Causal Identification for Complex Functional Longitudinal Studies
Authors: Andrew Ying
Venue: ICLR 2025 Poster (2025)
Keywords: Causal Inference
Stochastic Process
Longitudinal Data; Functional Data
Continuous Time.
Primary area: causal reasoning
TL;DR: We propose a novel causal identification framework for functional longitudinal data.

Abstract: Real-time monitoring in modern medical research introduces functional longitudinal data, characterized by continuous-time measurements of outcomes, treatments, and confounders. This complexity leads to uncountably infinite treatment-confounder feedbacks, which traditional causal inference methodologies cannot handle. Inspired by the coarsened data framework, we adopt stochastic process theory, measure theory, and net convergence to propose a nonparametric causal identification framework. This framework generalizes classical g-computation, inverse probability weighting, and doubly robust formulas, accommodating time-varying outcomes subject to mortality and censoring for functional longitudinal data. We examine our framework through Monte Carlo simulations. Our approach addresses significant gaps in current methodologies, providing a solution for functional longitudinal data and paving the way for future estimation work in this domain.

Published as a conference paper at ICLR 2025
CAUSAL IDENTIFICATION FOR COMPLEX FUNC- TIONAL LONGITUDINAL STUDIES
Andrew Ying
Irvine, CA 92606, USA ay syingg339e gmai com
ABSTRACT
Real-time monitoring in modern medical research introduces functional longitu- dinal data, characterized by continuous-time measurements of outcomes, treat- ments, and confounders. This complexity leads to uncountably infinite treatment- confounder feedbacks and ference methodologies cannot handle. Inspired by the coarsened data framework, we adopt stochastic process theory, measure theory, and net convergence to pro- pose a nonparametric causal identification framework. This framework gener- alizes classical g g-computation, inverse probability weighting, and doubly robust formulas, accommodating time-varying outcomes subject to mortality and censor- ing for functional longitudinal data. We examine our framework through Monte Carlo simulations. Our approach addresses significant gaps in current methodolo- gies, providing a solution for functional longitudinal data and paving the way for future estimation work in this domain.
data, which traditional causal in-
INTRODUCTION
The advent of real-time monitoring technologies in healthcare has led to the continuous-time mea- surement of outcomes, treatments, and confounders, which we term "functional longitudinal data." Here by "functional" we mean the first-generation data described in Wang et al. (2016), or termed curve data (Gasser et al.) [184]] Rice & Silverman, [991; Gasser & Kneip 1995), operating over time. For example, the Medical Information Mart for Intensive Care IV (MIMIC-IV) (Johnson al. 2023) freely accessible electronic health record (EHR) database that records ICU care data, including physiological measurements, laboratory values, medication administration, and clin- ical events. Another example is Continuous Glucose Monitoring (CGM) (Rodbard) 2016 Klonoff al. (017). increasingly adopted technology for insulin-requiring patients that provides insights into glycemic fluctuations. CGM offers a real-time, high-resolution stream of data, capturing the intricate fluctuations in interstitial fluid glucose levels every few minutes.
These examples illustrate the recent prevalence functional longitudinal data, highlighting the ne- cessity of causal framework, understanding treatment effects of paramount interest in these settings. However, there is a great lack of investigation of causal inference at the intersection of lon- gitudinal data and functional data. Even identifying causal parameters interest through observed data becomes highly nontrivial in this setting, due to the issue of uncountably infinite treatment- confounder feedbacks (Hernán & Robins 2020) within functional longitudinal data. Treatment- confounder feedbacks occur when treatments taken over time influence variables (confounders) that in turn affect future treatments. For example, in a medical study, a patient's is current medication (treatment) could affect their future health (a confounder). and their health might determine which medications they receive later. This mack-and-forth interaction over time creates a cycle that is difficult to disentangle when analyzing causal relationships. In functional longitudinal data, this feedback becomes even more complex because both treatments and confounders are recorded as continuous functions over time rather than at discrete time points.
Moreover, functional longitudinal data, modeled as infinite-dimensieee dimensional, continuous time stochastic processes, demand creasure-theoretic foundation to ensure mathematical rigor. This requirement introduces complexities far beyond the scope of classical causal inference, which typically assumes mnite-dimensional data and more elementary statistical tools. Apart from the mathematical rigor,

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from statistical level, traditional approaches for handling functional data often rely on paramet- ric or semi-pes cami-parmeetric modeling assumptions, such as smoothness or sparsity, to facilitate analysis and reduce dimensionality. However, these assumptions venience rather than being grounded in prior knowledge. As result, inferences drawn from such models may reflect the assumptions much as, more than, the data itself.
typically made for mathematical con-
To bridge this gap, we aim to propose a novel identification framework for functional longitudinal data with time-varying outcomes subject to mortality and censoring, who enjoys the nonparametric property, making it more flexible and adaptable to various datasets.
We first define causal quantity representing the mean of counterfactual outcomes under idealized randomized world. To connect the observed data distribution to this idcalized world, inspired by the coarsened data framework (Heitjan & Rubin 1991) and through the application of continuous-time stochastic process theory and measure theory, we upgrade classical causal assumptions to accommo- date functional longitudinal data nonparametrically. These together resolve the issue of uncountably infinite treatment-confounder feedbacks (Hernán & Robins, 2020) for functional longitudinal data. We generalize the well-known g-computation formula, inverse probability weighting formula, and double robust formula. We examine our identification framework through Monte Carlo simulations.
The paper organized as follows. In Section 2 we present literature review of related work. In Section 3 we define the notation and parameters of interest. Then we propose identification sumptions and generalize the well-known& g- g-computation, inverse probability weighting, and double robust formulas (Hernán & Robins 2020). Additionally, parametric. We conduct Monte Carlo simulations to examine our framework in Section [ Section 5 discusses future directions. While this paper builds population-level framework with numerical results, it does not explore estimation or associated inference, which is beyond the scope of this study and left for future research.
prove that our identification is non-
2 RELATED WORK
Causal Inference for Non-Functional Longitudinal Studies. Current causal frameworks for lon- gitudinal studies fall into two main categories: "regular longitudinal studies," where time advances in fixed intervals (Greenland & Robins, 1986 Robins 1986). and "irregular longitudinal studies," " where events occur at random but discrete time points (Lok] 2008 Røysland 2011|Ryygaan al.] 2022). "Regular longitudinal studies" are straightforward but limited to structured designs, while "irregular longitudinal studies," such as those by Rytgaard al. (2022), accommodate random visit times by modeling treatment and confounder processes as counting processes. These approaches sume finite
feedbacks and fnite-dimensional data, relying on stepwise paths and joint densities for causal identification.
However, modern medical studies often generate functional longitudinal data through continuous monitoring of treatments and confounders, seen in intensive care settings (Johnson et al. 2016 2018) and wearable devices for chronic disease management (Mastrototaro) | Klonoff 2005
Rodbard 2016). Existing frameworks, designed for discrete-time stepwise processes, are insuf- ficient for such infinite- -iieeiiooad data, highlighting the need for new causal inference tools that accommodate the complexities of functional longitudinal data.
Existing research on causal inference has examined functional data within observational studies, as highlighted in works by (Miao et al.) 2020 Zhang et al. 2021 Tan et al. 2022). These studies share similar data format with our analysis. However, our approach distinguishes itself by focusing on the time-dependent nature of longitudinal studies, where data evolve continuously over time. In contrast, the cited works primarily address "point exposure," which looks at the impact of single treatment or covariates measured at beginning of study, without accounting for how treatments or covariates may change and interact over longer period.
