Heterogeneous Treatment Effect in Time-to-Event Outcomes: Harnessing Censored Data with Recursively Imputed Trees
Authors: Tomer Meir
Uri Shalit
Malka Gorfine
Venue: ICML 2025 poster (2025)
Keywords: Heterogeneous treatment effect
Causal inference
Time to event data
Survival analysis
Instrumental variable
Rare disease
Multiple imputations
Primary area: general_machine_learning->causality
TL;DR: A non-parametric method for estimating heterogeneous treatment effects in survival data, overcoming prior limitations in heavy censoring and confounding. We demonstrate its superiority in both no-hidden-confounders and instrumental variable settings.

Abstract: Tailoring treatments to individual needs is a central goal in fields such as medicine. A key step toward this goal is estimating Heterogeneous Treatment Effects (HTE)—the way treatments impact different subgroups. While crucial, HTE estimation is challenging with survival data, where time until an event (e.g., death) is key. Existing methods often assume complete observation, an assumption violated in survival data due to right-censoring, leading to bias and inefficiency. Cui et al. (2023) proposed a doubly-robust method for HTE estimation in survival data under no hidden confounders, combining a causal survival forest with an augmented inverse-censoring weighting estimator. However, we find it struggles under heavy censoring, which is common in rare-outcome problems such as Amyotrophic lateral sclerosis (ALS). Moreover, most current methods cannot handle instrumental variables, which are a crucial tool in the causal inference arsenal. We introduce Multiple Imputation for Survival Treatment Response (MISTR), a novel, general, and non-parametric method for estimating HTE in survival data. MISTR uses recursively imputed survival trees to handle censoring without directly modeling the censoring mechanism. Through extensive simulations and analysis of two real-world datasets—the AIDS Clinical Trials Group Protocol 175 and the Illinois unemployment dataset we show that MISTR outperforms prior methods under heavy censoring in the no-hidden-confounders setting, and extends to the instrumental variable setting. To our knowledge, MISTR is the first non-parametric approach for HTE estimation with unobserved confounders via instrumental variables.

Heterogeneous Treatment Effect in Time-to-Event Outcomes: Harnessing Censored Data with Recursively Imputed Trees
Tomer Meir Uri Shalit 12
Malka Gorfine
Abstract
Ideal Data
Observed Data
Observed Data and Single Imputation
4
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Tailoring treatments to individual needs is a cen- tral goal in fields such as medicine. A key step to- ward this goal is estimating Heterogeneous Treat- ment Effects (HTE) the way treatments impact different subgroups. While crucial, HTE estima- tion is challenging with survival data, where time until an event (e.g.. death) is key. Existing meth- ods often assume complete observation, sumption violated in survival data due to right- censoring, leading to bias and inefficiency. Cui et al. (2023) proposed doubly-robust method for HTE estimation in survival data under no hid- den confounders, combining causal survival for- with augmented inverse-censoring weight- ing estimator. However, find it struggles un- der heavy censoring, which is common in rare- outcome problems such as Amyotrophic lateral sclerosis (AL.S). Moreover, most current meth- ods cannot handle instrumental variables, which are crucial tool in the causal inference arsenal. We introduce Multiple Imputation for Survival Treatment Response (MISTR), novel, general, and non-parametric method for estimating HTE in survival data. MISTR uses recursively imputed survival trees to handle censoring without directly modeling the censoring mechanism. Through ex- tensive simulations and analysis of two real- real-world datasets the AIDS Clinical Trials Group Proto- col 175 and the Illinois unemployment dataset we show that MISTR outperforms prior methods un- der heavy censoring in the no whwdee-coonoouu setting, and extends to the instrumental variable setting. To our knowledge, MISTR the first non- parametric approach for HTE estimation with un- observed confounders via instrumental variables.
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Time
Time
mmax
Observed Data
Observed Data and Single Imputation
Observed Data and Single
Trented
Latreated
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Event
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HTEA
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Combine HTE Estimators Single
Figure I. Our goal is to estimate heterogeneous treatment ef- fect, defined as the expected difference in survival times (or their transformation, see Eq. (1)) with and without treatment, condi- tional on set of covariates. We propose multiple-imputaaion based estimator that effectively leverages censored observations, outperforms existing methods, and is applicable in settings with instrumental variable adjustment for unobserved confounders.
1. Introduction
In the field of causal inference, the Heterogeneous Treat- ment Effect (HTE) characterizes the way individuals or subgroups respond to an intervention. For example when assigning treatments to patients, estimating the HTE allows us to take into account relations between patient character- istics and treatment responses and thus optimizing clinical outcomes (Kent et al., 2018; Collins & Varmus, 2015). Due to its wide applicability, HTE estimation is area of active research within the machine learning community (Athey & Imbens, 2016: Shalit et al., 2017; Wager & Athey, 2018; Athey et al., 2019; Künzel et al., 2019; Nie & Wager, 2021; Curth van der Schaar, 2021; Curth & Van Der Schaar, 2023; Curth et al., 2024). This work focuses on HTE in survival analysis, where the outcome of interest time to event. This is common example, selecting a cancer treatment that leads to longer survival. Survival outcomes are also of interest in commer- cial applications, for instance, when enacting interventions to reduce user churn. A major challenge associated with survival outcomes is censoring, as we explain below.
'Department of Data and Decisions Sciences, Technion, Haifa, Israel 'Department of Statistics and Operations Research, Tel Aviv University, Tel Aviy, Isracl. Correspondence to: Malka Gorfine <gorfinem@ tauex.tau.ac
case in medicine, when, for
Proceedings of the ndnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Heterogencous Treatment Effect in Time-to-Event Outcomes: Harnessing Censored Data with Recursively Imputed Trees
Let T R4 be the survival time, W& E {0, the binary treatment assignment, and X E X' a set of time-independent covariates. Our goal is to estimate the average effect of treatment W on the survival time T, given X = z. Using the potential outcomes framework (Rosenbaum & Rubin, 1983), let T and T represent the potential survival times under control and treatment, respectively, with the observed survival time T Tw. The objective is the conditional average treatment effect (CATE) on the survival time
the effect of claimant and employer incentives on reducing the duration of unemployment.
We demonstrate MISTR's s superior performance through a comprehensive simulation study and real-world data analy- sis, showing substantial improvements over Causal Survival Forest (CSF) (Cui et al., 2023) under high censoring rates, For instance, Figure 2 presents the estimated CATE plotted against the true CATE on random test set points across two simulation designs; see details in Section 6.
E - X =
We further leverage the modularity of our approach and extend MISTR to address unobserved confounding by intro- ducing MISTR-IV, which leverages IVs to correct for bias. Figure 3 demonstrates the improved accuracy of MISTR- IV over IPCW with IV. The effectiveness of MISTR-IV is further validated through practical applications to real- world data. In Table we present qualitative compari- son of MISTR with CSF (Cui et al., 2023) and the classic and widely used inverse probability of censoring weighting (IPCW) approach (Robins et al., 1994).
or transformation of the survival time.
In practice, survival datasets typically include censored observations, where the exact event time is unknown but falls within a known range (Klein& & Moeschberger, 2006; Kalbfleisch & Prentice, 2011). 1). We focus on right-censored data: e.g. a 5-year study of a cancer treatment might include many patients who survived after 5 years; for these patients only know that their survival time is at least 5 years. Censoring introduces significant methodological difficulties, since applying standard causal effect estimation methods to censored data leads to biased estimates.
Table I. Methods Applicability and Performances. V indicates minimal MSE (Figure 4), v denotes applicability, and x indicates
non-applicability.
In this paper, we introduce MISTR (Multiple Imputations for Survival Treatment Response), new estimator for HTE estimation in right-censored survival data. Inspired by Re- cursively Imputed Survival Trees (RIST) (Zhu & Kosorok, 2012), MISTR eliminates the need to estimate the censoring distribution. This in turn allows the flexible use of methods which otherwise would not be applicable to reght-censored data. We build upon this flexibility and show how one can create methods which are more robust, more accurate, and applicable to a broader settings compared with exist- ing methods. Figure 1 presents our approach schematically, where censored observations are efficiently imputed multi- ple times, treatment effects are estimated for each imputed dataset, and the resulting estimators are then combined.
Case
Settings
IPCW
CSF
MISTR
Cui et al. (2023) simulated settings
1-4
v
v
+
Censoring probability reaches low values Censoring depends on unobserved covariates
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x
x
v
+
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v
v
v
High censoring rate
8-9
r
Unobserved confounding with IV adjustment
201-204
Type CSF mse: 0.414, con 0.750
Type CSF mss: O 0.038, cor: 0.895
0.4
0.05-
Estimated effect
Estimated effect
0.00
Our motivating example is the public data from the AIDS Clinical Trials Group Protocol 175 (ACTG 175) (Hammer al., 1996), a randomized controlled trial (RCT) that com- pared four treatment strategies in adults infected with human immunodeficiency virus (HIV). The original dataset has a 75.6% censoring rate. We demonstrate the HTE estimation using MISTR and highlight its superiority. More gener- ally, the methods we develop in this work are applicable to both clinical trial and observational data, including both the case where all confounders are observed as well as the chal-
0.0
-0.05
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True effect
Type MISTR mse: 0.144, 0.893
Type MISTR mse: 0.031, cor: 0.93
Estimated effect
Estimated offect
0.2
0.00
-0.05
lenging setting of instrumental variables (IV) with censored survival outcomes (Angrist, 1990; Angrist & Evans, 1998; Tchetgen Tchetgen et al., 2015; Wang et al., 2022). We illustrate the setting that includes unobserved confounders using data from the Illinois Unemployment Insurance Exper- iments (Woodbury& & Spiegelman, 1987), where we estimate
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True effect
Figure 2. Estimated CATE based on CSF (Cui et al., 2023) (blue) and MISTR (green) vs. the true CATE. Results are shown for 5000 test points from Settings 8-9. See Section 6 for details.
2

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Heterogencous Treatment Effect in Time-to-Event Outcomes: Harnessing Censored Data with Recursively Imputed Trees
Tyoe 1,800
since CSF relies on estimating the censoring probability, it is subject to the same limitations associated with such approaches. We compare with CSF extensively throughout this work. We further note that unlike our proposed method, CSF cannot utilize IV.
15-
1.5
Extmmnted aftect
03-
Entimmted
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As highlighted by Dietterich (2000), incorporating random- ness can enhance the performance of tree-based methods, Building on this idea, Zhu & Kosorok (2012) proposed the recursively imputed survival trees (RIST), nonparamet- ric approach for estimating the probability of remaining event-free given a set of covariates, based on right- survival data. RIST employs extremely randomized trees, enabling more effective utilization of censored observations compared to other tree-based methods. This results in im- proved model fit and reduced prediction error.
-2.5
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Tre effect
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Tue effect
To estimate causal effects in the presence of confounding, IV methods can be employed (Angrist et al., 1996; Abadie, 2003; Hernán & Robins, 2006). An IV variable is associ- ated with treatment assignment, affects the outcome only through the treatment, and is independent of unmeasured confounders. survival data, IV methods have been widely studied. For example. Tchergen Tchetgen al. (2015) pro- posed regression- IV approach using additive hazard models, while Sørensen et al. (2019) and MacKenzie et al. (2014) applied IV methods within the Cox proportional haz- ard framework. Martinussen et al. (2017) introduced an IV-based approach for estimating structural cumulative sur- vival models to capture time-varying effects on the survival function, and Kjaersgaard & Parner (2015) developed a peeudo-observation approach that accounts for the paramet- ric contribution of covariates. In contrast, this work focuses on directly estimating HTE, which can be defined by differ- ent estimands, using nonparametric methods. This approach offers greater flexibility in capturing complex relationships between covariates and treatment effects and as we show can readily incorporate IV methods.
Figure 3. Estimated CATE based on IPCW-IV (black) and MISTR- IV (brown) versus the true CATE. Results are shown for 5000 test points from Settings 201-202. See Section 6 for details.
2. Related Work
A common approach for estimating HTE in the presence of censoring relies on estimating the censoring mechanism. Specifically, in IPCW (Robins et al., 1994) the censoring probability is first estimated, and the HTE is then estimated using non-censored observations, which are up-weighted by their censoring probability. A similar approach is used in meta-learners that handle right-censoring (Bo et al., 2024). However, a key drawback of relying on such modeling is that when the censoring probability is low, poorly estimated, or influenced by unobserved covariates, this can lead to biased HTE estimates.
Other methods focus on specific estimands in the uncon- founded setting. For instance, Henderson et al. (2020) de- veloped nonparametric Bayesian accelerated failure time models to estimate differences in expected log-failure times. Zhu & Gallego (2020) introduced a targeted maximum like- lihood framework for differences in survival probabilities, while Chapfuwa et al. (2021) analyzed differences in sur- vival time. In contrast, we propose a more general estimand and can accommodate to confounded settings.
3. Problem Setup
We consider a dataset of n independent and identically dis- tributed observations. For each observation i = 1,..., n, the survival time is denoted by T, E R4 and the correspond- ing right-censoring time by C2 E R.4.. The observed time is represented by T; = min(T,, Ct), and the event indicator is defined as 5; = T(T; < Ca), with I(-) being the indica- tor function. Each observation is represented by the tuple [X,, W,, T. J.). where X, E R' is a vector of p baseline covariates and W; E {0.1) indicates the binary treatment assignment. The potential outcomes of observation i are denoted by T and T,. Our objective is to estimate
Non-parametric methods are often used for their flexibil- ity in modeling complex relationships. Recently, Cui et al. (2023) proposed CSF, -arramemitiiic approach for HTE estimation with right-censored survival data. CSF extends the causal forest framework of Athey et al. (2019) by incor- porating an IPCW approach combined with doubly robust estimating equations. They demonstrated the superiority of their approach over multiple other methods including vanilla IPCW, S-leamner (Künzel et al., 2019), and an adaptation of the virtual twins method (Foster et al., 2011). However,
((x) E -(())))-((()
(I)
3

