness. The first result focuses the validity of Algorithm A4 when the censoring model is accurately estimated, while the second addresses its validity when the survival model is accurately estimated, leveraging the auxiliary result stated in Theorem A2.
Robustness when the censoring model is accurate
Theorem A3. Let {(x,, Tú Ci) be i.i.d. random samples from some distribution PX.T.C- Px Prix Peix, under Assumption 2.1. Let fojx(c (c x) be the probability density of Pcix. Consider an independent random test sample (X441;Tu+1) Px Prix- Let 1(x=+1: Deat) indicate the lower bound output by Algorithm A4, based on imput calibration data {[X,, Tú, E with T. = T, ^ C2 and E, 1(T; < C.). Then, this lower bound satisfies:
- {x7 - (Ixxx.xxx)) a(x)
Toti Aoo 2 L(Xat1i; Danl] 2 1 - a E
((x)
where
n a ffxxx E
00
fcix(c| (x)
fcix(c| (X)
dc
2
f A(X,T)
Á(X,T)
Above, the expectation is taken with respect random sample (X,T), for (X,T, PxI. and = TAC.
Intuitively, Theorem A3 tells us that the unite-sample coverage achieved by our method depends on how well we can estimate the censoring distribution, Paix- In the special case where fc)x(e .) = fcix (c x) for all x, which corresponds to statistically exact imputation, then (ffeix x fexx) and the saate-sample bound given by Theorem A3 becomes the same as the bound obtained by Candès al. (2023) under type-I censoring.
Proof of Theorem A3. The high-level idea of this proof is to connect the lower bound output by Algorithm A4 to the imaginary lower bound which would be obtained by applying Algorithm A1, the approach originally proposed by Candès et al. (2023), to an imaginary data set sampled from the same distribution but subject to type-I instead of right censoring. We know from Proposition 2.2 that, if the imputation phase of Algorithm A4 utilizes an accurate survival model, the two aforementioned methods become equivalent, and thus the desired result should follow from Theorem B. in Candès et al. (2023). We will now make this intuition precise.
Let Deat denote the ideal calibration data set containing the true censoring times C2 along with the corresponding values of X, and T;i i...
D = (xx, T, C.))
Let (Xn+1; Dia) denote the imaginary output that one would obtain by applying Algorithm A4 using Dr, instead of Dea in the calibration phase. Equivalently, i." 2Xx Daa) the lower bound that would be produced under type-I censoring by
37

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Doubly Robust Conformalized Survival Analysis with Right-Censored Data
Algorithm A1, the approach originally proposed by Candès et al. (2023). Then,
Tu+1 Aoo < L(Xx+11:eal Deal]
I P Ta+i A co < L (Xn-1: Dan)
+ Tati Aco < L(x Deal] -P L Dial)
$ P Ta+1 Aco < (Xn=n=
+ + +
(1)
1
Sa
ExxPx
2
((x) c(X)
(2)
1
Bx
1
+ n
E
xx) X) fcxx(e A(X,T)
fcix (c ())
de
2
((X)
c(X)
A(X,T)
where drv (Fx:fPxx..c) (( denotes the total variation distance between the distributions of Deat and D the inequality (1) follows directly from Theorem B.1 in Candès et al. (2023) applied with w(x) = 1/c(x) and ((x) = 1/((x). and the inequality (2) follows from Lemma A1.
Robustness when the survival model is accurate
Theorem A4. Consider a random sample (X, T') from some distribution Рx. = Px Prix. Let L(x; Deal) indicate the corresponding survival lower bound output by Algorithm A4. For any x E X, let (x()) denote the estimate of the a quantile of the conditional distribution Tf X utilized in the final phase of Algorithm A4, with the function do : X + R+ depending only on M Assume also that there exist a constant b >> 0, and a function qa X - RR,, such that, for any O,
P PT 2 (z) + c
1113 1 a - be, almost surely with respect to Px.
Then, for any constant & 0,
P p Dea)) 2 a
(( 1/11+0)
(b+ 11
(A10)
and, for any 8 E (0, 1), with probability at least 1 - -3
(E (x))) 1/8
P
T
L(x;
x 2 a
(A11)
BL/E
Proof of Theorem A4. The proof of this result is an immediate consequence of Theorem A2, since by design Algorithm A4 leads to Lxx Deal) (2) almost surely for any x E X.
A5.3.2. ASYMPTOTIC THEORY
Proof of Theorem 3.3. We begin by proving the unconditional result. For this, we consider cases separately, relying on Theorems A3 and A4 respectively.
Suppose Assumption 3.1 holds. In this case, the result follows immediately from Theorem A3, which tells us that, for any fixed and N.
P TTt1 2 L(Xn41: Dew)]
2 Tatı M 00 2 1(xx+11; :Da))
1 E 2 a
1
|
00 Scxx(c| X)
Sci((e| X)
Á(X,T) de
2
((X)
c(X)
2
A(X,T)
The proof is then completed by noting that the asymptotic limit of the right- fight-hand-side term above, for N, n + 20 is a under Assumption 3.1.
38

---

Doubly Robust Conformalized Survival Analysis with Right-Censored Data
Suppose Assumption 3.2 holds, In this case, can obtain the desired result by applying Theorem A4 with l which tells us that
P [TTn1 2 L(x: Deal) 2 - (b (0=
The proof is then completed by noting that the asymptotic limit of the right-hand-side term above, for N - is under Assumption 3.2.
Let us now turn to proving the conditional result. Under Assumption 3.2, applying Theorem A4 with = 1 tells us that, for fixed N and any 3 E (0, 1),
E [E(]]]
P P T Deal
x
1 - B.
