Enabling counterfactual survival analysis with balanced representations
Authors: Paidamoyo Chapfuwa
Serge Assaad
Shuxi Zeng
Michael Pencina
Lawrence Carin
Ricardo Henao
Venue:  (2021)
Keywords: survival analysis
time-to-event
counterfactual inference
causal survival analysis

Abstract: Balanced representation learning methods have been applied successfully to counterfactual
inference from observational data. However, approaches that account for
survival outcomes are relatively limited. Survival data are frequently encountered
across diverse medical applications, i.e., drug development, risk profiling, and clinical
trials, and such data are also relevant in fields like manufacturing (for equipment
monitoring). When the outcome of interest is time-to-event, special precautions
for handling censored events need to be taken, as ignoring censored outcomes may
lead to biased estimates. We propose a theoretically grounded unified framework
for counterfactual inference applicable to survival outcomes. Further, we formulate
a nonparametric hazard ratio metric for evaluating average and individualized
treatment effects. Experimental results on real-world and semi-synthetic datasets,
the latter which we introduce, demonstrate that the proposed approach significantly
outperforms competitive alternatives in both survival-outcome predictions and
treatment-effect estimation.

Under review as a conference paper at ICLR 2021
ENABLING COUNTERFACTUAL SURVIVAL ANALYSIS WITH BALANCED REPRESENTATIONS
Anonymous authors Paper under double-blind review
ABSTRACT
Balanced representation learning methods have been applied successfully to coun- terfactual inference from observational data. However, approaches that account for survival outcomes are relatively limited. Survival data are frequently encountered across diverse medical applications, Le., drug development, risk profiling, and clini- cal trials, and such data are also relevant in fields like manufacturing (for equipment monitoring). When the outcome of interest time-to-event, special precautions for handling censored events need to be taken, as ignoring censored outcomes may lead to biased estimates. We propose a theoretically grounded unified framework for counterfactual inference applicable to survival outcomes. Further, we formulate a nonparametric hazard ratio metric for evaluating average and individualized treatment effects. Experimental results real-world and semi-synthetic datasets, the latter which we introduce, demonstrate that the proposed approach significantly outperforms competitive alternatives in both survival-outcons predictions and treatment-effect estimation.
INTRODUCTION
Survival analysis or time-to-event studies focus on modeling the time of a future event, such as death failure, and investigate its relationship with covariates or predictors of interest. Specifically, we may be interested in the causal effect of given intervention or treatment on survival time. A typical question may be: will a given therapy increase the chances survival of an individual or population? Such causal inquiries on survival outcomes are common in the fields of epidemiology and medicine (Robins) 1986, Hammer et al. 1996 Yusuf et al., 2016). an important current example, the COVID-19 19 pandemic is creating demand for methodological development to address such questions, specifically, when evaluating the effectiveness of a potential vaccine or therapeutic outside randomized controlled trial settings.
Traditional causal survival analysis is typically carried out in the context of a randomized controlled trial (RCT), where the treatment assignment is controlled by researchers. Though they are the gold standard for causal inference, RCTs are usually long-term engagements, expensive and limited in sample size. Alternatively, the availability of observational data with comprehensive information about patients, such as electronic health records (EHRs), constitutes a more accessible but also more challenging source for estimating causal effects (Häyrinen et al. 2008 Jha et al. (2009). Such observational data may be used to augment and verify an RCT, after a particular treatment is approved and in use (Gombar et al. 2019 Frankovich al. 201T Longhurst et al. 2014). Moreover, the wealth of information from observational data also allows for the estimation of the individualized treatment effect (ITE), namely, the causal effect of an intervention the individual level. In this work, we develop novel framework for counterfactual time-to-evenl prediction to estimate the ITE for survival or time-to-event outcomes from observational data.
Estimating the causal effect for survival outcomes in observational data manifests two principal challenges. First, the treatment assignment mechanism is not known a priori. Therefore, there may be variables, known as confounders, affecting both the treatment and survival time, which lead to selection bias (Bareinboim & Pearl, 2012), i.e., that the distributions across treatment groups are not the same. In this work, focus selection biases due confounding, but other sources may also be considered. For instance, patients who are severely ill are likely to receive more aggressive therapy, however, their health status may also inevitably influence survival. Traditional survival analysis

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neglects such bias, leading to incorrect causal estimation. Second, the exact time-to-event is not always observed, Le., sometimes we only know that an event has not occurred up to a certain point in time. This is known as the censoring problem. Moreover, censoring might be informative depending on the characteristics of the individuals and their treatment assignments, thus proper adjustment is required for accurate causal estimation (Cole & Hernán 2004 Díaz 2019).
Traditional causal survival-analysis approaches typically model the effect of the treatment or co- variates (not time or survival) in a parametric manner. Two commonly used models are the Cox proportional hazards (CoxPH) model (Cox 1972) and the accelerated failure time (AFT) model (Wei 1992). which presume linear relationship between the covariates and survival probability. Further, proper weighting for each individual has been employed to account for confounding bias from these models (Austin 2007; 2014 Hernán et al. 2005). For instance, probability weighting schemes that account for both selection bias and covariate dependent censoring have been considered for adjusted survival curves (Cole Hernán 2004 Díaz, 2019). Moreover, such probability weighting schemes have been applied to causal survival-analysis under time varying treatment and confounding (Robins 1986; Hernán et al. 2000). See van der Laan & Robins (2003); T (2007); Van der Laan & Rose (2011); Hernán & Robins (020) for an overview. Such linear specification makes these models interpretable but compromises their flexibility, and makes it difficult to adapt them for high-dimensional data to capture complex interactions among covariates. Importantly, these methods lack a counterfactual prediction mechanism, which is key for ITE estimation (see Section 2).
