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RESEARCH ARTICLE
AteMeVs: An R package for the estimation of
the average treatment effect with
measurement error and variable selection for
confounders
Li-Pang ChenID
1
, Grace Y. Yi
2
*
1Department of Statistics, National Chengchi University, Taipei, Taiwan, ROC, 2Department of Statistical
and Actuarial Sciences, Department of Computer Science, University of Western Ontario, London, Canada
*gyi5@uwo.ca
Abstract
In causal inference, the estimation of the average treatment effect is often of interest. For
example, in cancer research, an interesting question is to assess the effects of the chemo-
therapy treatment on cancer, with the information of gene expressions taken into account.
Two crucial challenges in this analysis involve addressing measurement error in gene
expressions and handling noninformative gene expressions. While analytical methods have
been developed to address those challenges, no user-friendly computational software pack-
ages seem to be available to implement those methods. To close this gap, we develop an R
package, called AteMeVs, to estimate the average treatment effect using the inverse-prob-
ability-weighting estimation method to handle data with both measurement error and spuri-
ous variables. This developed package accommodates the method proposed by Yi and
Chen (2023) as a special case, and further extends its application to a broader scope. The
usage of the developed R package is illustrated by applying it to analyze a cancer dataset
with information of gene expressions.
1 Introduction
Bioinformatics has revealed that cancer stems from a genetic disorder, deiven by genetic varia-
tions that lead to the abnormally dysfunction of genes and their altered expressions (e.g., [1]).
Accurate assessment of gene expression levels becomes crucial for cancer diagnosis and treat-
ment. Chemotherapy is a commonly used approach in cancer treatment as it often effectively
eradicates malignant cells. In particular, the integration of targeted therapy with chemotherapy
is frequently used to control the growth, division, and spread of cancer cells (e.g., [2]). How-
ever, due to its lack of specificity in targeting cancer cells, chemotherapy drugs can impact
both cancer cells and healthy cells, which leads to significant side effects. One concern is
whether employing chemotherapy is more beneficial than taking alternative treatments that
avoid it. We are interested in studying whether taking chemotherapy has a causal effect on
increasing the survival of cancer patients.
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OPEN ACCESS
Citation: Chen L-P, Yi GY (2024) AteMeVs: An R
package for the estimation of the average treatment
effect with measurement error and variable
selection for confounders. PLoS ONE 19(9):
e0296951. https://doi.org/10.1371/journal.
pone.0296951
Editor: Suyan Tian, The First Hospital of Jilin
University, CHINA
Received: October 16, 2023
Accepted: December 20, 2023
Published: September 27, 2024
Copyright: ©2024 Chen, Yi. This is an open access
article distributed under the terms of the Creative
Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in
any medium, provided the original author and
source are credited.
Data Availability Statement: The full dataset is
available at the Kaggle website: https://www.
kaggle.com/datasets/raghadalharbi/breast-cancer-
gene-expression-profiles-metabric.
Funding: L.-P. Chen: National Science and
Technology Council G. Y. Yi: Natural Sciences and
Engineering Research Council of Canada; Canada
Research Chairs The funders had no role in study
design, data collection and analysis, decision to
publish, or preparation of the manuscript.
This research is partially motivated by the Molecular Taxonomy of Breast Cancer Interna-
tional Consortium (METABRIC) database, a Canada-UK Project that includes targeted
sequencing data of primary breast cancer samples collected by the Cambridge Research Insti-
tute and the British Columbia Cancer Centre in Canada [3]. The dataset with all gene expres-
sion names is publicly available on the Kaggle website (https://www.kaggle.com/datasets/
raghadalharbi/breast-cancer-gene-expression-profiles-metabric). One interesting question is
whether patients taking chemotherapy as a treatment (“hormone_therapy”: 1 is yes and 0 is
no) can increase the chance of the survival status (“overall_survival”: 1 is alive and 0 is dead).
