Estimating heterogeneous treatment eects versus building individualized treatment rules Connection and disconnection

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Estimating heterogeneous treatment eﬀects

versus building individualized treatment rules:

Connection and disconnection

Zhongyuan Chen and Jun Xie

Department of Statistics, Purdue University

150 N. University Street, West Lafayette, IN 47907

junxie@purdue.edu

October 5, 2022

Abstract

Estimating heterogeneous treatment eﬀects is a well studied topic in the statistics lit-

erature. More recently, it has regained attention due to an increasing need for precision

medicine as well as the increased use of state-of-art machine learning methods in the estima-

tion. Furthermore, estimating heterogeneous treatment eﬀects is directly related to building

an individualized treatment rule, which is a decision rule of treatment according to patient

characteristics. This paper examines the connection and disconnection between these two

research problems. Notably, a better estimation of the heterogeneous treatment eﬀects may

or may not lead to a better individualized treatment rule. We provide theoretical frame-

works to explain the connection and disconnection and demonstrate two diﬀerent scenarios

through simulations. Our conclusion sheds light on a practical guide that under certain cir-

cumstances, there is no need to enhance estimation of the treatment eﬀects, as it does not

alter the treatment decision.

Keywords: Heterogeneous treatment eﬀects, Individualized treatment rules, Mean squared error,

classiﬁcation error

arXiv:2210.01342v1 [stat.ME] 4 Oct 2022

1 Introduction

Estimation of heterogeneous treatment eﬀects (HTE) is a research problem commonly raised in

many ﬁelds, including politics, economics, education, and healthcare. Instead of an overall treat-

ment eﬀect, heterogeneity exists for subgroups or individuals within a population. We represent

HTE by the conditional average treatment eﬀect (CATE) function given a set of covariates. There

is a rich literature on estimating the CATE, mostly through regression methods, as well as more

recently developed machine learning algorithms [Kunzel 2019]. In theory, the accuracy of a CATE

estimator is measured by the expected mean squared error (EMSE), which is based on the quadratic

loss function. The convergence rate of the EMSE can be used to compare diﬀerent CATE estima-

tors.

Estimating heterogeneous treatment eﬀects is critical for making medical decisions, such as what

treatment to recommend. Building personalized treatment, also referred to as an individualized

treatment rule (ITR) [Qian and Murphy 2011], is usually done through estimating the CATE ﬁrst

and then deﬁning the optimal ITR as the sign function of the CATE (for two treatment options

denoted as 1 and −1). [Chen et al. 2022] provided a good introduction on building ITRs and

called the aforementioned method an indirect approach. Although related, estimating the CATE

and building optimal ITRs are separately studied in the literature. In this paper, we elaborate upon

the connection and disconnection between the two research areas with mathematical frameworks.

In Section 2, we review recent developments on the estimation of the CATE, including the use of

a machine learning method named X-learner [Kunzel 2019], which can improve estimation accuracy

with a faster rate of convergence than other conventional estimators. In Section 3, we connect

estimating the CATE to building an ITR and show that the optimal ITR is indeed the sign function

of the true CATE function. With this relationship, it is natural to expect that better estimators,

in terms of smaller EMSE, lead to improved ITRs. This, however, does not always happen due

to the mismatch between the loss functions of these two research problems. More speciﬁcally, we

use the quadratic loss function for estimation of the CATE but the 0-1 loss function for comparing

ITRs. In Section 4, we examine the disconnection through a mathematical framework. That is,

for many cases, improving the CATE estimators does not change the corresponding ITRs. Our

conclusion sheds light on a practical guide that in certain situations, there is no need to enhance

estimation of the CATE, as it does not alter the treatment decision. Section 5 provides simulation

examples to display the connection and disconnection.

2 Recent development on estimation of HTE

Suppose we have data from a two-arm clinical trial with (X, A, Y ), where Y∈Rdenotes a treat-

ment response variable (the larger value the better), X∈X⊂Rpis a set of covariates, and

A∈ A ={−1,1}denotes the treatment index corresponding to the control or treatment arm. We

assume (X, A, Y )∼ P, where Pis the distribution from a speciﬁc family. Denote the conditional

mean E(Y|X, A). Deﬁne

µ0(x) = E(Y|X=x, A =−1) and µ1(x) = E(Y|X=x, A = 1).

Then, the CATE function is τ(x) = µ1(x)−µ0(x). Let ˆτ(x) be an estimator of the CATE from

a set of independent random data from P. We are interested in estimators with a small expected

mean squared error (EMSE):

EMSE(P,ˆτ) = E[(ˆτ(X)−τ(X))2],

where the expectation is taken over ˆτand X, which are assumed independent of each other, e.g.,

ˆτis estimated from a training data set and Xdenotes a new data.

Various methods are available to estimate the CATE. The most commonly used one is to ﬁt re-

gression models for ˆµ1(x) and ˆµ0(x), and then ˆτ(x) = ˆµ1(x)−ˆµ0(x). As an addition to the rich litera-

ture, [Chen et al. 2022] considered very general regression models and applied dimension reduction

for high-dimensional covariates. Moreover, supervised learning algorithms are used to estimate ˆµ1

and ˆµ0by the machine learning community [Hu et al. 2021], including Bayesian Additive Regres-

sion Trees (BART) [Chipman et al. 2010] and Random Forest (RF) [Wager and Athey 2018]. One

of the most recent developments is an algorithm called X-learner [Kunzel 2019], which speciﬁcally

exploits structural properties of the CATE function for an improved estimator.

Following the results of [Kunzel 2019], we use the convergence rates of the EMSE to compare

diﬀerent CATE estimators. Suppose we observe independent and identically distributed data

(Xi, Ai, Yi)∼ P,i= 1, . . . , N, with mcontrol units and ntreated units, N=m+n. Let ndenote

the smaller sample size of the two treatment arms and assume mand nhave a similar scale. Most

of the estimation methods have a convergence rate that depends on the estimators of ˆµ1and ˆµ0.

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Estimatingheterogeneoustreatmenteectsversusbuildingindividualizedtreatmentrules:ConnectionanddisconnectionZhongyuanChenandJunXieDepartmentofStatistics,PurdueUniversity150N.UniversityStreet,WestLafayette,IN47907junxie@purdue.eduOctober5,2022AbstractEstimatingheterogeneoustreatmenteectsisawellstudie...

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