Causal Inference for Functional Data.
Existing Work for Functional Longitudinal Data. The only exceptions that investigated causal inference for functional longitudinal data are Ying (2024a) and Sun & Crawford (2022). However, Ying (202aa) only investigated single outcome, measured at the end of some medical studies, neither proving the nonparametric property nor conducting any numerical investigation. On the other hand, Sun & Crawford (2022) imposed stochastic differential equations with stringent parametric
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assumptions. This situation highlights significant gap in methodological advancements within the field. A related study (Ying) 2024b) explores a more general framework, building upon the methods and insights presented here. However, it does not include numerical examples or practical verifications, one of our contributions.
3 PROPOSED METHOD
3.1 PREPARATION
Consider a longitudinal study spanning from time O to oc:
A(t) and L(t) are two stochastic processes denoting the treatment administered and the measured confounders, respectively, at any given time t. At any time, A(l) and L(t) could be binary, categorical, continuous, or even functional. We denote A(l) = [A(+) :0 S
I) and L(() = (L(s) 0 S S (). with A and L representing the collections of treatments and confounders over the entire study.
We are interested in an outcome of interest (t),, as a subset of L(t), that is, (t) C L(t).
This notation was chosen purely for simplicity. We are not assuming (t)) must affect treatment assignment but instead allow this dependency to exist or not. This flcxibility is critical in many cases (e.g., disease progression), outcomes can influence treatment adjustments, and therefore acting as a confounder as well.
Let T be a time-to-eved endpoint, for instance, death, and C be the right censoring time. Define X min(T, C) the censored event time and A 1(T C) the event indicator. Therefore when A = 1, X = T and when A = 0, X = C. We also define N()) = 1(X < t) the counting processes of X.
Write the counterfactual time-to-event endpoint Ta and counterfactual covariates La (4). for any ā E A, where A encompasses all possible values of â. Therefore we have Xa = min a C¹) and Aa 1(Ta C). We assume that the future cannot affect the past, that is. 1 (Ta 2 t) (T > t) and La (t) = Læ (t) whenever a(t) = a' att). We also write TA (Ta la:ac E A} and L = (Ta ā a A).
The full data are {A, C, TA. LA} LA) and the observed data are {A, A, LJ. Note that on the observed data level, A() and L(tt) are not observed for & S X or defined for t < T. For easier notation in this paper, we offset A(t) A(X) and L(t) = L(X) for observed data whenever & > X. In this way, the stochastic processes A(t) and L(t) are well defined at any t > 0.
Define &. = = (LA(s), L(s), 1(X S 1(X S 8) Vo (}) as a filtration of informa- tion observed up to time t. Also we write F.- ooUngser and 9. 0{ A())) We define 9xx = Foc. We write Fo- and 9-- as the trivial sigma algebra for con- venience. Note that X is stopping time with respect to Ft. with Fx Fx ((A,,X,ATT))
We use P(drdódad)) (Bhattacharya & Waymire 2007 Durrett 2019 Gill & Robins
2001) represent the measure the path space induced by the stochastic processes. Note that this is not a density function. We use as the corresponding expectation.
In the context of MIMIC-III A(t) could represent antibiotics usage at time f, and L(t) may include range of clinical measurements, such as severity of illness scores, vital signs, laboratory values, blood gas values, urine output, weight, height, age, gender, service type, total fluid intake, and total fluid output at time t. The outcome (()) might measure illness progression influenced by antibiotics,
MMassrre theory is essential in continuous-time stochastic processes because it addresses challenges that density-based approaches cannot handle. Many processes, such as those with jumps or irregular paths, lack well-defined densities, yet measure theory allows work directly with their distributions. Additionally, stochastic processes often evolve in impractical, but measure-theoue -heorreticccccc methods naturally extend. It also enables rigorous definitions of key con- cepts like conditional probabilities and expectations, which are foundational in this field. Beyond practicality, measure theory aligns with the tradition and standard methodology in stochastic process theory, making it both necessary and convenient choice.
spaces (c.g.. path spaces), where defining densities
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such changes in severity scores time. T could represent the time to discharge mortality, with C' as the time the patient is censored, such as at the end of data collection. Counterfactual outcomes like Ta might represent the time to recovery under specific antibiotic regimen a, and (t) could represent the trajectory of severity scores under that treatment.
Similarly, in the context of CGM, (tt) represents insulin dosage time and [((t) includes glucose levels and immediate behavioral changes such as diet, medications, and physical activity at time The outcome (t) represents the glucose levels monitored in real time in response to insulin adjustments. 7 could represent the time to severe glucose event, with C' as the time the patient stops CGM usage. Counterfactual outcomes like Ta represent the time to stable glucose control under a specific insulin dosing regime a, and La (t) captures the counterfactual glucose trajectory.
We are interested in learning marginal mean of transformed potential outcomes including a time- to-event outcome and an outcome process under a user-specified treatment regime in the absence of censoring.
/ B[u(Ta, Gd)))
where y is some eser-specified function and G is priori defined (signed) measure on A, represent- ing a stochastic treatment regime. Here stochastic treatment regimes not prescribe a specific treat- ment value but instead define the probability of receiving each possible treatment. In other words, it assigns treatments randomly according to a specified probability distribution. Stochastic treatment regimes offer flexible approach for modeling treatments that either continuous challenging to precisely quantify. Unlike deterministic regimes, where treatment decisions are fixed, stochastic regimes introduce variability, enabling broader range of real-world applications. This approach is particularly beneficial in scenarios where treatments are not strictly prescribed but instead follow probabilistic guidelines are influenced by patient behavior external factors. Examples of v(-):
vTTa, (Ta t), for some time 0, identifies the effect of ā the survival probability. For instance, in MIMIC-III, this could represent the probability of a patient surviving beyond time under a specific antibiotic regimen ä. Alternatively, uTTa Ya) min(Ta. represents the restricted mean survival time.
Ya) (T) is the outcome measured at time for some 0. In CGM, it could correspond to the glucose level at time т under a specific insulin dosing strategy a. Alternatively, v(Ta, Ya) Ya (Ta) represents the outcome measured the time-to- event Ta- In MIMIC-III, this might capture the severity of illness or lactate level at the time of recovery or death. Finally, v(Ta, Ya) fo (()))a((dd)) represents the weighted averaged outcome over [0, r]. where w(t is sper-specified weight function.
We assume ((r(Ta, Ya)) is integrable against G. This exploration encompasses marginal means under static treatment regimes, as discussed in various literature (Rytgaard et al., 2022] Cain et al.] 2010 Young et al. 2011 Hernán & Robins 2020). This quantity be seen the mean of counterfactual outcomes under an idealized randomized world, where ă is randomized to follow a stochastic treatment regime G. Examples of G:
G 1(Ā - a) representing the averaged treatment outcome under specific regime is of interest. G = 1 (A = a) 1(Ā = ā') representing the averaged treatment effect of specific regime versus another á'.
For treatments like physical activity, which is inherently variable and challenging to quan- tify precisely, can be modeled using stochastic regimes. For example, rather than prescrib- ing a strict regimen of 30 minutes of exercise daily, a stochastic regime might increase the likelihood of patients engaging in activity based encouragements or incentives. In both case, G can be specified as a distribution instead of delta masses.