Fortunately, recent advances in machine learning, such as representation learning generative modeling, have enabled causal inference methods to handle high-dimensional data and to characterize complex interactions effectively. For instance, there has been recent interest in tree-based (Chipman
et al. 2010; Wager & Athey 2018) and neural-network-baee
(Shalit et al. 2017] Zhang et al.
2020) approaches. For pre-specified
horizons, the nonparametric Random Survival Forest (RSF) (Ishwaran et al.) (008)) and Bayesian Additive regression trees (BART) (Chipman et al.)
20100) have
been extended to causal survival analysis. RSF has been applied to causal survival forests with weighted bootstrap inference (Shen et al.] 2018 et al. 2020) while BART is extended to account for survival outcomes in Surv-BART (Sparapani et al. 2016), and AFT-BART (Henderson et al., 2020). See Hu et al. (2020) for extensive investigation of the causal survival tree-based methods. Alternatively, when estimating the ITE, neural-network based methods propose to regularize the transformed covariates or representations for individual to have balanced distributions across treatment groups, thus accounting for the confounding bias and improving ITE prediction. However, most approaches employing representation learning techniques for counterfactual inference deal with continuous or binary outcomes, instead of time-to-event outcomes with censoring (informative or non-informatiive. Hence, principled generalization the context of counterfactual survival analysis is needed.
In this work we leverage balanced (latent) representation learning to estimate ITE via counterfactual prediction of survival outcomes in observational studies. We develop framework to predict event times from a low-dimensional transformation of the original covariate space. To address the specific challenges associated with counterfactual survival analysis, we make the following contributions:
We develop an optimization objective incorporating adjustments for informative censoring, as well as a balanced regularization term bounding the generalization error for ITE prediction. For the latter, we repurpose a recently proposed bound (Shalit et al. 2017) for our time-to-event scenario. We propose a generative model for event times to relax restrictive survival linear and parametric as- sumptions, thus allowing for more flexible modeling. Our approach can also provide nonparametric uncertainty quantification for ITE predictions.
We provide survival-specife evaluation metrics, including a new noupaaametric hazard ratio estimator, and discuss how perform model selection for survival outcomes. The proposed model demonstrates superior performance relative to the commonly used baselines in real-world and semi-synthetic datasets.
We introduce a survival-specific semi-synthetic dataset and demonstrate an approach for leveraging prior randomized experiments in longitudinal studies for model validation.
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h,(t,e)
X
(xx)
P(Til X
Tv
Ho(r,e)
P[Tol X x)
(a) Counterfactual inference
(b) Non-infonnann informative
(c) Informative
Figure I: (a) Illustration of the proposed counterfactual survival analysis (CSA). Covariates = mapped into latent representation via deterministic mapping
((x:). The potential outcomes are sampled from l/ p(Ta| = æ) for A = a via stochastic mapping (r ?). where randomness is induced with flow-based transformation, é, of a simple distribution p(c), i.e., uniform or Gaussian. (b) and (c) show the proposed causal graphs for non-informative and informative censoring, respectively.
PROBLEM FORMULATION
We first introduce the basic setup for performing causal survival analysis in observational studies. Suppose we have N units, with N₁ units being treated and No in the control group (N = N₁ + No). For each unit (individual), we have covariates X, which can heterogeneous, e.g., a mixture of categorical and continuous covariates which, in the context of medicine, may include labs, vitals, procedure codes, etc. We also have a treatment indicator A, where A = 0 for the controls and A = 1 for the treated, well the outcome (event) of interest T. Under the potential-outcone framework (Rubin 2005). let Tu and Tj be the potential event times for given subject under control and treatment, respectively. In practice we only observe realization of the potential outcomes, Le., the factual outcome = T', while the counterfactual outcome Ti Ti-A is unobserved.
In survival analysis, the problem becomes more difficult because do not always observe the exact event time for each individual, but rather the time up to which we are certain that the event has not occurred: specifically, we have (right) censoring problem, most likely due the loss of follow-up. We denote the censoring time as C' and censoring indicator as d E {0, 1j. The actual observed time Y min C),, i.e., , the outcome is observed (non-censored) TA C' and 5 =
In this work, we are interested in the expected difference between the Ti and To conditioned on X for a given unit (individual), which is commonly known as the individualized treatment effect (ITE). Specifically, we wish to perform inference the conditional distributions of Ti and To. i.e. [(T]|)) and (T)) respectively, as shown in Figure Ial In practice, we observe N realizations of (Y,6, X, A) for observed time, censoring indicator, covariates and treatment indicator, respectively; hence, from an observational study the dataset takes the form D {1,, d,, 24, a.)) Below, we discuss several common choices of estimands in survival analysis.
