Doubly Protected Estimation for Survival Outcomes Utilizing External Controls for Randomized Clinical Trials

Thanks for the careful reviews. Here are our detailed responses to your questions.

Methods And Evaluation Criteria

1. The selection of the cutoff value $\tau$ for computing RMST is crucial in practice since the tail distribution after $\tau$ is neglected. Typically, the event rates at this cutoff value should exceed 10% to ensure sufficient data for model development. A common rule of thumb is to set $\tau$ to be around the 75%-80% quantile of observed event times. However, this choice varies case by case and should be informed by domain expertise with real-world data. A brief discussion on this will be included in the paper.

Experimental Designs Or Analyses

1. We have included additional set of simulation, where for RCT, the hazard is $\lambda_a(t\mid X) = \exp(-0.5a-0.2X_1-0.5X_2 -0.1X_3)$; for EC, $\lambda_0(t\mid X) = \exp(-0.1X_1-0.5X_2 -0.2X_3)$ (i.e., the coefficients are not the same anymore), and the covariates are generated by the multivariate normal distribution with pair-wise correlation 0.5 (i.e., non-independent features). The results are presented here (https://anonymous.4open.science/r/ICML2025-7977/fig4.png), and main findings stay the same as before.

2. Moreover, transformation on the covariates (e.g., polynomial transformation) can be considered as data pre-processing to capture the non-linear relationship between the covariates and the time-to-event outcomes. Furthermore, our method allows to use flexible survival models (e.g., survival random forest) for estimating the survival curves (e.g., $S_a(t\mid X)$), which could also capture the non-linear relationship.

Other Comments Or Suggestions: We have changed Assumption 3.2 to Informative censoring as $T^{(a)}\perp C\mid X, A=a, R=r$ for $a=0,1$ and $r=0,1$.

Questions For Authors: Thanks for catching this typo. We have changed Assumption 3.1 (ii) to "$0<\pi_A(X),\pi_R(X)<1$ in the support of $X$", where $\pi_A(X)=P(A=1\mid X, R=1)$ and $\pi_R(X)=P(R=1\mid X)$.

We thank reviewer for the through assessment. We here provide the detailed response to each of these points.

Experimental Designs Or Analyses:

1. We have refined and reorganized the experiment sections to align with the objectives of the simulation, including how to design the data generation mechanisms accordingly, competitors, evaluation metrics, results, and detailed discussions.

2. The names of real-world dataset have been changed to EVOLVE-1 study and REGAIN study and made consistent throughout. The EVOLVE-1 study (Evaluation of Galcanezumab v.s. Placebo in the Prevention of Episodic Migraine) serves as the randomized clinical trials (CGAG is its protocol name). REGAIN study (Evaluation of Galcanezumab v.s. Placebo in Patients with Chronic Migraine) serves as the external controls (CGAI is its protocol name). The references will be included in the paper.

3. The threshold $-0.1$ for the real-data is chosen by domain knowledge, that is, reducing the RMST at month $\tau=6$ by $0.1$ is considered clinically meaningful. Further, the PrSS could be computed under various other thresholds and similar conclusions could be drawn; see https://anonymous.4open.science/r/ICML2025-7977/fig3.png.

4. To interpret the results in Panel (C) of Figure 3, we could use an example to illustrate. Suppose that we aim to reach PrSS at most $0.6$ at month $6$, our method “adapt” only need to recruit $100$ patients for the placebo group (solid red line at month $6$), however, the benchmark method “aipw” needs at least $150$ patients for the placebo group (dash green line at month $6$). Therefore, our approach could attain similar levels of PrSS with fewer patients by leveraging the external controls and thus shorten the patient enrollment period, which could eventually accelerate the drug development for rare diseases.

