Estimating optimal treatment regimes in survival contexts using an instrumental variable Junwen Xia1 Zishu Zhan1 and Jingxiao Zhang2

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Estimating optimal treatment regimes in survival contexts using

an instrumental variable

Junwen Xia1, Zishu Zhan1, and Jingxiao Zhang2∗

1School of Statistics, Renmin University of China, Beijing, China

2Center for Applied Statistics, School of Statistics, Renmin University of China, Beijing, China

∗email: zhjxiaoruc@163.com

Abstract

In survival contexts, substantial literature exists on estimating optimal treatment regimes, where

treatments are assigned based on personal characteristics for the purpose of maximizing the survival

probability. These methods assume that a set of covariates is suﬃcient to deconfound the treatment-

outcome relationship. Nevertheless, the assumption can be limited in observational studies or ran-

domized trials in which non-adherence occurs. Thus, we propose a novel approach for estimating the

optimal treatment regime when certain confounders are not observable and a binary instrumental

variable is available. Speciﬁcally, via a binary instrumental variable, we propose two semiparamet-

ric estimators for the optimal treatment regime by maximizing Kaplan-Meier-like estimators within

a pre-deﬁned class of regimes, one of which possesses the desirable property of double robustness.

Because the Kaplan-Meier-like estimators are jagged, we incorporate kernel smoothing methods to

enhance their performance. Under appropriate regularity conditions, the asymptotic properties are

rigorously established. Furthermore, the ﬁnite sample performance is assessed through simulation

studies. Finally, we exemplify our method using data from the National Cancer Institute’s (NCI)

prostate, lung, colorectal, and ovarian cancer screening trial.

Keywords: Instrumental variable, Optimal treatment regime, Survival data, Unmeasured confound-

ing.

1 Introduction

In clinical settings, allocating all patients with the same treatment may be sub-optimal since diﬀerent

subgroups may beneﬁt from diﬀerent treatments. The objective of optimal treatment regimes is to

assign treatments based on individual characteristics, thereby attaining maximum clinical outcomes

arXiv:2210.05538v5 [stat.ME] 30 Oct 2023

or so-called values. Among the various clinical outcomes, survival probability holds particular sig-

niﬁcance in complex diseases such as cancer. Previous work was built on a critical assumption that

no unmeasured confounder exists so that the relationship between treatment and outcomes can be

deconfounded by observed covariates (Geng et al.,2015;Bai et al.,2017;Jiang et al.,2017;Zhou

et al.,2022). However, the application of the assumption in observational studies or randomized

trials with non-adherence issues is overly restrictive. For example, in an intervention experiment

aimed at assessing the eﬀect of a screening test or a drug, relatively healthy individuals may be more

likely to skip the intervention (Kianian et al.,2021;Lee et al.,2021). However, they are also more

likely to have better clinical outcomes, which makes the health condition an unmeasured confounder.

In an encouragement experiment or an observational study, the unmeasured confounding eﬀect be-

comes more serious. For example, socio-economic status would act as an important unmeasured

confounder, causally aﬀecting the patients’ access to treatment and their clinical outcomes (Louizos

et al.,2017).

In causal inference, an alternative variable called an instrumental variable (IV) can be utilized

to reveal causality between treatment and outcomes. An IV is a pretreatment variable that is inde-

pendent of all unmeasured confounding factors and exerts its eﬀect on outcomes exclusively through

treatment. Common IVs include assignments in randomized trials with possible non-adherence, ge-

netic variants known to be associated with the phenotypes, calendar periods as a determinant of

patients’ access to treatment, and treatment patterns varied by hospital referral regions. (Lee et al.,

2021;Ying et al.,2019;Mack et al.,2013;Tchetgen Tchetgen et al.,2015) There has always been

interest in incorporating an IV into survival contexts. IV approaches have been developed using a

speciﬁc semiparametric model, such as the additive hazards model (Li et al.,2014;Tchetgen et al.,

2015) and the Cox proportional hazards model (Tchetgen et al.,2015;Wang et al.,2022). However,

these model choices were mostly made to avoid theoretical diﬃculties rather than a prior knowledge

of survival distributions. And they focused on homogeneous treatment eﬀects that correspond to

static regimes. Another line starts with the seminar papers by Imbens and Angrist (1994), Angrist

et al. (1996). They focused on estimating local treatment eﬀects, deﬁned as treatment eﬀects for

compliers who would always adhere to their treatment assignments. In particular, Lee et al. (2021)

proposed a nonparametric doubly robust estimator to estimate the local treatment eﬀect on survival

probability. However, we are interested in the treatment eﬀect of the population when allocating

treatment. To the best of our knowledge, this article is the ﬁrst to go beyond specifying a survival

model to identify the treatment eﬀect (and assign treatment) using an IV.

