Causal Identification for Complex Functional Longitudinal Studies

Thank you for your detailed and constructive feedback, and for recognizing the strengths of this work. We appreciate the time you dedicated to our manuscript and the opportunity to address your concerns. Below, we respond to the key points you raised. ### Weaknesses 1. **Relation to "Causality for Complex Continuous-Time Functional Longitudinal Studies with Dynamic Treatment Regimes":** We acknowledge the connection with this paper, which indeed represents a subsequent work to the present one. While the later work generalizes identification results to include semiparametric frameworks, it lacks numerical examples and practical verifications. Our current submission provides a focused exploration of nonparametric identification, coupled with simulation-based numerical validation. Incremental contributions like this allow for a deeper understanding of specific aspects, offering clarity and actionable insights while building toward more comprehensive frameworks. 2. **Academic and Practical Significance of Solving More Complex Problems:** The introduction of variables such as \(T\), \(X\), and \(\Delta\) addresses complexities inherent in real-world medical applications. Medical studies often involve truncated follow-up due to death or censoring, which these variables explicitly model. Furthermore, in healthcare, interest frequently lies in the entire process of progression (e.g., \(Y(t)\)), rather than just a terminal outcome. By distinguishing between these components, our framework supports broader and more clinically relevant causal queries. 3. **Why is the Non-Parametric Property Important in a Functional Data Framework?** The non-parametric property aligns with the recent assumption-lean efforts in the causal inference community (see references below). Traditional approaches often rely on parametric or semi-parametric modeling assumptions, such as smoothness or sparsity, to facilitate analysis and reduce dimensionality. However, these assumptions are typically made for mathematical convenience rather than being grounded in prior knowledge. As a result, inferences drawn from such models may reflect the assumptions as much as, or more than, the data itself. For functional data, where the complexity of continuous, infinite-dimensional outcomes makes it even harder to justify any specific model, relying on parametric assumptions becomes especially unrealistic. Our framework deliberately separates modeling assumptions from identification, focusing purely on structural assumptions necessary for causal inference. This ensures that the framework extracts information only from the data, avoiding the risk of introducing unwarranted or misleading conclusions based on arbitrary assumptions. By adopting a non-parametric approach, we provide a more flexible and adaptable methodology that reflects the complexities of real-world data, particularly in healthcare settings where data rarely adhere to idealized models. This choice strengthens the framework’s robustness and relevance, particularly for functional data analysis. References: - Vansteelandt, Stijn, and Oliver Dukes. "Assumption-lean inference for generalised linear model parameters." *Journal of the Royal Statistical Society Series B: Statistical Methodology* 84.3 (2022): 657-685. - Vansteelandt, S., Dukes, O., Van Lancker, K., & Martinussen, T. (2024). Assumption-lean Cox regression. *Journal of the American Statistical Association,* 119(545), 475-484. 4. **More explanations on \(v\) and \(G\):** Yes, we are adding more explanations and examples around them. Thank you for pointing this out! 5. **"Can Equation (1) be understood as a general expression representing the average treatment effect, the averaged treatment outcome, or a transformed form of these?"** Yes, our characterization can accommodate all the cases you’ve mentioned here because we’ve allowed \(G\) to be a signed measure. This means it does not have to be positive, allowing, for instance, \(1(\bar{A} = \bar{1}) - 1(\bar{A} = \bar{0})\) (the difference of two delta measures), which represents the average treatment effect of always-treated vs. never-treated.