Estimands of Interest We begin by considering survival analysis in the absence of intervening treatment choice, A. Let F(t)z) & P(T f|X = 2) be the cumulative distribution function of the event (failure) time, t, given a realization of the covariates, x. Survival analysis is primarily concerned with characterization of the survival function conditioned on covariates S(t|z) & 1- - F Ft and the hazard function or risk score, X(t(ar), defined below. 5(1]=: is monotonically decreasing function indicating the probability of survival up to time t. The hazard function measures the instantaneous probability of the event occurring between {t, t + At} given T' > t and At - O. From standard definitions (Kleinbaum & Klein 2010), the relationship between cumulative and hazard function is formulated
P(t < T < dt X x
dlog S(t)x) dt
f((())
A(t[z) = lim dt-50
(1)
P(T x z)dt
S(([z)
From 1 we see that f(t)z) A P t = x) = Sttlz) is the conditional event time
density function (Kleinbaum & Klein 2010
Given the binary treatment A, we are interested in its impact on the survival time. For ITE estimation, are also interested in the difference between the two potential outcomes Tı, To. Let St(t)z) and Xa (t ()) denote the survival and hazard functions for the potential outcomes TA. i.e., Ti and To- Several common estimands of interest include (Zhao al. 2012 Trinquart al. 2016): difference
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in expected lifetime: ITE(t, 2) = f (S((t))) (u(tl)))d To|X = z), difference in survival function: ITE(1,,)) x) = Si - (1]2)) and hazard ratio: ITE((,,)) = ((o()))/\o(((p))
The inference difficulties associated with the above estimands from observational data are two-fold. First, there are confounders affecting both the treatment assignment and outcomes, which stem from selection bias, i.e., the treatment and control covariate distributions are not necessarily the same. Also, do not have direct knowledge of the conditional treatment assignment mechanism, i.e., P(A= a = z), also known as the propensity score. Let IL denote statistical independence. For estimands to be identifiable from observational data, we make two assumptions: (i) {Ti, IL A|X, ie.. no unobserved confounders or ignorability, and (ii) overlap in the covariate support 0 < P(A 1| = x) 1 almost surely if p(X = x) > 0. Second, the censoring mechanism is also unknown and may lead to bias without proper adjustment. We consider two censoring mechanisms in our work, (i) conditionally independent or informative censoring: T TL C| A, and (ti) random or non-informative censoring: T IL C. Note that for informative censoring, we also have to consider potential censoring times C1 and Co and their conditionals p(C²) x) and p(Co) X), respectively. Figure I shows causal graphs illustrating these modeling assumptions.
MODELING
To overcome the above challenges and adjust for observational biases, we propose a unified frame- work for counterfactual survival analysis (CSA). Specifically, we repurpose the counterfactual bound in [hhalit et al. (2017) for our time-to-event scenario and introduce a nonparametric approach for stochastic survival outcome predictions. Below we formulate a theoretically grounded and unified approach for estimating (i) the encoder function r = ((z). which deterministically maps covari- ates to their corresponding latent representation E IRd, and (ii) two stochastic time-to-event generative functions, h (-). to implicitly draw samples from both potential outcome conditionals to Ph. P X = z), for A (1, (1.0), and where ta indicates the sample from ph, (Ta X x) is for A = a. Further, we formulate a general extension that accounts for informative censoring by introducing two stochastic censoring generative functions, v-(-), draw samples for potential censoring times ca ~ p2,, (CalX = x)) The model-specifying functions, (ha()), (-),, (()). are parameterized via neural networks. See the Supplementary Material (SM) for details. Figure Ia summarizes our modeling approach.
Accounting for selection bias We wish to estimate the potential outcomes, ie., event times, which are sampled by distributions parameterized by functions (ha(-), (()), i...
t~ Ph. (TIX x., A, a)
La ~ p x
=
We obtain (3 from via the strong ignorability assumption, i.e., (To, Tı> 1L A|X (consis- tent with the causal graphs in Figure 1b and Icb and 0 < P(A = a|X x) < 1, and the consistency assumption, i.e,, T = Ta
a. A similar argument can be made for informative censoring based Figure 1c we can also write co Pr,l = x)). Given (3) model functions {-) {-)} and (-) for informative censoring can be learned by leveraging stan- dard statistical optimization approaches, that minimize loss hypothesis L given samples from the empirical distribution (y, 5,x, a) ~ >p(Y 5, X, A), i.e., from dataset D. Specifically, we write = (y (.). [C,, (ta: (ta: y, 5)]. where ln (fa, y. 8) is a loss function that measures the agreement of tu Ph, (TalX = z) (and ca ~ Pu. (Cal = z) for informative censoring) with ground truth {v. 5}. the observed time and censoring indicator. respectively.
For some parametric formulations of event time distribution Ph, a (TAlX = I). egg. exponential, Weibull, log-Normal, etc., and provided the censoring mechanism is non-informative, - -ln, (ta, V, 6)
is the closed form log likelihood. Specifically, -tn, 1 (ta, y, 6) & log pa, (Ta X = x) 8
log fr, x (z) + (1 - 5) log Sp, (2). which implies that the conditional event time density and survival functions can be calculated in closed form from transformations (-) (()) of x. See the SM for parametric examples of L accounting for informative censoring.
We further define the expected loss for given realization of covariates x and treatment assignment over observed times y (censored and cen-censoredd, and the censoring indicator 5 as (,, (z, a) A E ((). &,, (ta: y, 6) as in Shalit et al. (2017). For a given subject with covariates x and treatment assignment a, we wish to minimie both the Tactual and counterfactual losses, Lp and Ler,
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respectively, by decomposing C LB + LCF as follows
LF = Ble.a))p(AA.)),,,.
LCF = -.e.a)-p0--A.