The dataset contains Z-scores of m-RNA levels for 331 genes, where the Z-score is defined as
an expression level in the tumor sample the mean expression in reference sample
standard deviation of expression levels in reference sample ;
which is a continuous variable. In addition, some gene expressions may be confounded with
the outcome and treatment, shown in Fig 1.
Taking the causal inference paradigm, we formulate the question as the estimation of the
average treatment effect (ATE), defined as the difference between potential outcomes under
two treatments, where the two treatments refer to taking and not taking chemotheropy,
respectively.
Various causal inference methods, accompanied by R packages, are available in the litera-
ture. Examples include iWeigReg [4], SVMMatch [5], CausalGAM [6], and wfe [7]. The R
package mediation [8] is developed to conduct mediation analysis. Package qualCI [9] is used
to analyze causal inference with qualitative and ordinal information on outcomes. Matching-
Frontier [10] applies the matching method to handle the balance issue in causal inference.
In the framework of causal inference, the inverse probability weighted (IPW) estimation
method has been widely used to estimate average treatment effects due to its simplicity and
transparent interpretation (e.g., [11–13]). The method adjusts for the effects of measured con-
founders by re-weighting the data as if the weighted data were collected from randomized
Fig 1. An illustrative diagram of the causal relationship with possible confounders.
https://doi.org/10.1371/journal.pone.0296951.g001
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Competing interests: The authors have declared
that no competing interests exist.
controlled trials. The validity of the method requires two key conditions: (1) the treatment
model is correctly specified to consistently estimate propensity scores, and (2) the variables in
the treatment model are precisely measured.
In the presence of measurement error, directly applying those methods to the observed data
usually yields unreliable estimation results. As illustrated in Fig 1, in the study of the treatment
effect on the outcome, confounders (e.g., gene expressions) may possess complex features:
some are associated only with the treatment variable, some are solely related to the outcome
variable, some are connected to both treatment and outcome variables, and some are not rele-
vant to either treatment or outcome variable. Including irrelevant confounders in the analysis
or model building may lead to misleading results.
In the literature, variable selection for causal inference (e.g., [14–16]) or measurement error
correction (e.g., [17–22]) are discussed under various settings. However, in the concurrent
presence of both features, limited work has been carried out to estimate ATE except for [23].
Moreover, there is a lack of user-friendly R packages designed to facilitate causal estimation
for data with both measurement error and spurious variables.
In this paper, we develop an R package, called AteMeVs, which is desired to estimate the
average treatment effect with measurement error and variable selection for confounders. This
package, available at CRAN [24], is developed to implement a recent method proposed by
[23], which estimates the average treatment effect for noisy data that include both measure-
ment error and spurious variables needed to be excluded. The developed package contains a
set of functions that provide a step-by-step estimation procedure, including the correction of
the measurement error effects, variable selection for the estimation of propensity scores, and
estimation of ATE. Our functions contain multiple options for users to implement, including
different ways to correct for the measurement error effects, various penalty functions for vari-
able selection, and different regression models for characterizing propensity scores.
2 Notation and framework
2.1 Propensity score
In contrast to the variables indicated by Fig 1, we now introduce abstract symbols to classify
the associated variables differently. Let Tdenote the observed binary treatment (e.g., chemo-
therapy) with T= 1 if treated and T= 0 if untreated. Accordingly, we consider counterfactual
responses corresponding to the treatment status. For t2{0, 1}, let Y
(t)
represent the potential
outcome of the patient if the patient would have received T=t. As described in Section 1, the
goal is to assess the causality effect of the treatment (e.g., chemotherapy) on increasing the
patient’s survival, or equivalently, we are interested in estimating the ATE,
t0≜EðYð1ÞÞ EðYð0ÞÞ:
To facilitate an individual’s characteristics, we let Wdenote the p-dimensional vector of
pre-treatment confounders for the individual, which as an example, can be understood as gene
expressions in Fig 1. To reflect the possible dependence of Ton W, we consider the conditional
probability
p≜PðT¼1jWÞ;
also called the propensity score for the individual. The introduction of the propensity score
allows us to use the observed outcome, denoted by Y, and the treatment information to
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consistently estimate τ
0
(e.g., [25]), which basically is due to the property (e.g., [23])
t0¼ETY
p
Eð1TÞY
1p:ð1Þ
The validity of (1) hinges on the following standard assumptions in the causal inference
framework:
1. The strong ignorable treatment assumption (SITA): given the covariates W, potential out-
comes Y
(0)
and Y
(1)
are independent of T;
2. The stable unit treatment value assumption (SUTVA), also known as the consistency
assumption: each subject’s potential outcomes are not influenced by the actual treatment
assignment of other subjects. Therefore, the observed outcome, Y, for an individual is
assumed to be linked with potential outcomes via Y=TY
(1)
+ (1 −T)Y
(0)
;
3. The positivity assumption: the propensity score is between 0 and 1, i.e., 0 <π(W)<1 for
all W.