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Let C denote the set of effectively censored observations, i.e., observations with T. Tmaz and 6, = 0. Using RIST we estimate (4) for each effectively right-censored obser- vation. Based on these estimators we generate A imputed (2) event times, denoted Ti,., for each i E C, a 1, 1...., A. Each imputed dataset is constructed by combining the ath imputed times with the observed events, adjusting for Heterogencous Treatment Effect in Time-to-Event Outcomes: Harnessing Censored Data with Recursively Imputed Trees
for known function g()) For instance, with (T,)) min(T; h). where h is a constant, 7(-) represents the differ- ences in restricted mean survival time (RMST), and with g(Z,) = I(T, 2 h) it represents the differences in survival functions.
are random forest-based weights, (xr is the terminal node that contain in the bth tree, B is the number of trees, and denotes the cardinality (Athey et al., 2019).
Our goal is to estimate T(x) and derive an estimator for its variance using survival data, which include observations where T. may be right-censored.
The following assumptions are required for identifiability in the unconfounded case (Cui et al., 2023):
4. The MISTR Procedure
A.I Finite horizon. The outcome transformation g() ad- mits maximal horizon 0 < h such that
In principle, Equation (2) above provides a path for flex- ible estimation of causal estimands. However, for time- to-event outcomes subject to right censoring, where T; is only observed for a portion of the training sample, Eq. (2) is inapplicable. We propose an approach that relies on efficient imputation of event times for right-censored ob- servations and thus allows us to make use of this equation. The core of MISTR includes the following steps: initially, RIST is employed to impute event times for censored obser- vations. These imputed event times are then merged with the observed events to form a complete, uncensored dataset. Subsequently, r(x) is estimated using Eq. (2). This proce- dure is replicated A times, each time with different imputed event times, producing A estimates, Fi (ar), ...., FA(x), Our proposed estimate is the average of these A estimates.
g(t) g(h) for all allt? h.
A.2 Consistency. T. = T. almost surely.
A.3 Overlap.
The propensity score e(x)
Pr (W, 1 X x) follows Te e(x) 1 ne
for some 0 < qe S
A.4 Ignorable censoring. T; IT C. X,, W.
A.5 Positivity. Pr (C2 < tmaal X2, W,) S 1 nC for some
0 < nC 1. 1. A.6 Ignorability. ( 11 W/| X
Additionally, following Assumption 1, the effective non- censoring indicator is defined by 6th = (T; A h) S C.). This implies that if an observation reaches the finite hori- zon h, no information is missing for estimating the HTE estimand associated with h.
RIST consists of the following key steps (Zhu & Kosorok, 2012): The process begins by constructing M extremely randomized survival trees, where each split is determined by selecting K random covariates and choosing the best split that maximizes node separation, ensuring each terminal node contains at least Pemin observed failure times (Ishwaran et al., 2008). A conditional survival distribution
Consider first the unconfounded setting and as start assume a constant treatment effect т. Consider the partially linear
model of (Robinson, 1988): (Ti) = TW; + f(X,) + Ç where E(G XX, = 0, W; 11 S. XX, = Pr(W,- = 1|X,), and m(x)= = Elg(T,) = z] Then, it is easy verify that (TT)) Elg(T.) = ((W, ((W](X))) + G. In the absence of censoring, 7 can be estimated by the score function (Chernozhukov et al., 2018:
Pr(T, > 1 W, C,, T, > C.)= Pr(T, > || X,, W2) Pr(T. > G. X,, W,,C)
VE C.
(4)
Cui et al., 2023)
0, where m(.) and 2(-) are estimates of ml-) and e(-) derived via cross-fitting. Building on this estimator, heterogeneity in the treatment effect is then incorporated by assigning weights to each training sample according to its similarity to the test sample. Namely, the estimator of 7 r(x), denoted by F(x)), is the solution
Σ (W. (in--e(()))) -(x))) - -WW -
is then estimated for each censored observation. To han- dle censoring, a one-step imputation replaces censored data based on its estimated probability. This process generates M independent imputed datasets, each used to refit sur- vival tree. Steps are recursively repeated Q times, refining predictions before computing the final survival estimators, denoted by RIST
(,(7(x)) = Σ
-
m(,))
(x) - -((x,)1 =0
where
1 B I(X, E(X,E
(x(x)= =
(3) the pre-specified horizon h, Ti, A = min(T,,.. h) and 8h i.a = [[Tta Ah) & C.). This results in fully observed
B 1=
[o6(z)
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Heterogencous Treatment Effect in Time-to-Event Outcomes: Harnessing Censored Data with Recursively Imputed Trees
datasets. The estimators 71 (2x),.... .... (ar) then obtained using Eq. (2), and the final MISTR estimator of Tz is given by (x) = A Ea (x))
5. The MISTR-IV Procedure Addressing Confounding using IVs
Unobserved confounders are often present in observational data, meaning that Assumption A.6 does not hold. Let U, E IR, ;= 11=1..... 1,.... n1 be the samples of the unobserved con- founder that influence both the treatment and the outcome. If Assumptions A. I-A.5 hold the treatment effect remains identifiable provided there exist binary insturmental vari- able, denoted by Z4 (0,1), that satisfies the following assumptions (Wang al., 2022), where W; (Z;) denotes the potential outcome for treatment:
Algorithm The Proposed Approach- MISTR input [X,, W T, 6, /i= training set
input I E
input Q: number of iterations for RIST
input A: number of imputed datasets
input imaz: maximum time allowed for imputation input h: estimand horizon
I: Train M extremely randomized survival trees using the original data, and get estimators of the conditional survival distribution (Eq. (4)).
B.1 Treatment consistency. W; W.(2))
B.2 Independence. 74 Il Us X;
2: for to Q do
B.3 Instrumental Relevance. Z, 1 W; X,
3: Apply M imputations for effectively censored obser-
B.4 Sufficiency of U. (T, C.) IL (W,, Z,) U4).
vations using the most recent estimates of Eq. (4) from the current RIST..
B.5 Monotonicity. W 1) 2 Wi = 0).
4: One extremely randomized survival tree is fitted for each of the M imputed datasets, resulting in updated estimators of Eq. (4).
Under Assumption A.2, Assumption B.4 implies the exclu-
sion restriction Z, (T;, C.) (Wi, U, X.). Hence the absence of censoring, Eq. (2) replaced by
5: end for output RISTQ
6: for = to n do
(T(x)) = Σ (x(x)) { Z - T(xx)) } (o(T)) - -m(x)))
7: if' 6; and T: maa then
8: Estimate Pr(T; X, W,, T, C2 C;) using
(W, -xx)) = = (5)
RISTQ
9:
Sample A event times, Ti, a = 1, I,...,A
where T(X,,) Fi(Z[ We accommodate right-
10: else
censoring by multiple imputation.
11:
Ti.a < T,, a === I.....A
12: end if
Our MISTR estimator of HTE, along with its variance es- timator, can be seamlessly extended to account for unob- served confounding using the IV approach. Given a training set of independent and identically distributed observa- tions, {X,, W., Zg, T2 8.}. i 1,.... n. causal forest for IV (Athey al., 2019, Section 7.1) is applied each im- puted dataset at step 16 of Alg. 1. Notably, the imputation step remains independent of the IV. Note that unless we assume that every observation has the same treatment ef- fect r(x) when X; = I, our estimator represents the HTE among compliers (Imbens & Angrist, 1994; Abadie, 2003; Athey al., 2019).
end for
14: Tia Ah min( 11), 1 A, .....
15: 6%. ITTio, S. h)) == I,..., A,i 1,...,
16: Train A causal forests (Athey et al., 2019), each using different imputed dataset.
17: Get Fa (x) by (2) and ((F((z)) by the causal forest,
a= I....,A. output 7M (x) and T (x))
The variance of each f (x), denote by ((fa())) can be estimated by the causal forest (Athey et al., 2019) using an adaptation of the bootstrap of little bags (Sexton Laake, 2009). The average estimated variances is then v(x) A-1 -A (x)). Finally, the variance of (Mr))) can be estimated by Rubin's rule (Rubin, 1987):
6. Experiments
In this section we evaluate MISTR. Since in causal effect inference problems the ground truth cannot in general be known, we use simulated and semi-simulated experiments in both unconfounded and IV settings.
1+ --1
(x))
4-1 ~
6.1. Benchmark Cases
We start by comparing the performance of MISTR to exist- ing baselines. Cui et al. (2023) showed that their method
The MISTR procedure summarized in Algorithm 1.
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Heterogencous Treatment Effect in Time-to-Event Outcomes: Harnessing Censored Data with Recursively Imputed Trees
superior to the random survival forest (Ishwaran & Kogalur,
Setting 1
Setting
Setting 3
Setting
0.045
1.0
2019), S-learner (Künzel al., 2019), enriched random survival forest (Lu et al., 2018), and the IPCW causal forest. Therefore, we focus on comparing our method with the CSF method of Cui et al. (2023). A comparison to additional baselines such as T-learner (Bo et al., 2024) is available in Table S3. First, we utilize Cui et al. (2023)'s benchmark (Settings 1-4). Then, we present new scenarios that go be- yond the censoring probability assumptions required by CSF and IPCW approaches. These scenarios are characterized by censoring within specific time interval (Settings 5, 10) higher censoring rates (Settings 6, 8, and 9), and unknown censoring mechanism (Setting 7).
0.042
0.04
0.065
0.06
6.0
0.9
0.035
0.038
0.055
5.0
0.6
Setting
Setting
Setting
Setting
MSE 1.4
1.8
1.4
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1.0
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0
e x
0 &
07
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0.06
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SEM CS?
Meen VESTR
000055
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SCM
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0.04
For each scenario in settings 1-10, we generate a training set and two test sets: Random and Quantiles. The train- ing set comprises Merain 5000 samples, {X,, W,,T,, 5,}, with i = 1, I,.... "trrin- Each observation includes p = 5 independent covariates generated from a Uniform[0.1 dis- tribution. In Setting 7, we sample two additional covariates from the same distribution, which are used only for sam- pling the censoring time C, and are unobserved during training or testing. The Random test set is generated in the same way as the training set with Neett 5000. The Quan- tiles test set comprises 21 observations, {X,, W,, Ti, 6.). where i = 1,...,21. 21. Each X; includes p 5 covariates, all equal to a specific quantile value 0.00, 0.05, 0.95, 1. For example, for: = 2, Ø2j = 0.1 for j = ,....,,. 5. In Setting 7, two additional covariates, with values equal to the others, are included for sampling the censoring time C. e.g., for i = 2, #2j 0.1 for j = . 7. True HTE values for all sets are approximated as the mean of 20,000 Monte Carlo samples. The sampling distributions of C; and T3 are detailed in Table S1, and the sampled populations are illustrated in Figure $1. The maximum time maz. the horizon h, and the censoring rate are shown in Table S2.
Figure 4. Mean + SEM of the MSE for the estimated r(X:). es- timated by MISTR and by CSF (Cui et al., 2023) on Quantiles test set (Q) and on Random test set (R). Results are based on 100 replications and multiplied by 100 for readability. The corre- sponding values are shown in Table S3.
6.2. Semi-simulated MIMIC Setting
So far we have presented experiments based on independent baseline covariates. In practice however, baseline covari- ates are often correlated through both known and unknown relations. Next, we present an experimental setting using baseline covariates from a realistic dataset, with simulated treatment assignments and outcomes.
We use the MIMIC-IV (2.0) dataset (Johnson et al., 2022; Goldherger et al., 2000), based on 25,159 ICU admissions recorded between 2014 and 2020. Following previous work (Meir et al., 2022: Meir & Gorfine, 2025), we consider only admissions classified as "emergeney". distinguishing be- tween direct emergency and emergency ward (EW) cases. For patients with multiple admissions, only the latest admis- sion is included, with additional baseline covariates summa- rizing the patient's is admission history. A total of 36 baseline covariates are used, including patient characteristics, ad- mission history, and lab test results. Table S4 provides a summary of the baseline covariates distributions.
We apply MISTR (Alg. 1) and CSF as detailed in Appendix A.1 and compare their results. Results are based 100 replications for each of Settings 1-10. Within each replica- tion, the mean squared error (MSF) of the estimator of r (xx) with respect to its true value r(x) is calculated for both the Random and Quantiles test sets. The MSE and standard error of the mean (SEM) across the 100 replications are summarized in Figure 4. Figure 2 and Figure S3 compare the estimated effects versus the true effects for one Ran- dom test set from each setting. Evidently, in Settings 1-3, both approaches perform comparably. while Settings 4-10 highlight the superior performance of MISTR. Additionally, Figure S2 presents the mean, empirical SE, and estimated SE of the estimatesd 7 T(X)). for the Quantiles test set as estimated by both approaches across the 100 replications.
The treatment assignment (W) is sampled from Bernoulli(O.5) distribution. Additionally, we use the co- variate X(36) (standardized age), together with the first five lab test results (normal or abnormal binary indicators) X(1) - X(5) to simulate failure time. The failure time is sampled from a Poisson distribution with 'y such that A =
30 + 0.75(11 - w) E X(a) + (.75 0.45W.
Note that xy is only influenced by part of the available base- line covariates, and that the HTE involves a combination of binary baseline covariates and a continuous covariate. Censoring time is sampled from a Poisson distribution with X = 21, 23, 24.7, 26.5, 29 for Settings MIMIC 1-5, respec-
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Heterogencous Treatment Effect in Time-to-Event Outcomes: Harnessing Censored Data with Recursively Imputed Trees
ple T, and C, from Poisson distributions with parameters
Table 2. Mean squared error (MSE) and standard error of the mean
>j. and Ac; respectively, see Table S7 for details.
(SEM) of the estimated r(X.) estimated by MISTR and by CSF (Cui et al., 2023) on MIMIC test set. Results are based on
For each observation with baseline covariates X; we sample 20,000 random U; values. For each X,, U3 pair we then calculate 1 and X9; and sample a pair of times Ti T70 from these distributions, cap them by h, and caleulate the difference between the capped values to get one possible value of RMST HTE. We take the average of 20,000 such differences for each observation as the Monte Carlo estimate for its true RMST HTE. We set taa = and = 8.
replications and are multiplied by 100 for readability.
CSF
MISTR
Censoring [%]]
Setting
MSE
SEM
MSE
SEM
MIMIC 1
89.5
8.741
1.350
6.262
0.830
MIMIC 2
84.6
6,532
0.775
5.891
0.755
MIMIC 3
80.0
5.632
0.405
5.200
0.500
MIMIC 4
75.3
5.349
0.368
5.034
0.369
MIMIC
69.9
5.474
0.624
5.494
0.649
In order to evaluate MISTR-IV we introduce an additional baseline which we call IPCW-IV, referring
instru- mental forest (Athey et al., 2019) utilizing IPCW, where non-censored observations are up-weighted by their proba- bility of not being censored. Table 3 presents a comparison of the HTE as calculated by CSF, MISTR, IPCW-IV, and MISTR-IV. Figure 3 and Figure S9 present the true vs, es- timated HTE as calculated by IPCW-IV and MISTR-IV. As expected, CSF and MISTR, which assume unconfound- edness, produce biased estimates in the presence of unob- served confounding, whereas MISTR-IV is applicable and outperforms the IPCW-IV baseline.
tively, where the highest censoring rate is in MIMIC 1.
We randomly split the full dataset into 5 folds, numbered 1-5, each with ~ 5030 patients. The maximum time for the RIST is tmaz = 29, and the RMST horizon is h = 28. The sampled populations are shown in Figure S4. We compare the HTE results of MISTR with those given by CSF (Cui et al., 2023) based on 5 replications, such that each fold is used either for training testing. The mean and SE of the MSE across the 5 replications of the analysis are presented in Table 2. A comparison of the estimated effect versus the true effect for one test set is shown in Figure S5. Evidently, as observed in carlier experiments, at lower censoring percentages (e.g.. MIMIC-5), both CSF and MISTR exhibit similar performance. However, as censoring increases, MISTR outperforms CSF. We use this setting to demonstrate the sensitivity of the variance to the bag size l and to the number of imputations (Appendix B). Based on the results of these analyses, we recommend using A 2 100. For the variance, a U-shaped relationship is expected due to the trade-off between the "within-imputation" and "between- imputation" components. Consequently, we recommend selecting the bag size using cross-validation.
Table 3. Comparison of the mean absolute error (MAF) and stan- dard error of the mean (SEM) of the estimated r (X,) estimated by CSF, MISTR, IPCW-IV, and MISTR-IV on random test set with Neest = 5000 observations for the IV case. Results are based on 100 replications and multiplied by readability factor of 100.
CSF
MISTR MAE
IPCW-IV
MISTR-IV
Sctting
MAE
SEM
SEM
MAE
SEM
MAE SEM
200
21.29
0.50
20.91
0.49
26.28
0.47
14.33
0.32
200-a
25.45
0.56
25.04
0.55
31.50
0.58
16.32
0.37
200-b
29.17
0.56
28.91
0.54
42.46
0.82
21.12
0.48
201
27.61
0.56
27.28
0.55
35.84
0.66
18.24
0.42
6.3. Instrumental Variable Setting
202
29.03
0.57
28.70
0.57
33.27
0.56
17.50
0.40
203
18.25
0.40
16.30
0.38
27.70
0.55
12.82
0.29
In this sub-section we demonstrate the use and applicabil-
204
16.14
0.46
13.37
0.44
32.10
1.12
11.40
0.37
ity of MISTR-IV. For each scenario, we generate train-
204-a
18.28
0.52
15.52
0.51
36.19
1.01
13.00
0.41
ing set and Random test set each comprises Ttrain Mestt 5000 observations, (X,, W,, Z, T, 8,). with i == 1...., , Terain and i = L.... Titest, respectively. Each observation i comprises = 3 independent baseline covari- ates X,, and one hidden confounder U., both generated from a uniform distribution over [0, 1). The hidden confounder U; is used solely during data generation and remains unob- served during training and testing. The instrumental variable Z, is sampled from a (er5 distribution. For the treatment assignment, for each sample we first sample a variable W from the distributions defined for each setting given the baseline covariates. The treatment assignment is then Wi- = 1 if W > 0.5, and W, = 0 otherwise. We sam-
204-b
20.50
0.56
17.93
0.56
54.69
1.90
15.18
0.44
We further extended our simulation study to assess the sen- sitivity of MISTR-IV to weak instruments. Specifically, we re-ran Settings 200 (47% censoring) and 204 (88% censor- ing) with IV of varying strength. The settings remained unchanged, except for modification of the coefficient of Z in the model for W*, as detailed in Table S7. Results are pre- sented in Table and Figure S11. As expected, the MAE of all methods increases as the IV strength decreases. Nonethe- less, MISTR-IV outperforms the alternative approaches.
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Heterogencous Treatment Effect in Time-to-Event Outcomes: Harnessing Censored Data with Recursively Imputed Trees
7. Use Case HIV Clinical Trial
Table 4. Mean Squared Error (MSF) and standard error of the mean (SEM) for HIV datasets with additional censoring. Results are based on replications of censoring sampling.
We demonstrate the applicability of MISTR using the ACTG 175 HIV dataset (Hammer et al.. 1996). The data includes 2.139 HIV-infected patients, randomized into four treatment groups: ZDV, ZDV+ddI, ZDV+Zal, and ddI (see details in Appendix A.2). We use 12 covariates: 5 continuous and 7 binary. Population characteristies are presented in Table S9. An additional covariate available in the dataset, denoted Z30, is not included in the first analysis but is used in the second analysis for additional censoring sampling. An event occurrence was defined as the first of either a decline in CD4 cells count (an event indicating AIDS progression) or death (Hammer et al., 1996). Outcomes
CSF
MISTR
Setting
MSE
SEM
MSE
SEM
HIV-1
1.388
0.165
1.207
0.251
HIV-2
1.875
0.266
1.209
0.249
HIV-3
1.320
0.238
0.777
0.190
of the overall sample MSE across 10 replications of cen- soring sampling for CSF and MISTR are shown in Table 4. When looking at the patient level, both the SE and mean Est SE were smaller for MISTR in most patients. Results for HTE, SE, and mean Est SE are shown in Figure S8. Evi- dently, both methods yield similar results when censoring rate is low; however, MISTR outperforms CSF at a higher censoring rate and unobserved censoring mechanism.
given in months.
The main conclusion of Hammer et al. (1996) was that treat- ment with ZDI+ ZDV+Zal, or ddl alone is superior to treatment with ZDV alone. Here, we apply MISTR to demonstrate how can we optimize treatment by estimating the HTE and choosing the most effective treatment per pa- tient. We conduct three analyses with ZDV the baseline treatment (W=0), comparing it with each of the other three alternatives (W=1): ZDV vs. ZDV+ddI (HIV 1). ZDV vs. ZDV+Zal (HIV 2), and ZDV vs. ddl (HIV 3). Kaplan-Meier curves for the HIV-1, HIV-2, and HIV-3 scenarios are shown in the top row of Figure S7. In Appendix A.2, we compare the results of MISTR and CSF using the original data.
8. Use Case with Confounding- - Illinois -Ilinois Unemployment Insurance Experiments
To demonstrate the applicability of MISTR-IV to real- world data with unobserved confounding in the IV setting, we use the Illinois Unemployment Insurance Experiments (1 (1984-1985) (Woodbury & Spiegelman, 1987) public data. The experiments were conducted by the Illinois Department of Employment Security. Eligible claimants were randomly assigned to one of three groups: job searcher incentive (JSI), hiring incentive (HI), or control. In the ISI group, claimants were offered $500 bonus if they secured full-time em- ployment within 11 weeks and maintained it for at least 4 months. In the HI group, the bonus goes to the employer on the same conditions. The control group was unaware of the experiment. Since group assignment is random, we analyze each experiment separately, i.c., HI or JSI vs. control.
In the original HIV data, the rate of loss of follow-up censor- ing was relatively low. In such cases, MISTR and CSF pre- sented comparable results. However, a higher censoring rate is to be expected in many settings, especially in cases with long follow-up period. To demonstrate such a case, we ap- ply additional censoring to the HIV dataset. For each patient i we sample whether to apply additional censoring from a Bernoulli distribution with parameter p 0.6 + 0.25 Z30;, where Z30; is another covariate that is available in the data and indicates whether a patient started taking ZDV prior to the assigned treatment. This covariate is only used for sampling the censoring and is unobserved during the train- ing stage. For the patients for whom additional censoring applied, we sample the censoring time from a uniform distribution ranging from to the minimum between the cur- rent last follow-up time and the first 20% of the follow-up period, i.e., Uniform[1,.... min(T:; 5 [)]- This simu- lates a scenario where censoring is more likely to occur during a specific part of the follow-up period. We repeat the sampling process 10 times and construct 10 datasets with a different set of censored observations in each dataset.
Although group assignment was random, participation in the experiment was voluntary, leading to confounding (Sant' Anna, 2021). We use random group assignment instrumental variable (IV) Z and voluntary participation in the experiment as the treatment assignment W. A summary of group characteristics included in our analysis is shown in Table S12. We note that the HI group had a lower num- ber of claimants who agreed to apply, which suggests that confounding effects in this group are likely to be stronger.
We apply MISTR and CSF to the updated datasets using the same parem--asameters detailed in A.2 and calculate the mean HTE, empirical SE, and mean Est SE for each sample. We then compare the current results of each method to its respective baseline, which was calculated using the original data and was similar for both methods. The mean and SE
Our goal is to estimate whether claimant incentives reduce the duration of unemployment in presence of confounding. The outcome is measured as the time between the unemploy- ment claim and the rehire date. Claimants who reached 26 weeks or the maximum duration allowed for unemployment benefits in their state were considered censored.
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Heterogencous Treatment Effect in Time-to-Event Outcomes: Harnessing Censored Data with Recursively Imputed Trees
We apply CSE, MISTR, and MISTR-IV detailed in A.3 to estimate the expected HTE for each claimant. comparison of results across the three approaches and their correlation values are shown in Figure S12. When confounding effects are present, and due to the relatively low censoring rate (38%-43%) in this setting, MISTR and CSF yield similar and biased estimates for the HTE, with stronger bias in the HI group, as expected. In contrast, MISTR-IV is capable of addressing these confounding effects through the of IV.
the data contain a low censoring rate; however, with a higher percentage of missing data, MISTR outperforms prior meth- ods. In addition, we applied MISTR-IV to the Illinois un- employment dataset demonstrating its applicability in this more challenging setting.
A Acknowledgments
We wish thank the reviewers for their helpful suggestions. T.M. is supported by the Israeli Council for Higher Edu- cation (Vatat) fellowship in data science via the Technion; T.M. and U.S. were supported by ISF grant 2456/23; M.G. was supported by ISF grant 767/21 and Malag competitive grant data science (DS).
As is often the case when evaluating methods for causal ef- feet estimation, lacking ground truth makes any solid model evaluation and comparison challenging. Indeed, censoring and hidden confounding make the problem especially chal- lenging in our setting. Nonetheless, qualitative comparisons between the top 10% and bottom 10% of the population expected to benefit the most and the least from the treat- ment, as rated by CSF, MISTR, and MISTR-IV are shown in Table $13. Such comparisons may reveal differences between the model results that may be explained by domain knowledge and will help to guide model selection. We see that MISTR-IV arrives quite distinct conclusions regarding the populations most and least benefiting from the treatment. We leave full interpretation of these results to domain experts in the field.
Data and Code Availability
Codes available at https://githii com/ The ACTG 175 HIV data and the Illinois Unemployment Insurance Experiment data are pub- licly available. The MIMIC dataset is accessible at https: / //physionee org/ conttet//mmiiiiiii. wwwcc.iv.... 0/ and subjected to credentials.
Impact Statement
9. Discussion
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
In this work we presented MISTR, a novel approach for estimating HTE and its variance in survival data. MISTR is based multiple imputations of event times for censored observations, eliminating the need to estimate the censoring probability function and expanding the range of cases that can be effectively addressed.
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neous treatment effect in observational survival analysis.
Journal of Biomedical Informatics, 107:103474, 2020.
ISSN 1532-0464. doi: https://doi.ogg/101001// 10i..//..0...
103474.
Zhu, and Kosorok, M. R. Recursively Imputed Survival Trees. Journal of the American Statistical Association,
107(497):331-344 March 2012. ISSN 0162- 1459, 1537-
274X. doi: 10.
12