In particular, choosing 3 / / [{(X] completes the proof, because
P P T T
i(x: x 2 a -VBE((((X
21 VE(E(X))
A5.4. Double Robustness of Algorithm A5
A5.4.1. NON-ASYMPTOTIC THEORY
Robustness when the censoring model is accurate
Theorem A5 (Adapted from Theorem 3 in Gui et (2024)). Lei {(X;, Tı, C.) i.i.d. random samples from some distribution Px.T,c = Px Prix Peix, under Assumption 2.1. Assume that fa(x) is continuous in a and that there exists some constant 7/2 > such that 122,(z) S for Px-almost all x. Consider an independent random test sample (Xa+1, Tn+1) ~ Px- Pri. Let ixx,+1;; Daa) indicate the lower bound output by Algorithm A2, bused on imput calibration data D (i C;) with T, = T3 A C. Then, this lower bound satisfies:
Tatt 2 i(Xn+1; D(a))
ca (a(n))2)
+
42
max
11
fa Ra
-1)² 2
1
sup aE|0,1]
E
+
log(n)
Co
1
for any [0, 1], we define
(X((X) (X)
na
Proof of Theorem A5. Under the same assumptions, Theorem 3 in Gui et al. (2024) says that for any 8 E (0,1), 1), with probability at least 1 over D
P P 2 i(Xnt1;; Daa) Dial
[, 22 - -1)² log
ca ca(Xn+1) - Ea (Xn+1))fa
1 + + max
2 1
sup aE(0,
E
Our result then simply obtained by setting 8 1/n and marginalizing over D
39

---

Doubly Robust Conformalized Survival Analysis with Right-Censored Data
Theorem A6. Let {(X,, T,, C.)s= be i.idd random samples from some distribution Px.r.c==A Prix Peix, under Assumption 2.1. Let fc x(c x) be the probability density of Peix- Assume that fa(z) is continuous in and that there exists some constant Ga- >> 0 such that 1/(a(z) S Sta for Px-almost all x. Consider an independent random test sample (X++1, (n+1)TTn1) ~ Px Prix. Let L(X=+1: Deat) indicate the lower bound output by Algorithm A5, based on input calibration data {(X;, Ti, E with T. = Ti A C; and E; I(T; < C.). Then, this lower bound satisfies:
P PTaati 2 1(x++1: Peal)
max (1, #a
-1) 2
Ca Ca(Xn+1) Co (x 1 taa
1
- 12
- a
sup aE|u,1]
E
+
log(n)
n 2 fax(c X)
faxx(c] x)
dc
A(X,T) A(X,T)
Above, the expectation is taken with respect to a random sample (X, T) for (X,T, ~ PX.T.C and / AAC.
Intuitively, Theorem A6 tells us that the sinite-sample coverage achieved by our method depends on how well we can estimate the censoring distribution, Peix- In the special case where fcix(c x) = fcix(c æ) for all c, œ, which corresponds to statistically exact imputation, the finite-sample bound given by Theorem A6 becomes the same as the bound obtained by Gui al. (2024) under type-I censoring, reported Theorem A5.
Proof of Theorem A6. The high-level idea of this proof is similar to that of Theorem A3: we connect the lower bound output by Algorithm A5 to the imaginary lower bound which would obtained by applying Algorithm A2, the approach originally proposed by Gui et al. (2024), an imaginary data set sampled from the same distribution but subject type-I instead of right censoring.
Let Deat denote the ideal calibration data set containing the true censoring times C₁ along with the corresponding values of X; and T;; i.e.
D'd = (Ki: ((xx 3.))"
Let i.*(X (Xn+1;D(.) denote the imaginary output that one would obtain by applying Algorithm A5 using Di, instead of Demt the calibration phase. Equivalently, L "X Daa)) is the lower bound that would be produced under type-I censoring by Algorithm A2, the approach originally proposed by Gui et al. (2024). Then,
[Ttt < L(Xn+1:: [
S P
Tat1 < L (Xn+1 D(n))
+ Tutl < i(x Deal) [Tuti < <((((x+11:aa))
{P(Tnti P Tat1 < (X Diw)) + dTv (xtxe
where drv d (Pxx.f.cy) P( ((x)), denotes the total variation distance between the distributions of Deal and D Finally, the proof is completed by bounding IP [Tr+1 < 2xxn11; (Xn+1;Daa) Lemma A1.
Dan) with Theorem A5 and drv drv(P(x.f.c)p) (x. (x. with
Robustness when the survival model is accurate
Theorem A7. Consider a random sample (X, T) from some distribution Px,T = Px .Tix- Let L(x; Dea)) indicate the corresponding survival lower bound output by Algorithm A5. For any I E X. let (x( denote the estimate of the a quantile of the conditional distribution Tf | X x utilized in the final phase of Algorithm A4, with the function da : X + R2
40

---

Doubly Robust Conformalized Survival Analysis with Right-Censored Data
depending only on Main, Assume also that there exist a constant b 0, and a function qa X + R,,, such that, for any
<> O,
P 2 (x( +e + = 1 be, almost surely with respect to Px.
Then, for any constant 0,
P T L(X; [
2 (6- 1 (x)) 100+0)
(A12)
and, for any B E (0, 1), with probability at least 1 33
E E (X)]) 1/e
IP T 2 ((x;
x 21
(A13)
B1/e
Proof of Theorem A7. The proof of this result is an immediate consequence of Theorem A2, since by design Algorithm A5 leads to [a;; Dea)) S (x) almost surely for any a E X.
A5.4.2. ASYMPTOTIC THEORY
Proof of Theorem 3.6. This proof follows the same strategy as that of Theorem 3.3, but relying on Theorems A6 and A7 instead of Theorems A3 and A4.
We begin by proving the unconditional result. For this, consider cases separately, relying Theorems A6 and A7 respectively.
Suppose Assumption 3.4 holds. In this case, the result follows immediately from Theorem A6, which tells us that, for any fixed and N.
P Tatt 2 1(x Deal)
42 R
max (1, fua log(n))
a. - -1)²
(Xn+1)
+
+
1
sup
E
Ea (x )
n
The proof is then completed by noting that the asymptotic limit of the right-hand-side term above, for N, n - 20 is a under Assumption 3.4.
Suppose Assumption 3.2 holds. In this case, we can obtain the desired result by applying Theorem A7 with = 1, which tells us that
P [Tit1 2 L(X:Dau)) 2 a- (0-
The proof is then completed by noting that the asymptotic limit of the right-hand-side term above, for N - is under Assumption 3.2.
