Incremental Causal Effect for Time to Treatment Initialization

## Abstract

Abstract We consider time to treatment initialization. This can commonly occur in preventive medicine, such as disease screening and vaccination; it can also occur with non-fatal health conditions such as HIV infection without the onset of AIDS; or in tech industry where items wait to be reviewed manually as abusive or not, etc. While traditional causal inference focused on ‘when to treat’ and its effects, including their possible dependence on subject characteristics, we consider the incremental causal effect when the intensity of time to treatment initialization is intervened upon.
We provide identification of the incremental causal effect without the commonly required positivity assumption, as well as an estimation framework using inverse probability weighting. We illustrate our approach via simulation, and apply it to a rheumatoid arthritis study to evaluate the incremental effect of time to start methotrexate on joint pain.

## 1 Introduction

In the tech industry, companies often rely on manual reviewers to label content with “golden labels” in areas such as malware detection, email spam filtering, and abusive content management. These teams are frequently understaffed due to budgetary constraints and the overwhelming volumes of contents, leading to significant backlogs of review requests. To address this, increasing the number of reviewers would naturally reduce the backlog by accelerating the review process. If the number of reviewers were doubled, the likelihood of a request being reviewed at any given time would also double, as the requests are often selected for review at random.
Similarly in health studies, such as disease screening or vaccination, there is the policy related question about the intensity of intervention, and all of the above
relate to the causal problems considered by the so-called “incremental causal effects” (Kennedy, ; Naimi et al., ; Kim et al., ; Sarvet et al., ; Bonvini et al., ).

Causal inference centers around answering “what happens if a particular treatment assignment changes …” type of questions. An intervention on the treatment can change the value of a treatment, like from control to treated, or it can change its distribution, to be described in more details below. Causal inference literature started with investigating *static deterministic* interventions (Rubin, ; Hernán and Robins, ).
Here ‘static’ is in contrast with ‘dynamic’, where the (possibly time-varying) treatment assignment may depend on covariates (and prior treatments and outcomes) (Robins, ; Murphy, ; Robins, ; Moodie et al., ; Young et al., ; Haneuse and Rotnitzky, ; Rytgaard et al., ; Ying, ).
In contrast, under a static intervention the subjects are either all treated or all untreated; this is typically the case when considering an average treatment effect (ATE).
A second aspect about intervention is
‘deterministic’ versus
‘stochastic’.
Deterministic interventions assign a particular value of treatment to each individual, which may depend on the covariates, in the case of a deterministic dynamic intervention.
On the other hand, stochastic interventions
do not assign a particular value of the treatment, but rather a probability of receiving a particular treatment value (Cain et al., ; Díaz and van der Laan, ; Díaz and Hejazi, ; van der Laan et al., ). More specifically, a stochastic intervention assigns treatment randomly based on a probability distribution. Bonvini et al. () describes the example of a Bernoulli distribution for a binary point treatment, where a single parameter $p\in[0,1]$ indices such an intervention.
Finally, an intervention can be ‘pre-specified’ versus ‘otherwise’. For example,
Young et al. () described a pre-specified intervention as: “start combined antiretroviral therapy (cART) within 6 months
of CD4 cell count first dropping below x or diagnosis of an AIDS-defining illness,
whichever happens first”.
This is a pre-specified dynamic treatment regime as whether an individual starts treatment depends on their own health related history captured by the covariates.

An incremental intervention (Kennedy, ) is a special stochastic dynamic intervention (Hernán and Robins, ; Rytgaard et al., ) that is not pre-specified, but rather a function of the observed treatment distribution. A practical example given in Bonvini et al. () is for probationers to receive behavioral health serviced, in order to reduce their chances of re-arrest. The question of interest, to be elaborated below, is about shifting the odds of receiving such a treatment. This can be achieved via affecting probationers’ likelihood of attending services by, for example, providing transportation stipends. In public health, guidelines can be provided in order to affect people’s likelihood of getting disease screening or vaccination. The resulting causal effect of an incremental intervention is called the incremental causal effect. In general the incremental causal effect has the interpretation of a policy effect on the population, instead of the therapeutic effect on an individual.

For discrete time longitudinal studies, this leads to modifying the odds of receiving treatment at each time point, similar to the binary point treatment mentioned above Kennedy (); Naimi et al. (); Kim et al. (); Sarvet et al. ().

### 1.1 Positivity assumption

In addition to the causal interpretation as policy effects above, incremental causal effects have the notable advantage of avoiding the *positivity* assumption typically required of the more traditional causal estimands.

To identify estimands such as the average treatment effect from the observed data distribution, we typically need at least three common assumptions: consistency, positivity, and no unmeasured confounding or exchangeability (Hernán and Robins, ). Consistency is needed in order to bridge the observed data with the potential outcomes, as well as to exclude the possibility of interference where one subject’s treatment affects another subject’s outcome. No unmeasured confounding has been studied extensively in the literature, including instrumental variables (Angrist et al., ; Ying et al., ; Ying and Tchetgen Tchetgen, ) and proximal approaches (Miao et al., ; Tchetgen Tchetgen et al., ; Ying et al., ), as well as sensitivity analysis towards the violation of this assumption (Rosenbaum, ). Here we focus on the positivity assumption; that is, each subject in the population must have a positive probability of receiving each treatment level. In addition, for estimation approaches such as inverse probability weighting (IPW) or doubly robust approaches to have valid asymptotic inference, the stronger *strict positivity or overlap* assumption is needed; that is, the probability of receiving each treatment level needs to be bounded away from zero. These assumptions, while testable using observed data, can be violated in practice. Bonvini et al. () provided a recividism example where some probationers might be required to obtained the treatment under study by attending behavioral health services, therefore their probability of receiving treatment is one and not receiving treatment is zero. Similar cases can be found for vaccination where some people may not be suitable for receiving the vaccine. D’Amour et al. () showed mathematically that overlap is hard to satisfy when many covariates included in order to meet the unconfoundedness assumption that is also required in many causal inference approaches.
For studies involving time-varying treatment, covariates, and outcomes, the positivity assumption is even more of a challenge to meet.

Therefore, many traditional causal quantities of interest might be vetoed by absence of positivity alone after checking the data. On the other hand, incremental causal effects bridge this gap, avoids the positivity, and therefore can be investigated regardless of data restriction.

However, there is currently no investigation in the literature on incremental causal effects by modifying the distribution of time to treatment, especially under continuous time, to our best knowledge. All of the application examples stated at the start of this paper fall under time to treatment; that is, given a well defined time zero, how long a subject has to or choose to wait before getting the treatment.

In this paper, we generalize existing work on incremental causal effects to the continuous time to (initializing) treatment. We first define incremental causal effects by shifting hazard function via hazard ratio. We show that they can be identified by the observed data with only sequential randomization assumption and consistency assumption, avoiding positivity assumption. We construct a generic inverse probability weighted estimator accommodating numerous estimators for the hazard function, coupled with root-$n$ inference results via random weighting bootstrap. Our estimation tools can leverage off-the-shell software from survival analysis. We examine the finite-sample performance of these estimators via extensive Monte-Carlo simulations. We apply our estimator onto a rheumatoid arthritis study to evaluate the effect of methotrexate.