5. **Demonstration of Theorem 4** Thank you for highlighting the importance of Theorem 4. We agree that it is a key contribution of the paper, and we understand the desire for its further validation. However, it is important to note that Theorem 4 establishes a theoretical foundation rather than a practical computation. Specifically, the theorem demonstrates that our framework is dense in the probability measure space, which implies that in real-world scenarios, one cannot practically encounter a measure that violates it. As such, there is no established practice or meaningful way to numerically verify this property. Instead, Theorem 4 serves as the cornerstone for future work, ensuring that any estimation procedure—whether parametric or non-parametric for nuisance functions—can operate freely within our framework. This generality is crucial for enabling subsequent developments in functional data analysis for causal inference, including estimation, inference, and applications to real-world datasets like MIMIC-IV. While we acknowledge the importance of practical validation, the contribution of Theorem 4 is primarily theoretical, laying the groundwork for future empirical applications and methodological advancements. 6. **Definition of "Functional" Data** Thank you for your observation regarding the definition of "functional" data. We will incorporate the feedback from other reviewers and refine our definition, drawing from established references such as Wang, Chiou, and Müller (2016). Specifically, we will clarify that functional data refers to data arising from continuous-time processes, often characterized by high-dimensional, smooth trajectories or curves, and distinguish our focus within this framework. This will help to more rigorously position our approach while emphasizing its relevance to causal inference in complex scenarios. 7. **Assumption of Functional Continuity** The assumption of functional continuity reflects the population-level nature of our framework. The causal effect of a drug, for instance, operates in a continuous-time manner, even if real-world data are observed discretely. How sample-level data are recorded—whether densely, sparsely, or irregularly—is a topic for future work focusing on estimation. These considerations will include the minimum observation density required for consistent estimators and the potential shift to partial identification when non-dense observations are present. In this paper, our focus is on rigorously defining the estimand in a way that reflects the underlying continuous process at the population level. This definition is essential for future developments in estimation and inference, ensuring the framework remains robust and grounded in theoretical principles. For ethical concerns, Figure 1 will be removed according to another reviewer's suggestion to make room for more explanation.

We totally agree that the connection with these two papers can be made even clearer. * **Additions to "Causality for Functional Longitudinal Data" (referred to as "Ying (2024a)")**: * Ying (2024a) served as an initial stepping stone pushing the boundaries of causal inference into "functional longitudinal data," focusing on an oversimplified setting: a single, end-of-study outcome measured at one point in time. * In contrast, our paper, **as an extension pushing the boundary towards more realistic data**, introduces not only the newly added variable $T_{\bar{a}}$ but also incorporates a time varying outcome process $\bar Y_{\bar{a}}$. Both are time-varying processes yet they are different because $T_{\bar{a}}$ is a terminating 0-1 process that prevents the observation of other processes, whereas $\bar{Y}_{\bar{a}}$ is truly functional (or curve) data. Furthermore, we allow the data to be subject to right censoring ($C$). While the methodology may seem similar, this progression mirrors how incremental causal inference research has historically evolved—from simpler to more complex scenarios. Confirming the framework's applicability to a more complicated setting is both non-trivial and meaningful. * Our framework showcases nonparametric properties, which Ying (2024a) does not address. * We included an extensive simulation study, which is absent in Ying (2024a). * **Additions to "Causality for Complex Continuous-Time Functional Longitudinal Studies with Dynamic Treatment Regimes" (referred to as "Ying (2024b)")**: * Ying (2024b) extends the boundaries drawn in our paper to dynamic treatment regimes (DTR), a broader and more complex concept. However, this required substantial preparation, making the paper more difficult to understand. For instance, Ying (2024b) dedicates a full section (Section 3) to defining counterfactual outcomes under DTRs, a challenging extension. In contrast, our paper focuses on static treatment regimes (where the intervened distribution of $\bar{A}$ does not depend on covariates), a cleaner and more natural starting point. This mirrors how initial investigations in other contexts often begin with simpler concepts, such as average treatment effects for point exposures or marginal structural models for regular (discrete-time) longitudinal studies, both of which fall under static treatment regimes. * To prove nonparametric properties, Ying (2024b) relies on semiparametric theory and shows that the tangent space under its assumptions equals $L_0^2(P)$. In contrast, our paper takes a direct approach, proving that the subset of probability measures satisfying our assumptions is dense. * We included an extensive simulation study, which Ying (2024b) does not have. We hope this summary clarifies how our paper builds upon and differentiates itself from these foundational works.

The discussion period for reviewers ends on 12/02, please let me any follow up questions and we will be glad to answer.