Let us now turn proving the conditional result. Under Assumption 3.2, applying Theorem A7 with = 1 tells us that, for fixed N and any 3 E (0, 1).
P P T2 P Dex) x] 2 -
E
[[E(
2 1 - B.
3
In particular, choosing 3 /B [(X) completes the proof, because
P P T2 i(x;
|x]
2 1 - (C--VEE(((]] 2 21 -VE(((())


Reviews:
[{"review_id": "9iu98XSE80", "reviewer": "Reviewer_FTCz", "summary": "This paper proposes a new conformal inference approach specifically for right-censored data, aimed at constructing lower prediction bounds (LPBs) for survival times. The method is theoretically asymptotically doubly robust and demonstrates strong empirical results, offering more informative and reliable LPBs compared to existing alternatives.", "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": "Thank you for your review and your positive evaluation of our paper. We’re grateful for the opportunity to clarify some aspects of our empirical analysis, particularly since many important results are presented in the appendix and may have been unintentionally overlooked.\n\nThe appendix contains an extensive and carefully structured set of numerical experiments specifically designed to study how the performance of our method depends on the accuracy of the survival and censoring models. These results provide strong empirical support for the theoretical double robustness of our method—namely, that valid and informative predictive bounds can be obtained when either distribution is modeled accurately. We believe that a more detailed look at these experiments will confirm that our claims are fully substantiated by the data.\n\nThe main paper (Figure 1 and the discussion in Section 4) focuses on settings that vary the difficulty of estimating the survival model. These experiments show that our method approaches oracle performance when the survival distribution is easier to learn, while still outperforming existing alternatives in more challenging settings. These results are complemented by a broad set of additional experiments detailed in the appendix and only briefly summarized in Section 4 due to space constraints.\n\nFor example:\n- Figures A2 and A6 vary the number of training samples used to fit the censoring model, showing that coverage improves as the censoring model becomes more accurate, especially in difficult settings where the survival model may be unreliable.\n- Figures A5 and A9 vary the number of irrelevant covariates used in the censoring model, demonstrating the impact of overfitting on performance.\n- Figures A3 and A7 examine the joint effect of training sample size on both models, further illustrating our method’s double robustness.\n\nThese and many other results presented in Appendices A3 and A4 offer what we believe is a thorough empirical validation of our approach. While Section 4.3 of the main paper already points readers to these results, we now realize that some of this material may not have been sufficiently highlighted, and that even attentive readers could miss it.\n\nWe will take full advantage of the extra page allowed in the camera-ready version to address this. In particular, we plan to improve how these experiments are referenced and summarized in the main text, and will likely move an important figure (such as A2 or A3) into the main body to make the role of the censoring model even more visible. This will ensure that all readers, regardless of how closely they examine the appendix, clearly see the breadth and depth of the empirical results supporting this paper. Thank you again for this helpful feedback!", "claims_and_evidence": "The paper proposal of doubly robust conformal inference method for survival analysis is supported by their theory and strong empirical results.", "methods_and_evaluation_criteria": "The paper presents DR-COSARC (Doubly Robust Conformalized Survival Analysis under Right Censoring), which builds on recent work in conformalized survival analysis by focusing on more practical right-censored cases. It further incorporates doubly robust properties, making the proposed framework more reliable. DR-COSARC is designed with both fixed and dynamic cutoffs, enhancing its generalizability. The method is evaluated on both simulated and real-world datasets, and compared against several existing frameworks.", "theoretical_claims": "The paper theoretically proves (under the provided assumption) that the proposed DR-COSARC is doubly robust for both constant cutoffs (Theorem 3.3) and Adaptive custoffs (Theorem 3.6).", "experimental_designs_or_analyses": "While the experiments in Fig. 1 demonstrate that the proposed method performs well, its success is attributed to the framework’s ability to model the censoring distribution accurately. However, this aspect is not clearly demonstrated or fully substantiated.", "relation_to_broader_scientific_literature": "The contributions of the paper is related to the broader scientific literature for conformal inference.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "N/A", "other_comments_or_suggestions": "N/A", "questions_for_authors": "N/A", "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": "VW1Q7htZNt", "reviewer": "Reviewer_qnsR", "summary": "The paper addresses the problem of constructing lower prediction bound (LPB) for survival time under conditional independent right censoring. In particular, the paper extends the approaches in Candes et al. (2023) and Gui et al. (2024) for type I censored data, which assumes that the right censoring time is always observed, to the censoring scenario that are more commonly encountered in practice where the right censoring time is only observed for censored subjects. The proposed approaches first impute the missing censoring times from the estimated conditional censoring distribution, and then apply the methods in Candes et al. (2023) and Gui et al. (2024) for type I censored data. With an additional adjustment, the proposed LPBs enjoy a doubly robust (DR) property: the obtained LPB has asymptotically valid marginal coverage when either (i) the conditional censoring distribution is estimated sufficently well, or (ii) the conditional quantile of event time is consistently estimated and the true conditional event time distribution satisfies a smoothness condition. In addition, when (ii) is true, the proposed LPB also also has valid approximate conditional coverage. Simulation studies are done to compare the performance of the proposed LPBs with existing apoproaches. The results show that the proposed LPBs is robust in challenging settings. The proposed approaches are applied to construct LPB for survival times in seven publicly available datasets.\n\n## update after rebuttal\nI updated the overall recommendation from 2 to 4. Previously I choose 2 because there are errors in a math expression and the algorithm in the main text missing the important DR adjustment step which caused confusion. The authrors clarified that the error in the math expression is only a typo and they will revise the algorithms and writing of the paper. These resolves the above concerns. As for the assumption in the theory being a strong assumption, I agree that it is okay to keep it as it is for the ICML paper, without making the paper more technical. Also the authors mentioned that they improve on addressing the simulation results in the main text.", "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": "Thank you for your detailed and thoughtful review. We find your feedback very helpful. While some presentation issues may have caused confusion, they are easily resolved and do not reflect inherent flaws. We respond below and will incorporate your suggestions into the revised paper.