(4)
Let u _ P(A 1) denote the marginal probability of treatment assignment. We can readily decompose the losses in according to treatment assignments. The decomposed factual Lp and similarly, the decomposed counterfactual Ler = (1-w) u) f u- cCr In practice, only factual outcomes are observed, hence, for randomizde non-controlled experiment, we cannot obtain an unbiased estimate of LCF from data due to selection bias (or confounding). Therefore, we bound Ler and below following (Shalit et al. 2017
u. C. C- + (1-w) u) - -
A=
Corollary Assume (-) is an invertible map, and a Sn (z, a) E G, where G is a family of functions, p _ p4 [R|A- = a) is the latent distribution for group A = and a >> 0 is a constant. Then, we have:
L S (1 - u) L + u L + (PM A=1 PP -=
f S - + LA + a. (PM (-1 0 )
(5)
The integral probability metric (IPM) Müller 1997; Sriperumbudur et al. 2012) measures the distance between two probability distributions p and g defined over M, i.e., the latent space of R. Formally, (P, q) - sup IMr g(m) (p(m) - ((m) dm), where g : m + R, represents a M (Shalit et al., 2017) Therefore, model
class of real-valued bounded measurable funetions functions (+) (()) can be learned by minimizing the upper bound in (5) consisting of (i)
only factual losses under both treatment assignments and (ii) IPM regularizer enforcing latent distributional equivalence between the treatment groups. Note that if the data originates from a RCT it follows (by construction) that IPMg(rg . -=0)== 0.
Accounting for censoring bias Below we formulate an approach for estimating functions (-)-) and v-(.) for synthesizing (sampling) non-censored ta ~ Ph, (TAlX = x) and censored ca ~ Pu, (Ca X æ) times, respectively. While some parametric assumptions for P (Tx = x) yield easy-to-evaluate closed forms for (t, (z) that can be used as likelihood for censored observations, they restrictive, and have been shown generate unrealistic high variance samples (Chpfuwa et al. 2018).. So motivated, we seek a nonparametric likelihood-based approach that can model flexible family of distributions, with an easy-to- to-sam approach for event times ~ Ph. (Ta = z). We model the event time generation process with a source of randommess, p(e), e.g. Gaussian or uniform, which is obtained from neural-networkkhaae based nonlinear transformation. In the experiments we use planar flow formulation parameterized by (U,. Wn. by (Rezende & Mohamed 2015). however, other specifications can also be used. Note that Missouridou et al. (2018)) has previously leveraged normalizing flows for survival analysis, however, our approach very different in that focuses on formulating i) counterfactual survival analysis framework that accounts for informative times as a continuous variable instead of discretizing them. Specifically, we transform the source of randomness, e, using single layer specification follows
ion-informativ censoring mechanisms and confounding, and ii) model event
εn = Un tanh(Wie + bn): e Uniform 1) = hA(T, Ex). r= ((r)
(6)
where (Un, Wr) E Rdxd c) Rd, d is the dimensionality of the normalizing flow; each component of e is drawn independently from Uniform(0, 1), and én may be viewed as a skip connection with stochasticity in Further, (r εx) and ((z) are time-to-event generative and encoding functions, respectively, parameterized as neural networks. For simplicity, the dimensions of r and e set to d, however, they can be set independently if desired. In practice, are interested in generating realistic event-time samples; therefore, we account for both censored and non-censored observations by adopting the objective from Chaptuwa et al. (0118). formulated as
CCSA A B.y.A...))-=((.......)))). [8 (ly tal) + (1 - 8) (max y taa))]
where the first term encourages sampled event times f.2 to be close y. the ground truth for observed events, f.e. 5 = 1, while penalizing fa for being smaller than the censoring time when = 0. Further, the expectation is taken over samples (a minibatch) from empirical distribution p(Y, 6, X, A).
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Informative censoring We model informative censoring similar to 7 but mirroring the censoring indicators to encourage accurate censoring time samples Ca for 5 O, while penalizing ca for being smaller than for d (observed events). Specifically, of randomness like in (6) but parameterized by {Uz, W,,by, bu) and censoring generative functions v (x(r,&)) (v), parameterized neural networks, where Ca Pu, (C = x) formulated as
set an independent source
Le(v, ②) Bly.bax.al-ptu.........ve (1-6) 6) (Iv cal) + 8 (max(0, -ca)]]
(8)
Further, we introduce an additional lime-order-consistenny loss that enforces the correct order of the observed time relative to the censoring indicator, i.e., Co to if 6 and to Co i = I, thus
(ro 4) = X,),)))))p [ō (max(0, ca)) + (1 5) (max(0, Co tu))l (9)
Note that (rc v, +) does not depend the observed event times but only the censoring indicators. Finally, we write the consolidated CSA loss for informative censoring (CSA-INFO) by
aggregating 7 and as CT INFO & CCSA + be + ETC.
Learning Model functions (ha()), v())) are learned by minimizing the bound 5 via stochastic gradient descent on minibatches from D, with C for non-informative censoring and CFSA CCSA-INFO for informative censoring. Further, for the IPM regularization loss in (5), optimize the dual formulation of the Wasserstein distance, via the regularized optimal transport (Villani 2008
Cuturi 2013). Consequently, only require a - CCn, (3r, a) to be 1-Lipschitz (Shalit et al. 2017
and a 18 selected by grid search on the validation set using only factual data (details below).
4 METRICS
We propose a comprehensive evaluation approach that accounts for both factual and causal metrics. Factual survival outcome predictions are evaluated according to standard survival metrics that measure diverse performance characteristics, such concordance index (C-Index) (Harrell Jr al. 1984),
mean coefficient of variation (COV) and calibration slope (C-slope) (Chapfuwa et al. 2020). See the SM for more details these metrics. For causal metrics, defined below, we introduce nonparametric hazard ratio (HR) between treatment outcomes, and adopt the conventional precision in estimation of heterogeneous effect (PEHE) and average treatment effect (ATE) performance metrics (Hill 2011). Note that PEHE and ATE require ground truth counterfactual event times, which is only possible in jeetheisytthetic data. For HR, we compare our findings with those independently reported in the literature from gold-ston gold-standard RCT data.