In applications, πis frequently characterized by a parametric model:
g1ðpÞ ¼ W>g;ð2Þ
where γ= (γ
0
,γ
1
,,γ
p
)
>
is the vector of regression parameters of dimension p+ 1, with γ
0
representing the intercept; and g() is a link function. For ease of exposition and the inclusion
of the intercept, we slightly abuse the notation Win (2) by including 1 to the original p-dimen-
sional vector of confounders here and in the subsequent development. Common choices of g
() include the logit, probit, and complementary log-log functions, respectively yielding
•the logistic regression model:
p¼exp ðW>gÞ
1þexp ðW>gÞ;ð3Þ
•the probit regression model:
p¼ΦðW>gÞ;ð4Þ
with Φ() representing the cumulative distribution function of the standard normal distribu-
tion, and
•the complementary log-log regression model:
p¼1exp f exp ðW>gÞg:ð5Þ
2.2 The IPW estimator
With the setup in Section 2.1, the estimation of τ
0
can be carried out using the measurements
of a random sample, say {{T
i
,Y
i
,W
i
}: i= 1, . . .,n} of size n, where T
i
,Y
i
, and W
i
represent the
corresponding variables for subject iwith i= 1, . . .,n.
The estimation of τ
0
basically involves the following two steps. In the first step, we estimate
the propensity score π
i
=P(T
i
= 1|W
i
) for subject i= 1, ...,nbased on estimating parameter γ
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in model (2). Let S
i
(γ;W
i
) denote the likelihood score function obtained from subject ithat is
derived from fitting model (2). If the true value of W
i
is available, one may solve
X
n
i¼1
Siðg;WiÞ ¼ 0ð6Þ
for γto obtain a consistent estimate of γ, denoted b
g, provided usual regularity conditions.
Then we calculate π
i
with γin (2) replaced by the estimate b
g, and let bpidenote the resulting
value of π
i
.
In the second step, utilizing the property (1) yields a consistent estimate of the ATE τ
0
by
the following IPW estimator, as initiated by [25]:
bt¼1
nX
n
i¼1
TiYi
bpi1
nX
n
i¼1
ð1TiÞYi
1bpi
:ð7Þ
To mitigate unstable numerical results caused by extreme values of bpithat may be close to 0 or
1, [12] proposed a stable version of (7), which also offers a consistent estimator of τ
0
:
bt¼X
n
i¼1
Ti
bpi
!1X
n
i¼1
TiYi
bpiX
n
i¼1
1Ti
1bpi
!1X
n
i¼1
ð1TiÞYi
1bpi
:ð8Þ
We use (8) for the following development.
2.3 Irrelevant variables and measurement error
As noted in [23], the validity of (7) or (8) breaks down in the presence of two features of noisy
data: measurement error and irrelevant variables.
In applications, some variables (e.g., gene expressions) in W
i
can be subject to measurement
error. To reflect this feature, we write W
i
as ðX>
i;Z>
iÞ>so that all error-prone variables are
included in X
i
and all precisely measured variables go to Z
i
. Let X∗
idenote the observed surro-
gate measurement of X
i
.