\n## DR Adjustment\nYou're right that the DR adjustment is essential for establishing double robustness. This is discussed in Section 2 and shown in line 5 of Algorithm 3. The confusion arose because it is also part of the fixed-cutoff method but was accidentally omitted from Algorithm 2 (though present in Algorithm A4). We will fix this omission and also correct the outdated sentence comparing Algorithms 2 and 3. Empirically, the DR adjustment typically has a small effect, as the preliminary LPB very often is already lower than the uncalibrated quantile. Based on your comment, we will include in the appendix an additional figure showing this explicitly.\n## Convergence Rate of Censoring Model\nWe agree the convergence rate assumed for the censoring model in the theoretical analysis is quite strong, but we highlight this directly after Assumption 3.1, noting it suggests more data should be allocated to training than calibration. We view this as a reasonable and useful recommendation.\nThat said, it may be possible to relax the converge rate under stronger assumptions or with more technical effort. However, we do not see a compelling need nor an obvious way of doing so without making the paper far more technical than is appropriate for ICML.\nOur main goal is to establish the soundness of the method and highlight its double robustness—a key property not shared by existing alternatives. Whether the convergence rate assumption can be weakened seems secondary. In practice, what matters is whether the method performs well, and our results show that it does. (After all, it is unclear what it would even mean for a convergence rate to “hold” for a single finite dataset.)\nWe would of course welcome future work that seeks to tighten or relax this theoretical condition.\n## Censoring Time Imputation\nThank you—you are right about the typographical error in Equation (4); we will correct the integration limits. We will also revise the vague phrase “accurately estimates” to clarify that the synthetic sample matches the target distribution only if $\\hat{f}\\_{C \\mid X} = f\\_{C \\mid X}$.\n## Comparison with Naive CQR\nThere appears to be a misunderstanding. As stated on page 6, Naive CQR calibrates CQR to predict $\\tilde{T} = T \\land C$ instead of $T$, which leads to overly conservative bounds. This is very different from our DR correction, which uses the survival model’s uncalibrated quantile. We do not take the minimum with the Naive CQR bound, and doing so would be unjustifiably conservative. Indeed, our method produces more informative bounds than Naive CQR, as shown in Figures 1 and 2.\n## Design of Empirical Experiments\nWe conducted a thorough empirical evaluation, much of it in Appendix A3–A4. We will better reference these results in the main text and will add a new figure to further highlight double robustness.\nOur experiments explore the performance of our method in the face of key challenges like model misspecification, limited training data, and covariate dimensionality. For example:\n- Figures A2, A6: Vary censoring model training size; performance improves as quality increases.\n- Figures A5, A9: Add irrelevant covariates in censoring model; overfitting degrades performance.\n- Figures A3, A7: Vary both models’ training size; highlight double robustness.\n\nFigure A2 is particularly illustrative: with a poor survival model, performance improves rapidly as the censoring model improves. We will complement this with a similar figure where the survival model is more accurate and our method remains robust even if the censoring model is poor.\n## Different Censoring Models\nWhile the main text uses GRF for simplicity, Appendix A3.2 includes additional experiments with other censoring models (see p.18). We limited real-data results to GRF to avoid overwhelming the appendix, which already contains 16 supplementary figures and 7 tables. We believe these experiments sufficiently support our claims.\n## Survival Modeling in Benchmark Data Sets\nWe will clarify that our comment about “simpler” datasets refers to the fact that the uncalibrated survival model performs reasonably well, suggesting $T|X$ is easier to estimate in these benchmarks. However, our synthetic experiments show that when survival modeling is hard, alternative methods fail—whereas ours maintains valid coverage as long as either model is accurate. If the survival model is perfect, conformal inference may not be needed—but in practice, we don’t know if it is, and our method offers stronger protection than alternative approaches.\n## LPB Units\nLPBs are in different units across datasets and should not be directly compared across them.", "claims_and_evidence": "Overall the claims made in the submission is supported by clear and convincing evidence, but I found their statement of the algorithms and the claims for DR results potentially confusing and misleading. \n\nTheir algorithm first impute the missing censoring times and then applied the existing approaches for handling type-I censored data, and they showed in Proposition 2.2 that if the true conditional censoring distribution is known, the data with the imputed censoring times follows the same distribution as the hypothetical type-I censored data if all the censoring times were observed. This justifies the use of the imputed censoring times and the application of the existing approaches for type-I censored data. While reading, that makes me think that the proposed approach will rely on the conditional cenosring distribution to be estimated resonably well, so it is kind of surprising that their approach enjoys the DR property. It turns out that their algorithms also has a DR adjustment step, which takes the minimum of (i) the LPB from the existing approaches under type-I censoring and (ii) the estimated quantile function of the conditional event time distribution. Since (ii) has approximately valid marginal coverage if the quantile function of the conditional event time distribution is estimated sufficently well, by definition, the adjusted LPB has the DR property claimed in the paper. However, this key step of adjustment is barely mentioned in the main text and it is burried in Algorithm A4 and A5 in the appendix (except mentioned in algorithm 3 Step 4 and 5, but it replicates the DR adjustment step of Algorithm A5 in Step 3, and it is not higlighted to be an adjustment step for DR). So I think their current writing is potentially confusing. More discussion about this key step and the consequence and potential limitations of taking the minimum of two LPB is needed for clarity.", "methods_and_evaluation_criteria": "Yes, the proposed methods and/or evaluation criteria (e.g., benchmark datasets) make sense for the problem or application at hand.", "theoretical_claims": "I checked their theoretical results in Proposition 2.2, Theorem 3.3 and Theorem 3.6 and the corresponding assumptions, as well as some of the proofs in the appendix.  Besides the concerns about their statement of the algorithms and the DR results mentioned in the section of “Claims and Evidence”, I’m also concerned with the second condition in their Assumptions 3.1 and Assumotion 3.4, which incolves the sample size n for the calibration set. \n\nThe statement is about the limit of a quantity related to the estimation error of the conditional censoring distribution converges to zero faster than 1/n rate, as the sample sizes n and N goes to infinity. I guess N denotes the sample size of the data used to train the censoring model, but the definition of N seems to be not mentioned in the paper.  \n\nMy concern for this assumption is how strict it is. In practice, the integral term in their assumption that involves the estimation error of the conditional censoring distribution usually converges at no faster than N^{-1/2} rate, which is the parametric rate of convergence. So it seems that this condition requires that n is of order at least N^2, which is pretty large. The choice in the simulation section with sample size 1000 for each set seems to not follow this requirement.I think more discussion and examples of when this condition can be satisfies will enhance the theory part.  \n\nAlso more discussion about how the Assumptions 3.1, 3.2, 3.4, 3.5 compare with the assumptions in the literature will make the theory part stronger. \n\nThere is an error in equation (4). The limits of the integral should be from \\tilde T_i to infinity?\n\nDiscussion from line 180 – 184 (left) is inaccurate. The synthetic sample shares the same distribution as the ideal sample only if \\hat f_{C|X} is exactly equal the truth.", "experimental_designs_or_analyses": "I read the simulation in the main text and briefly browsed the additional results in the appendix. \n\nThe simulation studies is done under various situations, but few discussions are made on how the results relate to their theoretical results of double robustness. I think the paper would benefit from discussing under which scenarios, the proposed methods are (1) expected to have good estimate for the censoring model and/or event time model, (2) expected to show good performance according to their theory, and (3) expected to outperform the comparison methods, etc. For example, their paragraph on “Leveraging Prior knowledge on P_{C|X}” makes me think that this probably relate to how well the conditional censoring distribution can be estimated under these models. \n\nI’m also curious about how the performance of the proposed approaches compares with their counterparts without the DR adjustment step. \n\nIn simulation results, the lower bound for the proposed LPB is lower than the naïve CQR, which is not surprising as the DR adjustment step in the proposed algorithms takes the minimum of the two LPB where one of them is the naïve CQR. I suggest commenting on this result as it relate to the potential limitation of the proposed approach.\n\nFor application, the event time distribution is estimated under different models but the censoring distribution is always estimated using grf. Why only one censoring model is considered instead of multiple ones like for the event time distribution?\n\nThe discussion of the application results can be more informative. For example, in line 414 (left), “the comparatively simpler nature of survival analysis with these datasets”. Do you mean the true distribution of the T|X or C|X follow simplier models, or there are fewer covariates, or the model can estimate the conditional distributions T|X, C|X well so that the error rate condition required in your Assumptions are more likely to hold?\n\nFor the data analysis results in the appendix (Table A4, etc.), what is the unit of the LPB. I saw the numbers varied a lot across datasets. Does it mean there is more uncertainty in some than the other, or they are due to different units of the time-to-event in the datasets? How can we tell if they are informative?", "relation_to_broader_scientific_literature": "The paper contribute to the conformal inference literature by extending the existing approach for constructing LPB for time-to-event data under type-I censoring to the more general type of right censoring encountered in practice where the censoring times are not always obseved. The paper extend both the algorithms with fixed cutoffs and with adaptive cutoff for type-I censored data.", "essential_references_not_discussed": "I’m not aware.", "other_strengths_and_weaknesses": "Strength:\n\nOriginality for addressing the problem of conformal inference for right censored data under a more realistic censoring setting than the type-I censoring that has been studied in the literature.\n\nComprehensive simulations and real data analysis.\n\nWeakness:\n\nThe writing and flow of the paper can be improved. For example, perhaps Section 2.2 can be folded in 1.3.  The last paragraph in the right column in line 157 may be simplified by saying that (3) is the conditional density of C|X,\\tilde T, E=1.\n\nThe clarity of the writing may be improved if the authors use detailed and accurate wording. For example, what does “robust” and “delicated” mean in “tend to be more robust in more delicated scenarios” (line 088 left).", "other_comments_or_suggestions": "In line 181, Ci, i should be subscript.", "questions_for_authors": "1.\tHow the two proposed algorithms and the writing of the paper can be revised to highlight the key adjustment step for DR property and acknowledge the limitation that this adjustment step may cause?\n\n2.\tWhat is the cost of achieving the DR property with the adjustment step? Are there any justification that with this adjustment step the LPB is still informative enough? Does it worth to achieve DR at the cost of this adjustment step?\n\n3.\tCan the second condition of Assumptions 3.1 and 3.4 can be relaxed?\n\n4.\tHow can the simulation studies be better designed or how can the current simulation results to be better intepreted to show the advantage of the DR property of the proposed approaches?\n\nI think addressing the above questions will improve the clarity of the paper and make the paper much stronger.", "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": "Gejp2TbfzD", "reviewer": "Reviewer_Udgo", "summary": "This paper studies conformal inference for right-censored data, in which it generalizes prior works beyond the type-I censoring setting (i.e., when all the censoring times are observed). Its main idea is to impute the  censoring times by sampling from the estimated censoring mechanism, and obtain a \"synthetic\" data set whose censoring times are fully observed, on which prior methods can be applied to \nproduce calibrated lower prediction bounds. \n\nIt is shown that the proposed method enjoys a double-robustness property, achieving asymptotic coverage guarantees as long as either the distribution of $T|X$ or $C|X$ is well estimated. The proposed method is evaluated in extensive simulation and real-data studies, in comparison with existing methods.", "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": "Thank you for your thoughtful review and for your positive evaluation of our paper. We appreciate your careful reading, and we are happy to answer your two very insightful questions.\n\n## Imputation randomness\n\n*\"Since the missing censoring times are imputed by sampling from the estimated $C \\mid X$, the results depend on the realization of the random variables. I wonder how stable this algorithm is to different realizations of censoring times.\"*\n\nThis is an excellent question. Conceptually, we believe statisticians should not view randomness as inherently problematic. Imputing missing censoring times by sampling from their estimated conditional distribution given observed data is a natural and principled strategy. Even if the censoring times were observed directly, they would still be treated as realizations from a random process—our method simply mirrors that generative view using model-based imputation. This allows us to reduce a more complex model-free problem (survival analysis under general right censoring) to a more tractable one: survival analysis under type-I censoring, for which conformal methods are more easily applicable.