Nonparametric Hazard Ratio In a medical setting, the population hazard ratio HR(() between treatment groups is considered informative thus has been widely used in drug development and RCT (Yusuf et al. 2016 Mihaylova et al. 20122). For example, HR(t) < 1, > 1, or ~ indicate population positive, negative and neutral treatment effects at time respectively. Moreover, HR(t) naturally accounts for both censored and non- outcomes. Standard approaches for computing HR(() rely on the restrictive proportional hazard assumption from CoxPH (Cox|1972) which constituted as semi-parametric linear model X(t(a) = (ts(t) exp(a3). However, the constant covariate (time independent) effect is often violated in practice (see Figure 26). For CoxPH, the marginal HR between treatment and control can be obtained from regression coefficiet 3 learned via maximum likelihood without the need for specifying the baseline hazard (t): HRCoxPH(t)
X(ta=1) exp(B). So x(
motivated, we propose a nonparametric, model-free approach for computing HR((). in which we do not assume parametric form for the event time distribution or the proportional hazard assumption from CoxPH. This approach only relies on samples from the conditional event time density functions, ((1(a)) and (tola)) via la h-) from 6
Definition 1
We define the nonparametric marginal Hazard Ratio and its approximation, HR(t), as
(t) So(t) (t)) HR(() = (1) (:(1) ()))))
SPKM (t) mi(t) HÍR()) = 3TKM(() mo(t)
(10)
where for Ht(t) we leveraged to obtain 10 and St(t) & ds(t)/ dt. The nonparametric assump- tion for S(t) makes the computation of S (t) challenging. Provided that S(t) is monotonically decreasing function, for simplicity, we fit a linear function S(t) = m t c, and set ('(t) ze m. Note that the linear model is only used for estimating S'(() from the nonparametric estimation of S(1).

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Table Performance comparisons on ACTG-SYNTHETIC data, with 95% HR(t) confidence interval. The ground truth, test set, hazard ratio is (t = (0.9,0.)))
Causal
Factual Mcan COV
Method
EPERE
FATB
HR
C-Index (A=0, A=l)
C-Slope (A=0, A=1)
CoxPH-Unifora
NA
NA
(0.
NA
NA
NA
CoxPH-IPPW
NA
NA 0.,
NA
NA
NA
CoxPH-OW
NA
NA 58.00...
NA
NA
NA
Surv-BART
352.07
77.89
0.0[0.0, 0.0)
(0.706, 0.686)
0.001
(0.398, 2c)
AFT-Wcibull
367.92
133.93
0.7770.47.0.4)
(0.21. 0.267)
6,209
(0.707, 0.729)
AFT-log-Normal
377.76
157.64
(.47:0.47.004))
(0.675, 0.556)
6.971
(0.707, 0.729)
SR
369.47
88.55
0.38
0
(0.791, 0.744)
(0.985, 1.027)
CSA (pmposeil)
358.72
6.8
0. 0.45 (9.00.99...65)
(0.787, 0.7675
0.T37
(0.983, 7.026]
CSA-INFO (proposed)
344.3
31.19
(....)
(0.78, 0.764)
0.13
(0.999, 1.029)
Bias from (t) can be reduced by considering more complex function approximations for S(t), e.g-- polynomial or spline. For the nonparametric estimation of S(t) we leverage the model-free popu- lation point-estimate-based nonparametric Kaplan-Meier (Kaplan & Meier 1958) estimator of the survival function $PKM (t) in Chapfuwa et al. (2020) to marginalize both factual and counterfuctual predictions given covariates x. The approximated hazard ratio, HR((), is thus obtained by combining the approximations SPKM (1) and ma. A similar formulation for the conditional, Hf(()z)), can also be derived. See the SM for full details on the evaluation H(((t) and Ht(+)=). Note that for some AFT or CoxPH-basee parametric formulations, HR Ht)z) can be readily evaluated because ff (talaz) and S(La) (x) are available in closed form.
In the experiments, we will use HR(() to compare different approaches against results reported in RCTS (see Tables and 3 Further, will use HR([[z) to illustrate stratified treatment effects (see Figure 2). Note that though a neural-based survival recommender system (Katzman et al.] 2018) has been previously used to estimate H))())) (x). their approach does not account for confounding or informative censoring thus it is susceptible to bias.
Precision in Estimation of Heterogeneous Effect (PEHE) A general individualized estimation error is formulated as EPEHE V x [(TEE (x) ITE()))3) where ITE(2) is the ground truth, TTE(2) Er [7 (Zi) 7 (To) X x] and 7() deterministic transformation. In our experi- ments, )) is the average over samples from te ~ Ph. AT æz. Alternative estimands, e.g., thresholding survival times (TTa) A(TA > T), can also be considered as described above.
Average Treatment Effect (ATE) The population treatment effect estimation error is defined as
CATE JATE- - AÎE|, where ATE = Ex [TE(()] (ground truth) and ATE = Ex [ITE())).
EXPERIMENTS
We describe the baselines and datasets that will be used to evaluate the proposed counterfactual survival analysis methods (CSA and CSA-INFO). Pytorch code including the new semi-synthetic dataset (see below) will be made publicly available. Throughout the experiments, we use the standard HR(() for CoxPH based methods and (10) for all others. The bound in (5) is sensitive to a, thus we propose approximating proxy counterfactual outcomes {YCF. 8CF} for the validation set, according to the covariate Euclidean neighour (NN) from the training set. We select the a that minimizes the validation loss L = LF + Ler from the set (0, 0.1, 1, 10, 100).