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Heterogencous Treatment Effect in Time-to-Event Outcomes: Harnessing Censored Data with Recursively Imputed Trees
A. Experiments Details
In this appendix we provide further details about the experiments described Section 6.
A.1. Additional Details to Benchmark Cases
When applying MISTR (Alg. 1) and CSF to the benchmark cases (Section 6.1), i.e. settings 1-10, the imputation process involves STQ of Mirees = 1000 and Q = recursion steps, following the conclusions of (Zhu & Kosorok, 2012) regarding the trade-off between accuracy and computational burden. Then, we sample censored observation and calculate Tia A and aa Subsequently, we train A 200 causal forests, with 2000 trees in each forest, and infer (2) and (x)) for both the Random and Quantiles test sets. We compare these results of MISTR with the ones of CSF with 2000 trees. We estimate variance for each causal forest and the CSF using bags of size 8.
200 event times Ta for each
A.2. Addiotional Details to HIV Data Analysis
The hyper-paramtedd chosen for MISTR during the HIV data analysis of Section 7 were 2,500 imputations, a minimum of observed events per leaf, q = recursion steps, and = 200 outputs during the RIST training stage. Then, we used 2,000 trees in each causal forest, e = 18, and a minimal node size of 5, Additionally, CSF was applied with 2,000 trees, l 18, and minimal node size of 5.
A.3. Addiotional Details Illinois Unemployment Insurance Experiments
While conducting the analysis of Section 8, we applied MISTR and MISTR-IV, we use M 2500 ERTs with minimum of 18 observed events in each leaf, Q 3 imputation steps, and maximum time of lmaa 182 days. We use A 200 imputations, and thus A 200 causal forests and instrumental forests (in MISTR and MISTR-IV, respectively). For CSF, MISTR, and MISTR-IV, each forest consists of 2000 trees, with a minimum of 18 observations leaf. Finally, the horizon is set to 25 weeks, i.c., h = 175 days.
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Heterogencous Treatment Effect in Time-to-Event Outcomes: Harnessing Censored Data with Recursively Imputed Trees
B. Additional Experiments
In this Appendix, we present an additional discussion regarding the selection of two hyperparameters MISTR. The first the bag size E, which influences the variance calculation. The second the number of imputations required for accurate estimation of the HTE.
B.1. Sensitivity to Bag Size Parameter (?)
We utilize the MIMIC semi-simulated settings to demonstrate the sensitivity of the variance estimation the & parameter as follows. The effect of this parameter depends not solely on the parameter itself but on the ratio of & the number of trees in the causal forest (Sexton & Laake, 2009). Therefore, for this demonstration, we change the number of trees in each causal forest from 2000 to 300, and l ranges between and 100. The mean MSE and SE over 5 replications (of the five folds) are presented in Table S5.
In all settings MIMIC 1-5, the MSE and SE of MISTR are stable for different values of f, while the results of CSF fluctuate, with larger fluctuations as the percentage of censoring increases. addition, regardless of f, MISTR provides better results than that of CSF when censoring percentage is high, and comparable results as the percentage of censoring decreases.
The mean value of the total estimated variance is shown in Table S6 for different values of f. Table S6 includes comparison between the total estimated variance of CSF, and the total variance of MISTR, with separation to its between-imputation (BI) and within-imputation (WT) components. As expected, the BI component increases with the value of f, while the WI component decreases. Overall, the total estimated variance is minimal in the region of 8 < l S 20, for these settings with 300 trees in each forest. While the values may change, the U-shape of the total variance is expected in each setting. Thus, for different setting with a different number of trees, is possible conduct a that minimizes the total variance.
search to set the value of l
B.2. Sensitivity to the Number of Imputations
Lastly, we discuss the effect of the number of imputations required. We run MISTR once with A 2000 imputations and causal forests. Then, we sample without replacement out of the 2000 trained causal forest results, with & ranging between and SE of the MSE, in addition to the
2 and 200, and include them in the final HTE and variance estimation. The mean and SE of the estimation variance, are presented in S6 for each of the MIMIC 1-5 settings. The results are based on 50 repetitions for each value of k. As expected, for lower values of A, and for higher censoring percentage, the MSE and the estimation variance are both larger and with larger SE. As A increases, the MSE and estimated variance stabilize and we therefore recommend to use 2 100.
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Heterogencous Treatment Effect in Time-to-Event Outcomes: Harnessing Censored Data with Recursively Imputed Trees
C. Additional Results HIV Dataset Analysis
In this Appendix, we provide additional results from the analysis of the original HIV dataset (without the additional censoring presented in 7).
We demonstrate the applicability of MISTR using data from the AIDS Clinical Trials Group Protocol 175 (ACTG 175) (Hammer et al., 1996). ACTG 175 was double-blind, randomized controlled trial (RCT) that compared four treatment paths in adults infected with the human immunodeficiency virus (HIV) type I: monotherapy with zidovudine (ZID),, monotherapy with didanosine (ddI), combination therapy with ZDV and ddI, or combination therapy with ZDV and zalcitabine (Zal). The publicly available dataset includes 2,139 HIV-infected patients, randomized into four groups with the assigned treatments: ZIV, ZDV+ddl, ZDV+Zal and ddI. As in previous work (Tsiatis et al., 2008; Zhang et al., 2008; Lu et al., 2013; Fan et al., 2017; Cui et al., 2023), we use 12 baseline covariates, including 5 continuous baseline covariates: age, weight, Karnofsky score, CD4 cell count at baseline, and CD8 cell count at baseline, as well as 7 binary baseline covariates: gender, race, homosexual activity, history of intravenous drug use, hemophilia, antiretroviral history, and symptom status. An additional covariate available in the dataset, denoted as Z30, indicates whether a patient started taking ZDV prior to the initiation of the assigned treatment. This covariate is not included in the first analysis but is used in the second analysis for additional censoring sampling.
An event occurrence was defined as the first of either a decline in CD4 cells count, an event indicating AIDS progression, or death (Hammer et al., 1996). A reduction in CD4 cells count was considered as reduction of at least 50% from an average of two pre-treatment counts, with follow-up confirmation test after 3-21 days. Therefore, we consider outcomes of patients that occur within 28 days to be similar and conduct our analysis with resolution of months. This accounts for the possibility that diagnostic differences may arise from the logistical timing of follow-up meetings rather than any medical reasons. Population outcomes are described in Table S9.
As in Section 7, we conduct three analyses with ZDV as the baseline treatment (W=0), comparing it with each of the other three alternatives (W=1): ZDV vs. ZDV+ddI (HIV 1), ZDV vs. ZDV+Zal (HIV 2), and ZDV vs. ddl (HIV 3). Kaplan-Meier curves for the HIV-1, HIV-2, and HIV-3 scenarios are shown in the top row of Figure S7.
We apply MISTR to each of the cases, HIV-1, HIV-2, HIV-3, with Emax = 31 months, 2,500 imputations, minimum of 3 observed events per leaf, y = 3 recursion steps, and A = 200 outputs during the RIST training stage. Then, we use 2,000 trees in each causal forest, f = 18, and a minimal node size of 5. The horizon is set to h = 30 months. Additionally, we apply CSF with 2,000 trees, f = 18, and a minimal node size of 5. Finally, we estimate the expected HTE for the entire sample of 2,139 patients for each treatment alternative. We repeat the estimation using MISTR and CSF 10 times each. The mean HTE,, empirical SE, and mean estimated SE (Est SE) across 10 repetitions of all the samples are shown in the second, third, and bottom rows of Figure S7, respectively.
The HTEs for 1IV-1 treatment options were in the range of [0.71, 4.99] months and [0.59, 4.95 months estimated by MISTR and CSE, respectively. This means the 7VD+ddl is preferred over ZVD alone for all patients (no negative HTEs), albeit with varying benefits across patients. The Pearson correlation between the mean HTE of MISTR and CSF for HIV-1 was 0.995, i.e., both methods give similar HTE estimates, with smaller SE for MISTR, and with similar mean estimated SE. Similarly, for HIV-2, the HTES were in range of [1.00, 4.30) months and [0.91, 4.22] months for MISTR and CSF, respectively with correlation of 0.997; and for HIV-3 in range of [0.17, 3.89) months and (0.17, months for MISTR and CSF, respectively, with correlation of 0.996.
Choosing a treatment based the ATE alone, would have led to choosing treatment with ZVD+ddI for all patients. However, considering the HTES calculated by MISTR, we find that only for 1,209 patients the optimal treatment option is ZVD+ddl, while for 878 patients ZVD+Zal is preferred, and for 52 patients, ddl alone is preferred. Here, the optimal treatment for each patient is defined as the one for which the RMST HTE, calculated by MISTR, yields the highest RMST compared to the baseline treatment (ZDV), i.e., the highest among HIV- HIV-2, and HIV-3 scenarios. A sample of 12 patients in shown in Table S10, including 4 patients for whom the optimal treatment is either ZDV+ ZDV+Zal, or ddI. We did not find any patient in the given dataset for whom it is better to be treated by ZVD alone. The correlation between MISTR and CSF estimates is very high in this data.
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Heterogencous Treatment Effect in Time-to-Event Outcomes: Harnessing Censored Data with Recursively Imputed Trees
Table ST. Experiments settings. "AFT" denotes the accelerated failure time model. In all settings sample Perain 5000 and 5000 observations with independent baseline covariates X 1xp from standard uniform distribution. In Setting two additional convariates are used, only for sampling censoring time.
Setting
Parameter
Model
Value
log T 1.85 (.8I(X(1) 0.5) + 0.7.X 0.2X (3)
T
AFT
1
+ (047-0.4((X()) < 0.5) (,4 W +E
Ao W. = 2t expp exp 0.5 + 0.2X
C
Cox
+ +1.15 0.5/ < 0.5) 0.3X 0.5
W]
T
Cox
\r = 0.5t exp(X + (-0.5 (2)) W
2
C
Uniform
(0,3)
T
Poisson
Ar = (2) + X + 6 2(x 0.3) W
3
C
Poisson
Ao 12 + log(1 1 + exp (x
T
Poisson
À7 X(2) ((3) max X(1) 0.3)
4
C
Poisson
Àc = 14 log exp (3()))
T
Poisson
dT X + +((3) + +6 2 0.3)
5
for 0.6,
C
g(s ~ ((s~U(0,1) IJ)
((5)
1 + T(X < 0.5) otherwise.
T
Poisson
AT X + +(3) + 6 -
0.3) W
6
C
Poisson
Ac log(1 cxp(2X x(a)))
T
Poisson
Ay X +X(3) + ((X 0.3)
7
C
Poisson
cc = = 3 4X + 22(77)
T
Poisson
Ar = X + X(3) + + 2(X (1) 0.3) ..3..W
8
C
Poisson
Àc =
log T 0.3 - (x < 0.5) 0.5X + 0.2X
T
AFT
9
(1 (x8 0.5) 0.8X W +e
Ac = 21 exp 0.9 + 2X 0.5
(2) + 2 (3)
C
Cox
+ (1.15 + 0.5I < 0.5) - 0.3 x W]
T
Cox
Ar \T X) = 0.5l exp X (() + (-0.5 + X(2)) W for < 0.1,
10
00
C
g(s ~ Uniform(0, 1))
g(8)
c~ Uniform 00, 0.05]
otherwise.
Table Definitions Emax, h, and resulting censoring percentage for settings 1-10.
Setting
tmaz
h
Censoring [%]
0.8
0.7
15.3
2
0.8
0.7
29.6
3
12
11
11.3
4
4
3
21.0
5
7
6
73.4
6
7
6
76.2
7
7
74.0
8
6
92.7
9
0.8
0.7
92.1
10
0.8
0.7
69.9
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Heterogencous Treatment Effect in Time-to-Event Outcomes: Harnessing Censored Data with Recursively Imputed Trees
Table S.3. Mcan and SEM of the MSE for the HTE F();) estimated by MISTR, CSF (Cui et al., 2023), X-learner (Bo et al., 2024), and
T-leamnr (Bo et al., 2024) on Quantiles test set and Random test set. Results are based on 100 replications and are multiplied by for readability. "obs." denotes observations.
Quantiles (21 obs.)
Random (5000 obs.)
T-learner
X-learner
CSF
MISTR
T-learner Mean
X-learner
CSF
MISTR
Setting
Mean
SEM
Mean SEM
Mean
SEM
Mean
SEM
SEM
Mean
SEM
Mean
SEM
Mean
SEM
0.191
0.009
0.157
0.007
0.036
0.002
0.036
0.002
0,212
0.003
0.158
0.003
0.044
0.002
0.044
0.002
0.337
0.015
0,282
0.014
0.058
0.004
0.058
0.004
0.368
0.005
0.275
0.005
0.065
0.004
0.065
0.004
27.901 1.169
23.938
1.183
7.036
0.370
7.016
0.383
26.045
0.308
19,890
0.305
4.839
0.216
4,880
0.223
4.449
0.200
3.397
0.179
0.742
0.042
0.496
0.032
0.048 4.986
3.582
0.045
0.971
0.039
0.674
0.030
5.713 0.235
5.158
0.239
1.464
0.070
1.350
0.062
5.108
0.063
4.018
0.059
0.937
0.042
0.840
0.040
5.270 0.247
4.693
0.242
1.672
0.085
1.314
0.062
5.159
0.068
4.046
0.064
0.960
0.040
0,817
0.034
6,900 0.318
6,518
0.348
1.824
0.101
1.560
0.080
0.079 6.119
4.814
0.072
1.025
0.042
0.926
0.038
4.978 0.279
4.669
0.292
1.289
0.094
0.851
0.039
4.609
0.071
3.773
0.065
0.924
0.043
0.606
0.024
0.384 0.041
0.357
0.039
0.048
0.003
0.043
0.002
0.351
0.035
0.326
0.035
0.060
0.002
0.062
0.002
0.640 0.028
0.027 0.535
0.193
0.015
0.146
0.008
0.675
0.010
0.539
0.010
0.212
0.009
0.162
0.007
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Heterogencous Treatment Effect in Time-to-Event Outcomes: Harnessing Censored Data with Recursively Imputed Trees
Table S4. MIMIC dataset. Summary of baseline covariates of the overall sample and by each fold.
Overall
Fold
Fold
Fold
4
5
25159
5031
5031
5031
5031
5035
Night
No
11597 (46.09)
2341 (46.53)
2275 (2))
2325 (46.21)
2297 (45.66)
2359 (46.83)
admissiin, a (1)
Yes
13562 (53.91)
2690 (53.47)
2756 (54.78)
2706 (53.79)
2734 (54.34)
2676 (53.15)
Sex, n (1)
Female
12286 (48.83)
2449 (48.66)
2435 (48.40)
2472 (49.14)
(0)
2495 (49.55)
Male
(7)
2582 (51.32)
2596 (0)
2559 (50.86)
2596 (51 60)
2540 (50.45)
Direct
No
22388 (88.
4448 (88.41)
4488 (89.21)
4487 (89.19)
4478 (89.01)
4487 (89.12)
emergency. (9)
Yes
2771 (11.01)
583 (11.59
543 (10.79)
544 (10.81)
553 (10.99)
548 (10.88)
Previous admi ssion
No
23128 (91.93)
4599 (91,41)
4625 (9193)
4683 (92.49)
4604 (1))))
4647 (92.29)
this month, (%)
Yes
2031 (8.07)
432 (8.59)
406 (8.07)
378 (7.51)
427 (8.49)
388 (7.71)
Admiission age
mean (SD)
64.27 (17.88)
63.88(17.94) (17.8
63.88 (17.80)
(17.94)
Insurance, (3)
Medicaid
(5.66)
293 (5.82)
(5.82)
2KT (5.59)
284 (5.65)
(3.400
Medicare
(42.4)))
2112 (41 98)
2164 (43.01)
2070 (41 .14)
2128 (42. 30)
2130 (42, 30)
Other
2626 (52.22))
(52.20)
(51.16)
2680 (53.27)
2619 (52.06)
2633 (52.29)
Divorved
2041 (8.11)
409 (8.13)
429 (8.53)
(8.31)
398 (7.91)
387 (7.6)
Marital status, (%)
Married
11283 (44.85)
2248 (44.68)
2210 (43.93)
2234 40)
2297 (45.66)
2294 (45.56)
Single
8413 (33.44)
1673 (33,25)
1711 (34.01)
1692 (33.63)
1668 (33.15)
1669 (33.15)
Widowed
3422 (13.60)
701 (13.93)
681 (13.54)
687 (13.66)
668 (13,28)
685 (13.60)
Race, (5r
Asian
(4.114
201 (4.00)
192 (5.82)
217 (4.31)
199 (3.36)
225 (4.47)
Black
3542 (14.08)
656 (3004)
696 (13.83)
741 (14,73)
713 (14.17)
736 (14.62)
Hispanic
1326 (5.27)
(4.83)
266 (5.29)
260 (5.17)
(5.39)
286 (5.68)
Other
(6.64)
340 (6.76)
357 (7.10)
320 (6.36)
307 (6.10)
346 (6.87)
White
17587 (69.90)
3591 (71.38)
3520 (69.97)
3493 (69.43)
3541 (70.38)
34-2 (68.36) 36)
Admissions number, (%)
(6168)61.88
3046 (60.54)
3079 (61,20)
3125 (62.11)
3097 (61.56
3121 (61.99)
4116 416.366636
832 (16.54)
821 (16.32)
829 (16.48)
807 (16.04)
827 (16.43)
3+
5575 (22.16)
1153 (22.92)
[131 (22.48)
1077 (21.41)
1127 (22.40)
1087 (21.59)
Anion gap, n (%)
Abnormal
2297 (9.13)
477 (9:48)
474 (9.42)
435 (8.65)
465 (9.24)
446 (8.86)
Normal
(90.87)
4554 (90.52)
(90.58)
4596 (91.35)
4566 (90.76)
(91.14)
Bicarhomate, (%)
Abnormal
(24.35)
1254 (24.933
1217722.199 19)
1194 (23.73)
1242 (24.69)
(24.211
Normal
19033 (75.65)
3777 (75.07)
3814 (75.81)
3837 (76.27)
3789 (75.31)
3816 (75.79)
Calcium lotal,
Abnormal
7320 (29.09)
1475 (29.32)
1472 (2)
1436 (28.54)
1435 (28.52)
1502 (29.83)
Normal
17839 (70.91)
3556 (70.68)
3559 (70.74)
3595 (71 46)
3596 (7).48)
3533 (70.
Chloride,
Abnormal
4846 (19.26)
1007 (20.02)
935 (18.58)
981 (19.30)
(19.20)
957
Normal
(80.7780.74)
4024 (79.9%)
4096 (81.42)
4050 (80.50)
4065 (80.80)
4U78 (80.99)
Creatimine, n (%)
Abnormal
7117 (28.29)
1456 (28.943
(28.22)
1406 (27.95)
(27.77)
1408 (27.96)
Normal
18042 (71.71)
3575 (71 .666)
3581 18))
3625 (72.05)
3634 (72.23)
3627 (72.04)
Glucuse, n (%)
Abuormal
3278(65,16 (65.6)65.25)
(65.16)
3289 (65.37)
3269 (64.98)
3293 (95
3285 (65.24)
Normal
8743 (34.75)
1753 (34.84)
1742 (34.63)
1762 (35.02)
1736 (34.51)
1750 (34.76)
Magnesium, n (%)
Abnormal
(2.71
428 (8.51)
446 (8.87)
453 (9.00)
425 (8:43)
465 (9.24)
Normal
22942 (91.19)
4603 (91.49)
(91.13)
4578 (91.00)
4606 (91.55)
4570 (90.76)
Phosphaie, n (%)
Abnormal
6956 (27.65)
1421 (28.24)
1354 (26.91)
1367 (27.17)
1418 19)
1396 (27. 73)
Normal
18203 (72.35)
(71.76)
(73.09)
3664 (72.83)
(71.8)
(72.27)
Potassium, n (5)
Abnormal
2T05 (8:37)
418 (8.31)
431 (8.57)
428 (8.51)
(8.27
(8.1)818
Noemal
4613 (91.66)
(91.69)
4600 (91.43)
4603 (91.49)
(91.7)
4623 (91.82)
Sodium,
Abnormal
(11.6))
(12.42)2)
553 (10.99)
375
59% (11,99)
591 (TT.74)
Normal
22217 (88.31)
4406 (87 58)
4478 (89.01)
4456 (88.57)
4433 (88. 11)
4444 (88. 26)
Usea mitrogen, a (%)
Abnormal
10025 (39.85)
2038 (40.51)
1984 (39.44)
1977 (39.30)
(40.91)
Normal
15134 (50.15)
2993 (59.49)
3065 (60.92)
3047 (60.56)
3054 (60.70)
2975 (59.09)
Hematoenii n (%)
Abnormal
(781166881)
3506 (69.69)
(68.81)
3455 (68.67)
3458 73)
$429 (68. 10]
Normal
7848 (31.
1525 (30.31)
1568 (31.17)
1576 (31.33)
1573 (31 27)
1606 (31,90)
Homoglobin. n (%)
Abaormal
18345 (72.92)
(73,44)
3684 (73.23)
3670 (72.955
3668 (72.91)
3629 (72.08)
Normal
6814 (27.08)
1337 (26.58)
1347 (26.77)
1361 (27
1363 (27.09)
1406 (27 92)
MCH, (86)
Abnormal
6555 (26.05)
1335 (26.54)
1286 (25. 56)
1367 (27.17)
1262 (25.00 08)
1305 (25.92)
Normal
(73.95)
3696 (73.46)
(74.44)
3664 (72.83)
3769 (74.92)
3730 (74.08)
MCTIC, (18)
Abnormal
7756 (30.83)
1602 (31 84)
1581 (31.43)
1546 (30.73)
1 (29.82)
1527 (30.33)
Normal
17403 17)
3429 (68. 16)
3450 (68.57) 57)
3485 (69.27)
3531 (70. 18)
3508 (69 67)
MCV, (8)
Abnormal
5100 (20,27)
1052 (20.91)
99% (19.80)
1053 (20.95)
969 (19.26)
1030 (20,440040 (20.46)
Normal
20059 (79.73)
3979 (79.09)
4035 (80.20)
3978 (79.07)
4062 (80.74)
4005 (79.54)
Platolet count, n
Abnormal
7276 (28.92)
1496 (29.74)
[465 (29.08)
1503 (29 87)
[41T (28.05)
1403 (27.86)
Normal
17883 (71.08
3535 (70.26)
3568 (70.92)
3528 (70.13)
3620 (71.5)
3632 (72.14)
RDW. (%)
Abnormal
7275 (28.92)
1461 (29.053)
(29.28)
1447 (28.76)
(28.76)
1447 (28.74)
Normal
17884 (71 (88
3570 (70.96)
3558 (70.72)
3584 (71.24)
3584 (71 24)
3588 (71.26)
Red blood
Abnormal
19161 (6)
3843 (76.39)
3837 (76.27)
3856 (76.64)
3837 (76
3758 (75.23)
cells, (%)
Normal
5998 (23,84)
1188 (23.6) (23)61)
1194 (23.73)
1175 (23. (3)
1194 (23.73)
1247 (24.77)
White blood
Abuormal
10003 (39.76)
2031 (40.37)
1987 (39.50)
1994 (39.63)
1945 (38.66)
2036 (40.64)
cells,
Normal
15156 (60.24)
3000 (59.63)
3044 (60.50)
3037 (60.37)
3086 (61.34)
2989 (59.36)
18