\n\nWhile data randomness is fundamental to statistical inference in general, many conformal inference methods in particular introduce additional randomness into the observed data—most notably through random sample splitting between training and calibration sets. The random imputation step in our method is thus fully consistent with the broader conformal inference framework and, in fact, this randomness plays a key role in enabling formal coverage guarantees.\n\nThat said, your point about stability is well taken. To address it concretely, we will include a figure in the camera-ready version of the paper that empirically assesses the variability of the results obtained by applying our method to the same observed data set across different imputations of the latent censoring times. These experiments will show that, as long as the calibration sample size is at least moderately large (on the order of a few hundred observations), the sampling variability introduced by our method is minimal and does not meaningfully impact the informativeness of our predictive inferences. Moreover, as expected, this variability decreases as the calibration sample size increases.\n\nPrompted by your suggestion, we will also mention in the discussion that one could explore possible extensions of our method that further reduce this variability by leveraging recent developments in e-value-based methods. We expect that such extensions may reduce variance at the cost of increased conservativeness—an interesting tradeoff that we agree is worth flagging as a possible direction for future work.\n\n## Finite-sample theory\n\n*“(If I understand correctly) The proposed method (fixed version) does have finite-sample guarantees when $C \\mid X$ is known, which is perhaps worth emphasizing.”*\n\nYes, you are absolutely correct. Indeed, Theorem A3 in Appendix A5.3 provides a finite-sample guarantee for our method under the fixed-cutoff implementation, which becomes tighter if the censoring mechanism is known. We will highlight this more clearly in the main text.\n\nAdditionally, we have also established finite-sample validity for the adaptive-cutoff implementation in Theorem A5 (Appendix A5.4). These results are somewhat more conservative—not because the method is less effective in practice, but because the theoretical analysis is more technically involved. This is similar to what Gui et al. (2024) encountered in their analysis under type-I censoring.\n\nWe believe this distinction is worth clarifying in the camera-ready version: while the adaptive version of our method performs better empirically (as our results show), it is more challenging to analyze theoretically. This situation is not uncommon in modern statistical methodology, where practical performance can sometimes outpace what current theory can tightly capture. We’ll make sure to emphasize that the theory remains sound in both cases and the gap lies primarily in the sharpness of the finite-sample bounds, not in the validity or robustness of the approach itself.\n\nThank you again for your helpful comments. We believe your suggestions will help improve the clarity and impact of the final version.", "claims_and_evidence": "The main claim of the paper is that it provides calibrated lower prediction bounds for right-censored data, generalizing prior works beyond the type-I censoring setting. The method is said to be asymptotically valid if either the censoring mechanism or the survival distribution can be well estimated. The claim is supported by theoretical results (Theorems 3.3 and 3.6), as well as simulation results.", "methods_and_evaluation_criteria": "The proposed method makes it possible to construct LPB when the censoring times are not fully observed. This is achieved by imputing the missing censoring times by sampling from the estimated distribution of $C|X$. I wonder, however, how stable this algorithm is,\nconsidering that the missing censoring times are imputed with random variables.", "theoretical_claims": "I went over the proof outline, and did not spot any significant issue.", "experimental_designs_or_analyses": "The experimental design follows from settings in existing works; it also provides comprehensive comparison and sensitivity analysis.", "relation_to_broader_scientific_literature": "The main claim of the paper is that it provides calibrated lower prediction bounds for right-censored data, generalizing prior works beyond the type-I censoring setting. This greatly improves the applicability of the framework.", "essential_references_not_discussed": "NA.", "other_strengths_and_weaknesses": "NA.", "other_comments_or_suggestions": "Page 4, line 181, $\\hat C i$ -> $\\hat C_i$", "questions_for_authors": "1. Since the missing censoring times are imputed by sampling from the estimated $C|X$, the results depend on the realization of the random variables. I wonder how stable this algorithm is to different realizations of censoring times.\n2. (If I understand correctly) The proposed method (fixed version) does have finite-sample guarantees when $C|X$ is known, which is perhaps worth emphasizing.", "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": "bCSWeoKTXo", "reviewer": "Reviewer_5akg", "summary": "The paper proposes a doubly robust conformal inference method for constructing lower prediction bounds (LPBs) for survival times under right-censored data. By imputing unobserved censoring times using a machine learning model and calibrating survival models via weighted conformal inference, the method theoretically guarantees asymptotic validity if either the censoring or survival model is correctly specified. Extensive experiments on synthetic and real datasets demonstrate robustness in challenging scenarios where existing methods (e.g., Kaplan-Meier decensoring) underperform. The approach extends prior work on type-I censoring to handle more practical right-censoring settings.\n\n## update after rebuttal\nI updated the overall recommendation from 3 to 4 as my concerns have been adequately addressed.", "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": "Thank you for your thoughtful and generally positive review. We appreciate your recognition of the novelty and strength of our contributions and are happy to respond to your comments below.\n## Asymptotic vs Finite-Sample Theoretical Results\nYou're right that Theorems 3.3 and 3.6 provide asymptotic double-robustness guarantees. However, as shown in Theorems A3 and A5 (Appendix), we also establish finite-sample coverage results for both the fixed- and adaptive-cutoff versions of our method. While these bounds are somewhat loose and not directly useful for practitioners, they rely on weaker assumptions and are key to establishing asymptotic validity, offering some theoretical support for our method’s strong empirical performance.\nThe challenge of deriving tighter finite-sample guarantees reflects the inherent complexity of conformal inference under censoring—a challenge also noted in Candès et al. (2023) and Gui et al. (2024), even in the simpler type-I setting. In short, censoring makes theoretical analysis harder.