Relation To Broader Scientific Literature

1. First, the proposed method can be generalized to any estimand that is a function of the survival function $S_a(t)$ (e.g., mean or median of the survival time), not necessarily limited to one particular estimand as the EIFs are derived for $S_a(t)$. Let the estimand of interest be $\theta_\tau(t)=\Phi_\tau(S_a(t))$, the associated EIF for the estimand of interest can be obtained as $\psi_\theta(t) = d\Phi_\tau(q)/dq \cdot \psi_{S_a}(t)$ by Taylor expansion, where $\psi_{S_a}(t)$ is the EIF for $S_a(t)$, and the DR-learner is directly applicable to detect the outcome drifts in the estimand of interest. Thus, our proposed method should be useful to any integrative (causal or not) analysis for survival outcomes. We will emphasize this point in the paper.

2. Survival problems should be important in the machine learning (ML) community, such as dropout and customer churn. In many ML problems, there are heterogeneous data sources that can be integrated for the same task or domain adaptation; the critical issue is to handle data heterogeneity. Our proposed selective borrowing method offers a new perspective of the integrative analysis for the survival outcomes with a proper way for inference, which could be a valuable contribution to the general ML community. We will include such discussions in the paper as well.

We thank the reviewer for the comments. Here is a detailed response to your concerns.

Other Comments Or Suggestions

1. We will restate each theorem and lemma in the supplementary material. In the main text, we will cross-reference the proof of each theorem and lemma in the Appendix. "Theorem 3.4 is proved in Appendix A.1, Theorem 3.5 is proved in Appendix A.2, Theorem 3.6 is proved in Appendix A.3, Lemma 3.7 is proved in Appendix A.4, and Theorem 3.8 is proved in Appendix A.5".

2. We have already prepared the codes along with one implementation example for the proposed method in the supplementary material.

Questions For Authors

1. One short paragraph of double robust estimators will be included for the related work section: ”However, existing integrative methods are limited by the assumption of the Cox model, either on the cause-specific or subdistribution hazard scale, which requires to accurately model the survival probability. In recent years, semiparametric efficient and doubly robust estimators, which leverage the efficient influence function (Bickel et al., 1993; Tsiatis, 2006; van der Vaart, 2000; van der Laan & Robins, 2003), including estimation equation methodology (Hubbard et al., 2000; Robins & Rotnitzky, 1992; van der Laan & Robins, 2003) and targeted maximum likelihood estimation (van der Laan & Rubin, 2006; Rytgaard et al., 2022), have gained great popularity in many fields and are increasingly used to draw inference about treatment effects.”

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1. Bickel, P. J., Klaassen, C. A. J., Ritov, Y., & Wellner, J. A. (1993). Efficient and adaptive inference in semiparametric models, Forthcoming monograph.
2. Tsiatis, A. A. (2006). Semiparametric theory and missing data (Vol. 4). New York: Springer.
3. van der Vaart AW (2000) Asymptotic statistics, vol 3. Cambridge University Press, Cambridge.
4. van der Laan MJ, Robins JM (2003) Unified methods for censored longitudinal data and causality. Springer, Berlin.
5. Hubbard AE, van der L MJ, Robins JM (2000) Nonparametric locally efficient estimation of the treatment specific survival distribution with right censored data and covariates in observational studies. In: Statistical models in epidemiology, the environment, and clinical trials, Springer, Berlin, pp 135–177.
6. Robins JM, Rotnitzky A (1992) Recovery of information and adjustment for dependent censoring using surrogate markers. In: AIDS epidemiology, Springer, Berlin pp 297–331
7. van Der Laan, Mark J., and Daniel Rubin. "Targeted maximum likelihood learning." The international journal of biostatistics 2.1 (2006).
8. Rytgaard, Helene C., Thomas A. Gerds, and Mark J. van der Laan. Continuous-time targeted minimum loss-based estimation of intervention-specific mean outcomes. The Annals of Statistics 50.5 (2022): 2469-2491.

Thanks for the careful review and nice words. We hereby provide one-to-one responses to your concerns.

Methods And Evaluation Criteria

1. The current three settings represent three typical scenarios we often encounter in practice: Setting 1, where all the ECs are comparable after adjusting for covariate shift and should be included; Setting 2, where the unmeasured confounding is present, none ECs are comparable and the external data should not be used; Setting 3, where there is lack of conccurency, only 1/3 of ECs are comparable and that portion of external data should be borrowed.