---

Heterogencous Treatment Effect in Time-to-Event Outcomes: Harnessing Censored Data with Recursively Imputed Trees
Table S5. Comparison of the MSE of the estimated HTE F(X)) of MISTR with CSF (Cui et al., 2023) MIMIC test set. Different values are considered, with a total 300 trees the causal forests. Results are based on replications and are multiplicd by 100 for readability.
MIMIC 1
MIMIC
MIMIC
MIMIC 4
MIMIC
CSF
MISTR Mean
CSF
MISTR Mean
CSF
MISTR Mean
CSF
MISTR Mean SE
CSF
MISTR
Mean
SE
SE
Mean
SE
SE
Mean
SE
SE
Mean
SE
Mean SE
Mean
SE
e
2
8.47
2.36
6.25
1.82
6.97
2.14
5.88
1.67
5.81
1.18
5.19
1.13
5,47
0.55
5.02
0.81
5.69
1.60
5.48
1.46
9.39
3.59
6.25
1.86
7.02
2.34
5.89
1.66
5.94
0.83
5.20
1.13
5.57
0.84
5.03
0.81
5.83
1.50
5.50
1.45
8.92
3.53
6.22
1.81
7.01
2.05
5.87
1.66
6.28
1.00
5.20
1.13
5.52
0.78
5.04
0.80
5.65
1.55
5.49
1.49
8.63
2.76
6.22
1.83
7.24
2.24
5.88
1.67
6.25
0.87
5.20
1.11
5.54
0.70
5.01
0.83
5.70
1.18
5.50
1.47
10
8.63
2.43
6.22
1.83
7.28
2.14
5.91
1.67
5.97
0.85
5.21
1.15
6.04
0.97
5,04
0.81
6.02
1.35
5.51
1.47
20
9.06
3.12
6.22
1.83
8.07
2.63
5.89
1.68
6.03
0.95
5.20
1.15
6.14
1.02
5.03
0.85
5.54
1.49
5.50
1.44
50
9.69
3.37
6.26
1.84
7.74
2.02
5.88
1.64
6.71
1.39
5.14
1.09
6.30
1.18
5.03
0.82
5.87
1.88
5.50
1.46
100
10.86
3.52
6.25
1.80
7.96
1.90
5.99
1.74
6.83
0.84
5.21
1.11
6.26
1.26
5.00
0.85
6.48
1.54
5.55
1.45
Table S6. Comparison of the MSE of the estimated HIT F(X.) of MISTR with CSF (Cui al., 2023) MIMIC test Different values are considered, with a total of 300 trees in the causal forests. Results are based on replications and are multiplied by 100 for readability. BI denotes "between imputations", and WI denotes "within imputation".
MIMIC
MIMIC 2
MIMIC
MIMIC
MIMIC 5
CSF Total
MISTR BI
CSF
MISTR BI
CSF Total
MISTR BI
CSF Total
MISTR BI
CSF
MISTR
Total
WI
Total
Total
WI
Total
WI
Total
WI
Total
Total
BI
WI
2
7.62
4.11
0.88
3.23
5.19
3.90
0.74
3.16
4.52
3.69
0.56
3.12
3.44
3.38
0.39
2.99
2.96
3.21
0.27
2.94
4
5.53
3.66
0.91
2.74
4.33
3.44
0.76
2.68
4.40
3.30
0.57
2.73
3.37
2.96
0.40
2.56
2.89
2.74
0.29
2.45
6
5.72
3.49
0.92
2.57
4.07
3.29
0.75
3.15
2.54
2.79
2.85
0.42 2.42
2.57
2.63
0.31
8
5.09
3.48
0.95
2.52
4.71
3.25
0.79
2.45
3.11
3.11
0.62
2.49
2.35
2.80
0.45
2.35
2.52
2.57
0.32
2.25
10
4.95
3.45
0.97
2.48
4.32
3.25
0.81
2.43
2.88
3,05
0.61
2.43
2.82
2.75
0.44 2.31
1.96
2.52
0.33 2.19
20
5.27
3.43
1.03
2.39
4.04
3.23
0.90
2.33
3.27
3.11
0.71
2.40
2.85
2.74
0.54 2.20
2.11
2.52
0.40
2.13
3.63
3.46
1.26
2.19
3.02
1.08
3.16
2.18
1.87
2.82
0.73
2.09
1.99
2.64
0.60
100
4.20
3.75
1.71
2.03
2.77
3.47
1.52
1.94
2.19
3.31
1.34
1.96
3.49
3.01
1.18
1.83
2.01
2,79
0.99
1.79
Table S7. Survival and censoring time distributions.
Setting
dr
A
W
Type 200
AT 2X(1) + + -
Ac = 7
W = 0.50 + 0.5 0.2N(00
Type 200-a
AT 2X(1) X(2) +0++++(++ -W3)
AC 7
W 0.5U + 0.42 +.247+..200...1
Type 200-b
ÀT 2X(1) X22 +
W
AC = 7
W 0.50/ +
Type 201
AT 2X(1) X12) +20+4+2(X05
W.3)W
AC
W 0.5U + 0.35Z 12655++0.200 0.2N
Type 202
AT 2X(1) + X22 + 3770+++++3300
0.3)W
AC = 7
W = 0.5U + 0.352 10.35++000 0.2N
Type 203
ÀT 2X(1) X 2/ +5+2 -333
AC 6
W 0.50 +.52 0.22N(001)
Type 204
AT 2X(1) X(2) + +6++++( 0.3)W
AC = 4
W 0.50 + 0.5 0.2N(00
Type 204-a
ÀT 2X(1) -X2) + +6+++ 0.3)W
ÀC
W 0.50 0.4 0.2N
Type 204-b
ÀT = 2X(1) +X + 20 +6+2( 0.3)
AO = 4
W = 0.50 + 0.3 1 0.2N
19