\nImportantly, our experiments show that the adaptive version of our method consistently outperforms the fixed-cutoff version, despite requiring somewhat stronger assumptions for double robustness. This illustrates a broader pattern in modern data science: practical performance sometimes exceeds what theory can rigorously explain. We’ll clarify this point and make the finite-sample results more visible in the revised version.\n## Experiments with High-Dimensional Data\nOur synthetic experiments are designed to evaluate the performance of our conformal inference method under the most relevant practical challenges that may affect the quality of the survival and censoring models—including model misspecification, limited sample sizes, and increasing numbers of covariates. These challenges commonly arise in high-dimensional settings, but in our context they do not require using extremely large feature spaces in simulation.\nFigures A5 and A9 (referenced in Section A.3) vary the number of irrelevant covariates used to train the censoring model and show that performance degrades as more noise features are added—due to overfitting. This is consistent with double robustness and precisely represents the kind of failure mode that our framework is designed to mitigate. While a sparse learner could further improve performance in such settings (since the true censoring model is sparse), our focus is on imputation and conformal inference, not sparse modeling. Our method is designed to be broadly applicable and can be used with any underlying model, including sparse ones. We will highlight these experiments more clearly in the revised version.\n## Data Pre-Processing\nThe preprocessing steps in Appendix A4.1—such as merging rare categorical levels and removing extreme outliers—are standard procedures to ensure model compatibility and stability. We are not aware of any way these steps would introduce “bias” in our experiments. That said, we will clarify and briefly justify them in the final version to avoid confusion.\n## Computational Cost\nThanks for this suggestion. The imputation step is a key conceptual contribution of our method, but computationally it is very light. Once the censoring model is trained, sampling requires evaluating a one-dimensional integral (Eq. 4), which is either analytical or quickly approximated. This step is not a bottleneck. As with all comparable methods, the main cost lies in training the survival and censoring models. We’ll clarify this in the revised paper.\n## Practical Relevance\nWe agree that domain-specific evaluation—such as assessing the clinical value of lower predictive bounds—is important future work. Our goal in this paper was to develop and validate the core methodology, which is why we focused on benchmark evaluations.\nThat said, we believe our survival LPBs have the potential to be useful for practitioners. For example, in healthcare they may inform treatment prioritization based on survival likelihood under resource constraints (as noted by Candès et al., 2023). We will expand this motivation in our introduction to make the practical relevance more self-contained.\n## Split vs. Full Conformal Inference, and Possible Extensions using Cross-Validation\nWe chose split conformal inference for its simplicity, efficiency, and ability to support double-robustness guarantees under relatively mild assumptions. These same considerations motivated prior work on conformal survival analysis under type-I censoring (Candès et al., 2023; Gui et al., 2024).\nExtending our method to full conformal inference, cross-validation, or jackknife+ could potentially improve small-sample performance, but each would involve substantial computational and theoretical challenges—especially due to censoring. We agree these are potentially interesting directions for future work and will mention them in the revised discussion. Thank you for the suggestion!", "claims_and_evidence": "Overall, the claims made in the submission can be supported by theoretical proofs or experiments.", "methods_and_evaluation_criteria": "Comprehensive evaluation across 10 synthetic settings and 7 real datasets.", "theoretical_claims": "Theorems 3.3 and 3.6 are well-constructed, leveraging double robustness principles from causal inference. However, the proofs assume asymptotic consistency of model estimates, which may not hold with finite training data.", "experimental_designs_or_analyses": "Synthetic experiments convincingly validate robustness, but lack exploration of high-dimensional covariates (p≫1000).  \nReal-data preprocessing (e.g., merging rare factor levels) is not rigorously justified and may introduce bias.", "relation_to_broader_scientific_literature": "The work builds on conformal inference for survival analysis and connects to double robustness in causal inference.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "Strengths:  \nFirst method to extend doubly robust conformal inference to right-censored data.  \nClear theoretical-empirical synergy.  \n\nWeaknesses:  \nLimited discussion of computational costs for imputation and calibration.  \nReal-data experiments lack domain-specific evaluation (e.g., clinical utility of LPBs).", "other_comments_or_suggestions": "N/A", "questions_for_authors": "Why use split-conformal instead of full-conformal inference? Could cross-validation improve small-sample performance?", "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": "KZVTAxID11", "reviewer": "Reviewer_MQ29", "summary": "This work presents a novel conformal inference framework for survival analysis with right-censored data, motivated by the limitations of existing conformal methods for construction lower prediction bounds (LPBs) for general right-censored data. The core idea is to fit a censoring distribution and sample from the corresponding truncated distribution (“Decensoring”), changing the right-censored dataset into a semi-synthetic “decensored” dataset, where the existing methods apply. This paper proved in Theorem 3.3 and 3.6 that the marginal coverage of constructed LPBs on ideal assumption has asymptotic double robustness via a fixed or adaptive censoring-time filter technique. Experiments results have shown that this framework, leveraging an adaptive cutoff, yields the best LPB, compared to other calibration methods.", "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": "Thank you for your thoughtful review and encouraging evaluation.\n## Accuracy of the Censoring Model\nWe appreciate your interest in how the censoring model’s accuracy affects performance. However, we believe there may be some misunderstanding about how best to assess this in practice.\nRather than reporting standard imputation error metrics—which would be both difficult to interpret in our context and unobservable in real applications—we evaluate performance by varying key simulation parameters that directly affect model quality, such as training sample size (Figures A2, A3, A6, A7) and the number of irrelevant features (Figures A5, A9). These factors offer a more meaningful and observable way to assess the impact of censoring model accuracy.\nCollectively, these experiments support our central empirical claim: the method achieves valid inference as long as either the survival or censoring model is reasonably accurate. This double robustness is the core strength of our approach and is clearly demonstrated in our results.