Thank you for your detailed feedback. I appreciate the time you have taken to read the paper and provide thoughtful comments and questions. --- ### 1. Clarification on Data Type (Continuous vs. Discrete Measurements) - **Short Answer:** Our paper focuses on discretely measured data from continuous processes because, in practice, it is impossible to store or analyze infinite data. - **Long Answer:** In statistical inference, especially in the causal inference framework, there are typically two steps: **identification** and **estimation**: - **Identification** defines the parameter of interest and determines how, given infinite copies of data, one can mathematically identify the parameter. This is the focus of our paper. - **Estimation** involves constructing estimators using finite samples with desirable statistical properties, such as consistency or efficiency. While many studies integrate both steps, our paper specifically focuses on identification for **functional longitudinal data**. Beyond these two layers, for functional data derived from continuous processes, there is an additional layer: **representation error**. This refers to the gap between having infinite copies of discrete-time observations (what is practical) versus infinite copies of continuous-time observations (theoretical ideal). While representation error is an important aspect, it is tied to the estimation step and is therefore outside the scope of this paper. Note that this is pointed out by reviewer dyEJ's summary. - Representation error has been well-discussed in works like "Wang, Chiou, and Müller (2016)," which categorizes discrete-time observations into dense, sparse, or irregular regimes. Exploring how representation error diminishes under specific assumptions (e.g., number and frequency of observations) is indeed an interesting direction, but our paper focuses strictly on identification under the framework of **functional longitudinal data**. ### 2. Comparing Papers and Differentiation You summarized correctly that our paper differentiates itself by addressing more realistic data settings, incorporating nonparametric properties, and focusing on static treatment regimes. Additionally, I want to highlight a key feature that sets our paper apart: **the inclusion of a numerical study**, which the other papers lack. This provides empirical evidence supporting our theoretical findings and strengthens the practical relevance of our approach.

Thank you for your thoughtful and detailed feedback. Your comments provide valuable insights that will help us improve both the presentation and clarity of our manuscript. Below, we address your specific concerns and questions. Clarification of Summary We appreciate your summary and would like to clarify that our focus is on the underlying curve data \( X(t) \), as introduced in Wang et al. (2016). At the population level, our framework abstracts away the sparsity or regularity of sample-level observations. In future work, we plan to extend our framework to sample-level data, where factors such as sparsity or irregularity could influence the consistency of estimators. For example, investigating how the number of observed time points \( p_n \) scales with the sample size \( n \) in densely observed data could provide valuable insights. Weaknesses 1. Connection Between \( T \) and \( Y(t) \): We will improve the explanation of how \( T \) (the event time) relates to \( Y(t) \) (the outcome). Intuitively, \( T \) serves as a "Cemetery Point" where all other stochastic processes cease to evolve due to the individual’s death. For instance, if \( Y(t) \) represents disease progression and \( T \) represents death, \( Y(t) \) is fixed as \( Y(T) \) for \( t > T \). Both \( Y(t) \) and \( T \) are practically relevant, so we distinguish them in our framework. This distinction aligns with prior work such as Rytgaard et al. (2022). 2. Improving Readability: We will follow the concrete steps suggested by other reviewers, including simplifying notations, adding examples, and providing more intuitive explanations throughout the manuscript. This will make the connection between key components clearer. Questions 1. When \( A(t) \) is a Function: The results in our paper remain unchanged if \( A(t) \) is a function, vector, or scalar. This is because our framework is built using measure theory, which generalizes across these cases. Theorems and results retain their form regardless of the specific nature of \( A(t) \). 2. Why \( Y(t) \subseteq L(t) \): This is purely for notational simplicity. Including \( Y(t) \) separately would make the measures and derivations more cumbersome without adding clarity. The framework does not assume \( Y(t) \) must impact treatment assignment but allows this dependency to exist or not, reflecting scenarios like disease progression influencing treatment decisions.

