Doubly Robust Conformalized Survival Analysis with Right-Censored Data

Thank you for your thoughtful review and encouraging evaluation.

Accuracy of the Censoring Model

We 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. Rather 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. Collectively, 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.

Discussion of Concurrent Work

The 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.

Empirical Performance Relative to Naive-CQR

This 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).

Writing of Sections 1.3 and 2

Thank 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.

Computational Cost

We 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.

Other Comments or Suggestions

While 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.

Fairness of Holding Out Calibration Data for Uncalibrated Method

This 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. In 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.

Model Misfit and the Motivation for Conformal Inference

In 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.

Varying the Censoring Proportion

As 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. That 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.

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.

Asymptotic vs Finite-Sample Theoretical Results

You'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. The 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. Importantly, 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.

Experiments with High-Dimensional Data

Our 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. Figures 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.

Data Pre-Processing

The 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.

Computational Cost

Thanks 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.

Practical Relevance

We 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. That 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.

Split vs. Full Conformal Inference, and Possible Extensions using Cross-Validation

We 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). Extending 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!

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.

DR Adjustment

You'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.

Convergence Rate of Censoring Model

We 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. That 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. Our 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.) We would of course welcome future work that seeks to tighten or relax this theoretical condition.

Censoring Time Imputation

Thank 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 $\hat{f}\_{C \mid X} = f\_{C \mid X}$.

Comparison with Naive CQR

There 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.

Design of Empirical Experiments

We 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. Our experiments explore the performance of our method in the face of key challenges like model misspecification, limited training data, and covariate dimensionality. For example:

• Figures A2, A6: Vary censoring model training size; performance improves as quality increases.
• Figures A5, A9: Add irrelevant covariates in censoring model; overfitting degrades performance.
• Figures A3, A7: Vary both models’ training size; highlight double robustness.

Figure 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.

Different Censoring Models

While 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.

Survival Modeling in Benchmark Data Sets

We 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.

LPB Units

LPBs are in different units across datasets and should not be directly compared across them.

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.

Imputation randomness

"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."

This 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.

While 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.

That 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.

Prompted 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.

Finite-sample theory

“(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.”

Yes, 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.

Additionally, 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.

We 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.

Thank you again for your helpful comments. We believe your suggestions will help improve the clarity and impact of the final version.

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.

The 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.

The 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.

For example:

• 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.
• Figures A5 and A9 vary the number of irrelevant covariates used in the censoring model, demonstrating the impact of overfitting on performance.
• Figures A3 and A7 examine the joint effect of training sample size on both models, further illustrating our method’s double robustness.

These 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.

We 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!