To characterize the relationship between X∗
iand X
i
, we consider the classical additive error
model (e.g., [26,27])
X∗
i¼Xiþei;ð9Þ
where the error term e
i
is independent of {T
i
,X
i
,Z
i
,Y
i
} and follows N(0, S
e
) with covariance
matrix S
e
.
Model (9) is the most commonly used in the literature; it facilitates situations where the
observed value fluctuates around the true value with an error term, and the degree of measure-
ment error in X∗
iis reflected by the value of S
e
. In this paper, we consider the following four
cases for S
e
:
Case 1:S
e
is known;
Case 2:S
e
is unknown and estimated from repeated surrogate measurements fX∗
ij :j¼
1;;ni;i2Rgfor a subset, say R, of {1, 2, ,n} with jRj¼mand m<n;
Case 3:S
e
is unknown and estimated from repeated surrogate measurements fX∗
ij :j¼
1;. . . ;nigof X
i
for i= 1, ,n, where n
i
2 that may or may not depend on i;
Case 4:S
e
is unknown and estimated from an external validation sample ffXk;X∗
kg:k2Vg,
where Vis index set for the subjects in the validation sample.
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Case 1 is useful for addressing data issues in which the existence of measurement error is
acknowledged, yet the magnitude of this error has not been quantified. In this case, we often
conduct sensitivity analyses to assess the sensitivity of inference results to varying degrees of
measurement error, where user-specified values for S
e
are typically used to describe different
scenarios of measurement error in X
i
(e.g., [27,28]). This case was also considered by [23].
Cases 2 and 3 complement each other in addressing two scenarios with repeated surrogate
measurements. In contrast to the availability of replicates, Case 4 assumes the availability of an
external validation dataset.
The second issue of noisy data concerns irrelevant variables in the data. As shown in Fig 1,
some gene expressions have no connection with chemotherapy or patient’s survival status. To
reflect this feature, we write Wi¼ ðW>
Ii;W>
IIiÞ>for i= 1, ,n, where WIiincludes the informa-
tive confounders associated with T
i
and Y
i
, and WIIicontains noninformative confounders.
We further write WIi¼ ðZ>
Ii;X>
IiÞ>and WIIi¼ ðZ>
IIi;X>
IIiÞ>so that Zi≜ðZ>
Ii;Z>
IIiÞ>represents the
subvector of error-free confounders in W
i
and Xi≜ðX>
Ii;X>
IIiÞ>is the subvector of error-prone
confounders in W
i
. Let pZand pXdenote the dimension of Z
i
and X
i
, respectively. We let X∗
i¼
ðX∗>
Ii;X∗>
IIiÞ>denote the observed version of X
i
, where X∗
Iiand X∗
IIiare the observed measure-
ments of XIiand XIIi, respectively.
3 Estimation methods
Here we describe methods for estimating the average treatment effect τ
0
, with the features of
variable selection and measurement error accommodated for each of the four cases described
in Section 2.3. The main idea comes from Section 3.1 of [23], which is developed under Case 1
described in Section 2.3. Sections 3.2-3.4 extend the development in Section 3.1 to respectively
handle Cases 3.2-3.4 described in Section 2.3.
3.1 Implementation steps for Case 1
First, we describe the algorithm for Case 1 where S
e
is user-specified. The algorithm of esti-
mating τ
0
contains the five steps, summarized as follows. For details, see Section 3.1 of [23].
Step 1. Simulation:
We simulate a sequence of artificial surrogates, denoted fX∗
iðk;cÞ:k¼1;;K;c2C;
i¼1;;ng, where Kis a user-specified positive integer, C¼ fc1;c2;. . . ;cMgis a sequence
of Mnon-negative values taken from ½0;cMwith a given cMand ψ
1
= 0, and
X∗
iðk;cÞ ¼ X∗
iþffiffiffiffi
c
peik with eik independently generate from Nð0;SeÞ:ð10Þ
Step 2. Estimation of Treatment Model Parameters:
Parameter γin model (2) is estimated by solving (6) with X
i
replaced by X∗
iðk;cÞ, and let
b
gðk;cÞdenote the resulting estimate. Calculate b
gðcÞ ¼ K1PK
k¼1b
gðk;cÞfor c2C.