With the rapid development of IVs in causal inference, recently, there has been much interest in

using an IV to estimate optimal treatment regimes, but existing studies are not suitable for survival

contexts. Cui and Tchetgen Tchetgen (2021b) introduced an IV to estimate the optimal treatment

regime from a classiﬁcation perspective. At the same time, Qiu et al. (2021a) studied the problem

by stochastic regimes. Zhang and Pu (2021), Han (2021a), and Qiu et al. (2021b) discussed the IV

assumptions made in the literature (Cui and Tchetgen Tchetgen,2021b;Qiu et al.,2021a). In cases

where these assumptions are questionable, alternative assumptions such as the assumption A (Han,

2021a), the assumption made in Theorem 3.1 (Cui and Tchetgen Tchetgen,2021a), and the lower

bound of the value function (Cui and Tchetgen Tchetgen,2021b;Han,2021b;Pu and Zhang,2021;

Chen and Zhang,2021) may be employed.

The prostate, lung, colorectal, and ovarian (PLCO) cancer screening trial (Team et al.,2000), a

two-arm randomized trial examining screening tests for PLCO cancers, inspires this work. Between

November 1993 and July 2001, ten centers recruited participants across the U.S. and collected the

data up to 2015. We will focus on determining the optimal assignment of ﬂexible sigmoidoscopy

screening (a screening test for colorectal cancer) to increase life expectancy. It is important to per-

sonalize the ﬂexible sigmoidoscopy screening as not all individuals will experience the desired beneﬁt

(Tang et al.,2015). Additionally, there is non-adherence to the screening in the trial, which could be

inﬂuenced by a range of unmeasured confounding factors (Lee et al.,2021;Kianian et al.,2021). For

example, relatively healthy individuals may be more likely to skip the screening. When unmeasured

confounders are present, the estimated optimal treatment regimes based on the assumption of their

absence become unreliable. An IV that eﬀectively compensates for the bias caused by unmeasured

confounding eﬀects can be employed to address this issue. In our case, the assigned treatment serves

as a suitable IV.

Our study extends the aforementioned literature on survival contexts by adapting and generalizing

the value search method (Jiang et al.,2017;Zhou et al.,2022;Zhang et al.,2012), which derives the

optimal treatment regime by maximizing a consistent estimator of the value function under a class of

regimes. Speciﬁcally, we introduce a novel inverse-weighted Kaplan-Meier estimator (IWKME) and a

corresponding semiparametric estimator for the optimal treatment regime that permit the utilization

of an IV to overcome unmeasured confounding. Additionally, we propose their doubly robust versions

to enhance resistance to model misspeciﬁcation. The contributions of this article are summarized as

follows. First, we propose novel estimators to identify the treatment eﬀects and assign treatments

for the time-to-event data, which go beyond specifying a Cox proportional hazards model or other

semiparametric models. Compared with the estimators in Cui and Tchetgen Tchetgen (2021b) who

considered the non-censored data, we add an extra term to ensure that our doubly robust estimator

remains consistent even when the conditional survival function of the censoring time is misspeciﬁed,

which distinguishes our estimator from theirs. As far as we know, it is the ﬁrst doubly robust

estimator to identify the treatment eﬀects using an IV in survival contexts. Previous doubly robust

estimators are based on the no the unmeasured confounding assumption (Bai et al.,2017) or focus on

the treatment eﬀect among compliers (Lee et al.,2021). Second, we consider the smoothing approach

to improving the performances of the estimators. The value function is non-smoothed with respect to

the parameters of the treatment regimes, which yields computational challenges in the optimization.

The kernel smoothing technique solves the challenges eﬃciently and maintains consistency as our

asymptotic result illustrates. Third, we prove theoretical guarantees for our proposed estimators.

Speciﬁcally, we provide an asymptotic bound for our estimator, which highlights the advantages of

the doubly robust estimator: it is not only consistent if one of the models is correctly speciﬁed,

but can also incorporate some semiparametric models or nonparametric models to have a root-n

consistency. This allows us to choose the machine learning method when estimating the models.

The remaining sections of the article are arranged as follows. Section 2 presents the mathematical

foundations and estimators for utilizing an IV in determining optimal treatment regimes, including

the doubly robust and kernel-smoothed estimators. Section 3 oﬀers the asymptotic properties of these

estimators. The ﬁnite sample performance is explored through simulations in Section 4. Section 5

illustrates the application of the prostate, lung, colorectal, and ovarian (PLCO) cancer trial dataset.

Proofs and additional numerical studies can be found in the supporting information. In addition,

our proposed method is implemented using R, and the R package otrKM to reproduce our results is

available at https://cran.r-project.org/web/packages/otrKM/index.html.