\n## Discussion of Concurrent Work\nThe paper by Davidov et al. (2025) is concurrent with ours. It appeared on OpenReview one week before the ICML submission deadline—after our submission—and prior to that, only an anonymous version was available, without author names. This made it both too recent and too ambiguous to cite appropriately. Moreover, even the anonymous version was first posted within four months of the ICML deadline, and we became aware of it only when our paper was already essentially finalized. While the two papers address related goals, the methods are conceptually very different. It would be interesting to see a detailed empirical comparison in future work.\n## Empirical Performance Relative to Naive-CQR\nThis seems to be a misunderstanding. Table A7 shows our method yields higher (more informative) LPBs than Naive-CQR on most datasets—except COLON, where performance is similar. This aligns with Figure 2 and expectations: Naive-CQR does not properly handle censoring and tends to be too conservative (although not always).\n## Writing of Sections 1.3 and 2\nThank you for the suggestion—we will expand the related work section in the camera-ready version. In Section 2, we intentionally highlight Candès et al. (2023) and Gui et al. (2024), as our method directly builds on and generalizes their work. While their names appear multiple times, each instance clarifies this connection. We believe the structure is appropriate but would welcome specific suggestions if further streamlining is needed.\n## Computational Cost\nWe agree this deserves mention. The imputation step adds negligible overhead—once the censoring model is fitted (as also required by existing methods), sampling is extremely fast. Compared to model fitting, the additional cost is minimal. We will clarify this in the revised version.\n## Other Comments or Suggestions\nWhile Section 4.1 focuses on experiments using a generalized random forest (grf) for the survival model, we do evaluate other models in the appendix, as described in Section 4.3. We will clarify this point in the main text and fix the minor inconsistencies and typos you pointed out.\n## Fairness of Holding Out Calibration Data for Uncalibrated Method\nThis is a fair question. While the uncalibrated method could use more training data, we applied all calibration techniques to the same pre-trained model to ensure a clean comparison. Using different training sets would introduce some confounding.\nIn any case, as shown in Figures A3 and A7, the uncalibrated method performs poorly in harder settings regardless of the training sample size, so using more data would not meaningfully improve its results. We’ll clarify this in the final version.\n## Model Misfit and the Motivation for Conformal Inference\nIn practice we wouldn’t know if the pre-trained survival model is accurate enough, and that’s exactly the motivation for conformal inference. Our goal is to provide robust, distribution-free guarantees even when the survival model is unreliable. Our framework is designed to achieve valid inference as long as either the survival or censoring model is reasonably accurate. The simulations were intentionally constructed to reflect this uncertainty and demonstrate our method’s double robustness.\n## Varying the Censoring Proportion\nAs the censoring rate increases, the censoring model becomes easier to estimate, while the survival model becomes harder to learn—highlighting the value of our method’s double robustness. Since only one model needs to be accurate, coverage should be maintained across a range of censoring levels.\nThat said, extreme censoring can make our survival bounds less informative (i.e., more conservative), even if their coverage remains valid. This is a general challenge in survival analysis, not specific to our approach. We appreciate this question, and space permitting, will include this discussion and potentially an additional supporting figure in the final version.", "claims_and_evidence": "Mostly. See below.", "methods_and_evaluation_criteria": "Yes", "theoretical_claims": "Math proofs seem valid but are not fully checked.", "experimental_designs_or_analyses": "1.\tAblation of “decensoring” in model evaluations: \nThe novelty in methodology of this work mainly relies on the “decensoring” of calibration dataset added to the established fixed/adaptive cut-off strategies. The claim that “as long as $f{c|x}$ is reasonably accurate reasonably accurate, our two-step method is anticipated to yield approximately valid inferences” in section 2.3.3 is not backed with information about how accurate they are in experiments. Even if double robustness does not require perfect censoring distribution theoretically, the discussion on how the performance of pre-trained censoring distribution would impact the conformal LPB is meaningful in practice.", "relation_to_broader_scientific_literature": "“Decensoring” or, more broadly, modelling both censoring and time-to-event is not a new idea. The key contribution of this work is to establish an integrated theory and methodology with the established conformal survival analysis.", "essential_references_not_discussed": "An important contemporary work is founded here \nhttps://openreview.net/forum?id=JQtuCumAFD\nThey introduced two methods: a “focused” method as dropout and “fused” method as a similar imputation method. According to their claim, their LPBs are finite-sample valid in section 1.3, also approximately valid in section 3.1.", "other_strengths_and_weaknesses": "Strength: Detailed information with the major conformal methods in terms of pseudo-codes is nice. \n\nWeaknesses：\nIf I understand Table A7 correctly, for SOTA models (random survival forests) in real-world observation datasets, DR-COSARC is not as competitive as Naive CQR.\n\nThe writing of this paper needs to be improved. \n1.\tRelated work should be expanded and reference to censoring time imputation should be added, while repeated citations of Candes et al. (2023) and Gui et al. (2024) in Section 2 should be simplified. Use a preliminary section if necessary.\n\n2.\tWhile the computational cost of conformal prediction methods can be justified by the need for informed decision-making in fields like healthcare and finance, the potential computation cost in the imputation step and the conformal step should be mentioned as potential limitations of scalability in the Discussion.", "other_comments_or_suggestions": "1.\tThe authors only mentioned DeepSurv (Katzman et al., 2018) and Random Survival Forest (Ishwaran et al., 2008) in the introduction and they only evaluated Random Survival Forest in Section 4.1. \n2.\tThe implementation of random survival forests is not consistent. “e.g., implemented via the R package ranger” in section 2.3.3, while it becomes “randomForestSRC” in section 4.1.\n3.\tTypo in line 181.\n4.\tTypo in the title of Algorithm A5.", "questions_for_authors": "I have some practical questions:\n\n1.\tIs it fair to hold out 20% calibration data in the uncalibrated method? How are pretrained models obtained?\n\n2.\tAs a main takeaway of this work, this framework is competitive in simulated cases where both pre-train models are badly fitted: How do we know they fit badly? \n\nThe censoring distribution modeling would flip the event indicator. How does the censoring rate in the training dataset affect the performance of pre-train models (and thus the proposed framework)?", "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: After a very fruitful discussion between the authors and the reviewers, the reviewers unanimously favored accepting this paper (all five of the reviewers gave a score of a 4). I am thus also recommending this paper for acceptance.
Decision: Accept (spotlight poster)