Step 3. Extrapolation:
For j= 0, 1, 2, . . .,p, let b
gjðcÞdenote the jth element of b
gðcÞ; fit a regression model to
fðc;b
gjðcÞÞ :c2Cgand extrapolate it to ψ=−1; and let ~
gjdenote the resulting extrapolated
value of γ
j
, the jth element of γ. Write ~
g¼ ð~
g0;~
g1;. . . ;~
gpÞ>.
Step 4. Variable Selection:
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Minimize the penalized quadratic loss function
‘PðgÞ≜‘ðgÞ nX
p
j¼1
rlðjgjjÞ
≜1
2ðg~
gÞ>Vnðg~
gÞ nX
p
j¼1
rlðjgjjÞ ð11Þ
with respect to γ, where ρ
λ
() is a user-specified penalty function with a tuning parameter λ,
and V
n
is a user-specified positive definite weight matrix.
Step 5. Estimation of ATE:
Write b
g¼ ðb
g>
I;b
g>
II Þ>with b
gI¼ ðb
g>
xI;b
g>
zIÞ>and b
gII ¼ ðb
g>
xII;b
g>
zIIÞ>corresponding to the non-
zero and zero components in b
g, respectively. With unimportant variables XIIiand ZIIiexcluded
from the initial model (2), the final treatment model is taken as
g1ðpiÞ ¼ W>
IigI;ð12Þ
where gIis the vector of model parameters associated with important covariates WIi.
For k= 1, ,Kand c2C, calculate an estimate, say, btðk;cÞ, of τ
0
using (8) with bpi
replaced by the propensity score for subject i, determined by the selected treatment model (12)
with XIireplaced by X∗
Iiðk;cÞ, the subvector of X∗
iðk;cÞthat corresponds to X∗
Ii. Then calculate
btðcÞ ¼ K1X
K
k¼1btðk;cÞ:
Finally, fit a regression model to fðc;btðcÞÞ :c2Cgand extrapolate it to ψ=−1. The result-
ing value, denoted as bt, is taken an estimate of τ
0
.
3.2 Implementation steps for Case 2
Consider Case 2 where repeated measurements fX∗
ij :j¼1;;ni;i2Rgof X
i
are available
for jRj≜msubjects, and surrogates X∗
ij and X
i
are linked via the measurement error model
X∗
ij ¼Xiþeij for i2Rand j¼1;;ni;
where for i2R;ni2;e
ij
follows N(0, S
e
) with unknown covariance matrix S
e
; and the e
ij
are independent of {T
i
,X
i
,Z
i
,Y
i(1)
,Y
i(0)
}.
With the repeated measurements, using the method of moments, we estimate S
e
by
b
Se¼X
i2RX
ni
j¼1ðX∗
ij X∗
iÞðX∗
ij X∗
iÞ>
X
i2Rðni1Þ;ð13Þ
where X∗
i¼n1P
ni
j¼1
X∗
ij.
To estimate τ
0
, we repeat the five steps described in Section 3.1 with S
e
in (10) replaced by
(13).
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3.3 Implementation steps for Case 3
Now we consider Case 3 described in Section 2.3, where S
e
is unknown but repeated surrogate
measurements fX∗
ij :j¼1;;nigare available for all the subjects in the sample, with n
i
2
for i= 1, ,n.
Adapting the development in [27,29] (p.107), we modify Step 1 in Section 3.1 as follows.