2 Methodology

2.1 Notation and data structure

Consider a binary treatment indicator A∈ {0,1}, an unmeasured confounder (possibly a vector-

value) U, a binary IV Z∈ {0,1}used to debias unmeasured confounding eﬀects, and a set of fully

observed covariates L(a vector with pdimensions). An individual treament regime is a function map

d(·)∈ D from patients’ baseline covariates Lto treatment 0 or 1 so that the patient with baseline

covariates L=lwould receive treatment 0 if d(l) = 0 or treatment 1 if d(l) = 1. For simplicity,

assume D={dη:dη(L) = I{˜

L⊤η⩾0},∥η∥2= 1}, where ˜

L= (1,L⊤)⊤and ∥η∥2=qPp+1

i=1 η2

However, it is also applicable to any other Dindexed by ﬁnite-dimensional parameters. Let T

be the continuous survival time of interest, with the conditional survival function ST(t|Z, L, A) =

P(T⩾t|Z, L, A) and the corresponding conditional cumulative hazard function ΛT(t|Z, L, A) =

−log{ST(t|Z, L, A)}. Due to limited time and refusals to answer, the survival time Tmay not be

observable; in such cases, the censoring time Creplaces T. Thus, the observation is ˜

T= min{T, C}

and the indicator of the censoring status δ=I{T⩽C}. In this article, we assume the complete

data {(Ui,Li, Ai, Zi,˜

Ti, δi), i = 1,...,n}are independent and identically distributed across i.

The counting process is deﬁned as N(t) = I{˜

T⩽t, δ = 1}, and the at-risk process is denoted

by Y(t) = I{˜

T⩾t}. The counting process of the censoring time is deﬁned as NC(t) = I{˜

T⩽

t, δ = 0}; the at-risk process of the censoring time is denoted by YC(t) = I{˜

T⩾t}. It is notable

that YC(t) = Y(t), and we will alternatively use Y(t) and YC(t) for better clariﬁcation. Potential

results are denoted with a superscript star. Given treatment A=a, let T∗(a) denote the potential

survival time, N∗(a;t) = I[min {T∗(a), C}⩽t, T ∗(a)⩽C] denote the potential counting process,

and Y∗(a;t) = I[min {T∗(a), C}⩾t] denote the potential at-risk process. Under the regime dη,

deﬁne the potential survival time as T∗(dη(L)) = T∗(1)I{dη(L) = 1}+T∗(0)I{dη(L) = 0}, the

potential counting process as N∗(dη(L); t) = N∗(1; t)I{dη(L) = 1}+N∗(0; t)I{dη(L) = 0}, and

the potential at-risk process as Y∗(dη(L); t) = Y∗(1; t)I{dη(L)=1}+Y∗(0; t)I{dη(L)=0}. We

can also utilize the potential censoring time C∗(a) to deﬁne the potential counting process and the

at-risk process. These two deﬁnitions will produce an identical estimator with a slightly diﬀerent

assumption. Further details and explanations can be found in Remark 2.1.

We now introduce some notations to facilitate our analysis of the asymptotic behavior. For two

sequences of random variables {Xn}∞

n=1 and {Yn}∞

n=1, the notations Xn≲pYnor Xn=Op(Yn)

indicate that for any ϵ > 0, there exists a positive constant Msuch that supnP(|Xn/Yn|> M )⩽ϵ.

The notation Xn=op(Yn) means that |Xn/Yn|converges to 0 in probability. The expectation of

a random variable Xis either PXor EX, and the sample mean of Xwith nsamples is PnX. We

will alternatively use PXand EX for better clariﬁcation. For a function gof a random variable X,

deﬁne ∥g(X)∥=qR{g(x)}2dP (x).

The survival function P(T > t) is of interest in the classic time-to-event data, while in the

context of the optimal treatment regime, potential survival function S∗(t;η) = P[T∗(dη(L)) > t]

under regime dηis of primary interest. For a predetermined t,S∗(t;η) as a value function provides

a basis for the deﬁnition of the optimal treatment regime.

dopt

η= arg max

S∗(t;η) = I(˜

L⊤ηopt ⩾0),(1)

where

ηopt = arg max

∥η∥2=1

S∗(t;η).(2)

In this article, despite the optimality of the regime being deﬁned by the largest t-year survival

probability, other value functions, such as the restricted mean (Geng et al.,2015) and the quantile

(Zhou et al.,2022) of the survival time, are also available. We defer the discussion in Section 6.

2.2 An estimator of the potential survival function with unmeasured

confounders

A consistent estimator for S∗(t;η) is necessary to estimate the optimal treatment regime. The

uninformative censoring assumption is usually utilized to describe the correlation between the survival

time and the censoring time. That is, the potential survival time is conditionally independent of the

censoring time given the observed covariates.

A1: (Uninformative censoring). T∗(a)⊥⊥ C|Z, L, A for a= 0,1.

Let SC{s|Z, L, A}=P(C⩾s|Z, L, A) denote the conditional survival function of censoring time

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EstimatingoptimaltreatmentregimesinsurvivalcontextsusinganinstrumentalvariableJunwenXia1,ZishuZhan1,andJingxiaoZhang2∗1SchoolofStatistics,RenminUniversityofChina,Beijing,China2CenterforAppliedStatistics,SchoolofStatistics,RenminUniversityofChina,Beijing,China∗email:zhjxiaoruc@163.comAbstractInsurviv...

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Estimating optimal treatment regimes in survival contexts using an instrumental variable Junwen Xia1 Zishu Zhan1 and Jingxiao Zhang2

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