3.2 IDENTIFICATION ASSUMPTIONS
We have defined the parameter of interest Intuitively if treating treatment process A as selec- tion process (Heitjan & Rubin 1991). is the mean of v(Ta, Y,) when A were to follow and there is censoring. To create such pseudo-population, note that for any sequences of partitions
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[O ool]?= where have partition Ax o0| numbers of the form to < ... < tk = oo, we loosely have the following decomposition
ool over oc] is finite sequence of K
P(dzdódadl)
(2)
K-1
II F ( Fr - ())(()))
(3)
Fo(t)41 Fr 1
(4)
((d((t)+1))) P(da(t++1)
(5)
where we temporarily write Fr and Fe as the distribution functions of T and C. We intervene treatment distribution at each time lg approximate the pseudo-population where A were to follow G as:
P o0) (ddddddddd
K-1
II Fr(ts+1 JANN D (1 (1-)) 1=0 P(dl(t)+1)) (G(aa(t)-)))((t)
(8)
Here informally, one might understand this intervention as we replace the censoring distribution (4)
and treatment distribution (5) between (f,, (yaa) by no censoring in (7) and targeted treatment distribution as in (8). For readers unfamiliar with intervention based causal inference language, refer to Rytgaard al. (2022) Definitions 1 & 2). A more formal and mathematically rigorous decompositions are given in Section A the appendix.
To eliminate confounder bias, need make sure there is no unmeasured confounders. We adapt the commonly known "coarsening at random" (Heitjan & Rubin 1991) assumption into:
Assumption (Full conditional randomization). The treatment assignment is independent of the all potential outcomes and covariates given history, in the sense that there exists a bounded function e(t, n) > 0 with So e(t, 1)dt - as 1 - 0, such that for any E [0, 00), 0, 13 > O.
sup | P(dt.dla A(t + .)) dl FillTv) < a(t. n))
(9)
acA
where || 1/rv the total variation norm over the path space's signed measure space.
This assumption claims that, the treatment distribution, or equally, the probability of coarsening, in a small period time around I, only depends on the observed data up to time and independent of further part of counterfactuals. This assumption says in approximating sense that there is no common cause between treatment decision between time [t. + n) and all future counterfactual confounders. Intuitively and unofficially,
might this saying dddt, A(t + n), F.) Ro di FF,,, or approximately, (Ta, Taa 1 Á(t + 5)
We also need an assumption over the censoring mechanism to eliminate the censoring bias. We consider the well-known conditionally independent censoring assumption (Tsiatis) 2006 Andersen
al. 2012). Define the full data censoring time hazard function
\ T, A, T.)
lim IP(C S + dt|C > 1, 1,T, A,T./..
(10)
dr
The following assumption requires that the full data censoring time hazard at time only depends on the observed data up to time t.
Assumption (Conditional independent censoring). The censoring mechanism is said to be condi-
tionally independent if
7 L)
lim P(C S t Ct > 1,7 dt
A,) t, > f)
(11)
Note that in order to overcome the continuous time issue, here we impose Assumption D over an infinitesimal period of time. This type of idea is also adopted in Assumption 2 Note that how Assumption 2 is given on the intensity process whereas Assumption iss on the conditioning event. This is because does not have intensity process for general stochastic process.
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With Assumptions 1 and 2 are able to show that whenever | c0|1 + 0, PAN [0,ce].G approximates a aseudo-measun where treatment distribution are intervened by uncountable times into following G, where the mesh | oo] of partition A oo] is defined as max max(maax;=..--. -o. (15++ 1z), 11) representing the maximum gap length of the partition:
Proposition 1 (Intervenable), Under Assumptions Aand the measures IP Fax co converges to the same (signed) measure Ps == dla yG (dā joa in the total variation norm on the path space as the meshes shrink to zero, regardless of the choices of partitions, that is,
(0 (drrbbdddd)) - P(dxadla (aaradda))G((d))aaa (Ga))) lıv - 0,
(12)
whenever A o0|1 - O.
We refer FG as the target distribution. This proposition has helped us use the intervened ob- served data distribution PAx ( to identify the target counterfactuals distribution
(G( in asymptotic sense. The following assumption links the observed variable with the counterfactuals.
Assumption 3 (Full consistency). For any I,
T= L() = (t).)
(13)
The full consistency assumption links the observed outcome and the potential outcome via the treat- ment actually received. It says that if an individual receives the treatment A = a. then his/her observed outcome Y matches Ya.
The following assumption ensures that the observed data can identify the target distribution. Assumption (Positivity).
PG P.
(14)
With the above assumptions, we are able to generalize the well-known identification formulas: g- computation, inverse probability weighting, and double robust formulas, into functional longitudinal data. Note that our assumptions can be weaker but chosen for ease to interpret.
3.3 IDENTIFICATION FORMULAS
Below we show how we generalize the well-known g-computation, inverse probability weighting, and double robust formulas for "functional longitudinal data." For readers unfamiliar with the con- cepts, we refer Hernán & Robins (2020). We also prepare a review for these formulas in discrete- time longitudinal data in Section B in the appendix.
Definition (G-computation process). Under Assumptions andd define
Hc(t) R& (X, Y) (.
(15)
as a projection process, which is apparently a Fo-mmartingle. We call Ho(t) the g-computation process. Note that
Ho(co) = vix->(XXX))) P). HG(0-) = -x)((xX)))
(16)
The g-computation process intuitively serves as consecutive adjustment of the target v(X, Y) from oo to 0. It represents mix of original conditional distributions of covariate process together with the intervened treatment process G, from end of study to the beginning. Following this adjustment to the beginning of study, we have:
Theorem 1 (G-computation formula). Under Assumptions 7
2
and
is identified via
g g-computation formula as
J E(u(Ta, (a))) = HG((-).
(17)
Definition 2 (Inverse probability weighting process). Under Assumptions
and define
d dP
Qe(t)= =
4
(18)
as the Radon-Nikodym derivative at any time which is apparently a F-martingale. We call (=(()
the inverse probability weighting process. Note that
dP 8xc
d.c
Qc( = E
Qc(0-)= =1
(19)
d PP
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The IPW process intuitively serves continuous adjustment of the treatment process Á from 0 to oc, using which as weights one may create a pseudo population as the whole process were to follow Pg. It reweights the observed data distribution P into F& from the beginning of the study to the end. Following this reweighting throughout the longitudinal study, we have:
Theorem (Inverse probability weighting formula). Under Assumptions
and 4
is iden-
tified via an inverse probability weighting formula as
J 1E [[ (d)))(da) [Qx()))((XX))]
(20)
For any two Ia-adapted processes (t) and Q(t), and a partition Ax [0, oo], we write
00x (H,00))(H1100000 (H,Q)= -H,Q)=E 20() { H( (()+)))) -HO) / H(O)G(da(O))
K
(21)
We also define E(H, Q) as the limit of oxx (H, (H,Q) in probability whenever it exists. We have: Theorem 3 (Doubly robust formula). Under Assumptions 7 2 3 and 4 for any 91 "a-aappted pro- cesses (t) and Qtt) at the law where E(H, Q), as the limit of Eax (0.co) (H, Q) in probability, exists and
lim |Ax 100
((E Q)) = E()(H.)))