For any c2Cand i= 1, ,n, we generate n
i
variates independently from the standard nor-
mal distribution, and let {d
ij
(ψ): j= 1, ,n
i
} denote them. Calculate diðcÞ ¼ 1
niP
ni
j¼1
dijðcÞand
cijðcÞ ¼ dijðcÞ diðcÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X
ni
l¼1fdilðcÞ diðcÞg2
s:
Then for k= 1, ,K, we define
X∗∗
iðk;cÞ ¼ X∗
iþffiffiffiffi
c
ni
sX
ni
j¼1
cijðcÞX∗
ij;ð14Þ
where X∗
i¼n1
iP
ni
j¼1
X∗
ij. Estimation of τ
0
can then be proceeded following the five steps in Sec-
tion 3.1, with X∗
iðk;cÞin (10) replaced by X∗∗
iðk;cÞin (14).
3.4 Implementation steps for Case 4
We now consider Case 4 described in Section 2.3. In this case, we have the main study data,
given by nfYi;Ti;Zi;X∗
ig:i2Mowith M¼ f1;;ng, and an external validation sample
ffXk;X∗
kg:k2Vgwith size jVj≜m, where the index sets Mand Vdo not overlap. We assume
that X∗
kand X
k
are related via (9) for k2V. Further, we make the transportability assumption,
considered in [30].
With the availability of X
k
and X∗
kin V, we can empirically estimate S
X
= var(X
k
) and
SX∗¼varðX∗
kÞ, and denote the resulting estimators by
b
SX¼1
jVjX
i2V
XiXi
ðXiXiÞ>
and
b
SX∗¼1
jVjX
i2V
X∗
iX∗
i
X∗
iX∗
i>;
respectively, where Xi¼1
jVjP
i2V
Xiand X∗
i¼1
jVjP
i2V
X∗
i.
Consequently, by the additivity of covariance matrices in (9), we estimate S
e
by
b
Se¼b
SX∗b
SX:ð15Þ
Then estimation of τ
0
is carried out following the five steps in Section 3.1, with S
e
in (10)
replaced by the estimator b
Sein (15).
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4 Implementation details
To implement the estimation procedures described in Section 3, we need to first decide the
choice of relevant quantities, including Kand Cin Step 1, the regression model for extrapola-
tion in Steps 3 and 5, and the penalty function together with the tuning parameter in Step 4. In
the following subsections, we discuss the choice of each quantity individually.
4.1 Choice of Kand Cin step 1
Integer Kdetermines the repetition of simulated data X∗
iðk;cÞfor each given ψ. To reduce
Monte Carlo errors, a larger value of Kis expected to produce a more stable result of b
gðcÞ. On
the other hand, a larger value of Krequires a substantially longer computational time. There-
fore, a suitable choice of Kis driven by the trade-off between the computation time and the
accuracy of the results. Empirical experience (e.g., [23,31–33]) suggests setting Kto be 50, 100,
200, or 500 may be reasonable for many applications.
Regarding the choice of C, one may take cMto be 1 or 2, and divide the interval ½0;cM
equally into Msub-intervals, where Mmay be taken as 5, 10, or other positive integers. Then C
is the set of the resulting cut points.
4.2 Choice of extrapolation function in steps 3 and 5
[26] (Section 5.3.2) suggests to use one of the following functions, denoted by φ(u) with
parameters β
0
,β
1
,β
2
, and β
3
, to approximate the true extrapolation functions in implementing
Steps 3 and 5:
1. the quadratic function
φðuÞ ¼ b0þb1uþb2u2;ð16Þ
2. the linear function
φðuÞ ¼ b0þb1u;ð17Þ
3. the rational linear function
φðuÞ ¼ b0þb1
b2þu:ð18Þ
To increase flexibility, we add the following function to approximate the extrapolation
functions for the implementation of Steps 3 and 5:
4. the cubic function
φðuÞ ¼ b0þb1uþb2u2þb3u3:ð19Þ
4.3 Choices of penalty function in step 4
In implementing Step 4 in Section 3.1, we consider the following commonly used penalty func-
tions ρ
λ
(u) that are included in the R package ncvreg:
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1. the least absolute shrinkage and selection operator (LASSO) penalty [34]:
rlðuÞ ¼ ljuj;ð20Þ
2. the smoothly clipped absolute deviation (SCAD) penalty [35]:
r0
lðuÞ ¼ lfIðulÞ þ ðaluÞþ
ða1ÞlIðu>lÞg;ð21Þ
where I() is the indicator function, u
+
= max{u, 0}, and a= 3.7. Here r0
lðuÞis the first order
derivative of the penalty function ρ
λ
(u) with tuning parameter λ.