The remainder of the article is organized as follows. We review related work in Section 2. We introduce notation, the causal estimand, and key assumptions for identification in Section 3.1, where we construct an inverse probability weighting identification approach. In Section 3.2, we propose an inverse probability weighted estimator, together with inferential results yielding confidence intervals. In Section 4.1, we conduct extensive simulations examining finite-performance of our estimator and inference results. In Section 4.2, we apply our framework to answer what happens if we change the probability of receiving methotrexate on rheumatoid arthritis. We end the paper with a discussion in Section 5.

## 2 Related Work

Kennedy () proposed incremental intervention which fully resolved the positivity issue mentioned above.
In the binary treatment case for subjects with degenerate probabilities (of zero or one) of receiving treatments, perturbing the odds does not change their degenerate probabilities, therefore the positivity can be seen as always satisfied. A disadvantage of the incremental causal effect, quoting Kennedy (), is: “First, we expect incremental causal effects to play a more descriptive than prescriptive role compared to other approaches. Specifically, they give an interpretable picture of what would happen if exposure were increased or decreased in a natural way, but will likely be less useful for informing specific treatment decisions.” As will be seen, contrary to what is said here, the incremental causal effect in our setting can yield useful policy-making decisions.

Díaz and van der Laan (), Haneuse and Rotnitzky (), Díaz and van der Laan (), Young et al. (), Díaz, Hoffman and Hejazi (), and Díaz, Williams, Hoffman and Schenck () considered general interventions that depend on the observed treatment process, and are referred to as modified treatment policies or interventions that depend on the natural value of treatment, the latter referring to a treatment that a subject would ‘naturally’ receive in the absence of it being assigned by an intervention. An example from Haneuse and Rotnitzky () concerns the surgery time for a patient, and asks the question of what if every patient’s operating time is reduced by say 5 minutes. Note that each patient’s (observed) operating time depends potentially on their individual covariates as well as additional unobserved attributes. In this sense the observed operating time contains more information than what is captured by the measured covariates. In a similar vein, the natural value of treatment also emphasizes the belief that it contains more information than what is captured by the measured covariates (Stensrud et al., ).
Because these interventions involve more information than the measured covariates, they are different from the pre-specified dynamic intervention introduced in the previous section (Haneuse and Rotnitzky, ).
Our work here also involves the observed treatment distribution, and considers the continuous time to treatment initialization that has not been previously studied in the literature.

Time to treatment or treatment initialization, or to a large extent similarly, time to treatment termination, has been consider in the dynamic treatment setting (Johnson and Tsiatis, ; Yang et al., ; Nie et al., ).
In particular, the first two references (Johnson and Tsiatis, ; Yang et al., ) considered continuous time to treatment termination. All of the above are for deterministic interventions.

Continuous time causal inference in longitudinal studies for stochastic interventions, a broader setting than time to treatment initialization studies, is general challenging, and
limited work has been done in the field (Rytgaard et al., ; Ying, , , ; Sun and Crawford, ).
Among these, Ying (, ) and Sun and Crawford () only considered static intervention, Ying () obtained only identification results without any estimation procedure, and none of them (Rytgaard et al., ) considered incremental causal effects .

Finally, the literature on causal inference until recently have paid less attention to the positivity assumption than, for example, unmeasured confounding. Petersen et al. (), Zhu et al. (), and Léger et al. () have recently investigated the importance and implications of this often-overlooked assumption. Hwang et al. () and Matsouaka and Zhou () aimed at relaxing the positivity assumption. Our paper here aims at a new class of estimands which completely avoid positivity, and this in turn opens up more causal questions that one can answer in a study.

## 3 Proposed approach

### 3.1 Estimand and Inverse Probability Weighted Identification

Suppose we observe $\{Y,T\wedge\tau,\Delta=\mathbbm{1}(T<\tau),L\}$, where $Y$ is an outcome of interest measured at time $\tau$, $T$ is time to (initializing) a certain treatment, and $L$ are baseline covariates. We assume $T$ to be absolutely continuous, and write $\lambda(t|l)$ and $\Lambda(t|l)$ as its hazard function and cumulative hazard function at time $t$ given $L=l$, respectively.
As reviewed in the literature above, the hazard function is a natural quantity used to describe the intensity of time to treatment (Yang et al., , ).
Nonetheless we also write $f(l)$, $f(t|l)$, and $f(y|t,l)$ as the (conditional) densities or probability functions.

We may define the potential outcome $Y_{t}$ (Neyman, ; Rubin, ; Holland, ) if the treatment were initialized at time $t$. Since $Y$ is measured at time $\tau$, whenever $t\geq\tau$ we denote $Y_{t}=Y_{\tau}=Y_{\infty}$ for someone who is never treated.

Unlike the previous literature,
we are interested in answering the question like, “what would be the expected outcome $Y$ if the hazard function $\lambda(t|l)$ were to be doubled?”
For the example mentioned at the start of the introduction, the tech company may consider hiring twice as many vendors for the manual review.
In general, we may consider shifting the hazard by a positive quantity $\theta(t,l)>0$. We consider this to be a natural ‘hazard preserving’ stochastic intervention; that is, to multiply the hazard function for time to treatment by a positive quantity.
We note that when $\theta$ is a constant, we have a *proportional hazards* intervention. This has the obvious advantage of simple interpretation, like hiring double the number of vendors in the example above.
The fact that it coincides with the familiar Cox model is another added advantage. More generally, we might consider recommendation of intensified disease screening for certain high risk groups described via $l$, or after a certain age $t$.

For the rest of the paper we denote $T(\theta)$ as the random draw following the hazard function $\theta(t,l)\cdot\lambda(t|l)$. For example, suppose the time to a certain treatment initialization, $T$, follows the Cox proportional hazards model including a single continuous covariate $L$, with the baseline hazard corresponding to the hazard of a Weibull random variable. Specifically, the hazard function of the time to the treatment initialization at time $t$ given $L=l$ is $\lambda(t|l)=\lambda_{0}(t)\exp(0.2l)$, where the baseline hazard is $\lambda_{0}(t)=0.9t^{0.5}$.
Figure 1 shows the effects of shifting the hazard $\lambda(t|l)$ by a constant positive quantity $\theta$, where we plotted the resulting hazard functions and density functions of the time to the treatment initialization $T(\theta)$ under interventions with various constant $\theta$ values.

Figure: Figure 1: Hazard and density functions of the time to treatment initialization $T(\theta)$ under interventions with various constant $\theta$ values.
Refer to caption: extracted/5866916/figures/intervention.png

The corresponding potential outcome under $T(\theta)$ is then by plugging in $T(\theta)$ into $Y_{t}$ as $Y_{T(\theta)}$. The incremental causal effect is defined as

$$ $\psi(\theta)=\operatorname{\mathbb{E}}(Y_{T(\theta)}).$ (1) $$

In particular when $\theta\equiv 1$, $\psi(1)=\operatorname{\mathbb{E}}(Y_{T(1)})=\operatorname{\mathbb{E}}(Y)$ corresponds to the factual distribution of $T$ that we have observed.