---

Heterogencous Treatment Effect in Time-to-Event Outcomes: Harnessing Censored Data with Recursively Imputed Trees
Table S8. Comparison of the MSE of the estimated HTB ((X,) using CSF, MISTR, IPCW-IV, and MISTR-IV on a random test set with
Freest 5000 observations for case. Results are based 100 replications. Results are multiplied by readability factor 100.
CSF
MISTR
IPCW-IV std
MISTR-IV Mean
Type
Mean
std
Mean
200
6.71
2.77
6.55
11.07
4.08
3,49
1.57
200a
8.90
3.42
8.74
3.44
16.00
6.14
4.50
200b
11.18
3.72
11.08
3.68
29.63
12.19
7.37
3.39
201
10.67
3.59
10.48
3.54
21.05
7.95
6.08
2.51
202
11.73
3.67
11.59
3.69
18.57
5.92
5.95
2.58
203
5.40
1.81
4.59
1.63
12.85
5.21
3.21
1.36
204
4.16
2.19
2.91
1.85
16.53
10.81
2.23
1.52
204a
5.12
2.44
3.82
2.19
20.73
10.83
2.96
1.82
204b
6.10
2.71
4.79
2.36
49.61
39.74
3.89
2.33
Table HIV RCT patients characteristics.
Overall
ZDV
ZDV+Zal
ZDV+ddl
ddl
n
2139
532
524
561
Age (Yaars)
mean (SD)
35.2 (8.7)
35.2 (8.9)
35.4 (8.8)
35.2 (8.7)
35.1 (8.5)
Weight (Kg)
mean (SD)
75.1 (13.3)
(13.2)
74.7 (13.2)
(1.96)
(139
Karnolsky
(SD)
95.4 (5.9)
95.4
95.7 (S.9)
95.5 (5.8)
95.1 (5.9)
CD4 (cells/mm)
mean (SD)
350.5 (118.6)
353.2 (114.1)
352,8 (115.5)
348.7 (130.2)
347.5 (114.4)
CD8 (cells/ m/m3m)
mean (SD)
986.6 (480.2)
987.2 (475.2)
984.1 (452.8)
1004.3 (488.0)
971.9 (502.7)
Gender, n (%)
0 (Female)
368 (17.2)
(18.8)
89 (17.0)
88 (16.9)
(16.2)
(Male)
(82.8)
432 (81.2)
435 (83.0)
434 (83.1)
470 (83.8)
Homosexual activity, n (%)
0 (No)
725 (33.9)
191 (35.9)
176 (33.6)
176 (33.7)
182 (32.4)
1 (Yes)
1414 (66.1)
341 (64.1)
348 (66.4)
346 (66.3)
379 (67.6)
Race, (56)
(White)
1322 (71.2)
376
(71.4)
384 (73.6)
388 (69.2)
(Non-white)
617 (28.8)
(29.3)
150 (28.6)
(26.4)
(30.8)
Symptoms, n (%)
0 (No)
1769 (82.7)
443 (83.3)
435 (83.0)
426 (81.6)
465 (82.9)
1 (Yes)
370 (17.3)
89 (16.7)
89 (17.0)
96 (18.4)
96 (17.1)
History of drug use,
0 (No)
1858 (86.9)
469 (88.2)
448 (85.5)
449 (86.0)
492 (87.7)
(Yes)
281 (13.1)
63 (11.8)
76 (14.5)
73 (14.0)
69 (12.3)
Hemophilia, n (%)
(No)
1959 (91.6)
490 (92.1)
478
479 (91.8)
(91.3)
I (Yes)
180 (8.4)
42 (7.9)
46 (8.8)
(8.2)
49 (8.7)
Antiretroviral history, (%)
0 (Naive)
886 (41.4)
223 (41.9)
(40.5)
213 (40.8)
238 (42.4)
1 (Experienced)
1253 (58.6)
309 (58.1)
312 (59.5)
309 (59.2)
323 (57.6)
ZVD -30, n (%)
0 (No)
962 (45.0)
241 (45.3)
230 (43.9)
(44.8)
257 (45.8)
(Yes)
1177 (55.0)
291 (54.7)
294 (56.1)
288 (55.2)
304 (54.2)
Last Follow-up (Days)
mean (SD)
879,1 (292.3)
801.2 (326.9)
905,8 (274.9)
916.2 (264.2)
893.5 (285.3)
Last Follow-up (Months)
mean (SD)
30.9 (10.4)
(1.9
32.2 (9.4)
31.4
Observed
(No)
1618 (75.6)
351 (66.0)
415 (79.2)
419 (80.3)
433 (77.2)
Event, n (%)
(Yes)
521 (24.4)
181 (34.0)
109 (20.8)
103 (19.7)
128 (22.8)
20

---

Heterogencous Treatment Effect in Time-to-Event Outcomes: Harnessing Censored Data with Recursively Imputed Trees
Table S10. HIV original dataset sample- patients HTE (months): sample of 12 patients is presented. The first rows represent patients who will benefit the most from the ZDV+ddl treatment, the 4 rows in the middle represent patients who will benefit more from the ZDl+Zal treatment, and the last 4 rows represent patients who will benefit more from the ddl treatment. When the percentage of missing data is low, as in this case, the mean estimates from CSF and MISTR over 10 repetitions are similar, with MISTR showing smaller SE. The mean estimated SE similar both methods.
ZDV vs. ZDV+ddI
ZDV vs. ZDV+Zal
ZDV ddl
CSF
MISTR
CSF
MISTR
CSF
MISTR Mean
Patient ID
Mean HTE
Mean
Mean HTE
Mean Est SE
Mean HTE
Mean
Mean
Mean
Mean HTE
Mean
Mean
SE
SE
SE
SE
Est SE
Est SE
HTE
Est
SE
Est SE
HTE
Est SE
140125
2.91
0.91
0.13
2.91
0.96
0.01
2.74
0.88
0.12
2.74
0.85
0.02
2.67
0.77
0.10
2.68
0.86
0.01
I 110581
3.12
1.28
0.25
3.15
1.25
0.02
2.90
1.35
0.15
3.02
1.39
0.02
1.59
1.34
0.22
1.73
1.37
0.02
10924
3.03
0.72
0.12
3.06
0.71
0.01
2.10
0.85
0.10
2.14
0.84
0.01
2.38
0.75
0.08
2.26
0.79
0.01
940521
1.99
0.71
0.07
2.11
0.76
0.02
1.74
0.68
0.10
1.82
0.66
0.01
1.54
0.82
0.11
1.59
0.80
0.01
81144
3.49
0.95
0.11
3.56
0.92
0.01
3.89
0.94
0.12
3.77
0.89
0.01
3.07
0.93
0.14
3.02
0.91
0.01
11449
0.72
0.51
0.10
0.88
0.49
0.01
1.03
0.36
0.07
1.07
0.42
0.01
0.51
0.51
0.06
0,64
0.47
0.01
990071
3.44
1.32
0.12
3.77
1.39
0.02
3,96
1.50
0.11
4.15
1.40
0.02
3.07
1.17
0.21
3.14
1.23
0.01
60617
1.16
0.16
2.42
1.23
0.02
3.13
1.26
0.19
3.17
1.32
0.02
1.82
0.24
1.83
1.33
0.02
150272
2.72
0.88
0.19
2.68
0.87
0.01
2.69
0.80
0.16
2.72
0.79
0.01
3.03
0.76
0.11
2.97
0.78
0.01
220429
3.24
0.91
0.18
3.20
0.90
0.01
2.84
0.84
0.17
2.89
0.87
0.01
3.43
0.78
0.12
3.36
0.83
0.01
211229
1.65
0.61
0.10
1.63
0.62
0.01
1.52
0.71
0.11
1.53
0.75
0.02
1.96
0.62
0.06
1.84
0.60
0.01
270842
0.79
0.10
2.86
0.79
0.01
2.09
0.76
2.23
0.77
0.01
2.92
0.08
2.97
0.77
0.01
Table S11. HIV dataset with higher percentage of censoring hased on an unobserved covariate. Mean and SE of the HTE (months) of the 12 patients of Table $10. Results calculated over repetitions. The estimates of MISTR closer to the original data baseline than the mean estimates of CSF
ZDV vs. ZDV+ddl
ZDV vs. ZDV+Zal
ZDV vs. ddl
CSF
MISTR
CSF
MISTR
CSF
MISTR Mean
Patient ID
Mean HTE
Mean
Mean
Mean
Mcan
Mean Est SE
Mean HTE
Mcan
Mcan
Mean Est SE
Mean
SE
SE
SE
SE
SE
SE
Est SE
HTE
Est SE
HTE
Est SE
HTE
HTE
Est SE
140125
2.25
1.49
1.33
2.84
1.10
1.29
1.50
1.31
1.22
1.86
1.03
0.86
2.42
1.35
1.48
2.30
1.06
0.83
110581
2.34
1.63
1.36
2.73
1.42
1.57
1.18
1.50
0.86
1.82
1.46
0.62
0.46
1.85
1.39
1.54
1.62
0.76
10924
2.98
1.20
0.82
3.01
1.21
0.92
1.27
1.30
0.97
1.91
1.08
0.61
2.06
1.18
1.08
2.24
1.22
0.90
940521
1.14
0.78
0.65
1.80
0.93
0.76
1.38
1.07
0.47
1.79
1.05
0.47
0.93
1.07
0.71
1.61
1.20
0.48
81144
2.39
1.18
0.65
2.87
1.09
0.74
3.11
1.47
1.06
2.66
1.09
0.78
2.43
1.31
0.58
2.39
1.08
0.53
11449
0.67
0.67
0.76
1.49
0.82
0.77
0.88
0.69
0.47
1.28
0.81
0.49
0.37
0.77
0.53
1.07
0.84
0.40
990071
2.25
1.62
1.29
2.73
1.42
1.51
1.68
1.41
0.87
2.08
1.37
0.52
2.51
1.70
1.16
2.14
1.46
0.75
60617
1.48
1.53
1.42
2.61
1.35
1.51
1.37
1.55
1.01
1.86
1.38
0.75
0.98
1.66
1.57
1.80
1.44
0.98
T50272
2.05
1.14
0.74
2.60
1.11
0.95
2.08
1.30
1.20
2.00
1.08
0.87
2.00
1.03
0.71
2.07
1.08
0.63
220429
2.58
1.09
0.70
2.91
1.17
0.97
2.37
1.39
1.56
2.24
1.17
0.96
2.17
1.17
0.82
2.28
1.14
0.73
211229
1.43
0.74
0.45
1.88
0.88
0.47
0.97
0.94
0.48
1.51
0.86
0.40
1.18
0.80
0.52
1.46
0.90
0.44
270842
2.28
1.14
0.75
2.56
1.02
1.02
1.37
1.00
0.90
1.74
0.99
0.87
1.88
1.07
0.97
1.94
0.98
0.69
21

---

Heterogencous Treatment Effect in Time-to-Event Outcomes: Harnessing Censored Data with Recursively Imputed Trees
Table S12. Illinois unemployment insurance experiments claimant characteristics. "HI" denoted hiring incentive group. "JSI" denotes job search incentive group.
Overall
Control
HI
JSI
12057
3932
3953
4172
Age (Years)
(SD)
33.0 (8.9)
33.0 (8.9)
33.1 (9.0)
32,9 (8.9)
Male, n (%)
No
(45.0)
1783 (45.3)
1826 (46.2)
1822 (43.7)
Yes
6626 (55.0)
2149 (54.7)
2127 (53.8)
2350 (56.3)
Race Black, n (%)
No
8934 (74.1)
2865 (72.9)
2942 (74.4)
3127 (75.0)
Yes
3123 (25.9)
1067 (27.1)
1011 (25.6)
1045 (25.0)
Race White, (%)
No
4299 (35.7)
1447 (36.8)
1394 (35.3)
1458 (34.9)
Yes
7758 (64.3)
2485 (63.2)
2559 (64.7)
(65.1)
Race Other, n (%)
No
10881 (90.2)
3552 (90.3)
3570 (90.3)
3759 (90.1)
Yes
1176 (9.8)
380 (9.7)
383 (9.7)
413 (9.9)
IV group, (%)
Control
3932 (32.6)
3932 (100.0)
Experiment
8125 (67.4)
3953 (100.0)
4172 (100.0)
Treatment Assignment, (%)
No
5963 (49.5)
3932 (100.0)
1374 (34.8)
657 (15.7)
(Voluntarily)
Yes
6094 (50.5)
2579 (65.2)
3515 (84.3)
Observed Event, (%)
No
4911 (40.7)
1690 (43.0)
1612 (40.8)
1609 (38.6)
Yes
7146 (59.3)
(57.0)
2341 (59.2)
(61.4)
Last Follow-up time
mean (SD)
137.5 (92.9)
143.1 (92.0)
137.2 (93.9)
132.5 (92.6)
Table S13. Qualitative comparison between the top 10% and bottom 10% of the population expected benefit the most and the least from the treatment, as rated by CSF, MISTR, and MISTR-IV.
Top 10%
Bottom 10%
Experiment
Covariate
CSF
MISTR MISTR-IV
CSF
MISTR MISTR-IV
Median Age (Years)
23.0
23.0
23.0
46.0
47.0
39.0
JSIE
Male (%)
56.2
58.0
66.7
80.9
80.5
75.1
White (%)
100.0
99.5
88.0
46.4
45.0
47.5
Median Age (Years)
34.0
34.0
34.0
36.0
36.0
42.0
HIE
Male (%)
0.0
0.0
0.0
100.0
96.7
79.5
White (%)
95.6
96.6
73.5
56.1
46.1
72.9
22

---

Heterogencous Treatment Effect in Time-to-Event Outcomes: Harnessing Censored Data with Recursively Imputed Trees
Type 1 15.3% censaring at
Type 2 29.6% censoring at
10a
10*
10a
W-1,
10a
W-O, D-1
101
W-O, Dw0
10a
1
10a
0.2
0.6
0.8
1.0
1.2
1.6
18
2.0
2.2
2.4
2.0
2.8
3.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.6
1.8
2.0
2.2
2.4
2.6
Type 3- 11.3% censoring h
Type 4-2.0% censoring at
10a
10
104
10a
10
10-1
10
10-2
110
11.0
12
18.0
19.0
20.0
0.0
1.0
2.0
3.0
4.0
5.0
6.0
8.0
9.0
10.0
1
Type 5 73.4% censoring h
Type 6 76.2% censoring at
10
10a
10a
10A
0.0
1.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
0.0
1.0
2.0
3.0
4.0
5.0
6.0
8.0
9.0
10.0
Type 7 74.0% censoring at h
Type 8 92.7% censoring at
10
10a
10a
10a
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
Type 9 92.1% censoring at h
Type 10 69.9% censoring at h
10a
10a
10
10a
10-1
10
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0.1
0.3
0.5
06
0.7
108
02
16
11
12
1.3
1.5
Figure $1. Observed events and censoring for experiments scenarios 1-10.
23