(22)
we have
= B() SA =
(23)
provided that either H = He or QQ
As one can see, the doubly robust formula provides extra protection against possible misspecification on either the g-computation process or the IPW process.
3.4 No RESTRICTIONS ON THE OBSERVED DATA DISTRIBUTION: A NONPARAMETRIC FRAMEWORK
For functional data, where the complexity of continuous, infiite-dimennsioal outcomes makes it even harder to justify any specific model, relying on parametric assumptions becomes especially unrealistic. In this subsection, we demonstrate that our identification framework imposes no re- strictions on the observed data. Our framework deliberately separates modeling assumptions from identification, focusing purely structural assumptions necessary for causal inference. This en- sures that the framework extracts information only from the data, avoiding the risk of introducing unwarranted or misleading conclusions based arbitrary assumptions. This property is advanta- geous for researchers and practitioners because nonparametric frameworks are flexible and require minimal assumptions, making them robust and adaptable diverse datasets. This aligns with the recent assumption-lean efforts in the causal inference community (Vansteelandt & Dukes 2022 Vansteelandt et al. (024)).
We demonstrate this by proving that, for any given observed data distribution, we can identify sequence of full data distributions-each satisfying Assumptions D 2 and such that their corresponding distributions on the observed data closely approximate the mitial observed data dis- tribution. That is, we write the set of all observed data distribution as P and its subset satisfying Assumptions I and as M, then we show that M is a dense subset of P in the total variation norm.
Up to now, we have used P to represent both the distribution on the sample space and the path space. In this subsection, we use F' to denote the distribution on the observed data (A, X, A, L) and F to denote the distribution on the full data (A, C, TA, LA). We have
Theorem 4. When the path space consists of all piece-wise continuous processes, for any measure P over the observed data A, there exists sequence of measures P the full data (A, C, A. LA) satisfying Assumptions 3 whose inductions on the observed data converges to TP in the total variation norm.

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Technically, we have not achieved full nonparametric paradigm. However, deem that the regu- larity condition "the path space is piece-wise continuous processes" is general enough for practical considerations. For example, both multivariate counting processes and continuous processes like Brownian process satisfy this regularity condition. It is noteworthy that achieving this "almost non- parametric" nature is the best can hope for. This realization was confirmed in (Gill al., 1997] Section 9) for "coarsening at random" assumption, even though our framework exhibits certain dis- tinctions.
EXPERIMENT RESULT
In this section, employ Monte Carlo simulations to empirically assess how the identification works. We decide to evaluate the performance of the g-computation formula only, for two reasons:
1. The g-computation formula is the only one that can be easily approximated through raw simulated data, whereas inverse probability weighting (and hence the doubly robust for- mula) cannot be directly approximated without estimation or computation. In fact, in causal inference with longitudinal data, the true causal effects are often not analytically com- putable. Instead, they are approximated numerically using methods like the g-computation formula through sampling like we outline below, with very large sample sizes, a standard practice for benchmarking estimator performance;
2. On the population level, the values of the three formulas all the same, equaling ( Therefore, approximating g-computation formula is sufficient for our purposes.
To that end, we need to go through 4 steps:
1. Come up with reasonable data generating process;
2. Compute the parameter of interest IT (or equivalently, the left-hand side of g- g-computation formula in Theorem [ according to this data generating process;
3. Simulate according to this data generating process;
4. Approximate the right-hand side of g-computation formula in Theorem Jssin the simu- lated data.
Step I: To sharp the focus and ease the computation, we consider a simple setting where there is no mortality or censoring (T C 30). or other measured confounding process, except for the outcome process itself. A more complicated scenario including mortality and censoring, and other confounding process, is considered in Section i the appendix. We take glucose levels as the out- come and insulin levels the treatment. Both glucose and insulin levels exhibit smooth, continuous changes over time. Gaussian processes are particularly well-suited for modeling such smooth and continuous temporal processes. For t E [0, 1]. consider potential outcome process Y, (t) capturing potential logarithm of glucose levels, following a Gaussian process with mean process as
E(Y = -a(t).
(24)
and covariance process
Cov = 3//c-al Vt, E [0,
(25)
This ensures the joint dependence among (t) and negative treatment effect of logarithm of insulin level a. For instance, Ya (L) can be log of blood glucose level. Define (Ta., a Ya) as the integral of Y, over time / [0, 1], that is,
/
(Ta, Vn) Va) (t)tt.t.
(26)
Suppose the targeted treatment regime G is Gaussian measure with process 0.5 and jointly independent normal variables at any time points. That is, the intervened A follows a Gaussian process with mean process
E(A(t)) =
0.5, -05.,
(27)
and covariance process
Cov[A((), A(s) = = 3/c-ol, Vl, s E (0, 1).
(28)
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representing increase of insulin level, possibly due some insulin intake.
Step: : Then we can show that ① (or equivalently, the left-hand side of g-computation formula in Theorem I equals zero, that is,
= To (t(t) G(da) (aata=//o - V/ a(t)dtt G(da)= = /. / ((t)) = S. dt = 0.
f B(u(Ta, (a)Taa.a)))Gda))
/
/ E(Ya(t))dtG(a)
(29)
(30)
Step In practice, we observe a stochastic process at finite points. We consider evenly splitting t [0, 1] into a grid of size K ++11 1: Ax 1] = {to = 0, = 1/K,...,fe
(K - - K, tx = 1), and for I i n, according to G specified in Step we simulate i.i.d. samples (t) according to
G specified in Step 1 at A (0, 1] as
A,(ta)
to 0.5
311 to
3igk to 3ltk +1
A.(A)
f1 - 0.5
s111-110 tol
3ix 1-41
~
(,(x-11) Ai(tx)
fx-1 0.5 1K 0.5
allx-1 fol tK -tal
alk 4 sic -iil
all KK-1l
aiK 'K-1l
By according to the distribution of (t) specified in Step and consistency, we generate (()) at AK [o, 1] as
Y(ll)) Y(G)
Aclta) Artti)
1x-1 11--1-9
fol
~
Y((fx-1) Y((k))
A((xx1))
19-1 tol
1--- 9
Ui (x-1l
A:(xx)
Pol
e 111
e 1ix--x
Step 4: The integral of Y((ls) over [0, 1] is K
(/OW)/(K + The approximate of the right- hand side of g- g-computation formula E ER (K(t))((KK +1/n.
We vary the grid sizes (K = 10, 50, 250) to examine how a denser grid improves the approximation. This approach simulates the scenario where the mesh | [0. 1] is shrunk to zero. Additionally, we vary the sample sizes (n 100,500, 2500) to explore how larger samples enhance the ap- proximation, leveraging the law large numbers better approximate the right-hand side of the g-computation formula. We repeat the process R 10,000 times. The resulting 10,000 approxi- mations of >m Σ K (KK + 1)/n are presented in boxplots in Figure where append biases.
The simulation results demonstrate that the g-computation formula can adequately approximate ① even with moderate sample and grid sizes. Increasing the sample size while keeping the grid size fixed enhances the accuracy and reduces the variance of the approximation. In contrast, increasing the grid size while keeping the sample size fixed does not consistently improve accuracy or reduce variance. However, simultaneously increasing both the sample and grid sizes significantly improves accuracy and reduces variance in the approximation.