###### Assumption 1 (Consistency) .

$$ $Y_{T\wedge\tau}=Y.$ (2) $$

This assumption links the observed outcome and the potential outcome via the treatment regime. It says that if an individual receives the treatment at time $T$, then his/her observed outcome $Y$ matches $Y_{T}$. The second assumption is typically referred to as the “sequential randomization” assumption (SRA), “sequential exchangeability”, or “no unmeasured confounders” assumption.

###### Assumption 2 (No Unmeasured Confounding) .

The time to decide treatment and treatment decision is independent of the potential outcomes given the covariate,

$$ $T\perp Y_{t}~{}|~{}L,~{}~{}~{}\forall t\in[0,\tau].$ (3) $$

It requires that time to treatment initialization is determined solely by the collected baseline information $L$.

We prove that (1) can be identified:

###### Theorem 1 (IPW Identification) .

Under Assumptions 1 and 2, the incremental causal effect $\psi(\theta)$ can be identified by the observed data distribution via

$$ $\psi(\theta)=\operatorname{\mathbb{E}}(Y_{T(\theta)})=\operatorname{\mathbb{E} }\left\{\theta(T,L)^{\Delta}e^{-\int_{0}^{T\wedge\tau}(\theta(t,L)-1)d\Lambda( t|L)}Y\right\}.$ (4) $$

This identification generalizes the classical IPW identification because the weights employed $\theta(T,L)^{\Delta}e^{-\int_{0}^{T\wedge\tau}(\theta(t,L)-1)d\Lambda(t|L)}$ are the Radon-Nikodym derivatives (or known as likelihood ratio) between the targeted incremental intervention distribution and observed data distribution (both can be found in the appendix), as the classical inverse probability weights.

As noted in the Introduction, while positivity is a fundamental assumption in causal inference for common estimands such as the ATE, here we do not need positivity for identification of the incremental causal effect from the observed data distribution.
More specifically, compared with the proof of identification here, traditional pre-specified stochatic dynamic interventions (including deterministic static interventions as special cases) operationalize by replacing $\lambda(T|L)e^{-\int_{0}^{T}\lambda(t|L)dt}$ by some pre-specified conditional distribution of $T$.
This necessitates to impose an additional positivity assumption. Or in other words, if all users with some particular history $L$ can never be unlocked at time $t$ or any person with medical history $L$ can never receive vaccine at time $t$, then we certainly cannot investigate questions like what happens if they do.

###### Assumption 1 (Consistency) .

###### Assumption 2 (No Unmeasured Confounding) .

###### Theorem 1 (IPW Identification) .

### 3.2 Inverse Probability Weighted Estimation and Inference

Suppose we observe a random sample of size $n$
denoted as $\{Y_{i},T_{i}\wedge\tau,\Delta_{i}=\mathbbm{1}(T_{i}<\tau),L_{i}\}_{i=1}^{n}$. Inspired by Theorem 1, if we were to know the truth $\Lambda(t|L)$, then one may plug it in and obtain an estimate of $\psi(\theta)$ by $\frac{1}{n}\sum_{i=1}^{n}\theta(T_{i},L_{i})^{\Delta_{i}}e^{-\int_{0}^{T_{i}
\wedge\tau}\{\theta(t,L_{i})-1\}d\Lambda(t|L_{i})}Y_{i}$. However in practice, $\Lambda(t|L)$ is hardly known to an analyst and must be estimated. Write the estimate as $\hat{\Lambda}(t|L)$, we may define the plug-in IPW estimator as

$$ $\hat{\psi}(\theta)=\frac{1}{n}\sum_{i=1}^{n}\theta(T_{i},L_{i})^{\Delta_{i}}e^ {-\int_{0}^{T_{i}\wedge\tau}\{\theta(t,L_{i})-1\}d\hat{\Lambda}(t|L_{i})}Y_{i}.$ (5) $$

With some regularity conditions on the nuisance estimate $\hat{\Lambda}(t|L)$ given in the appendix, we may derive the following asymptotic results on $\hat{\psi}(\theta)$.

###### Theorem 2 (Consistency) .

Under Assumptions 1, 2, and some regularity conditions, $\hat{\psi}(\theta)$ converges to $\psi(\theta)$ in probability, that is, for any constant $\varepsilon>0$,

$$ $\operatorname{\mathbb{P}}(|\hat{\psi}(\theta)-\psi(\theta)|>\varepsilon)\to 0,$ (6) $$

when $n\to\infty$.

###### Theorem 3 (Asymptotic Normality) .

Under Assumptions 1, 2, and some regularity conditions, the root-$n$ scaled centered difference $\sqrt{n}\{\hat{\psi}(\theta)-\psi(\theta)\}$ is asymtotically linear and thus converges to a normal variable weakly, that is,

$$ $\sqrt{n}\{\hat{\psi}(\theta)-\psi(\theta)\}=\frac{1}{\sqrt{n}}\sum_{i=1}^{n} \Xi_{i}+o_{P}(1)\to\mathcal{N}(0,\operatorname{Var}(\Xi_{1})),$ (7) $$

in distribution.

We use bootstrap to construct variance estimator of $\hat{\psi}(\theta)$ and build Wald-type confidence intervals.

###### Theorem 2 (Consistency) .

###### Theorem 3 (Asymptotic Normality) .

## 4 Experiment result

### 4.1 Simulation

In this section, we investigate the finite-sample performance of our estimator. We set $\tau=2$ and generate $n$ i.i.d. copies of $(L_{i},T_{i},Y_{i})$ as follows:

$$ $L_{i}\sim\text{Unif}(0,1),$ (8) $$

$$ $\operatorname{\mathbb{P}}(T_{i}>t|L_{i})=\exp\{-\exp(0.25L_{i})t\},$ (9) $$

$$ $Y_{i}\sim\mathcal{N}(\exp(1-1.5L_{i}-(2-T_{i})),0.5^{2}),$ (10) $$

where we observe $\{L_{i},T_{i}\wedge 2,\Delta_{i}=\mathbbm{1}(T_{i}<2),Y_{i}\}_{1\leq i\leq n}$.

We examine the performance of the IPW estimator $\hat{\psi}(\theta)$ and its variance estimate when $\theta(t,l)\equiv 1/3,1/2.5,1/2,1/1.5,1.5,2,2.5,3$ by reporting biases, empirical standard errors (SEE), average estimated standard errors (SD) based on $B=200$ multiplier bootstrap (van der Vaart and Wellner, ; Kosorok, ), and coverage probabilities (95% CP) of Wald type 95% confidence intervals using $R=1000$ simulated data sets of size $n=200,1000,5000$. The results are given in Table 1.