Baselines We consider the following competitive baseline approaches: (i) propensity weighted CoxPH (Schemper et al., 2009 Buchanan et al. 2014 Rosenbaum & Rubin 1983); (fi) IPM (5)
regularized AFT (FFg (log-Normal and Weibull) models; (Mv1) IPM (5) regularized deterministic semi- supervised regression (SR) model with accuracy objective from (Chapfuwa et al. 2018). as a contrast for the proposed stochastic predictors (CSA and CSA-INFO); and (2v) survival Bayesian additive regression trees (Sury-BART) (Sparapani et al. 2016). For CoxPH, we consider three normalized weighting schemes: (i) inverse probability weighting (IPW) (Horvitz Thompson
1952 Cao et
2009), where IPW; 2,
+ 1-2," ii) overlapping weights (OW) (Crump et al. 2006 Li et al. 2018 0s
where OW; = a; (1 - &) (1 a.) ê;; and dii the standard RCT uniform assumption. A simple linear logistic model é; = a (2r;; w), is used an approximation, é; to the unknown propensity score
P(A = X 2). See the SM for details the baselines.
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Table 3: Performance comparisons on FRAMINGHAM data, with 95% HR(() confidence interval.
Test set NN assignment of yCF and OCr yields biased HR(t) = 1. (7. while previous large
scale longitudinal RCT studies estimated HR(()= - 0. (.75 YYsuuf et al. 2016).
Causal
Factual Mean COV
Method
HR(()
C-Index (A=0, A=1)
C-Slope (A=0, A=1)
CoxPH-Uniform
1.69
NA
NA
NA
CoxPH-IPW
1.09
NA
NA
NA
CoxPH-OW
0.88 08)
NA
Surv-BART
(4.99.4. Be8)
(0.629, 0.630)
0,003
(0.232, 0.084)
AFT-Weibull
1.09 (10...00).00
(0.734, 0.395)
8.609
(0.857. 0.89)
AFT-log-Normal
1.55
(0.68, 0.56)
10.415
(0.979. 0.732)
SR
(0.601, 0.57)
0
(0.491, 0.63)
CSA (proposed)
g.04
(0.763, 0.728)
0,161
(0.891, 0.81)
CSA-INFO (proposed)
0.81 (0.77.083)
(0.752, 0.651)
0.156
(0.907, 0.881)
Datasets We consider the following datasets: (1)
Table 2: Summary statisties of the datasets.
FRAMINGHAM, is EHR- longitudinal car- diovascular cohort study that we use to evaluate the
FRAMINGHAM
ACTG
ACTO-SYNTHETIC
Events (%) Treatment (%)
26.0
26.9
48.9
10.4
49.5
55.9
effect of statins future coronary heart disease out-
3,435
1,054
2,139
comes (Benjamin et al. 1994]; (&i) ACTG, is a longi-
23
Missing (%)
0.23
1.41
1,38
tudinal RCT study comparing monotherapy with Zi- dovudine or Didanosine with combination therapy in
(days)
7,279
1,231
1,313
HIV patients (Hammer et al. 1996); and (fii) ACTG-
SYNTHETIC, is a semt-synthetic dataset based on ACTG covariates, We simulate potential outcomes according Gompertz-Coo Cox distribution (Bender et al. 2005) with selection bias from a simple logistic model for P(A 1|X- x)) and AFT-based censoring mechanism. The generative process detailed in the SM. Table 2 summarizes the datasets according to (1) covariates of size p; (ii) proportion of non-censored events, treated units, and missing entries in the N p covariate matrix; and (100) time range max for both censored and non-censored events. Missing entries are imputed with median mode if continuous or categorical, respectively.
Quantitative Results Experimental results for two data-sets in Tables and annd illustrate that AFT- based methods are high variance, inferior in calibration and C-Index than accuracy-basse -based methods (SR, CSA, CSA-INFO). Surv-BART is the least calibrated but low variance method. CSA-INFO and CSA outperform all methods across all factual metrics, whereas CSA-INFO is better calibrated, low variance but slightly lower C-Index than CSA. Note that we fit CoxPH using the entire dataset; since it does not support counterfactual inference, we do not present factual metrics. By properly adjusting for both informative censoring and selection bias, CSA-INFO significantly outperforms all methods in treatment effect estimation according HR(() and ÉPEHE, across non-RCT datasets, while remaining comparable to AFT-Weibul on the RCT dataset (see the SM). Further, RCT-based results on ACTG data in the SM illustrate comparable HR(t) across all models except for AFT-log-Nonen and Surv-BART, which overestimate, and SR, which underestimates risk. For RRT datasets (ACTG-SYNTHENTIC and FRAMINGHAM), CoxPH-OW has clear advantage over all CoxPH based methods, mostly credited to the well-behaved bounded propensity weights E [0, 1]. Interestingly, the FRAMINGIIAM observational data exhibits a common paradox, where without proper adjustment of selection and censoring bias, naive approaches would result in counter-intuitive treatment effect from statins. However, there is severe confounding from covariates such as age, BMI, diabetes, CAD, PAD, MI, stroke, etc., that influence both treatment likelihood and survival time. Table 3. demonstrates that CSA-INFO is elearly the best performing approach. Specifically, its HR(t), reverses the biased observational treatment effect, demonstrate positive treatment from statins, which is consistent with prior large RCT longitudinal findings (Yusuf et al.) 2016).