5 CONCLUSION
In this work, we proposed on a novel theoretical framework for causal inference under functional longitudinal studies. We introduced three methodological paradigms for causal identification: the g-computation formula, inverse probability weighting formula, and doubly robust formula. This framework, noted for nonparametric foundation, substantiates and expands upon the estimand-based causal framework introduced by Ying (2024a). It incorporates considerations for tinevvayying out- comes and addresses complexities such as death and right censoring, marking a significant advance- ment in the analysis of functional longitudinal data and enhancing the toolkit for causal inference in this area.
Our focus is on the underlying curve data (Wang 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 sample-level data, where factors such sparsity or irregularity could

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Simulation Results of the G -C -Cmmputation Formula
0.2
0.0
-02
Increasing Sample Sizes
100, K 100 Bias =1.34e-3
- 130 K 50 Bias = =-1.79e-3
100, 252 Bias = =-144e-3
0.0
-02
Bies 0.38c-3
Blas 0.83e-3
Biss 0.28c-2
0.1
-0.1
2500 K=10 =0.31e-3
n 2500 K 50
2500 K 250 =-0.1e-3
Increasing Grid Sizes
Figure 1: Simulation Results of using g-computation formula by varying grid sizes in K 10, 50, 250 and sample sizes n 100, 500, 2500, for R = 10000 repeats. We plot boxplots and give biases.
influence the consistency estimators. For example, investigating how the number observed time points Pn scales with the sample size in densely observed data could provide valuable insights.
There are significant theoretical and methodological opportunities, given the limited investigation on functional longitudinal data, for the machine learning, functional data analysis and causal in- ference communitics. To list a few, first, adapting our framework to accommodate scenarios where Assumption may not hold, including contexts involving time-dependent instrumental variables and iime-dependent proxies (Ying et al. (023), warrants rigorous exploration. Following the same spirit, dependent censoring can be considered, for instance, generalizing proxy method like Ying! (2024c). Second, the positivity Assumption i longitudinal studies faces practical challenges due to the po- tential scarcity of subjects adhering to specific treatment regimes within observed populations. One might consider using semiparametric models such as marginal structural models (Robins) 1998]
Røysland 2011)) and structural nested models (Robins 1999 Lok 2008). Other solutions include dynamic treatment regimes (Fitzmaurice et al. 2008 Young et al., 2011 Rytgaard et al. 2022) and incremental interventions (Kennedy) (017). Third, establishing the efficiency bound for our quantity of interest by leveraging semiparametric theory, represents engaging challenge. Fourth, partial identification using discrete-time observations a promising direction. Finally, developing a comprehensive estimation framework remains of ultimate interest.
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Andrew Ying. Proximal survival analysis to handle dependent right censoring. Journal of the Royal Statistical Sociery Series B: Statistical Methodology, pp. qkac037, 2024c.
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A FORMAL FORMULAS
For any partition Ax [0, oo], we have the following decomposition P(dzdódadd)
(31)
K-1
II PT < = ((t) (ty < 1 (,, " ((
(32)
P(T > (1)1 (1y < 1=+1, G, ((1St)+1,8=0)
(33)
PPC ~ [1+1100))))))))))+)))) ( R, (((:5050)))
(34)
PP >[C'>400 [[0=))++|((()))))((()+)) s, ((5t)118511)
(35)
((t+))) P)
(36)
(37)
We intervene treatment distribution each time tj approximate the pseudo-population where A were to follow G as:
PAR x [...d
(38)
K-1
II P ()) ((tte j=
(39)
[1 ( 1111, 0)]
(40)
Mdd(tt::)N(((+4)))..)
(41)
G[da(t)+))lā(tt)}
(42)
B A REVIEW OF IDENTIFICATION FORMULAS FOR REGULAR LONGITUDINAL STUDIES
The following is review of existing methods and rewritten by language. Consider longitudinal study with data collected at fixed times t = 0, O,...,.. K. To align with the notation used in this paper, let T K. Here, A(t) and L(t) can only change discrete time points associated filtrations are defined as F = o (AA(k,, L(k)), Ft- (A(kk 1), L(k: 1)). and gy = af (A(k), L(k 1)). For finite set of random variables, the measure on the path space induced by IP corresponds to a multivariate distribution. Denote the observed data density or probability mass function as p(ā, 1), and use p(-l.) for conditional densities or probabilities.
k. The
The observed data likelihood can be uniquely factorized based on the temporal sequence of events:
p(a, II
()) 1),1( 1)]]
(43)
where 1(-1)= a(-1) for notational convenience.
To identify (T). the following positivity assumption is imposed:
g(a) Il pla())a 1), L(j 1)},
(44)
almost surely over L. Alternatvell, stricter version requires:
pla(i)la(i L(j 1)} 0,
(45)
almost surely over L(j 1) for any a. For example. in the case of a binary treatment, if g(A) 1(A = 0) (i.c., "always under control"), the positivity assumption ensures that for any patient history L., the probability of remaining under control is nonzero.
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Given the positivity assumption, B{a(j)lā{} - 1)). leading to the target distribution:
substitute pla(j)lā(j 1),1(j 1)1 in 43 with
K
pa 1) = II (D1()),()) - 1)lpsla())a( - 1)}}
(46)
==0
This distribution enables the identification of Eg (x(())))) where EEg represents the expectation under pc. Additional assumptions are required to interpret (c(v)((()) causally:
f B()((a(())) Bs((((((K)))
(47)
The first assumption consistency:
= L(K)=L((K)
(48)
almost surely. This states that the observed outcome matches the potential outcome under the re- ceived treatment regime A = a.
The second is the sequential randomization assumption (SRA). also referred as "no unmeasured confounders." It posits that the treatment A(k) at time & is conditionally independent of the potential outcome La (K) given past treatment and covariate history:
A(k) 1 (a(K) A(k L(k- 1),
(49)
for any a E A and k: 0, K. This ensures exchangeability across treatment groups within strata defined by observed covariates. Note that alternative frameworks relax this assumption (Tch- etgen Tchetgen et al. [2088 2020 Ying et al. 2023), but they are beyond the scope of this discussion.
The g-computation formula (Greenland & Robins 1986; Robins 2000) and inverse probability
weighting (IPW) (Hernán et al. 2000; 2002) are two commonly used approaches for identification.
The g-computation formula identifies (47 through iterative conditional expectations:
J ((v(La(K)))())d (((KK) II
(50)
Alternatively, the IPW formula uses pseudoweights derived from the Radon-Nikodym derivative:
/ B(v(La(K))) = E
{(A))(L(K)
(51)
- P -
P(A())A(
Lastly, the doubly robust (DR) formula combines both approaches (Bang & Robins 2005
Van der
Laan & Robins 2003):
B(v(Ta(()))
(52)
K
=E (Qo(K))(L(K)) Σ (o(())H((() - Qo(k- - 1) Ho(())la(())A((+ -
(1))daa)))) 1))da(k))
(53)
For additional details,
Robins 1986 1997)
Hernán & Robins 2020
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C
PROOFS
C.I PROOF OF PROPOSITION 0
We temporarily define & $/c, = ([L((s), (())
1T S <) : <() ()) the censoring free filtration
and R,, = (()a(s)), 1(T. S s): S <t}) as the counterfactual filtration:
I Ax 20 c ((drradaa)G(daj)) TV K-1 II at ==
<tttal(((tt)))) (1)0,ee (00
PP( 1
- P(ddr,dd,
K-1
II ==
G(dā(t)+1) - 1
P(T S 1++1 $ /c,,,) |( P(T P(T 14-1 ( ) sc,,)
- P(dzaadla
+o(1)
II ((d I(xa 8, = s =t-r1 2 j=0
P(T.
hut.::ahhhhhppaa.........uaaasaa Fa.t)
PP G(dā))
K-1
II G(da(t)+))a((())) 1(z, s 61 =0) 1=
(P(Ta S bz
P(Ta > t 1s+1, a ( Pdl
P(dia(ty+1)) > t, Fat)
-(T < Stst1|
Hiy (10) P(Ta > t Fa 1-0 ((d Fa.,,)] +.