---

Heterogencous Treatment Effect in Time-to-Event Outcomes: Harnessing Censored Data with Recursively Imputed Trees
Setting Estimattd Values
Setting Estimated
Sctting Estimated Values
Setting Estimatce Values
Setting Estiaated Volues
0.10
10-
0.08
Oun
0.05
Dars
C2F
Oun Curs
4
Oun
0.00
HTE
5.1
HTE
0.04
Tue
Tuee
-0.5
Oy
-02
0.00
-010
0.50
X. percentile
X-i percentile
percentile
percentile
X.i percentile
Setting E-MSE
Secting E-MSE
Setting E-MSE
Setting E-MSE
Setting E-MSE
0.20
CS7
Ours
0.15
Oun
cS
60
Ours
1.50
1.21
C57
Out
CSF
Out
E-MSE
0.15
E.MSE
0.10
E-MEE
40
SMSE
1.00
EMSE
10
0.05
0.25
0:50
0.25
0.75
1.00
0.50 percentile
0.75
1000
0:50
0.75
1.00
X. porcenttie
percantile
X.1 porcentile
Setting Estimated Values
Setting Estimeted Velues
Setting Estimated Values
Setting Estimated Values
Setting Estimated Values
0.10
02
0.02
0.2
0.2
HTE
0.0
HTF
0.00
0.00
Thee
-0.02
Thee
-0.1
Our
Dars EV
0.2
Ours
Okr
-0.10
Ours
0.50
0.25
0.50
0.75
1.00
0.50
0.75
1.00
0.25
0.50
6.75
1.00
0:00
0.25
0.50
0.75
0.25
0.50
0.75
1.00
Xii percentile
X-i percentile
percentile
Xii percentile
X.i percentile
Setting E-MSE
Secting 7 E-MSE
Setting E-NSE
Setting E-MSE
Secting 10 E-MSE
Out
C9F
2um
C5F
10
Durt
0.3
Okm
CFF
Our
E.MSE
E.HSF
E-MSE
E-MSE
23
E-MSE
13
0.2
6.1
0.2
0-
0.0
0.50
0.25
0.50
0.35
1.00
0.00
0.25
0.50
0.75
1.00
0.00
0.25
0.50
0.75
1000
0.50
0.25
0.56
0.55
1.00
0.50
0.25
1.00 0.50
0.75
X. percentile
X.
percentile
X.l
percentile
X.1
percantilc
X.1
percentile
Figure S2. First and third panel rows: percentiles estimated effect calculated by CSF (Cui ct al., 2023) (blue line) and by MISTR (green line) compared with the true effect (black line). The colored area represent one empirical SE range for CSF (blue) and MISTR (green). The two dashed green lines represent one estimated SE range of MISTR. Second and fourth panel rows: percentiles estimated MSE (E-MSE) calculated for CSF (Cui et al., 2023) (blue line) and for MISTR (green line).
24

---

Heterogencous Treatment Effect in Time-to-Event Outcomes: Harnessing Censored Data with Recursively Imputed Trees
Typc 1 CSF- msc: 0,018, car, 0,960
Type CSF mSc. 0.015, cor: 0,984
Type
CSF- msc: 1,603, car. 0.962
Typo 4 CSF mSc 0.361, car. 0.955
Typc CSF- msc: 0.217, 0.8:
0.15
Estinated effect
Estimated etrace
0.13
Eatimated affect
Estimated effect
0.6
effect
0.10
0.2
0.05
-0.05
0.09
800
82
0.00
0.00
0.05
6:10
0.5
-0.1
0.0
91
-05
0.0
05
1.0
00
0.2
0.4
06
-62
0.0
0.2
6.4
True effect
True effect
True effect
Tru effect
True effect
Type 1 MISTR mis:: 8.018, con 0.965
Type
MISTR- - mss: 0.013, con 0.986
Type: 3 MISTR mise: 1.502, con 0.966
Type MISTR mee: 0.208, con 0.974
Type MISTR mise 0.18, can 0.1
Estimated effect
Eztiated eftect
Estimated offect
Esimated effect
0.6
effect
0.4
0.03
0.3
0.05
02
-0.05
0.00
0.00
0.00
6.10
0.15
0.0
0.5
0.0
100
0.0
0.2
0.4
0.2
0.0
0.2
04
True effect
True effect
Thue effect
True effect
Thue effect
Eipe 6 CSF
inc: 0,502, con: 0,912
Type 7 CSF
mpe: 0.411, con 0.937
1ype
CSF
mse: 0.414, car: 7.750
Type CSF
mse: 0.038, con 0.895
1ype CSF
msc: 0.090, con 0.9
0.4
0.4
Estimetedd effect
Estimated effect
0.2
Estimated
Latimated
0.0
0.0
0.0
-0.2
-02
-0.2
-02
-0.05
0.00
8:55
True effect
True effect
True effect
True effect
effect
Type
0.213,
0.934
Type
mea: 0,247, con
Type
Type MISTR meer 031, con
Type 10 MISTR meer con
0.4
0.4-
Estimatedd effect
Fatmatoe effect
Fellmated
Estimated
0.0-
00.05
-0.2
-0.2
True effect
Figure Estimated effect vs. true effect calculated by CSF (Cui et al., 2023) (first and third rows) and by MISTR (second fourth rows). The data contains 5000 observations that was randomly sampled based on the distributions of types 1-10.
25

---

Heterogencous Treatment Effect in Time-to-Event Outcomes: Harnessing Censored Data with Recursively Imputed Trees
MIMIC 1 89.5% censoring at
MIMIC 2 84.6% censoring at h
h
10º
Tmax
102
W=1, D=1
W-1. D-0
W=0,
10a
10
109
10
MIMIC 3 80.0% censoring at
MIMIC 4 75.3% censoring at h
10a
10a
10
10
109
109.
ON WOONTTOOOOONOOOOOOOOOOOOO RERERINNANAMOO
ONN
MIMIC 69.9% censoring
10a
10
10a
Figure Observed events and censoring for MIMIC simulation scenarios
26

---

Heterogencous Treatment Effect in Time-to-Event Outcomes: Harnessing Censored Data with Recursively Imputed Trees
0.736
8:50
Polimetedd altect
Extenattes eftect
Extimattd effect
Extomattes ettect
0.00
6:00
-0.35
-0.22
-0223
-1.9
-2:00
-10-
-1.00
-15
-1.
-3.23
-1.2
0.0
-30
-0.
0.
-0.
-1.
60
Tue effect
Thue erfect
Tue effect
The effect
Tue effect
0.795
HBSC
Perimweed wltwet.
Extimatedd eftect
Estimates effect
Entrematees oftect
0.00
0.00
0.00
-0.35
-0.2
Extimated
-0.23
-3.25
-0.30
-1.0-
-1.9
-15
-12
-3.23
-3.2
-0.5
0.0
-300
-05
02
-0.
25
-1.
-0.5
60
Tue offect.
True effect
Tue offett
Tiue effect
True offect
Figure Estimated effect vs. true effect calculated by CSF (Cui al., 2023) (upper) and by MISTR (lower). The data contains one test fold of circa 5000 observations that was randomly sampled from the MIMIC dataset.
MIMIC
MIMIC:
MIMIC
MIMIC
MIMIC'S
eerlance
Mean EST Varlance
Mcan EST Variance
B-
MSE
MSF
MSE
MSE
MSE
10a
10
102
10
102
10
102
10:
302
Figure S6. Mean and SE of the MSE (blue) and estimated variance denoted as "EV" (black) for different values of A. Results are based on 50 repetitions for each value of A and the MIMIC remi-simulated scenarios.
27

---

Heterogencous Treatment Effect in Time-to-Event Outcomes: Harnessing Censored Data with Recursively Imputed Trees
HIV 1 KM, 11.8% LOF at h
HIV 2 KM, 11.9% LOF at h
HIV KM, 11.4% LOF at h
0.9
0.9
0.9
P(t -T)
Pit T)
0.8
0.8
P(f
0.8
Tmax
Tmax
Tmaa
0.7
0.7
0.7
2DV+
ZDV+Zal
ddl
0
200
400
600
800
1000
200
400
600
800
1000
0
200
400
600
800
1000
T [days]
T [days]
T [days]
HIV 1 Estimated HTE cor: 0.995
HIV 2 Estimated HTE cor: 0.997
HIV 3 Estimated HTE cor: 0.996
6
5-
5-
MISTR (Months)
MISTR (Months)
MISTR (Months)
4
3
2
1
0
-1
-1
0
0
7
CSF (Months)
CSF (Months)
CSF (Months)
HIV 1
SE comparison
HIV 2
SE comparison
HIV 3 SE comparison
0.5
0.5
0.5
0.4
0.4
0.4
CSF SE is lower
CSF SE is lower
CSF SE is lower
MISTR SE
MISTR SE
MISTR SE
0.3
0.3
0.2
0.2
0.2
0.1
MISTR SE is lower
0.1
MISTR SE is lower
0.1
MISTR SE is lower
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.0
0.1
0.2
0.3
0.4
0.5
0.0
0.1
0.2
0.3
0.4
0.5
CSF SE
CSF SE
CSF SE
HIV1 1 Mean Est SE comparison
HIV
Mean Est SE comparison
HIV 3 Mean Est SE comparison
2.00
2.00
2.00
MISTR Mean Est SE
1.75
CSF SE is lower
MISTR Mean Est SE
1.75
CSF SE is lower
MISTR Mean Est SF
1.75
CSF SE is lower
1.50
1.50
1.50
1.25
1.25
1.25
1.00
1.00
1.00
0.75
0.75
0.75
MISTR lower
MISTR SE lower
MISTR is lower
0.50
0.50
0.5
1.0
1.5
2.0
1.0
1.5
2.0
0.5
1.0
2.0
CSF Mean Est SE
CSF Mean Est SE
CSF Mean Est SE
Figure S7. Kaplan-Meier (KM) curves for each of the scenarios HIV-1, HIV-2, and HIV-3 (top row). Mcan (sccond row), SE (third row), Mean Est the HTE 1-3 treatment alternatives of all 2,139 samples as calculated by CSF and by MISTR over 10 repetitions. Both methods present similar results in terms of the estimated effect and Est SE, with smaller SE for MISTR.
28

---

Heterogencous Treatment Effect in Time-to-Event Outcomes: Harnessing Censored Data with Recursively Imputed Trees
HIV mse: 0.596, 0.865
HIV 2 mse: 1.214, 0.716
HIV 3 0.404, 0.907
5
HTE current (CSF)
4-
HTE current (CSF)
HTE current (CSF)
3-
3-
3
0
01
5
HTE baseline (CSF)
HTE baseline (CSF)
HTE baseline (CSF)
HIV mse: 0.242, 0.867
HIV 2 mse: 0.788, cor: 0.720
HIV 3 0.282, 0.835
HTE current (MISTR)
HTE current (MISTR)
HTE current (MISTR)
4-
4
3-
3-
3
0
2
5
2
3
4
HTE baseline (MISTR)
HTE baseline (MISTR)
HTE baseline (MISTR)
HIV 1 SE comparison
HIV 2 SE comparison
HIV 3 SE comparison
2.0
CSF SE Is lower
2.0
CSF SE Is lower
2.0
CSF SE lower
MISTR SE
MISTR SE
MISTR SE
1.5
1.5
1.5
1.0
1.0
1.0
0.5
0.5
0.5
MISTR SE is lower
MISTR SE is lower
MISTR SE is lower
0.0
0.0
0.0
0.0
0.5
1.0
1.5
2.0
0.0
0.5
1.0
1.5
2.0
0.0
0.5
1.0
1.5
2.0
CSF SE
CSF SE
CSF SE
HIV 1 Mean Est SE comparison
HIV 2
Mean Est SE comparison
HIV 3 Mean Est SE comparison
2.5
2.5
MISTR Mean Est SE
MISTR Mean Est SE
MISTR Mean Est SE
2.0
CSF SE lower
2.0
CSF is lower
2.0
CSF lower
1.0
1.0
1.0
MISTR SE is lower
MISTR SE is lower
MISTR SE is lower
0.5
0.5
0.5
0.5
1.0
1.5
20
2.5
0.5
1.0
1.5
2.0
2.5
0.5
1.0
1.5
2.0
2.5
CSF Mean Est SE
CSF Mean Est SE
CSF Mean Est SE
Figure $8. Mcan HTE for datasets HIV-1, HIV-2, and HIV-3 with additional censoring that is based an unobserved covariate for CSF (top row) and MISTR (second row), with each method compared to its own haseline estimated using the original data. SE comparison (third row) CSF and MISTR over repetitions. Mean Est comparison (last row) CSF and MISTR over repetitions. In most the samples, SE and Mean Est SE of MISTR is lower.
29

---

Heterogencous Treatment Effect in Time-to-Event Outcomes: Harnessing Censored Data with Recursively Imputed Trees
Type 200 47.4% censoring at
Type 201 46.6% censoring at
10a
10a
h
Tmax
W=1, D=1
W=1, D=0
10a
W=0, D=1
10a
W=0, D=0
10a
10a
100
100
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
11.0
12.0
13.0
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
11.0
12.0
13.0
T
T
Type 202
47.4% censoring at
Type 203 66.3% censoring at
10a
10a
102
102
10
10a
109
109
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
11.0
12.0
13.0
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
11.0
12.0
13.0
T
Type 204 88.5% censoring at
10a
10?
10
100
0.0
1.0
2.0
3.0
4.0
5.0
60
70
8.0
9.0
10.0
11.0
12.0
13.0
T
Figure S9. Observed events and censoring for IV simulation scenarios 200-204, The data contains
5000 observations that was
randomly sampled based on distributions types 200-204.
30

---

Heterogencous Treatment Effect in Time-to-Event Outcomes: Harnessing Censored Data with Recursively Imputed Trees
Type 200 ESE mww. 2,459, coar: 0.847
Type 200 NISTR-mae 2.576, con 0.M44
Type 200 IPEW V 5.11, can: 3.361
Type 200 MRSTH IV mreee 1996, cor 0.923
125
1:25
1.5
1.25
1.00
3.00
1.00
1.00
Detimete afieet
Extimttd ofredd
Extimated effecct
Estimatedd efficet
2.7
0.5
2.75
2.75
9.5
030
0.50
0.50
0.5-
0:5-
3:25
0:25-
200
0.00
0.0
2.0
-025
-025
-015
-025
-130
-030
-030
-130
-55
Nyo 202 CSF moc 11.944 car: 0.843
Type 201 MISTR- moc: 11.334, con: 0.851
1:pe 201 IPCW V- moc:
9.544, can 0.648
Tepo 201 DSSTR N. mac: 990. cor 0.909
1.--
1.5
Extimetedd effect
Extimatedd effect
Estimurad afficet
Extimnnd affece
25
2.5
25
0.5
-05
-0.
-05-
-65
10
15
-65
3.0
-63
10
-25
2.0
10
202
CSP
0.536
202 MISTR
13.516, 0,938
Troe IPCW 10 358
0.537
Trpe N
333, 0.932.
3.25
3.00
3.00
1.00
3.00
Estimated affect
Estimatedd alfect
Extimatedd affect
Edtimatedd offect
0.50
00
00
60
00
Thee effect
Thee effect
Thue effect
True effect
Type
Type
Trpe
3.00
1.00
1.50
3.00
0.35
0.15
0.75
2.75
Estimattdd offect
Exkimaated affect
Extimaated effect
Extimatted offect
822
3.25
300
210
2:0
3.0
0.25
Thue effect
The effect
Thus effect
Thue effect
lype man: 0,546
Type
1ype V. can:
Type N cor 0.240
Eitimceed ceneeeedellee affect
cestmmreeddttee witeect
Extimaaeed offeurt
Esilmated offfcct
0.0
-02-
022
-02
02
-24
-24
Than offect
Then effect
rffect
Than offect
Figure S10. Estimated effect vs. true effect calculated by, from left to right, CSF of (Cui et al., 2023), MISTR, inverse propensity censoring weighting with instrumental forest (IPCW-IV), and MISTR-IV. The data contains n 5000 observations that was randomly sampled based on the distributions of types 200-204.
31