**Table 1: Simulation results of the IPW estimator. We report Bias ($\times 10^{-3}$), empirical standard error (SEE) ($\times 10^{-3}$), average estimated standard erros (SD) ($\times 10^{-3}$) and coverage probability of Wald type 95% confidence intervals (95% CP) of $\widehat{\psi}(\theta)$ by $B=200$ multiplier bootstrap, for $n=200$ sample sizes and $R=1000$ Monte Carlo samples.**
|  | $\theta(t|l)\equiv$ | 1/3 | 1/2.5 | 1/2 | 1/1.5 | 1.5 | 2 | 2.5 | 3 |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
|  | Bias | -1.348 | -1.086 | -0.769 | -0.4 | 0.121 | 0.117 | 0.085 | 0.048 |
| $n=200$ | SEE | 6.906 | 6.413 | 5.893 | 5.381 | 4.591 | 4.572 | 4.718 | 4.937 |
|  | SD | 6.928 | 6.476 | 6.001 | 5.516 | 4.612 | 4.525 | 4.616 | 4.792 |
|  | 95% CP | 95.0 | 95.4 | 95.1 | 95.4 | 94.7 | 93.8 | 93.3 | 93.6 |
|  | Bias | -0.189 | -0.177 | -0.158 | -0.132 | -0.071 | -0.06 | -0.054 | -0.054 |
| $n=1000$ | SEE | 3.121 | 2.91 | 2.682 | 2.444 | 2.03 | 2.018 | 2.099 | 2.222 |
|  | SD | 3.205 | 2.988 | 2.75 | 2.5 | 2.064 | 2.037 | 2.095 | 2.192 |
|  | 95% CP | 96.0 | 95.7 | 95.3 | 95.1 | 95.9 | 95.4 | 95.2 | 94.0 |
|  | Bias | -0.004 | 0.013 | 0.035 | 0.057 | 0.065 | 0.055 | 0.051 | 0.045 |
| $n=5000$ | SEE | 1.479 | 1.382 | 1.272 | 1.152 | 0.937 | 0.932 | 0.969 | 1.023 |
|  | SD | 1.44 | 1.342 | 1.235 | 1.122 | 0.927 | 0.916 | 0.943 | 0.988 |
|  | 95% CP | 94.5 | 94.4 | 94.8 | 94.6 | 94.9 | 95.0 | 95.0 | 94.3 |

As the simulation results illustrate, $\widehat{\psi}(\theta)$ perform well with small biases, across different constant functions $\theta(t,l)$, thus confirming our theoretical results. As expected from theory, variance estimates approach the Monte Carlo variance as sample sizes increase. Similarly, Wald type confidence intervals of $\psi(\theta)$ attain their nominal levels as sample sizes become larger.

### 4.2 Evaluating Incremental Causal Effect of Methotrexate on Rheumatoid Arthritis

We illustrate our framework by estimating incremental causal effects of the anti-rheumatic therapy Methotrexate (MTX) among patients with rheumatoid arthritis, using data from Choi et al. (). As a disease-modifying antirheumatic drug, MTX can slow joint damage and the progress of the disease. It is one of the most effective
medications for treating inflammatory types of arthritis. This, as well as its long track record and inexpensive price tag, explains why it’s usually the first drug prescribed for rheumatoid arthritis. Here we consider the average of reported number of tender joints at one year of follow-up $Y$ as the outcome, an important measure of disease progression.

Methotrexate use was recorded at each monthly clinic visit. At each visit, we classified methotrexate exposure status as ever-treated or never-treated, i.e., once a patient starts methotrexate therapy, he or she was considered on therapy for the rest of the follow-up. Our analysis includes individuals who survived and were followed up more than 12 months, with 1010 patients meeting our inclusion criteria. Figure 2 shows the probability of receiving Methotrexate treatment since patients’ first visit to the center, estimated by the Kaplan–Meier estimator $\hat{\operatorname{\mathbb{P}}}(T\leq t)$, with its pointwise 95% confidence intervals. As the Figure illustrates, nearly 12% of the patients initiated MTX usage at the first visit and around 25$ of the patients were treated at the 12 month visit.

Figure: Figure 2: Estimated probability of receiving Methotrexate treatment since patients’ first visit to the center $\hat{\operatorname{\mathbb{P}}}(T\leq t)$, with pointwise 95% confidence intervals.
Refer to caption: extracted/5866916/figures/KM_T.png

We used a Cox proportional hazards model (Cox, ) to estimate the conditional cumulative hazard function of the time to initialize Methotrexate $\Lambda(t|L=l)$, given the following variables $L$: age, sex, past smoking status, education level, rheumatoid arthritis duration, rheumatoid factor positive, health assessment questionnaire, patient’s global assessment, erythrocyte sedimentation rate, prednisone use, and number of tender joints (measured at baseline). The proportional hazards assumption was rigorously assessed in the appendix and determined to be valid.

We consider estimands $\psi(\theta)$ by letting $\theta$ being constant functions and varying $\theta\in[0.2,5]$. These are the incremental causal effects when the hazard rate for initiating MTX at any given time is scaled by $\theta$ across all patients. Such an analysis can illustrate how varying levels of aggressiveness or conservatism in prescribing MTX might influence average disease progression.

Plugging the estimated conditional cumulative hazards $\hat{\Lambda}(t|L)$ into our IPW estimator, we estimated the incremental treatment effect curve with $\hat{\psi}(\theta)$, where we also computed the pointwise 95% confidence intervals (with $B=200$ multiplier bootstraps). The results are shown in Figure 3. If the hazards function of the time to Methotrexate treatment initialization were increased proportionally for all individuals, the average number of tender joints at one year of follow-up would drop. More specifically, the average number of tender joints at one year of follow-up would decrease by 1.23% from 1.90 if the hazards doubled ($\hat{\psi}(2)$=1.87, 95% CI: 1.81–1.93), and by 5.42% if the hazards were multiplied five-fold ($\hat{\psi}(5)$=1.79, 95% CI: 1.71–1.88). Conversely, the average number of tender joints would increase by 0.83% if the hazards were halved ($\hat{\psi}(0.5)$=1.91, 95% CI: 1.86–1.97), and by 1.51% if the hazards were reduced five-fold ($\hat{\psi}(0.2)$=1.92, 95% CI: 1.86–1.98). The decline in the average number of tender joints as the hazards ratio increases from 0.2 to 5 is consistent with the protective effect of Methotrexate found in Choi et al. ().

Figure: Figure 3: Estimated incremental causal effects $\hat{\psi}(\theta)$, if the hazards were multiplied by factor $\theta$, with pointwise 95% confidence intervals.
Refer to caption: extracted/5866916/figures/jc_result.png

## 5 Conclusion

In this paper, we extended the concept of incremental causal effects to continuous-time treatment initiation. By defining these effects through shifting the hazard function via the hazard ratio, we demonstrated that they can be identified using only sequential randomization and consistency, without requiring the positivity assumption. We developed a flexible inverse probability weighted estimator, accommodating various hazard function estimators, and provided root-$n$ inference. The finite-sample performance of our framework and its potential real data value were assessed.

