STABLE ESTIMATION OF SURVIVAL CAUSAL EFFECTS

Thank you for your insightful comments and suggestions. We've taken your feedback into careful consideration and made the following clarifications and improvements:

1. Population vs Individual Level: Our method addresses the average treatment effect (ATE) rather than individual or heterogeneous treatment effects (HTE). Conditioning on the covariate is necessary to adjust for confounding when estimating both things.

2. Distribution of Expectations: We realize the lack of clarity regarding the distribution of expectations in the text. All our expectations are calculated concerning the observable distribution O=(X,A,E,T). We'll explicitly state this in the paper for better understanding.

3. Discrepancy in Bias vs. Outcome Regression (OR) and Relative RMSE: We display Relative RMSE because RMSE itself changes significantly over time, i.e., when we go from estimating the survival curve at time t=2 to time t=10, so we can’t really see the whole story on one set of axes. The choice to compare everything to OR gives us a reasonable scale to work with.

4. Assumptions Regarding Independent Censoring: Our method does not assume independent censoring, which sets it apart from some related works. We've highlighted in the covariate balancing section that our asymptotic bound is theoretically stronger and provably optimal even without this assumption compared to Xue et al. 2023. Additionally, we will try to clarify this further in the paper, emphasizing the strength of our approach in not relying on this assumption.

We appreciate your constructive feedback, and these clarifications and adjustments will be incorporated to enhance the paper's clarity and completeness.

1. The probability P. We could not find in the manuscript where P has been wrongly attributed to density. All of our notations refer to probability because the space of time T, assignment A and event E are all discrete.
2. Optimality of the variance of $\hat \psi$. The variance of $\hat \psi$ is indeed optimal as we have mentioned after Theorem 3.5. To clarify this further, we will discuss more the connection to the influence function of $\hat \psi$. In essence, our method has similar asymptotic properties to other semi-parametric efficient methods for estimating the ATE because our Riesz representer estimator is a consistent estimator of the inverse propensity weight, as pointed out in section 3.
3. Causal Survival Forest. Thank you for pointing out this relevant paper. We will add the new experimental result.
4. Hirshberg & Wager (2021) able to handle high dimensional data. We chose to state our asymptotic results in nonparametric terms for simplicity, but high-dimensional asymptotic results analogous to those in Hirshberg & Wager (2021) follow from the same arguments.

Thank you for dedicating time to assess our paper and provide invaluable feedback. We appreciate your insights into the presentation and structure of our work. We agree and have taken your suggestions seriously to enhance the readability of our paper.

1. Abstract. We acknowledge your recommendation and will revise our abstract as follows:

We study the problem of estimating survival causal effects, where the aim is to characterize the impact of an intervention on survival times using debiased machine learning. Historically, parametric or semiparametric models (e.g. proportional hazards) were popular, but they come with problematic levels of bias. Recently debiased machine learning approaches become popular, especially in applications to large datasets. However, despite their appealing theoretical properties, these estimators tend to be unstable because the debiasing step involves the use of the inverses of small estimated probabilities. This problem is exacerbated in survival settings where propensities are a product of treatment assignment and censoring probabilities. We propose a covariate balancing approach to estimating these inverses directly, sidestepping this problem. The method works by estimating the propensities not by their functional forms but by their roles as the Riesz Representers of our linear functional parameter. The result is an estimator that is asymptotically optimal in theory and stable and efficient in practice. In particular, under overlap and asymptotic equicontinuity conditions, our estimator is asymptotically normal with negligible bias and optimal variance. Our experiments on synthetic and semi-synthetic data demonstrate that our method has competitive bias and smaller variance than debiased machine learning approaches.

2. Introduction and Related Work. We will make the following changes: Paragraph 3 in the introduction which talks about the advantages and the instability problem of previous debiased machine learning methods, will be streamlined, and will incorporate paragraph 1 and 2 in related works. Paragraph 4 in the introduction which talks about covariate balancing will incorporate paragraph 3 in related works.

3. Section 2.3. We understand your concerns about the notation and its complexity in this section. These notations establish the framework for Section 3. We will move this subsection to section 3 and try to offer more contextual explanations for the readers.

4. Approach. We appreciate your feedback on this critical section. We firmly believe that the core innovation of our paper lies in the novel mathematical framework presented in Section 3. We will provide more contextual explanations and cut back on the theoretical details.

5. Experiments: We will bring the data generating process details from the Appendix to the main text.

6. Discrete time vs Continuous time SA: We chose to focus on discrete survival analysis due to its relevance in various real-world applications [1]. For example, In healthcare, patients may get checked up every few months so that we can not know exactly when the event happens, therefore discretized time is more robust. While our method primarily targets discrete survival analysis, we acknowledge the significance of continuous-time models. Theory-wise, the distinction between the 2 models, more specifically the fact that the discrete hazard is a conditional mean, while the continuous hazard is not, poses challenges in unifying the theoretical framework. We will tackle continuous SA in future work.

7. Asymptotic normality: Let Q be (estimate - estimand) / estimated standard error. The Kolmogorov-Smirnov statistics (the maximum difference between 2 CDFs) of Q produced by the methods vs. a normal distribution on the Synthetic dataset are: for Outcome Regression (OR): 0.93, for Double robust (DR): 0.25, for Balance: 0.06. We will also include a histogram of Q based on [2], which shows that Balance in fact approximates the normal distribution best.

[1] Multivariate Survival Analysis and Competing Risk.

[2] https://docs.doubleml.org/stable/guide/basics.html#regularization-bias-in-simple-ml-approaches