---

Heterogencous Treatment Effect in Time-to-Event Outcomes: Harnessing Censored Data with Recursively Imputed Trees
200 CSF 2,850, can 0.942
Syp 23C HSTR 2,376, cor: 0494
Typw 200 IPCN N-mm:: 8.15,, cor 0.384
Syow 202 NISTA IV- IM-maar 1,508, can 0.921
Pettmated wftwet
Estimatedd affect
Estimated effect
Estimatad effectt
-02-
-35
-1.0
-1.
-0.5
60
0%
16
-1.
-0.5
60
16
-1.
-54
65
1.
-10
-2.
5%
16
Tue offect
Tue offect
Tue offect
Tue effect
Pelimetdd altver
Extimteed affeet
Estimatedd effect
Estimated effect
20-
60-
3.--
3.0
-22
-05
-02
-25
-1.
-10
-1.0
-1.
-3.2
-3.2
Tue effect
Tue effect
Tue effect
Tue effect
Type 200b
1pw 2036 ment
Typa IPON
1ype 2006
mame 2.972, sun
10
10-
10-
10-
Petimeted aftwet
25-
Extimetedd offect
Extimatedd effect
3.-
Eatimatadd offectt
2.5
60-
60-
5.0-
3.0-
-05
-0.5
-1.0
-10
-1.
-1.0
Tue
Tue
Tue
Trpe 204 CSF 9se:2.122 2,324, can 0.546
Spe 204 MISTR. mic. 1 1884, cor. 0 0.654
Typs 204 IPON IV moc. 6.853, cov. 0.486
Tye 204 NISTR mnse 0,606, con. 0.849
3.
30-
1.--
Petimeted affect
Extmmeed affecct
Estmaredd affecct
2.5
Extimated afficct
-1.5
-3.
-1.8
-1.
Typo 204a CSF mss: 6,280 con 0.754
Tpe 204a MISTR.. rio: 4.969, con 0.772
35e 204a IPW mse. 18.057, cor 0.044
Type 2043 MISTR ISSTR.VV-mee mss 2,173, con 0,583
Falmattd affecct
Esliated affecct
Estimated affect
Estimated affect
-2.2
-0.3
05
1.0
-2.3
-033
60
05
1.0
-1.3
-25
65
1.0
-30
-25
25
1.0
Trun
Trun effect
Tun
Trnn
Typc 2046
3,469, 0.564
2046
Tpe (CW
0.087
2042
2.358, 0.453
Patimated edfect
Extimated affect
Estimated affect
Eatimated affect
135
-03-
-03
-05
-1.
-1.0
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Figure SIL. Estimated effect vs. true effect calculated by. from left to right, CSF of (Cui al., 2023), MISTR, inverse propensity censoring weighting with instrumental forest (IPCW-IV), and MISTR-IV. The data contains n 5000 observations that was randomly sampled based on the distributions of types 200-200b and types 204-204h.
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Heterogencous Treatment Effect in Time-to-Event Outcomes: Harnessing Censored Data with Recursively Imputed Trees
JSI CSF vs. MISTR Cor: 0.987
JSI MISTR-IV vs. MISTR cor: 0.841
JSI MISTR-IV vs. CSF cor: 0.816
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HI CSF vs. MISTR cor: 0.994
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HI MISTR-IV vs. CSF cor: 0.662
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Figure $12. Illinois unemployment insurance experiments results. Comparison of the HTE calculated by all three methods: CSF, MISTR. and MISTR-IV, for the setting (top row) and the best fit regression line between each two approaches is shown in black dashed line. "HI" denoted hiring incentive group. "JSI" denotes job search incentive group
setting (bottom row). The line y = is shown dashed red line.
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Reviews:
[{"review_id": "7p9aFK6p1G", "reviewer": "Reviewer_5m23", "summary": "The paper proposes MISTR—a novel non‐parametric approach for estimating heterogeneous treatment effects (HTE) in time-to-event (survival) data, where right censoring is prevalent. MISTR tackles censoring by employing multiple imputations through Recursively Imputed Survival Trees (RIST) to generate several “complete” datasets, on which causal forests are then used to estimate HTE and its variance. The method is extended to handle unobserved confounding via instrumental variables (IV), leading to the MISTR-IV variant. Extensive simulation studies and real-world analyses (e.g., HIV clinical trial data) support the authors’ claims of improved performance, especially under heavy censoring.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you very much for the careful review, thoughtful comments, and your support for the acceptance of our paper. The following is our point-by-point response.\n\n* Explain the equations presented in the main body of the paper.\n\nWe apologize for this omission. The revised version of the paper will include additional explanations for the equations presented in the main text.\n\n* How does MISTR perform when the assumption of ignorable censoring is only approximately met?\n\nIn survival analysis, Assumption A.4, stating that ${\\widetilde{T}}_i \\perp C_i \\mid X_i, W_i$, is very widely used and is considered standard. In our context, see for example, Cui et al. (2023), Bo et al., (2024), Golmakani \\& Polley, (2020) Iswaran \\& Kogalur, (2019)  and Lu et al., (2018).  The reason for its ubiquity is that this assumption is essential, as for each individual only one of the two times (event or right-censoring) is observed. Consequently, their joint distribution is unidentifiable. This assumption is empirically untestable and must be accepted or rejected based on subject-matter knowledge.  When there is a reason to believe this assumption may be substantially violated, for example due to the occurrence of other well-defined events, several strategies can be considered. One approach is to treat such events as competing risks and perform a competing-risks analysis. Alternatively, if one has a good knowledge of the specifics of the violation, one may employ parametric or semiparametric models that explicitly specify a joint distribution of the event and censoring times, such as copula-based models.\n\n* What methods do you suggest for assessing the strength and validity of the instrumental variables used, and how does MISTR-IV behave with weak instruments?\n\nFollowing this comment we extended our simulation study to assess the sensitivity of MISTR-IV to weak instruments by running Setting 200 (47\\% censoring) and Setting 204 (88\\% censoring) with IV of varying strength. We repeat the analysis of Section 6.2, modifying only the coefficient of $Z$ in the model of $W^{*}$, as reported in Table 1. The results are reported in Table 2 and Figure 1 in the following link: \n https://drive.google.com/file/d/1tT_ROACNebxVOC09ty2ng_q4luaBD8Ay/view?usp=sharing  .\nAs expected, the mean absolute error of all methods increases as the instrument weakens. Nonetheless, MISTR-IV outperforms the alternative approaches.\n\nValidating instrumental variable strength in partially linear IV regression is a challenging topic and an active field of research (Windmeijer, (2025),  Burauel (2023), Florence (2012), Hahn and Hausman (2002), Stock et al. (2002)). In linear models, IV strength is commonly evaluated using the effective first-stage F-statistic of Montiel, Olea, and Pflueger (2013). Windmeijer (2025) extends this approach to the Generalized Method of Moments (GMM) framework by proposing the Robust F-statistic. In the future we plan to investigate the best way to incorporate it in our approach. \n\nThe validity of the instrumental variable requires the standard IV assumptions: 1. Exclusion restriction: the instrument affects the outcome only through its influence on treatment assignment; 2. Independence: the instrument is independent of any unobserved confounders; 3. Relevance: the instrument is correlated with the treatment assignment. The relevance assumption can be empirically assessed by examining the correlation between the instrumental variable $Z$ and the treatment assignment $W$. In contrast, the exclusion restriction and independence assumptions cannot be tested directly and must be justified using domain knowledge.\nFor example, in the Illinois Unemployment Insurance Experiment, the proposal to join the experiment was randomized, supporting the independence assumption. However, validating the exclusion restriction requires establishing that the proposal itself did not influence the outcome directly regardless if the individual chooses to participate or not - a condition that is inherently unidentifiable from the observed data. Another approach gaining prominence recently is falsification tests, or negative controls (Eggers et al. 2024). Constructing such tests for censored time-to-event data is an interesting avenue for future research \n\n* The paper does not make strong theoretical improvements to the literature.\n\nIndeed, we view our main contribution as introducing new methods. While our design choices -- particularly addressing heavy censoring -- are strongly motivated, we do not claim a theoretical result. Instead, we demonstrate the advantages of our method through extensive experiments, including several with real-world data, and through empirical comparisons with existing approaches in realistic survival analysis settings.", "claims_and_evidence": "Claim 1) Superior Performance Under Heavy Censoring: The authors claim that MISTR outperforms existing methods like Causal Survival Forests (CSF) and IPCW-based approaches when censoring rates are high. This claim is supported by detailed simulation studies across multiple benchmark settingsand is further validated on real-world datasets.\n\nClaim 2) The paper asserts that MISTR-IV is the first non-parametric method that can estimate HTE in survival data in the presence of unobserved confounding using IVs. Comparative experiments (e.g., Table 3 and corresponding figures) demonstrate reduced bias in settings with confounding.\n\nClaim 3) Avoidance of Direct Censoring Mechanism Estimation: By leveraging multiple imputation via RIST, the method bypasses the need to explicitly model the censoring distribution—a step that often introduces bias when the censoring mechanism is complex", "methods_and_evaluation_criteria": "Yes, simulation and real-world data from an aids clinical trial RCT is used to validate findings.", "theoretical_claims": "No theoretical proofs or formal justifications are made for claims. The paper heavily relies on citations of previous methods to justify its math, but there are several instances when it would be helpful for the author to formally explain how they reached an equation or solution.", "experimental_designs_or_analyses": "Experimental design is sound.", "relation_to_broader_scientific_literature": "The method builds directly on earlier work such as RIST (Zhu & Kosorok, 2012) and causal forests (Athey et al., 2019), and it compares favorably against CSF (Cui et al., 2023). The authors position their contribution clearly against the backdrop of existing literature by highlighting that previous methods often require explicit censoring probability estimation, whereas MISTR’s imputation strategy avoids this potential pitfall.", "essential_references_not_discussed": "NA", "other_strengths_and_weaknesses": "Dependence on Ignorable Censoring Assumption:\nThe methodology still hinges on the assumption of ignorable censoring. In many practical applications, censoring may be non-ignorable or depend on unmeasured factors. The paper does not thoroughly explore how violations of this assumption might impact the estimates, which could be a significant drawback in real-world applications.\n\nThe paper does not make strong theoretical improvements to the literature, and does not justify the theoretical claims it does make.", "other_comments_or_suggestions": "It would be helpful to explain the equations presented in the main body of the paper (in addition to a formal proof in appendix)\n\nEditing score to a 4, questions were sufficiently answered and this paper makes a meaningful contribution.", "questions_for_authors": "How does MISTR perform when the assumption of ignorable censoring is only approximately met? Are there diagnostic tools or adjustments you would recommend?\n\nWhat methods do you suggest for assessing the strength and validity of the instrumental variables used, and how does MISTR-IV behave with weak instruments?", "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": "AcjuuR8x0p", "reviewer": "Reviewer_WUcd", "summary": "The authors propose a tree-based method for estimating Heterogeneous Treatment Effects (HTE) in survival analysis, and further extend it by incorporating instrumental variables to account for unobserved confounders. The authors conduct thorough and detailed experiments on both synthetic and real-world datasets to validate the effectiveness of the proposed method.", "questions": "", "limitations": "", "rating": 1, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you very much for your thorough review and insightful feedback. \n\n* Why does replacing censoring rate estimation in methods such as CSF  with the conditional\nsurvival distribution proposed by RIST  mitigate the impact of extreme cases? \n\nThank you for raising this important issue. It is well known that inverse probability weighting estimators, while often consistent, can suffer from high variance due to instability in the weights. The CSF method combines IPCW with a doubly robust approach to avoid discarding observations with $\\delta_i^h = 0$ and to mitigate bias of the censoring distribution. However, despite this robustness, the use of unstable weights can still result in substantial variance. In contrast, our proposed multiple imputation approach avoids IPW altogether. Our extensive numerical studies clearly demonstrate the superior performance of MISTR compared to CSF. We agree that this point should be better explained in the revision.\n\n* How IV and MISTR are integrated\n\nWe agree and apologize for not clearly explaining it. Consider first the unconfounded setting and as a start assume a constant treatment effect $\\tau$. Consider the partially linear model of Robinson (1988): $g(\\widetilde{T}_i) = \\tau W_i + f(X_i) + \\zeta_i$ where $E(\\zeta_i | W_i , X_i)=0$, $W_i \\perp \\zeta_i$, and $E(W_i|X_i ) = \\Pr(W_i=1|X_i)$. Then, it is easy to verify that $g(\\widetilde{T}_i) - E\\{g(\\widetilde{T}_i)|X_i \\} = \\tau \\{ W_i - E(W_i|X_i) \\} + \\zeta_i$. In the absence of censoring, $\\tau$ can be estimated by the score function provided in line 199 of the paper. Eq. (2) (of the paper) then goes beyond constant treatment effects and accommodates a heterogeneous effects function $\\tau(x)$.\nIn the confounded setting, we relax the independence assumption between $\\zeta_i$ and $W_i$, while the instrument $Z_i$ is independent of $\\zeta_i$ given $X_i$, along with Assumptions B.1--B.5. Hence in the absence of censoring, Eq. (2) is replaced by\n\n\n$$\nS^{IV}_n(\\tau(x)) = \n$$\n\n$$\n\\sum_{i=1}^n \\alpha_i(x) \\cdot \\left(Z_i - \\widehat{h}(X_i) \\right\\) \\cdot \\left[ g(\\widetilde{T}_i) - \\widehat{m}(X_i) - \\tau(x) \\{ \\left(W_i - \\widehat{e}_i(X_i) \\right) \\}  \\right]  = 0\n$$\n\n\nwhere $\\widehat{h}(X_i)=E(Z_i|X_i)$. We accommodate right-censoring by multiple imputation, and in Step 16 of Algorithm 1 causal forests are applied using either $S_n(\\tau(x))$ or $S^{\\sf IV}_n(\\tau(x))$, corresponding to the MISTR and MISTR-IV estimators, respectively. We'll include a detailed explanation in the revised version.\n\n* What is the author’s original theoretical contribution?\n\nWe view our main contribution to be the introduction of new methods. MISTR and MISTR-IV are new nonparametric estimators for HTE, designed for settings without and with unobserved confounding, respectively. Our methods build upon the foundations of RIST and Causal Forests (Athey et al., 2019), effectively merging their strengths to yield estimators with lower variance. This combination results in a notably flexible framework. As evidence of this flexibility, we show how the core approach can be easily modified to accommodate instrumental variables (resulting in MISTR-IV), extending its applicability in a way that is often challenging for other nonparametric estimators. While rigorous theoretical guarantees for our methods are currently lacking, extensive numerical studies, including using real-world datasets, demonstrate that MISTR and MISTR-IV consistently outperform state-of-the-art non-parametric alternatives, such as CSF without unobserved confounding and IPCW-IV with unobserved confounding, in terms of estimation efficiency.\n\n* could papers [3]--[8] also be applied to survival analysis problems? \n\nPaper [3] uses linear semiparametric or parametric regression models to relate the outcome and covariates, whereas our approach is fully nonparametric. Moreover, applying linear models to time-to-event data typically requires outcome transformation or additional constraints. Papers [4] and [5] address non-monotone missingness mechanisms, while right censoring is a special case of monotone coarsening (see Section 9.3 of ``Semiparametric Theory and Missing Data'', Tsiatis A.A., 2006)\n\nPaper [6] focuses on HTE under sample selection without exclusion restrictions, and paper [7] deals with estimating mean functionals under non-ignorable non-response, where missingness depends on unobserved values. Paper [8] tackles both latent confounding bias—stemming from unmeasured variables influencing both treatment and outcome—and collider bias, which arises from non-random sample selection affected by both treatment and outcome. In contrast to these works, right-censored data provide partial information—the event has not occurred up to the censoring time—which must be incorporated into estimation. As such, these works are not directly applicable to our setting, though adapting them to the specific settings and constraints of survival analysis could be a valuable direction for future research.", "claims_and_evidence": "There are several issues with the contributions claimed by the authors:\n\n1. The authors claim that existing methods such as CSF [1] have limitations in estimating censoring rates under extreme cases. However, doubly robust methods can address this issue to some extent. Moreover, the authors' use of the conditional survival distribution proposed in RIST [2] lacks a theoretical foundation for why it would be more accurate than censoring rate estimation in extreme cases. Similar to estimating censoring rates, I think this probability would also encounter issues like extreme values, which could lead to inaccurate estimates under such issues.\n\n2. Although the authors claim that combining IV with the proposed MISTR method can address confounding bias caused by unobserved confounders, they do not explain how IV is used to correct for bias and how it differs from previous IV methods. To me, it seems that the authors have merely applied existing IV theories and methods to MISTR without providing novel contributions.\n\n[1] Cui, Yifan, et al. \"Estimating heterogeneous treatment effects with right-censored data via causal survival forests.\" Journal of the Royal Statistical Society Series B: Statistical Methodology 85.2 (2023): 179-211.\n\n[2] Zhu, Ruoqing, and Michael R. Kosorok. \"Recursively imputed survival trees.\" Journal of the American Statistical Association 107.497 (2012): 331-340.", "methods_and_evaluation_criteria": "The proposed MISTR method builds upon the theory of RIST [1], which is intuitively sound but lacks formal theoretical proof. The combination of MISTR with IV is not thoroughly described, and it also lacks theoretical proof. The datasets and evaluation metrics used in the experiments are reasonable, and the experiments themselves are sufficiently detailed. \n\n[1] Zhu, Ruoqing, and Michael R. Kosorok. \"Recursively imputed survival trees.\" Journal of the American Statistical Association 107.497 (2012): 331-340.", "theoretical_claims": "All the claims and methodologies presented in the paper lack formal theoretical proof.", "experimental_designs_or_analyses": "The experimental design is reasonable and thorough, with detailed descriptions of the experimental setups.", "relation_to_broader_scientific_literature": "The proposed MISTR method essentially adopts the existing characteristics of RIST for imputing censored labels [1] and uses the imputed data to estimate the HTE, lacking an original theoretical contribution. Furthermore, the subsequent combination of IV and MISTR is merely an application of existing IV theory and methods [2-7] to MISTR, without offering any novel contribution.