2. As suggested, we consider two additional simulation settings, and the results are presented here (https://anonymous.4open.science/r/ICML2025-7977/fig1.png).

• (Setting 4) different covariate effects: for RCT, $\lambda_0(t\mid X)=\exp(-0.2X_1 -0.2X_2 -0.2X_3)$; for EC, $\lambda_0(t\mid X)=\exp(-0.5X_1-0.5X_2-0)$
• (Setting 5) different time-varying hazards: for RCT, $\lambda_0(t\mid X) = t \exp(-0.2X_1-0.2X_2 -0.2X_3)$; for EC, $\lambda_0(t\mid X) = 2t \exp(-0.2X_1-0.2X_2 -0.2X_3)$.

3. Both the proposed estimator and TransCox (Li et al. (2023b)) could handle differences in covariate effects and time-varying hazards. However, TransCox is only valid under the Cox model. If the conditional survival curve $S_a(t\mid X)$ is not a Cox model (e.g., under Settings Two and Three in the paper, if we integrate out $U$ (or $\delta$) out in the hazards, the model generation will no longer be Cox model), TransCox will have large biases whereas our proposed estimator still controls the bias due to its double robustness and achieves improved performance.

Experimental Designs Or Analyses

1. The penalty term is chosen to be the adaptive lasso (Zou, 2006). The conditional survival curves $S_a(t\mid X)$ and $S^C(t\mid X)$ for the event and censoring are modeled by the Cox PH model, and the propensities $\pi_R(X)$ and $\pi_A(X)$ are modeled by SuperLearner from the SuperLearner R package, which is an ensemble model of the Logistic Regression and Random Forest. These details will be added to the Simulation section.

Other Comments Or Suggestions

1. The comments on the tables and figures are well-received. We have updated the colors for figures by the Nature Platte. Also, the shapes of points are changed to be different for estimator and scenario. The updated figures are provided here (https://anonymous.4open.science/r/ICML2025-7977/fig2/fig2_1.PNG; https://anonymous.4open.science/r/ICML2025-7977/fig2/fig2_2.PNG; https://anonymous.4open.science/r/ICML2025-7977/fig2/fig2_3.PNG).

Questions For Authors

1. The conditional hazard function should be indexed with a. The original table only includes the hazard functions that have been modified under each considered setting. To avoid confusion, we explicitly list the hazard function as $\lambda_a(t \mid X, R)$ under each setting.

• (Setting 1) $\lambda_a(t \mid X, R) = \exp(-0.5a - 1_p^T X \cdot 0.2)$;
• (Setting 2) $\lambda_a(t \mid X, R) = \exp[-0.5a - 1_p^T X \cdot 0.2 + 3\{U + \mathbf{1}(R=0)\}]$;
• (Setting 3) $\lambda_a(t \mid X, R) = \exp(-0.5a - 1_p^T X \cdot 0.2 + 3\delta \mathbf{1}(R=0))$.

2. Under Setting 2, we include $U$ in the hazards for RCT as well to keep the variability of hazards the same level across two datasets. In particular, for $R =1$, $U \sim N(0,1)$ with zero-mean is included in the hazard model, whereas for $R=0$, $U+1 \sim N(1,1)$ with non-zero mean is included, which is expected to introduce more outcome drift for the external controls.

3. Lack of concurrency will be discussed further in the section of Introduction, where FDA drafted the Guidance documents on the use of external controls: "Lack of concurrency could occur when RCTs and ECs are collected in different time periods or under varying healthcare settings. Therefore, directly integrating ECs with RCTs without any adjustment could introduce biases into the treatment estimation". The guidance reference from FDA will be added to the paper

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1. Li, Z., Shen, Y., and Ning, J. Accommodating time-varying heterogeneity in risk estimation under the cox model: A transfer learning approach. Journal of the American Statistical Association, 118(544):2276–2287, 2023b.
2. H. Zou. The adaptive lasso and its oracle properties. Journal of the American statistical association, 101(476):1418–1429, 2006.