C.2 PROOF OF THEOREM 1
Since Pe (ddddddddd)) is limit of measures in total variation norm of
P PAx (...
(54)
whenever AA [t, oo] O, have
/ (1,,0,a,1)
- Belf(X,
(55)
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for any bounded functions f(x, 5, a, 1). Therefore, we have
He(0-) / (a))((a) Ga))
/ 1(z, D) P(dzdódadl) - - Ø) Pan,
[ (((,, () ()s
/ 1(z, 5) Pa ( - / B(u(Ta, (a))
+ +
K-1
/ p(x Ø)
II j=0
P(T S ty+1|0 t,41, (y+ F,, )=06
P(T > 1/-1/1 > 1
P(d ((t
[ - 1(r v 1,+1, (( / (d) + +0(1)
(a))
where (1) converges to zero when Ax]0, - 0. By Assumptions less than equal to
2 3
and the above term
/ (1 P(T P(T I (ta
P(T> (x(a(le). Fix paa
K-1
1--1(= s 11-1(ss4K1 - [ ())
P(T (/,
j=0
P(T > ty+1 (t(++)), ||0
[ 1(x < 6= (() - Ga)))(a)) + +0(1),
which by Assumptions 12 and equals
/ ((o I tk (xla(tk), Mil
P(Ta > txlā(tx) (u F(n.))
1 1 tk. da (aat(t((x)aa(
II P(T (i
P(T > 1ب+1 (((;+1)), h|g3::hhhhhhhhoooooo.ooooooonohh
P((())))))))))))))))))))))))))) cc,,)
1(x < ty+1, = ( - B(u(Ta, Ya)) + t0())
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which by Assumptions 12 and is less than equal
(( P(Ta S lx Ftx 1) ((
P(Ta > tkl {(x 11xx Sou ((daatt) (x-1)
K-1
- (xa 5 tk: da = []](a 1)
II
PTT s (1
f=U
P(T > 14+1 (()))))) Ic,,) (7 P(d(( G),,,)
- 1(x S ( - Ya)G((d) + o(1).
By iterating the above process for 0 S S K - 2, we arrive the conclusion.
C.3 PROOF THEOREM
2
The proof is immediate by noting that
E [X) = -EcM((, = E(v(Ta, (d))
(56)
by Theorem
C.4 PROOF OF THEOREM 3
We first prove the theorem when H Hgg Indeed, as the limit and expectation can interchange, can show that
Eu (0,00] He. Q)]]
E (X) - Hc((x))}
E (Q(z) / (G(CH))(((((())))))))) -Hat))))))s((()))
K
E (o(t) (e(t))))G((((()))))))))))) -Ha(s)
0
+
Σ
1=1
(oit) { He Helt)))G((((()))))(())) - Ha(t)))
E
(Q()) {
Heltt))) Es [He((s+))1(1]] }))
E
Σ (((HH 1²)a
K-1 Σ 1=1
< ksup kssp (())(t)(()))
(1m41-t5 11 + 0,
when Ax [0, xc]1 + 0, where we have used the fact that H (t) is marttinine and Assumptions
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We now proceed to the case when Q Qg- Indeed, the limit and expectation can interchange, we can show that
E
(0.00)
(H, [H)
k
( Qo((s-1) H(tt) (((1)((()))(((1)))
E
(b(0) - ((O)r
E
((66))((4--Q444) ( +o|
E
K
R {
Qa(()-1) -
K
Σ BG
Qe(ts-1) / [((
(+(()=))s((t==)ll t²-1)a
1=0
ksup (t))))
E(g 1) 0,
when Ax(0, where we have used the fact that Qs(() is PP-martingale and Assumptions
C.5 PROOF OF THEOREM 4
We first simplify our setting by ignoring censoring and absorbing event time T' into L as well. This is because the conditional independent censoring assumption (Assumption 2 is known to be nonparametric. Our observed data become (A, L) and full data become (A, L))-
We first proves that Assumption] I does have restrictions on the observed data. We proceed with a constructive proof. For any partition Ax [0, co]. we define a measure on the full data path space. In fact, one has the knowledge on the decomposition
P(dad() = II ((dd)t)
(57)
j=u
K-1
=
(58)
Intuitively (d (t (t)))))((())a))))))a))) P(dā(t)+1)la()) ((1)) is close to ((d((()+))la(t)).(a)) Therefore, one may define measure on the full data path space by
are close P(dia ((dia(ty++))la(tt)) whereas the other term
IP (dx = II (()
II P (()aa))) (59)
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Then without loss of generality one may construct among La. One can also define
P 00 (la) by assuming joint independence
K-1 II P(dd(t)++)))Ft.).
P P o0 (dal (
(60)
Then for any sequences of partitions with the mesh going to zero, one may construct sequence of measures and show that this sequence of measures Cauchy by a triangular inequality and Assump-
tion [ following similar logic as previous proofs. Therefore the sequence converge to a measure pe, which is independent of the choice of partitions.
Next we need to show that Tp induces P on the observed data and IP" satisfies Assumption The first is trivial because any (0,00, [0,c0) induces P on the observed data, then so is their limit. To prove the second, for any time and E >> 0, one might smartly choose partition A [O, oo] with P Axx [0,] close enough to pd and 4, 7 Ax|0, oe). This can be done because the convergence point is independent of the choice of partitions. We have
E
- 1p [al + n), FillTv))
S B (I [d/A| - 1x 10, (d/Al
+ +E" (I) 1 (d[al.
P Ax 20 [d[a/A(t + n)) F.llTT) + +E (II 20 |Alt + 17), F.] - 1p [dl. A(tt m), &illTv
< 2|| P - -P 20] 111v
R {1) 1 10 (dAl(FF))))))) Pp 1x (d d[ 1), P.llTT)
< 4|| PF - 1 201 1|Tv
+ E A [0.cc] {1l P 1x (d (d.)
P (0,0) (dTa|A(( + 1)) F. }:]]
The first term can be chosen to be sufficiently small. We rewrite the second term
| ( 1 (dl
Al + Fillrv]
-11 / FTW (dx
= sup
- - [di + 1)) F. (dl)
I sup
sup =
f((a)(((t++)) f
(d + Rd
-P 00 (dT((F))) P AK (000) (da(t+++
sup fill - gi1 / f([a)g(((++ +n) P ( (d/
= sup
[I AK [0,00] [dā(t (t + m)(a(t), TA - -P dā + n) P(dF.)
= sup MI ا f(Ta)o(a(t + m)) P (0))0 ((F,))
sup
K-1
P(da(t + R.) - P(ds,)
j=D
K
<
E
P(da(t + n)) P(da(t + n))
1=
TV
< (tt,,)).