\n\n[1] Zhu, Ruoqing, and Michael R. Kosorok. \"Recursively imputed survival trees.\" Journal of the American Statistical Association 107.497 (2012): 331-340.\n\n[2] Wang, Linbo, et al. \"Instrumental variable estimation of the causal hazard ratio.\" Biometrics 79.2 (2023): 539-550.\n\n[3] Tchetgen, Eric J. Tchetgen, et al. \"Instrumental variable estimation in a survival context.\" Epidemiology 26.3 (2015): 402-410.\n\n[4] Burgess, Stephen, Dylan S. Small, and Simon G. Thompson. \"A review of instrumental variable estimators for Mendelian randomization.\" Statistical methods in medical research 26.5 (2017): 2333-2355.\n\n[5] Hansen, Bruce. Econometrics. Princeton University Press, 2022.\n\n[6] Wooldridge, Jeffrey M. Introductory Econometrics: A Modern Approach 6rd ed. Cengage learning, 2016.\n\n[7] Imbens, Guido W., and Donald B. Rubin. Causal inference in statistics, social, and biomedical sciences. Cambridge university press, 2015.", "essential_references_not_discussed": "The authors provide a thorough overview of the related work in survival analysis and cites most of the key studies.", "other_strengths_and_weaknesses": "The strengths of the paper lie in the thoroughness of the experiments. The authors conduct extensive experiments on both synthetic and real-world datasets, and provides detailed description of the experimental settings. However, several significant weaknesses are evident:\n\n1. The paper lacks theoretical foundations, with no theoretical proofs supporting the proposed content.\n\n2. The method proposed in the paper lacks originality. The MISTR method essentially adopts the existing characteristics of RIST for imputing censored labels and uses the imputed data to estimate the HTE, without offering an original theoretical contribution. Moreover, the combination of IV and MISTR is merely an application of existing IV theory and methods to MISTR, without presenting any novel contribution.\n\n3. The description of how MISTR is integrated with IV is unclear.", "other_comments_or_suggestions": "The following are my comments after reviewing the authors' responses in the rebuttal and discussion stages: \n\nSome concerns have been addressed: \n1. How IV and MISTR are integrated is clarified, which is missing in the original manuscript.\n2. RIST's advantages over censoring rate estimation methods are clarified. \n\nHowever, a major concern remains unresolved:\n\n**The proposed method lacks a theoretical foundation.** The authors have repeatedly argued that the paper focuses more on methodological contributions rather than theoretical contributions, and therefore ”theoretical results are a bonus but not required” as Reviewer 7fsC suggested. **However, theoretical results are of course crucial and necessary for a causal inference method.** My continued emphasis on the lack of a theoretical contribution is not to suggest that the paper does not present new theory—I certainly understand that, in addition to proposing new theories, proposing correct new methods is also an important contribution. Rather, I am pointing out that the method proposed in this paper, regardless of its lack of novelty, lacks a theoretical foundation. **The authors have not provided any theoretical proof or even a discussion** (such as the theoretical discussion regarding the correctness of the RIST method in [1]) **to demonstrate the correctness of the proposed method** (i.e., the combination of the existing methods). As a result, **readers are unable to assess whether the effectiveness demonstrated in the experimental validation holds only on specific datasets, making it hard to judge when the proposed method is effective or when it may fail.** This, in turn, limits the applicability of the proposed method.\n\nTherefore, while I fully agree that correctly combining existing methods to solve a practical problem can be considered as an important contribution. As I mentioned in my rebuttal comments, **the authors' approach of merely combining existing methods without investigating the correctness of this combination—such as whether the conditions for identifiability and consistency change after the combination—clearly represents an insufficient contribution.**\n\nIn conclusion, although I greatly appreciate the authors' writing and experiments, and I am very grateful for their responses and discussions, my recommendation still leans toward rejection.\n\n[1] Zhu, Ruoqing, and Michael R. Kosorok. \"Recursively imputed survival trees.\" Journal of the American Statistical Association 107.497 (2012): 331-340.", "questions_for_authors": "The following questions are based on my previous comments, and I encourage the authors to carefully review them.\n\n1. Why does replacing censoring rate estimation in methods such as CSF [1] with the conditional survival distribution proposed by RIST [2] mitigate the impact of extreme cases? I believe this probability might still be susceptible to similar extreme value issues.\n\n2. I could not find any details in the paper on how IV and MISTR are integrated. Could the authors clarify this aspect?\n\n3. It seems that the authors simply apply the existing RIST method for imputation, then uses the imputed data to estimate HTE, followed by the application of existing IV theory to address confounding bias from unobserved variables. What is the author's original theoretical contribution?\n\n4. In addition to the theories and methods addressing censoring within survival analysis, there are approaches that consider censoring from the perspectives of selection bias and missing data [3-7], including methods that use IV to simultaneously address confounding bias due to unobserved variables [8]. Could these methods also be applied to survival analysis problems?\n\nOverall, I appreciate the experimental contributions of this paper; however, the theoretical contributions are quite limited, which may make it unsuitable for publication in ICML.\n\n[1] Cui, Yifan, et al. \"Estimating heterogeneous treatment effects with right-censored data via causal survival forests.\" Journal of the Royal Statistical Society Series B: Statistical Methodology 85.2 (2023): 179-211.\n\n[2] Zhu, Ruoqing, and Michael R. Kosorok. \"Recursively imputed survival trees.\" Journal of the American Statistical Association 107.497 (2012): 331-340.\n\n[3] Heckman, James J. \"Sample selection bias as a specification error.\" Econometrica: Journal of the econometric society (1979): 153-161.\n\n[4] Malinsky, Daniel, Ilya Shpitser, and Eric J. Tchetgen Tchetgen. \"Semiparametric inference for nonmonotone missing-not-at-random data: the no self-censoring model.\" Journal of the American Statistical Association 117.539 (2022): 1415-1423.\n\n[5] Wang, Sheng, Jun Shao, and Jae Kwang Kim. \"An instrumental variable approach for identification and estimation with nonignorable nonresponse.\" Statistica Sinica (2014): 1097-1116.\n\n[6] Heiler, Phillip. \"Heterogeneous treatment effect bounds under sample selection with an application to the effects of social media on political polarization.\" Journal of Econometrics 244.1 (2024): 105856.\n\n[7] Li, Wei, Wang Miao, and Eric Tchetgen Tchetgen. \"Non-parametric inference about mean functionals of non-ignorable non-response data without identifying the joint distribution.\" Journal of the Royal Statistical Society Series B: Statistical Methodology 85.3 (2023): 913-935.\n\n[8] Li, B., Wu, A., Xiong, R., & Kuang, K. (2024). Two-stage shadow inclusion estimation: an IV approach for causal inference under latent confounding and collider bias. In Forty-first International Conference on Machine Learning.", "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": "lgK5oUQ0JP", "reviewer": "Reviewer_7fsC", "summary": "The authors consider the problem of estimating an individual treatment effect from survival data, where some of the observations may be right-censored. They propose a two-stage approach, MISTR, for this problem. In the first stage, they use recursively imputed survival trees (RIST) to impute the censored survival times in the data. Using these imputed times, they can apply a known result where the the treatment effect is given as the root of the \"Robinson-style partialling-out score function\" in the absence of censoring. They also show how to extend their approach to the case of unobserved confounding using instrumental variables. Finally, they compare their result against the recently proposed causal survival forest (CSF) method on synthetic and real datasets, showing comparable performance in low censoring regimes and when the assumptions of CSF are met, but improved accuracy in the presence of high censoring or when some of the modeling assumptions of CSF are violated.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you very much for the detailed review, constractive comments, and your support for the acceptance of our paper. The following is our point-by-point response.\n\n\n*  The max time for RIST and the RMST horizon are very close, so  it seems that the imputation procedure should have a very minimal effect. \n\nRight censoring can occur at *any* time between 0 to $t_{max}$. As a result, the censoring rate may be high even when $h$ is very close to $t_{max}$, and therefore imputation has a substantial impact.\n\n* The main weakness of the paper is the lack of baseline methods in the experiments. \n\nGolmakani \\& Polley, (2020)  propose two algorithms for constructing super learners in survival data prediction where the individual algorithms are based on proportional hazards. Bo et al., (2024) studied two meta-learning algorithms, T-learner and X-learner, each combined with three types of machine learning methods: random survival forest, Bayesian accelerated failure time model and survival neural network. These additional baselines will be incorporated in the revised version. \n\nThe existing literature for this problem is not sparse and there are  baseline methods included in Cui et al., (2023). In particular, Cui et al. showed that their method is superior to the random survival forest (Iswaran \\& Kogalur, 2019), S-learner (Kunzel, 2019), enriched random survival forest (Lu et al., 2018), and the IPCW causal forest. Therefore, we focus on comparing our method with the CSF method of Cui et al. (2023). This point will be added to the revised version of the paper.\n\nWe believe that comparing to methods that ignore right censoring or causal inference principles adds little value, as their limitations are well established in the literature, even if not in our exact setting.\n\n* It would be helpful to include a derivation or explanation for some of these in an appendix, especially the estimator for the target quantity in equation (2).\n\n\nThank you for bringing this up. The revised version will include detailed explanations, as suggested.\n\n* It also seems that equation (4) is only valid when $t > C_i$.\n\nThank you for noticing this mistake, it will be corrected.\n\n* The analysis on the Illinois unemployment insurance dataset seemed incomplete. Metrics or other analysis should be provided to indicate that the MISTR-IV results are actually better.\n\n\nThank you for this important comment. Indeed, as the reviewer points out, lacking ground truth makes any solid model evaluation and comparison challenging. Indeed, censoring and hidden confounding make the problem especially challenging in our setting. Nonetheless, following the reviewer's suggestion we have added qualitative comparisons between the top 10\\% and bottom 10\\% of the population expected to benefit the most and the least from the treatment, as rated by CSF, MISTR, and MISTR-IV; see Table 3 in the following link: \nhttps://drive.google.com/file/d/1tT_ROACNebxVOC09ty2ng_q4luaBD8Ay/view?usp=sharing \n . Such comparisons may reveal differences between the model results that may be explained by domain knowledge and will help to guide model selection. We see that MISTR-IV arrives quite distinct conclusions regarding the populations most and least benefitting from the treatment. We leave a full interpretation of these results to domain experts in the field.", "claims_and_evidence": "The main weakness of the paper is the lack of baseline methods in the experiments. There are only two baselines, CSF (in every experiment) and IPCW-IV (in the instrumental variable settings). The authors mention two meta-learner methods which can handle right-censoring (Golmakani & Polley, 2020 and Bo et al., 2024) in the first paragraph of the Related Work section. It seems that these should be applicable to the setting of this paper, and these should therefore be included as comparison methods. If the existing literature for this problem is sparse and there are not many existing baseline methods, some naive approaches could also be included to demonstrate the value of the proposed model. Specifically, *any* method for survival analysis which estimates a conditional survival distribution can be used for this problem as follows:\n- Directly estimate the survival function and include the treatment assigment as a covariate, or estimate the survival function for the treated and untreated groups separately.\n- Using the estimated survival functions, estimate $\\tau(x)$.\n\nThis approach ignores all of the causal inference minutiae and therefore should be expected to have worse performance. But it would still be valuable to confirm that this is actually the case, and show the value of directly accounting for the causal nature of the problem.\n\nSimilarly, one could also check naive baselines which ignore the specifics of survival analysis (i.e., the presence of censored data) and apply generic methods causal regression problems to the dataset where the censored observations are simply dropped, or where the censoring is ignored and the censored time is treated as the actual event time. Again, we would expect such approaches to have inferior performance because not accounting for censoring can lead to bias, but confirming this empirically would greatly strengthen the paper especially since the existing baselines are sparse.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "Most of the theoretical claims are statements of certain estimators. It would be helpful to include a derivation or explanation for some of these in an appendix, especially the estimator for the target quantity $\\tau$ in equation (2).\n\nIt also seems that equation (4) is only valid when $t > C_i$, is this true?", "experimental_designs_or_analyses": "The synthetic datasets were effectively designed for simulating the different challenging aspects of the problem, e.g. high censoring, unmeasured confounding, etc.\n\nI found the HIV clinical trial experiment to be an exceptionally convincing setup. The accuracy of causal inference methods can be difficult to evaluate on real data since the ground truth is not known. To address this problem, the authors first computed estimates for the target function using the full dataset, which contained only mild censoring. Their method and the baseline CSF obtained similar results. To show the value of their method, the authors used a held-out covariate to increase the censoring in the data, then showed that the estimate using their method remained relatively stable as the censoring increased, while the quality of the estimate of CSF degraded more significantly. I am not sure if this technique is standard as I am not an expert in causal inference, but this setup is one of the best I have seen for validation when the ground truth is unknown.\n\nThe analysis on the Illinois unemployment insurance dataset seemed incomplete. The results show that MISTR without accounting for the IV gave similar results as the baseline CSF, and that both of these deviated from MISTR-IV. However, no metrics or other analysis was provided which would indicate that the MISTR-IV results are actually *better*. Since the ground truth is not available in this setting, some qualitative analysis of the results would be helpful.", "relation_to_broader_scientific_literature": "The relationship to prior work is clearly discussed. The authors focus primarily on the most relevant existing baseline, causal survival forests (CSF, Cui et al. (2023)), and emphasize that their method makes two main improvements: (1) they do not require to estimate the censoring probability, which leads to greater robustness and accuracy, and (2) their method can handle the instrumental variable (IV) setting.", "essential_references_not_discussed": "I am not aware of any critical references that have been missed.", "other_strengths_and_weaknesses": "I found the paper to be clear and well-written.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "In the last paragraph of Section 6.2, it is stated that \"The maximum time for the RIST is $t_{max} = 29$, and the RMST horizon is $h=28$.\" The RMST is computed as $g(\\widetilde{T}_i) = \\min(\\widetilde{T}_i, h)$. The max time for RIST and the RMST horizon are therefore very close, so if my understanding is correct, the imputed points can only impact the RMST value by a small amount (when the imputed value is between 28 and 29) and most of the imputed values will just default to the RMST horizon. In this case, it seems that the imputation procedure should have a very minimal effect. Can the authors comment on this?\n\n---\n\nAfter the rebuttal, the authors have addressed all of my questions. I have raised my score accordingly.\n\nI also read the other reviews and rebuttals. In particular, it seems that Reviewer WUcd is concerned by the lack of a theoretical contribution, and because the proposed method seems combinatorial in nature. As this paper is focused on introducing a novel method, theoretical results are a bonus but not required, and the empirical validation of the proposed method is convincing.\n\nRegarding the nature of the method as a combination of existing techniques, the review instructions state \"For example, originality may arise from creative combinations of existing ideas.\" As the authors mention, it can be generally challenging to accommodate instrumental variables with nonparametric estimators. Especially given the strength of the empirical results, deriving a flexible nonparametric framework which is compatible with IVs should constitute a creative combination.", "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:

Meta-review: This paper introduce a new method to estimate treatment effects with censored data, focusing on the case with heavy censoring and providing flexibility to incorporate an IV. Reviewers were sharply divided. Two reviewers appreciated the practicality of the method and strength of the empirical evaluation. One felt that the lack of theoretical guarantees for the method is a deal-breaker. I agree that the lack of theoretical guarantees is a significant weakness. The authors claim, essentially, that this is not a theory paper. However, verifying basic properties (e.g., the method is consistent for the desired treatment effect, showing that confidence intervals are valid) is very desirable for causal inference methods that are being used to derive statistical conclusions, even if the point of the paper is not to push the state of the art in theoretical guarantees (eg having the fastest convergence rates). That said, given the unusually strong empirical motivation and evaluation, I do believe that the evidence presented in the paper is sufficient to produce an insight of value to the community -- modeling the censoring mechanism is not the best way to handle heavily censored data in causal inference tasks -- that future work will be likely to build on. In that sense, although this paper does not by itself present a complete story for the method being proposed (it would be a clear accept if that were the case), it is nevertheless a valuable contribution.
Decision: Accept (poster)