Qualitative Results Figure 2a demonstrates that CSA-INFO matches the ground truth population hazard, HR(t), better than alternative methods on ACTG-SYNTHETES data. See the SM for ACTG and FRAMINGHAM. Figure 2b shows sub- poppllttulaoonn log hazard ratios for four patient clusters obtained via hierarchical clustering on the individual log hazard ratios, log HR(()ar, of the test set of FRAMINGHAA data. Interestingly, these clusters stratify treatment effects into: positive (2), negative (1 and 3), and neutral (4) uub-populations Moreover, the estimated density of median log HR (t)x) values in Figure 2c illustrates that nearly 70% of the testing set individuals have log HR(f)z) < O. thus may benefit from taking statins. Further, we isolated the extreme top and bottom quantiles, HR([)æ) < 0.024 and HR([)x) 1.916, respectively, of the median log HR(()z) values for the
8

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Under review as a conference paper at ICLR 2021
6.00
(17)
(25k)
A12
0.0
CLA-NFCC
(45)
(202)
Average (513)
100
RCT
806
640
104
640
102
301 1213
1ime (days)
Time (lays)
media log lilttbk
(a) ACTG-SYNTHENTIC HR((t)
(b) FRAMINGHAM log HR(+) (c) FRAMINGHAM log HR pdf
Figure 2: (a) Inferred population HR(()) compared against ground truth (EMP) on ACTG-SYNTHETIC data. ASA-INNFObaare (b) cluster-specific average log (t x) curves and (c) estimated density of median log H)(+ values on the test set of the FRAMINGHAM dataset. Clusters assignment were obtained via hierarchical clustering of individualized log Ht)+ traces.
test set of FRAMINGHAM, as shown in Figure 2ec After comparing their covariates, we found that individuals with the following characteristics may benefit from taking statins: young, male, diabetic, without prior history (CAD, PAD, stroke or MI), high BMI, cholesterol, triglycerides, fasting glucose, and low high-density lipoprotein. There seem to be consensus that diabetics and high-cholesterol patients benefit from statins (Cheung et al. 2004 Wilt et al. 2004). See SM for additional results.
CONCLUSIONS
We have proposed unified counterfactual inference framework for survival analysis. Our approach adjusts for bias from two unknown sources, namely, confounding due to covariate dependent selection bias and censoring (informative or non-informative). Relative to competitive alternatives, we demon- strate superior performance for both survival-outcome prediction and treatment- estimation, across three diverse datasets, including semi-synthetic dataset which introduce. Moreover, we formulate a model-free nonparametric hazard ratio metric for comparing treatment effects or leveraging prior randomized real-world experiments in longitudinal studies.

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Under review as a conference paper at ICLR 2021
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13


Reviews:
[{"review_id": "v3wYBvhfQHG", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 4, "confidence": 5, "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": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "Summary:\nThis paper provides an approach for causal inference in observational survival dataset in which the outcome is of time-to-event type with right-censored samples. The method consists of a representation learning component to reduce selection bias and a survival analysis component which is modeled with normalizing flows. \n\nPros:\n- The paper addresses an important and interesting question.\n- The manuscript is well-written and easy to understand.\n\nCons:\n- My main concern is the minimal originality and significance of this work. That is, the representation learning component is directly taken from (Shalit et al., 2017) and the objective function for the survival analysis component is directly taken from (Chapfuwa et al., 2018). Also, the use of normalizing flows for modelling time-to-event kind of targets is not well motivated. \n- The literature review only points to several publications but does not go in depth into why/how the proposed method differs from those. There are also some important references that are missing. For example, Miscouridou et al. (2018) also use normalizing flows for survival analysis and it is necessary that the authors discuss how their work differs from theirs.\nMiscouridou, X., Perotte, A., Elhadad, N., & Ranganath, R. (2018). Deep survival analysis: Nonparametrics and missingness. In Machine Learning for Healthcare Conference\n- the paper suffers from many inaccurate statements; a few examples follow:\n1. In the third paragraph of the Introduction section, the authors mention that “the treatment assignment mechanism is not known a priori. Therefore, there may be variables, known as confounders, affecting both the treatment and survival time, which lead to selection bias”. This is wrong; even if we know the treatment assignment policy a priori, we still might have selection bias … these two are independent.\n2. In the last sentence of paragraph four of Introduction, the authors say that the methods (cited above) that account for confounding bias (by re-weighting) lack a counterfactual prediction mechanism. This is wrong because trivially, all methods have a prediction mechanism in place, and the ones that do account for confounding bias can predict counterfactuals accurately as well.\n3. The authors should note that representation learning does not **remove** confounding bias; it might only **reduce** it. Also, reweighting does not **remove** confounding bias either; it just **accounts for** it.\n\nMinor:\n- In section 2, under *Estimands of Interest*, the authors state that “$\\lambda(t | x)$ is defined below” but it’s never defined.\n- In section 3, under *Accounting for selection bias*, the authors state that we can go from Eq. (1) to (2) because identifiability holds (since “$X$ is a sufficient set from $A$ into $T$.”). Could you please explain how this is different from Ignorability? I.e., $\\{ T_0, T1 \\} \\perp A | X$ ?", "extra_scores": {}, "extra_text": {}}, {"review_id": "ncuqnOmWwsy", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "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": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "**Summary and key claims**\n\nThis paper repurposes the balanced representation learning framework for estimating treatment effects, originally proposed in (Shalit et al., 2017), for the survival prediction setup. The proposed model deals with selection bias induced by confounded treatment variables, and in addition, deals with censoring bias and informative censoring.\n\n*The key contributions claimed by the paper are:*\n- Developing a loss function incorporating adjustments for informative censoring and selection bias.\n- Developing a generative model for event times based on planar normalizing flows. \n- Proposal of survival-specific evaluation metrics, including a new nonparametric hazard ratio estimator.