Assumption tions.
is irrelevant here because it is imposed on the stochastic process but not the distribu-
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For Assumption 4 necessary condition for it to hold is that IP is well supported the path space conditioning on any filtration. For this to happen, note that for any P. one can find a well-supported F' satisfying Assumptions [ 2 that (1 =) IP +- IP' is well-supported and hence satisfies Assumption 4 Since addition will not break Assumptions and and approximates IP.
we have found (1 - E) + IP
satisfying Assumptions]
D ADDITIONAL EXPERIMENT RESULT
In this section, we employ Monte Carlo simulations to empirically assess how the identification works. To that end, we need to go through steps:
Come up with a reasonable data generating process;
2. Compute the parameter of interest (T) (or equivalently, the left-hand side g-computation formula in Theorem [ according to this data generating process;
3. Simulate according to this data generating process:
4. Approximate the right-hand side of g-computation formula in Theorem I| using the simu- lated data.
Step 1: For / E [0, 1], consider potential outcome process Ya( and potential covariate La(() following Gaussian process with mean process as
B(Ya(t))) (aa(t) (((a(())) 0.5a(t))
(61)
and covariance process
(ov((Y,(),, (,), Ya (s)) Lo)())()),La))))
(
-1-6/--- /2)
Cov (Cor)Latt), Ya(s)) Lor(La(t).La(s)) 6-1---1-11/2
3 /2
(62)
This ensures the joint dependence among (t) and (t) and non-zero treatment effect of a. Gen- erate the event time following Cox model
P(T. > t|Ya, La) exp (-texp (0.2a(t) 0.8Ya 2L (t))}.
(63)
and an independent censoring process
PPCC P(C'>))= eexp(-t), when t < 1
(64)
(0,
when />>
where time I represents administrative censoring. Define v(Ta, Ya) as the integral of Y, over time E [0, 1], that
TsAl
v Ya)
Ya(t)dt.
(65)
where Ta capped at because beyond that all subjects are censored. Suppose the targeted treat- ment regime G is Gaussian measure with mean zero process and jointly independent normal vari- ables at any time points. That is, the intervened follows Gaussian process with mean process
B(A(t)) O,
(66)
and covariance process
Cov Cov(A()), A(s) = 331--l /2, VI. [0,
(67)
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Step: Then show that ① (or equivalently, the left-hand side of g g-computation formula in Theorem I equals zero, that is,
f
- [
/
Tami
B(u Ga)))(da)=
Ya(t)dt G(a)
(68)
B(TO)AL E
Ya(()dl G(a)
(69)
E(TOJAL
E(Ya(t))d(((O) = = B((b)) a(t)dtG(a)
(70)
E(TOA1
((())(((()) / =.
(71)
We used the fact that, through our data generating process, the treatment does not have effect on survival time.
Step In practice, we observe a stochastic process at finite points. We consider evenly splitting
[0, into a grid of size K + 1: Ax [0. 1] (fo = f = 1/K,... ... .tx-1 = (K - 1) K, lx = 1), and for S i n, according to G specified in Step we simulate i.idd samples (t))) according to
G specified in Step at Ax [0, as
Ac(to)
to 0.5
A.(
f1 - 0.5
e loi
-1
~
(,(x-1))
fx-1 0.5 AK
alix-1 fol 1-38k-A -tal
Blk--
all IK-1l
A:(tx)
311 'K-1l
By according to the distribution of (t()) specified in Step 1 and consistency, we generate (i(t) and (t(t) following a similar manner. Generate the event time following a Cox model
P(T: AA Y,, L.) exp exxe (0.2A(() (!) (L))
(72)
and independent censoring process
P(C, > 1) - Sexp(-(), when / 1)=11, 0,
(73)
when 1,
Step 4: The integral of (tx)) over [0, is EOSKST (itt)/(KK + 1). The approximate of the rinh side of g-computation formula is Σ EOCKST, Yitt)/((K 1)/n.
We vary the grid sizes (K 10, 50, 250) to examine how denser grid improves the approximation. This approach simulates the scenario where the mesh | [0. 1] shrunk to zero. Additionally, we vary the sample sizes (n 100,500, 2500) to explore how larger samples enhance the ap- proximation, leveraging the law large numbers to better approximate the right-hand side of the g-computation formula. We repeat the process R 10,000 times. The resulting 10,000 approxi- K mations of Σi Σ k= (K( + +1/nn are presented in boxplots in Figure where append biases.
The simulation results demonstrate that the g-computation formula can adequately approximate (1) even with moderate sample and grid sizes. The biases are larger than that in Section 4 possibly be- cause we are focusing on a more difficult task. This time, however, increasing the sample size while keeping the grid size fixed does not enhance the accuracy. Increasing the grid size while keeping the sample size fixed does improve accuracy or reduce variance in this case. Again, simultaneously increasing both the sample and grid sizes significantly improves accuracy and reduces variance in the approximation.
22

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Published as a conference paper at ICLR 2025
imuuation Results of the G CCmmputation Formula
0.1
100, K 10
100, 50 -..44-3
100, 250
Increasing Sample Sizes
=1.84e-3
0.0
5000 K 10 =24:25e25-
500, 50 =9.4-ee
500, 250
=2.4e-3
-022
=5000. -10 =24.08--
2500. K 50 =9.30e-3
25000 250 =2.23e-3
Increasing Grid Sizes
Figure 2: Simulation Results of using g-computation formula by varying grid sizes in K 10, 50, 250 and sample sizes 100, 500, 2500, for R 10000 repeats. We plot boxplots and give biases.