Thank you for your constructive feedback and thoughtful questions. We appreciate your recognition of the contributions and potential impact of our work. Below, we address the key points you raised. Weaknesses 1. Numerical Experiment Simplification: We acknowledge the limitations of the current simulations and will incorporate additional scenarios accounting for mortality and censoring in the revised manuscript. 2. Causal Structure and \( Y(t) \subseteq L(t) \): This notation was chosen purely for simplicity. Including \( Y(t) \) separately would make the measures and derivations more complex without adding clarity. We are not assuming \( Y(t) \) must affect treatment assignment but instead allow this dependency to exist or not. This flexibility is critical as in many cases (e.g., disease progression), outcomes can influence treatment adjustments, and therefore acting as a confounder as well. 3. Intuitive Explanations for Propositions: We agree with this suggestion and will include intuitive explanations and examples following definitions and propositions, addressing similar feedback from other reviewers. Questions 1. Interventional Distribution (Equations (7)-(10)): We will expand this section to bridge the conceptual gap from Equations (2)-(6) to (7)-(10). Specifically: - Equations (4)-(5) describe the probability of censoring within or beyond \([t_j, t_{j+1}]\). - Intervening to a pseudo-world where censoring always happens after \(t_{j+1}\) leads to terms like \( {1 - 1(x \leq t_{j+1}, \delta = 0)} \). - Similarly, treatment distributions are intervened into \(G\) as shown in Equation (10). For additional context, we will reference Rytgaard et al. (2022) in the reference of our paper, particularly Definitions 1 and 2, to clarify intervention-based causal inference. 2. Extension to Dependent Censoring: Yes, our framework can be extended to handle dependent censoring. If the dependency is explained by observed factors, this is already addressed under Assumption 2. For unobserved dependency, methods like proxy variables (e.g., Ying, A. (2024). Proximal survival analysis to handle dependent right censoring. Journal of the Royal Statistical Society Series B: Statistical Methodology, qkae037.) can be adapted to extend our framework. Similarly, this approach could be generalized for unmeasured confounders.

Thank you for your detailed and thoughtful feedback. Your comments provide valuable guidance for improving our manuscript. Below, we address the key points you raised. The draft is under changing now to incorporate all reviewers' suggestions but here is a preliminary reply for what we will do and also answer some of your questions. General Revisions We will implement all suggested improvements in the "Weaknesses" section to enhance clarity and accessibility: 1. Terms such as "g-computation" and "counterfactual time-to-event endpoint" will be clearly defined with examples. We will also add reviews of some terms in the classical discrete-time case in the appendices. 2. The literature review will be expanded to include detailed textual explanations, replacing Figure 1. 3. Section 3.1 will be streamlined with concrete examples (e.g., for 𝜈 and 𝐺) and unnecessary notations removed. Specific Responses to Weaknesses Experiments Limited to Simulation Data: The primary aim of our paper is to address the identification problem for functional longitudinal data. As you and other reviewers noted, this is a novel area in the field, and our focus in this work is strictly on theoretical identification rather than estimation or inference. Given the 10-page limit and the complexity of the problem, our approach represents an incremental but crucial step in developing a solid foundation for future estimation frameworks. As described in the paper, the g-formula, inverse probability weighting (IPW), and doubly robust (DR) formulas yield identical results at the population level under identification, all equal to (1). Since all three are theoretically equivalent on the population level, verifying the g-formula suffices as a sanity check for our purposes. Additionally, the g-formula is the most computationally feasible for simulation because its implementation does not require knowledge on the measures themselves, unlike IPW or DR (unlike in discrete-time and non-functional cases, one have the knowledge and can compute the density, here we need to compute measures, which is far from trivial). This also addresses Question 2: At the population level, the three formulas are theoretically equivalent under our framework. Their finite-sample differences stem from estimation, which will be a focus of future research. With that said, we realized that our simulation can be oversimplified, as noted by other reviewers as well. Therefore, we will make our simulations more complex by adding more covariates, separating from the outcomes, adding mortality and censoring as well. Clarity Regarding the Statement on Density Functions (Line 141): In infinite-dimensional spaces, the concept of density is not applicable as in finite-dimensional Euclidean spaces. Instead, one must resort to measure-theoretic approaches. For example, the measure 𝑃(𝑑\bar{𝑎}𝑑\bar{𝑙})) referenced in the paper pertains to the probability distribution over the paths of stochastic processes, which is characterized directly via measures rather than densities. We will revise this section to clearly explain why density functions are unsuitable in this context and why measures are used instead.