There are many open questions. First, one can consider augmenting the inverse probability weighted estimator through semiparametric theory to attain efficiency, using nonparametric estimation for nuisance parameters. This is currently under our investigation. Second, the idea can also be extended to complex longitudinal studies, including time-varying confounders and survival outcomes, or more generally, marked point processes (Rytgaard et al., ), càdlàg processes (Ying, ). Third, one might consider when Assumption 2 fails but either instrumental variable assumption (Ying et al., ; Ying and Tchetgen Tchetgen, ; Ying, ) or proxy variable assumption (Miao et al., ; Tchetgen Tchetgen et al., ; Ying et al., , ) holds. Last, conditional incremental effect (McClean et al., ) can also be of interest.

## Appendix A Proofs

### A.1 Proof of Theorem 1

We have

$$ $\displaystyle\psi(\theta)$ $\displaystyle=\operatorname{\mathbb{E}}(Y_{T(\theta)})$ (11) $\displaystyle=\operatorname{\mathbb{E}}[\operatorname{\mathbb{E}}\{ \operatorname{\mathbb{E}}(Y_{T(\theta)}|T(\theta),L)|L\}]$ (12) $\displaystyle=\int\operatorname{\mathbb{E}}(Y_{t}|T(\theta)=t,L=l)f(l)\{\theta (t|l)\lambda(t|l)e^{-\int_{0}^{t}\theta(s|l)\lambda(s|l)ds}\}^{\delta}$ (13) $\displaystyle~{}~{}~{}\{\operatorname{\mathbb{E}}(Y_{\tau}|T(\theta)\geq\tau,L =l)e^{-\int_{0}^{\tau}\theta(t|l)\lambda(t|l)dt}\}^{1-\delta}dtdld\delta$ (14) $\displaystyle=\int\operatorname{\mathbb{E}}(Y_{t}|L=l)f(l)\{\theta(t|l)\lambda (t|l)e^{-\int_{0}^{t}\theta(s|l)\lambda(s|l)ds}\}^{\delta}$ (15) $\displaystyle~{}~{}~{}\{\operatorname{\mathbb{E}}(Y_{\tau}|L=l)e^{-\int_{0}^{ \tau}\theta(t|l)\lambda(t|l)dt}\}^{1-\delta}dtdld\delta$ (16) $\displaystyle=\int\operatorname{\mathbb{E}}(Y_{t}|T=t,L=l)f(l)\{\theta(t|l) \lambda(t|l)e^{-\int_{0}^{t}\theta(s|l)\lambda(s|l)ds}\}^{\delta}$ (17) $\displaystyle~{}~{}~{}\{\operatorname{\mathbb{E}}(Y_{\tau}|T\geq\tau,L=l)e^{- \int_{0}^{\tau}\theta(t|l)\lambda(t|l)dt}\}^{1-\delta}dtdld\delta$ (18) $\displaystyle=\int\operatorname{\mathbb{E}}(Y|T=t,L=l)f(l)\{\theta(t|l)\lambda (t|l)e^{-\int_{0}^{t}\theta(s|l)\lambda(s|l)ds}\}^{\delta}$ (19) $\displaystyle~{}~{}~{}\{\operatorname{\mathbb{E}}(Y|T\geq\tau,L=l)e^{-\int_{0} ^{\tau}\theta(t|l)\lambda(t|l)dt}\}^{1-\delta}dtdld\delta$ (20) $\displaystyle=\operatorname{\mathbb{E}}\left\{\theta(T,L)^{\Delta}e^{-\int_{0} ^{T\wedge\tau}(\theta(t,L)-1)d\Lambda(t|L)}Y\right\}.$ (21) $$

where the first equation follows by definition, the second by iteration of expectation, the third by definition of expectation, the fourth by the definition of $T(\theta)$, the fifth by Assumption 2, the sixth by Assumption 1, and the last by the Radon-Nikodym Theorem.

Another intuition of checking this is through densities: The observed data density is

$$ $\displaystyle f(Y,T,L)=f(L)\{\lambda(T|L)e^{-\int_{0}^{T}\lambda(t|L)dt}f(Y|T, L)\}^{\Delta}\{e^{-\int_{0}^{\tau}\lambda(t|L)dt}f(Y|T>\tau,L)\}^{1-\Delta}.$ (22) $$

The target incremental intervention density is

$$ $f(Y,T,L)=f(L)\{\theta(T|L)\lambda(T|L)e^{-\int_{0}^{T}\theta(t|L)\lambda(t|L) dt}f(Y|T,L)\}^{\Delta}\{e^{-\int_{0}^{\tau}\theta(t|L)\lambda(t|L)dt}f(Y|T\geq \tau,L)\}^{1-\Delta}.$ (23) $$

The density for the potential outcomes is

$$ $f(Y_{T(\theta)},T,L)=f(L)\{\theta(T|L)\lambda(T|L)e^{-\int_{0}^{T}\theta(t|L) \lambda(t|L)dt}f(Y_{T}|L)\}^{\Delta}\{e^{-\int_{0}^{\tau}\theta(t|L)\lambda(t| L)dt}f(Y_{\tau}|L)\}^{1-\Delta}.$ (24) $$

Integrating $Y_{T(\theta)}$ yields the parameter of interest (1). By Assumptions 1 and 2, the density for the potential outcomes is equal to the target incremental intervention density. Therefore, to identify (1) through the observed data density, one can use Radon-Nikodym weights to reweight $Y$.

### A.2 Proof of Theorem 2

Here we assume the regularity condition that $\hat{\Lambda}(t|L)$ is uniformly consistent for $\Lambda(t|L)$, that is,

$$ $\operatorname{\mathbb{P}}(\sup_{t}|\hat{\Lambda}(t|L)-\Lambda(t|L)|> \varepsilon)\to 0$ (25) $$

as $n\to\infty$. Then we have

$$ $\displaystyle=\left|\hat{\psi}(\theta)-\psi(\theta)\right|$ (26) $\displaystyle=\left|\frac{1}{n}\sum_{i=1}^{n}\theta(T_{i},L_{i})^{\Delta_{i}}e ^{-\int_{0}^{T_{i}\wedge\tau}(\theta(t,L_{i})-1)d\hat{\Lambda}(t|L_{i})}Y_{i}- \psi(\theta)\right|$ (27) $\displaystyle\leq\left|\frac{1}{n}\sum_{i=1}^{n}\theta(T_{i},L_{i})^{\Delta_{i }}\left\{e^{-\int_{0}^{T_{i}\wedge\tau}(\theta(t,L)-1)d\hat{\Lambda}(t|L_{i})} -e^{-\int_{0}^{T_{i}\wedge\tau}(\theta(t,L_{i})-1)d\Lambda(t|L_{i})}\right\}Y_ {i}\right|$ (28) $\displaystyle+\left|\frac{1}{n}\sum_{i=1}^{n}\theta(T_{i},L_{i})^{\Delta_{i}}e ^{-\int_{0}^{T_{i}\wedge\tau}(\theta(t,L_{i})-1)d\Lambda(t|L_{i})}Y_{i}-\psi( \theta)\right|$ (29) $\displaystyle\leq\left|\frac{1}{n}\sum_{i=1}^{n}\theta(T_{i},L_{i})^{\Delta_{i }}\left\{e^{-\int_{0}^{T_{i}\wedge\tau}(\theta(t,L_{i})-1)d\hat{\Lambda}(t|L_{ i})}-e^{-\int_{0}^{T_{i}\wedge\tau}(\theta(t,L_{i})-1)d\Lambda(t|L_{i})}\right \}Y_{i}\right|+o_{P}(1).$ (30) $$