23


Reviews:
[{"review_id": "n35Ys4D743", "reviewer": "Reviewer_Fipq", "summary": "In this paper, authors consider causal inference on time-varying data (functional longitudinal data). They generalize the classical g-computation, inverse probability weighting and doubly robust formulas to the time-varying setting subject to censoring and mortality. The g-computation formula is simulated using Monte-Carlo on a toy dataset, achieving promising results.", "questions": "Line 141, authors mentioned \"note this is not a density function\". Then please specify what this is. \n\nLine 325, why is it sufficient to evaluate the approximation of the G-computation formula? I don't think on population level, the values of three formulas are numerically equal. Even if they are equal, they may have quite different finite-sample behaviors.", "limitations": "", "rating": 5, "confidence": 2, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "Treatment effect on functional longitudinal data seems to be an understudied subject. This research nicely fills the gap of existing works. \n\nThe resulting G-computation can be quite straightforwardly approximated using observations under simulation settings.", "weaknesses": "The way this paper is written obscures its main ideas (at least to a general, non-expert reader). There are many terms and phrases used without clear explanation (e.g., \"g-computation\", \"counterfactual time-to-event endpoint\"). This restricts the range of potential readers of this paper.  \n\n**The experiments are limited to only simulation data and only validate the G-computation formula**. \n\nThe literature review of this paper (section 2) does not seem to provide much information of existing works as without proper explanation, readers may be unclear what \"temporal aspect\", \"point exposure\" and \"end-of-study outcome\" means. I recommend removing Figure 1 and expanding on each of the subsections, providing more details of existing works. \n\nThe preparation in Section 3.1 is quite long. Without concrete examples, it is hard for readers to understand what they actually mean. I suggest skip some unnecessary notations, and explain them as the paper progresses. \n    - Some symbols are better explained with examples. For instance, authors could give an example of nu and G, around equation (1).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "iWrBkRebXL", "reviewer": "Reviewer_2NEN", "summary": "This paper proposes a causal identification framework that bridges classical causal inference framework, continuous-time longitudinal analysis and functional data analysis. In this framework, the parameter of interest is the marginal mean of counterfactual outcomes under a measure that allows randomly assigned treatments, with absence of censoring. Leveraging the tools in stochastic process, the authors then demonstrate the identification results for three classical estimation strategies in causal inference: g-computation, IPW and doubly robust estimation. The authors further claims that the identification framework also has non-parametric property.", "questions": "1. Why is the interventional distribution ((7)-(10)) formulated in this way? Specifically, I’m curious about where the term $\\{1 -   \\mathbb{1}(x \\leq t_{j+1}, \\delta=0) }$ comes from.\n2. Can this framework be extended to dependent censoring?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "This paper establishes a new causal identification framework for continuous-time longitudinal studies with functional data, and provides clear and concise theoretical demonstration. I believe that this framework will be of interest to causal inference and machine learning communities.", "weaknesses": "1. The numerical experiment might be an over-simplification of the survival analysis scenario since neither mortality nor censoring are taken into consideration. \n2. What is the causal structure that the framework is focusing on? Specifically, why set $Y(t)$ (outcome of interest) to be a subset of $L(t)$ (measured confounders)? I might misunderstood but are we assuming that previous outcome will impact the current treatment assignment (since confounders, from my understanding, will impact treatment assignment)?\n3. I guess it would be helpful to attract readers in a wider community if more intuitive explanation could be added after stating definitions/propositions.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "c6oIxqLPeN", "reviewer": "Reviewer_dyEJ", "summary": "The paper is challenging to follow, so I may have misunderstood some parts. My understanding is that the \"functional longitudinal data\" investigated here are conventional functional data, as described by Wang et al. (2016), which can be measured intensively, sparsely, or irregularly. However, this paper focuses solely on the ideal (hypothetical) setting where continuous-time measurements are available for each experimental subject, resulting in infinite-dimensional data. If this interpretation is correct, the goal of this paper is to explore causal identification for infinite-dimensional functional (time-varying) outcomes that are subject to mortality and censoring by generalizing the classical g-computation, inverse probability weighting, and doubly robust formulas.\n\nReference: \nWang, Chiou and Müller (2016). Functional data analysis. Annual Review of Statistics ands its application.", "questions": "Could you elaborate on the situation when A(t) is a function? \n\nWhy should Y(t) be a subset of L(t), and what does it mean?", "limitations": "", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 1, "contribution": 2, "strengths": "The approach is nonparametric and it accommodates functional treatment processes A(t) and functional confounders L(t), as well as functional response Y(t).", "weaknesses": "The paper is hard to follow and the connection of the event-time T to the outcome Y(t) is unclear.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "eS5uadVFcO", "reviewer": "Reviewer_WrGA", "summary": "This paper presents a nonparametric causal identification framework for functional longitudinal data, exemplified by the MIMIC-IV dataset, which includes complexities such as events like death. By leveraging stochastic process theory and measure theory, the framework generalizes g-computation, inverse probability weighting, and doubly robust formulas, effectively handling time-varying outcomes with mortality and censoring. Monte Carlo simulations verifies it.", "questions": "Please provide additional explanations for the questions raised in the Weakness section.", "limitations": "", "rating": 3, "confidence": 2, "soundness": 2, "presentation": 1, "contribution": 2, "strengths": "This paper addresses the problem of causality identification in complex scenarios inherent in functional longitudinal data, a highly general and advanced form of data. If it can overcome the criticisms and shortcomings pointed out by other reviewers and get published, it could serve as a milestone, taking one step forward in functional data analysis for causal inference.", "weaknesses": "This paper is more abstract than necessary, making it harder to read than other papers. Even after spending more time on it than on other review papers, the preliminaries, equations, and theorems presented in the paper are not easily understood at once. As a result, I had to refer to the cited papers, and I found two papers that share many aspects with this one. One is \"Causality for Functional Longitudinal Data,\" which is cited by (Ying, 2024) in this paper, and the other is \"Causality for Complex Continuous-Time Functional Longitudinal Studies with Dynamic Treatment Regimes,\" which is not cited in this paper. The title of this paper is \"Causal Identification for Complex Functional Longitudinal Studies,\" suggesting that the most significant update in this paper is the \"complex\" aspect in the title. The paper \"Causality for Complex Continuous-Time Functional Longitudinal Studies with Dynamic Treatment Regimes\" seems to have updated the dynamic treatment regimes element one step further.\n\nSpecifically, in the preparation section, the time-to-event endpoint $T$, $C$, $X$, and $\\Delta$ are newly defined, assuming a more complex situation where the study may be forcibly terminated due to an event like death before the study is completed. This updates Theorems 1, 2, and 3 from the previous paper. I would like to ask about the academic and practical significance of solving the more complex problem introduced by those new variables. The most important contribution of this paper seems to be the introduction of the non-parametric property through Theorem 4. Why is the non-parametric property important in a functional data framework? The paper states that it makes the model more flexible and adaptable to various data, but doesn't the continuous functional data, which is more extensive, lead to increased computational and implementation complexity, a common drawback of non-parametric models? Doesn't this also create issues with the interpretability required for healthcare data analysis?\n\nOne of the most important equations in this paper seems to be Equation (1). The rest of the paper is dedicated to finding another representation of Equation (1). However, it is not easy for readers to immediately understand what Equation (1) means and why we should be interested. Additionally, it is not straightforward to grasp what $\\mathbb{G}$ represents. By referring to the paper by Ying (2024), I could somewhat understand $\\mathbb{G}$ through the following example:\n*When the causal outcome under a specific regime $\\bar{a}$ is of interest, for instance, all patients were under treatment, the point mass (delta) measure $\\mathbb{G} = \\mathbb{1}(\\bar{A} = \\bar{a})$ can be considered.*\nIncluding this example in this paper would help in understanding. Furthermore, providing a concrete example of what $\\nu$ represents would help comprehend Equation (1). Can Equation (1) be understood as a general expression representing the average treatment effect, the averaged treatment outcome, or a transformed form of these?\n\nIf the non-parametric property is a major contribution of the paper, it should be demonstrated through more concrete experimental examples, such as using the MIMIC-IV data mentioned in the introduction. The experimental section currently numerically verifies Theorem 1, which has already been proven in the Appendix, but a demonstration of Theorem 4 seems more necessary. However, under Theorem 4, it is only mentioned that \"we have not achieved the full nonparametric paradigm.\"\n\nAdditionally, there is a need to clearly and definitively define loosely defined “functional” data. The abstract of this paper describes it as \"characterized by continuous-time measurements,\" while another cited paper describes it as \"characterized by continuous-time processes and high-dimensional measurements.\" I believe \"continuous\" alone is not sufficient to be called functional. What is the rationale for developing a framework that assumes functional continuity in the model even though real-world healthcare data does not have mathematically rigorous time continuity and does not observe over an infinite time? (The previous paper assumed up to time $\\tau$, but this paper assumes up to $\\infty$.) What is the justification for this assumption?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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:

Decision: Accept (Poster)