\n\n**Originality and significance**\n\nOverall, I think that the paper is a straightforward application of the balanced representation method in (Shalit et al., 2017) to survival outcomes. It does not seem like survival prediction is any different from the conventional ITE setup with respect to the representation learning aspect of the model, hence I don't think that the paper contributes methodologically to the problem of handling selection bias. Moreover, the censoring terms in the loss function are also very similar to those introduced previously in Chapfuwa et al ICML 2018 paper. Based on this, I think that the extent of technical contribution in the paper does not pass the acceptance threshold.\n\nI was expecting some more analysis on the interplay between censoring bias, selection bias and the effect of both on causal identifiability. Unfortunately, it seems that authors chose to address each of these impairments/biases separately using existing solutions, and the resulting model is simply an amalgamation of existing ideas. \n\nThe \"generative modeling\" part of the model is somehow alien to the original problem of estimating treatment effects on survival outcomes. It is not clear why planar flows were specifically used and why flows are needed at all since the complexity needed in modeling is in the relation between features and survival parameters and not the complexity of the survival distribution itself. The usage of normalizing flows on the output layer is fine but seems to me unnecessary; this reinforces my impression of the model being all over the place.\n\nOn the positive side, I think that the problem of estimating ITEs on survival is very important and rarely addressed. Most ML models for ITEs focus on real-valued targets, but this is rarely the relevant setup in practice as survival is often the measure of treatment efficacy in medicine. I also think that the idea of comparing the estimated HRs of RCTs with the ones recovered by the model is a very smart way to evaluate counterfactual inferences, and can be a useful evaluation metric for future papers.\n\n\n\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "lJk5ROlM9I2", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 7, "confidence": 4, "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": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper is very well written. The motivation and formalism is also clear with every step in the argument properly justified. The extension of individualized treatment effects to survival data is in some sense straightforward as both areas are quite mature and can be unified with aggregated loss functions dealing with biases of different type. The proposed solution, metrics and datasets proposed for this problem are compelling though and I believe will serve as a benchmark for further studies on treatment effects and survival data.\n\nOne question I have is on Corollary 1. I don't see a meaningful difference between this statement and that given by (Shalit et al, 2017), nor a proof in the Appendix. Why the separate statement?\n\nThe experiments are somewhat underwhelming. Survival BART has also been developed for treatment effect estimation [1]. This benchmark is certainly more relevant. Similarly, Survival-based deep learning architectures have been developed which could have been considered as well [2]. At least considering these, perhaps modelling each treatment group separately or including treatment as an additional feature should be considered to understand where the source of gain comes from. As presented, since almost all benchmarks consider linear interaction between features I would guess that improvements come from non-linear modelling rather than bias reduction from censoring and selection bias. \n\n[1] Hu, Liangyuan, Jiayi Ji, and Fan Li. \"Estimating Heterogeneous Survival Treatment Effect via Machine/Deep Learning Methods in Observational Studies.\" arXiv preprint arXiv:2008.07044 (2020).\n[2] Lee, Changhee, et al. \"DeepHit: A Deep Learning Approach to Survival Analysis With Competing Risks.\" AAAI. 2018.", "extra_scores": {}, "extra_text": {}}, {"review_id": "dAyVns7P2pW", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 7, "confidence": 4, "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": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "Disclosure: I found this paper online during review process https://arxiv.org/abs/2006.07756\n\n\nThis is a comprehensive paper with interesting application of counterfactual inference under survival analysis setting. Overall, my recommendation is to accept. \n•\tIt nice that the proposed nonparametric approach in this paper can adjusts for bias from confounding due to covariate dependent selection bias and censoring (informative or non-informative). \n•\tUnder three criterions [concordance index (C-Index) (Harrell Jr et al., 1984), mean coefficient of variation (COV) and calibration slope (C-slope) (Chapfuwa et al., 2019)] and three datasets [FRAMINGHAM, ACTG, semi synthetic ACTG], compared proposed method with 7 seven others, including survival Bayesian additive regression trees (Surv-BART) (Sparapani et al., 2016) [using nonparametric Kaplan-Meier based estimator] and Cox proportional hazard model (using real HR form, using three normalized weighting schemes). \n•\tP6, equation (10), the nonparametric form is a natural adoption of KM estimator. We know S^{‘}=-f(t). Wondering the motivation of choosing a linear approximation to S, and curious would the cardiovascular and HIV data adopted happen to be with S not so curved? Could you shed light on these?\n•\tP3, assumption of “no unobserved confounders or ignorability” sounds strong. Understand the mathematical challenge if relaxing it. Maybe for future research.\n•\tThe overall presentation is nice. The organization of a few places might be improved to help first time reader to follow, eg. ITE initially defined on P3 without example till two paragraphs below, h_{A} first mentioned with no prior definition and no explicit math relation to p(T|X), briefly specify “Do (A=a)” is for effect of intervention.  \n•\tMinor issues like align the symbols used across the paper, eg. add subscript when define S^{‘}_{i},  m_{i}, ... i=0, 1 to increase clarity. \n", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: Summary: This paper provides an approach for causal inference in observational survival dataset in which the outcome is of time-to-event type with right-censored samples.  To this end, the paper adapts the balanced representation learning approach proposed in (Shalit et al, 2017) to the context of survival analysis.
The paper adapts an approach that uses flexible models to learn nuisance models, common in machine learning.

The authors validated their approach via simulation study and a set of application datasets: a EHR-based cohort study of cardio-vascular health, an RCT dataset of HIV patients, and a semi-synthetic dataset.

The main concerns of reviewers were due to perceived lack of originality relative to the original proposal in (Shalit et al, 2017)

Decision: Reject