Note that

$$ $\displaystyle\left|e^{-\int_{0}^{T_{i}\wedge\tau}(\theta(t,L_{i})-1)d\hat{ \Lambda}(t|L_{i})}-e^{-\int_{0}^{T_{i}\wedge\tau}(\theta(t,L_{i})-1)d\Lambda(t |L_{i})}\right|$ (31) $\displaystyle=\left|e^{-\theta(T_{i}\wedge\tau,L_{i})\hat{\Lambda}(T_{i}\wedge \tau,L_{i})+\int_{0}^{T_{i}\wedge\tau}\hat{\Lambda}(t|L_{i})d\theta(t,L_{i})}- e^{-\theta(T_{i}\wedge\tau,L_{i})\Lambda(T_{i}\wedge\tau,L_{i})+\int_{0}^{T_{i }\wedge\tau}\Lambda(t|L_{i})d\theta(t,L_{i})}\right|$ (32) $\displaystyle\leq\Bigg{|}\left\{e^{-\theta(T_{i}\wedge\tau,L_{i})\Lambda(T_{i} \wedge\tau,L_{i})+\int_{0}^{T_{i}\wedge\tau}\Lambda(t|L_{i})d\theta(t,L_{i})}+ o_{P}(1)\right\}$ (33) $\displaystyle\left\{\theta(T_{i}\wedge\tau,L_{i})\left|\hat{\Lambda}(T_{i} \wedge\tau,L_{i})-\Lambda(T_{i}\wedge\tau,L_{i})\right|+\int_{0}^{T_{i}\wedge \tau}\left|\hat{\Lambda}(t|L_{i})-\Lambda(t|L_{i})\right|d\theta(t,L_{i}) \right\}\Bigg{|}$ (34) $\displaystyle\leq C\sup_{t}\sup_{L_{i}}\left|\hat{\Lambda}(t|L_{i})-\Lambda(t| L_{i})\right|.$ (35) $$

### A.3 Proof of Theorem 3

Besides the uniform consistency condition we assumed in the previous theorem, we further assume that $\hat{L}ambda(t|L)$ is asymptotically linear for $\Lambda(t|L)$, that is,

$$ $\hat{\Lambda}(t|L)-\Lambda(t|L)=\frac{1}{n}\sum_{i=1}^{n}\xi_{i}(t,L)+o_{P}(1)$ (36) $$

for some random process $\xi_{i}(t,L)$, where $o_{P}(1)$ converges to zero as $n\to\infty$ uniformly over $t$. Then by a simple Taylor expansion, we have

$$ $\hat{\psi}(\theta)-\psi(\theta)=\frac{1}{n}\sum_{i=1}^{n}\Xi_{i}(\delta)=\frac {1}{n}\sum_{i=1}^{n}\theta^{\Delta_{i}}\hat{\operatorname{\mathbb{P}}}(T_{i}>t |L_{i}\}^{\theta-1}|_{t=X_{i}}Y_{i}+\frac{\partial\psi(\theta)}{\partial \Lambda(t)}\xi_{i}(t,L)|_{t=X_{i}}-\psi(\theta),$ (37) $$

and therefore by central limit theorem we have reached the conclusion.
∎

## Appendix B Methotrexate Data Processing

The raw dataset included the information of 1240 patients. For all the patients, we extracted 11 baseline covariates described in Section 4.2, as well as $T$, the time to receive Methotrexate (MTX) treatment, and the number of tender joints at month $\tau=12$, which is the outcome $Y$ of interest. Some patients exited the study before reaching month $\tau=12$, i.e., one year of follow-up, resulting in no recorded $Y$ for these individuals. After excluding them, 1010 patients met our inclusion criteria.

Denote $L_{1,i}$ to $L_{11,i}$ as the baseline covariates, $T_{i}$ as the time to receive MTX, and $Y_{i}$ as the number of tender joints at one year of follow-up for each patient ($i=1,2,\dots,1010$). Define $\Delta_{i}=\mathbbm{1}(T_{i}<\tau)$. For the $i$th patient, the individual data consists of $\{Y_{i},T_{i}\wedge\tau,\Delta_{i},L_{1,i},\dots,L_{11,i}\}$.

For constructing the Cox proportional hazards model, we tested the proportional hazards assumption by examining the Schoenfeld residuals and cumulative martingale residuals. For simplicity, we used age_0, sex, smoke_0, edu_0, duration_0, rapos, haqc_0, gsc_0, esrc_0, onprd2_0, and jc_0 as the abbreviations for the covariates age, sex, past smoking status, education level, rheumatoid arthritis duration, rheumatoid factor positive, health assessment questionnaire, patient’s global assessment, erythrocyte sedimentation rate, prednisone use, and number of tender joints (measured at baseline), respectively. Figure 4 gives the Schoenfeld residuals plots of time-dependent Cox regression and the results of Schoenfeld individual tests and global Schoenfeld test. Table 2 shows the results of the test of proportionality for fitting an Cox-Aalen model. Figure 5 gives the cumulative martingale residuals plots for each covariate. The results show that the Cox proportional hazards model we constructed is appropriate.

Figure: Figure 4: Results of Schoenfeld residuals examination.
Refer to caption: extracted/5866916/figures/schoenfeld.png

**Table 2: Results of the test of proportionality for fitting an Cox-Aalen model.**
|  | $\sup{|\hat{U}(t)|}$ | p-value |
| --- | --- | --- |
| age_0 | 16.80 | 0.158 |
| sex | 6.30 | 0.248 |
| smoke_0 | 10.30 | 0.406 |
| edu_0 | 5.52 | 0.932 |
| duration_0 | 10.60 | 0.306 |
| rapos | 4.68 | 0.294 |
| haqc_0 | 9.52 | 0.320 |
| gsc_0 | 6.28 | 0.774 |
| esrc_0 | 8.40 | 0.078 |
| onprd2_0 | 7.98 | 0.140 |
| jc_0 | 14.00 | 0.094 |

Figure: Figure 5: Cumulative martingale residuals plots for each covariate.
Refer to caption: extracted/5866916/figures/martingale.png

## Peer Reviews

confidence: 3
contribution: 2
initial_rating: 3
presentation: 1
questions: Can you explain which pieces of this work are novel/provide a contributions list?  From the introduction, it looks like the move to continuous time to treatment is new.  However, the abstract makes it sound like avoiding the positivity assumption is also novel, while the paper makes it seem like that was something that was already shown as a property of incremental causal effects.

I'm not used to seeing L chosen as the variable for covariates/potential confounders. (I've seen W, X, V, C....) It's not a problem, but is the choice of L based on any particular subset of the literature?

As per my confusion in the Weaknesses section, can you clarify what you mean by an incremental intervention being one that is "a function of the observed treatment distribution"?

In the second paragraph of the introduction, you state that, in a static intervention, "the subjects are either all treated or all untreated", in contrast to a dynamic intervention, where treatment could depend on covariates.  This makes it sound like the entire population is either all treated, or all not treated.  This is a valid scenario to consider (e.g., a state government policy that then affects everyone living in that state), but you describe static interventions as being "typically the case when considering an average treatment effect (ATE)", in which case, you typically need examples of both treated and untreated subjects.  I would assume that you meant "each subject is either treated or untreated, assigned independently of their covariates", but that's not particularly close to what you wrote, so I must be misunderstanding something.  Actually, reading the abstract of Bonvini et al (2021), they describe ATE as relating to "the effect of everyone deterministically receiving versus not receiving treatment" - as in, the counterfactual question.  Is that what you're referring to here?

Especially in medical examples (such as the MTX arthritis example), individual covariates can change over time.  Does your model take into account that an individual's covariates L could change over the timesteps before they get treated, which could in turn affect the probability of treatment?

I'm not following the explanation at the beginning of Related Work about how incremental intervention avoids making the positivity assumption.  Summarizing Kennedy (2019), you say that, for subjects with 0 or 1 probability of treatment, we can see positivity as always satisfied "because perturbing the odds does not change their degenerate probabilities".  How does that follow?

In the line before Theorem 1, it says "We prove that ([1]) can be identified".  What is ([1]) referring to here?  Are you referring to Theorem 1?  Assumption 1?  Equation 1?

Are there any baselines you can use for comparison in the experimental results?  Some other effect estimation method, or at the very least some naive baseline that could provide some calibration for the experimental results?
soundness: 3
strengths: While there are some clarity issues with the narrative of the paper, the individual sections and descriptions in the paper are clear and easily readable.  The literature review on incremental causal effects and time to treatment is very thorough, situating the paper nicely in the literature.  The examples chosen, especially the rheumatoid arthritis example, are strong, and the analysis of the rheumatoid arthritis experiment (the reasoning about doubling the hazard decreasing joint pain) is especially compelling and highlights very nicely how this method could be used in practice.
summary: In many settings, whether or not a subject receives a treatment at any given time point may be a function of their covariates.  In these settings, we can reason about the time to treatment from when an individual becomes treatment-eligible to when they actually receive treatment.  Such a model can allow us to reason about how changes to covariates can affect time to treatment and, through treatment, some relevant outcome.  This model has the benefit of not requiring the positivity assumption.  The authors define this model in terms of hazard functions and provide an estimator.  They then assess their model on both synthetic and empirical data and demonstrate how it can be used to inform policy/medical practice decisions.
weaknesses: The biggest weakness of this paper in my eyes is that the problem being solved is not clearly defined.  Some of this seems to be due to language issues (while the occasional grammatical issue or awkward phrase don't generally impede understanding, there are a few parts where the intended meaning isn't clear), and some of this is due to the lack of a clear motivating example".  Specifically:

- In the introduction, the authors define an incremental intervention as an intervention "that is not pre-specified, but rather a function of the observed treatment distribution."  Without defining what is meant by "the observed treatment distribution", I assume that it means which units in the sample population received treatment and which didn't (P(T)), or maybe the conditional probability distribution in the sample population (P(T|L)).  I'm having a hard time understanding what this means, even after having read the whole paper.  The example in this section, as well as the MTX use case in the experiments, seem to deal with an intervention on treatment (such as being assigned to a behavioral health program or being prescribed MTX) that is, presumably, informed based on the individual's covariate values.  So this is an intervention that is a function of the observed covariates, not "of the observed treatment distribution."  Am I misunderstanding something about your approach here?

The organization of the introduction seems a bit backwards to me.  The example in the third paragraph (probationers being assigned to behavioral health services) is a great motivating example for the general "time to treatment initialization" problem setting.  The first paragraph contains two good examples, but the narrative about "time to treatment" is not made clear.  For example, the first 4 sentences of the paper talk about a tech team struggling to keep up with review requests and asks the reader to consider the effect of doubling the number of reviewers on the processing time of requests.  Coming into this paper with causality in mind, this sounds an awful lot like reasoning about intervening on a treatment (the number of reviewers) and measuring the effect on an outcome (the processing time).  However, as becomes clear later in the paper, the "processing time" is not, in fact, the outcome, but the time until treatment.  And if that's time until treatment, then I suppose treatment = somebody reviewing a request.  But then I'm not sure what the outcome is....backlog size??

If you want to open with that example, you should start by clearing explaining how it maps to your problem. An example flow, assuming I'm understanding the problem correctly: "We're interested in understanding how long it takes people on a tech team to respond to review requests.  The time until review is not static, but depends on many features, such as how many reviewers the help desk has at that time. 
 After a system outage, the number of requests has increased, creating a large backlog.  The scheduler wants to decrease the size of this backlog, which they plan to do by decreasing the time until review.  The number of reviewers has a large effect on the time until review, so the scheduler decides to double the number of reviewers, which then doubles the likelihood of a request being reviewed at any given time, as requests are often selected for review at random.  This process - reasoning about percentage changes to covariates to determine their effect on time to treat and, thus, the effect of treatment - is called "incremental causal effects"."

The second paragraph of the introduction also oddly placed.  Digging into the different types of interventions is interesting, and the distinctions brought up are quite relevant, but without having a clear problem statement yet, it's unclear how to fit your proposed method into that framework.  I think you're missing a paragraph in the introduction where you define (not technically, but in straightforward language) your problem statement. (i.e., the time until treatment for each subject is based on some measured covariates; the outcome is an effect of that treatment and starts be recorded as soon as treatment is applied to that subject; we want to reason about how changes in the probability of treatment function affect outcome).

Section 1.1 is focused on, and named after the positivity assumption, and highlights bypassing the positivity assumption as a key advantage of incremental causal effects.  However, this section, from what I can tell, never actually explains how it bypasses positivity.  (Also, the phrase "avoids the positivity" is weird - reword that) It's only the introduction, so I don't expect an in-depth explanation yet, but given that it's a whole section in the intro about positivity, at least a sentence giving an intuition about why we can ignore positivity would help.  Following on from that about positivity, it looks like it's addressed in the first paragraph of the Related Work section.  However, the explanation in the related work is not very clear or detailed (and again, especially given how prominently positivity was just highlighted in the introduction, I expected a deeper/clearer explanation).

Some terminology explanation is missing.  Line 171 defines $\lambda(t|l)$ and $\Lambda(t|l)$ as just "its hazard function and cumulative hazard function at time t given L = l, respectively."  I assume "its" here refers to T.  However, from what I can see, you never actually define either hazard function, despite them being fairly core to your method.

I like the setups chosen for both the synthetic and empirical experiments, but the lack of a baseline makes interpreting the results near-impossible.  For example, in the simulation results, you say that your results illustrate that the incremental causal effects "perform well with small biases."  I'm struggling to see how you came to that conclusion from Table 1 alone.  Looking at the numbers in the "Bias" row, they look low, but are they actually low for that problem?  Did you use additional visualizations to come to the conclusion that these numbers represent good performance?

Between the clarity issues throughout and the difficulties in interpreting the experimental results, I don't feel comfortable voting for acceptance.  If these issues are adequately addressed, though, I'm open to increasing my score.

---

confidence: 3
contribution: 3
initial_rating: 6
presentation: 3
questions: 1- How is it possible to analyze the estimator with finite sample data? For example, is there a high-probability guarantee for it?

I am not completely familiar with the area covered in the paper, and I’m uncertain about its contribution to the field. I may revise my score after considering the feedback from other reviewers.
soundness: 3
strengths: 1- The paper addresses an important problem and proposes an algorithm to solve it.

2- The proposed approach has been analyzed both theoretically and empirically.
summary: This paper extends INCREMENTAL CAUSAL EFFECT to continuous-time treatment. To this end, the author shows that the target quantity is identifiable under certain assumptions, excluding the well-known positivity assumption. An estimator is then proposed that is consistent. The effectiveness of the estimator has been validated through empirical experiments.
weaknesses: 1- Some definitions in Subsection 3.1, such as the hazard function and related concepts, are not clear to the reader. It would be beneficial to provide more detail, as there is still enough space available.

---

confidence: 3
contribution: 2
initial_rating: 6
presentation: 3
questions: 1. If there is unmeasured confounding, is it natural to apply sensitivity analysis or proximal causal inference in this setting? How might these approaches integrate with your proposed IPW estimator?
2. Are the regularity conditions outlined in Theorem 2 and Theorem 3 considered trivial or standard in common survival models?
3. Is there any pattern in the estimator performance (bias, variance, or stability) as $\theta$ changes?
soundness: 3
strengths: 1. This paper extends incremental causal effects, which do not rely on the traditional positivity assumption, to a new setting. This advancement allows for new approaches to studying time-to-treatment problems in fields such as public health and policy-making.
2. Theoretical guarantee is provided.
3. The presentation and flow of this paper are clear.
summary: This paper studies a novel setting in causal inference: incremental causal effect of intervening the continuous time to treatment initiation by shifting the hazard function. It introduces an IPW estimator with proofs of consistency and asymptotic normality. It is also validated through empirical simulations and a real-world study.
weaknesses: 1. Additional experiments could provide deeper insights into the behavior and robustness of the proposed approach under different scenarios. For example exploring different shift interventions, hazard functions, and comparative analysis against any alternative estimators.

Minor comments:

2. The clarity of Theorems 2 and 3 could be improved by stating all conditions and notations explicitly.
2. In the simulation, it can be made more clear to state the true effect and whether the outcome is censored in the DGP.
3. Typos: in line 59, line 340, and Theorem 3 proof.

---

confidence: 4
contribution: 2
initial_rating: 6
presentation: 3
questions: See weaknesses.

1. Line 65: The paper states, "In general, the incremental causal effect has the interpretation of a policy effect on the population, instead of the therapeutic effect on an individual." This reminds me the field of reinforcement learning (RL), which is specifically designed for learning policy rewards. Could there be potential benefits in using RL to learn incremental causal effects?
2. Could you provide a concrete example to illustrate the difference between "time to treatment initiation" and "continuous time to initiating treatment"? These terms appear several times in the paper, but their distinction remains unclear to me. Since this paper focuses on the continuous version, clarifying this difference could help motivate the choice.
3. Line 340: typo 25$ should be 25%.
soundness: 2
strengths: 1. The paper addresses a gap in causal inference by extending incremental causal effects to continuous time-to-treatment initiation, an area that has not been extensively studied before.
2. The authors provide consistency and asymptotic normality theorems for their estimand, adding theoretical rigor to their approach.
3. The paper is well-written and easy to follow. I particularly like the related work section which offers a comprehensive background that is beneficial for readers.
summary: This paper proposes a novel methodology for estimating incremental causal effects in continuous-time settings, with a specific focus on time-to-treatment initiation. This approach intervenes on the intensity (hazard function) of treatment initiation without relying on the positivity assumption. By shifting the hazard function through a multiplicative factor theta, the authors develop an identification strategy using inverse probability weighting. Theoretical justification, along with both synthetic and real-world experiments, is provided.
weaknesses: 1. A major strength claimed by the authors is that the new estimand avoids the positivity assumption. However, as noted in related work (line 125), "Kennedy (2019) proposed an incremental intervention that fully resolved the positivity issue." Since this paper also focuses on incremental causal effects, this aspect of the contribution may lack novelty. 
2. In the synthetic experiment, only a single feature L is used, following a simple uniform distribution. The experiment would be more robust if it included multiple features and more common distributions, such as Gaussian. Additionally, the experiment would benefit from comparisons with other baseline methods, as the paper currently presents only the performance of their model without comparative analysis.
3. In Section 4.2, the authors mention that the decreasing trend in the average number of tenders aligns with findings from a 2002 paper. While it is always impossible to know the true causal evidence in real-world cases, a quantitative comparison of the estimated causal effects with results from other studies would strengthen the paper beyond trend consistency alone.

Meta-review: The paper introduces a continuous-time version of the "time to treatment" problem, which I found to be particularly relevant in applications such as AI for healthcare and is, to the best of my knowledge, understudied. It can be valuable for the ICLR community to be exposed to this. On the negative side, the original discrete-time formulation covers the bulk of conceptual questions, and it is can be seen from the paper, the technical aspects of the continuous-time version are relatively unsurprising, with the estimation method being a pipeline of standard problems and solutions - usual consistency / convergence in probability / asymptotic normality / bootstrap + Wald CIs. All of this is nice to have, but not the type of result that is the focus of ICLR (but not out of scope either, to be clear). As I've mentioned, the problem itself is still not given as much attention as I think it deserves, and this counts - although one may wonder whether there is some opportunity lost in getting into more subtleties of the problem given that the solution for the given framing is relatively classical - this is also manifested by having quite some space left in the manuscript itself. Reviewers raised all sorts of clarification questions, which I think were valid but also addressable. 

Discussion was admittedly underwhelming despite probing, which was disappointing . Reviews were reasonably informative though and, along with the rebuttals, were useful for the assessment. I found the comment about reinforcement learning (reply to reviewer "roun") to be interesting, and I didn't fully understand it: e.g. to which extent the randomness inheriting to this problem clashes with the usual deterministic policies of optimal single-agent problems. Maybe this could be elaborated upon in any particular version of this problem.

Decision: Accept (Poster)