3. the minimax concave penalty (MCP) function proposed by [36]:
r0
lðuÞ ¼ ðlu=aÞþð22Þ
with a= 3.
4. the Elastic Net [37]:
rlðuÞ ¼ lfð1aÞu2þajujg;ð23Þ
with α2[0, 1]. If α= 1, then (23) reduces to (20); when α= 0, then (23) gives the L
2
-norma
penalty for the ridge regression.
4.4 Determination of tuning parameter
To achieve satisfactory performance of the selection procedure, we may consider one of the
following criteria for choosing a suitable value for the tuning parameter λ:
1. Bayesian Information Criterion (BIC)
Given a grid Λof possible values for the tuning parameter λ, and for λ2Λ, let
b
gðlÞ ¼ argming‘PðgÞ
and let df
λ
denote the number of non-zero elements of b
gðlÞ. We define
BICðlÞ ¼ 2‘ðb
gðlÞÞ þ 2ðlog nÞdfl:
Then the optimal tuning parameter λ*is chosen as the minimizer of BIC(λ):
l∗¼argminl2LBICðlÞ:
2. V-fold cross validation (CV)
The original dataset is first divided into Vsubsamples with an equal size, where Vis a user-
specified positive integer, such as V= 5. The rth subsample is taken as the testing set and
the remaining (V−1) subsamples are merged as the training set.
Applying (11) to the training set gives us an estimator of γ, denoted γ
(−r)
(λ). Then we evalu-
ate ℓ(γ) in (11) at γ=γ
(−r)
(λ) based on the rth testing set, and let ℓ
(r)
(γ
(−r)
(λ)) denote the
resulting value. Finally, we compute
CVðlÞ ¼ 1
VX
V
r¼1
‘ðrÞðgð rÞðlÞÞ:
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The optimal tuning parameter λ*is then determined by the minimizer of CV(λ):
l∗¼argminl2LCVðlÞ:
This approach can be realized by employing the function cv.ncvreg in the R package
ncvreg.
5 Syntax of R package
In this section, we present our developed R package, AteMeVs, which implements the estima-
tion procedures described in Section 3, together with the details in Section 4. The developed R
package utilizes the available R packages: MASS and ncvreg. The former is used to generate a
multivariate normal distribution to address (10), and the latter is used to implement penalty
functions as outlined in Section 4. Below, we describe the syntax of the developed functions
that implement the step-by-step estimation procedure in Section 3.
SIMEX_EST
The function SIMEX_EST implements Steps 1-3 in Section 3.1, given by
SIMEX_EST(data, PS = “logistic”, Psi = seq(0,1,length = 10),
px = p, K = 200, extrapolate=“quadratic”, Sigma_e, replicate =
“FALSE”, RM = rep(0,px)).
The arguments in this function include
•data: an n×(p+ 2) matrix of a dataset. The first column records the observed outcome, the
second column displays the values for the binary treatment, and the remaining columns
store the observed measurements for the confounders.
•PS: a specification of a link function g() in (2).logistic refers to the logistic regression
function (3),probit reflects the probit model (4), and cloglog gives the complementary
log-log regression model (5).
•Psi: the specification of Cin Step 1.
•px: the dimension of X.
•K: a positive integer Kin Step 1.
•extrapolate: the extrapolation function in Step 3. quadratic reflects the quadratic
polynomial function (16),linear gives the linear polynomial function (17),RL is the ratio-
nal linear function (18), and cubic refers to the cubic polynomial function (19).
•Sigma_e: the covariance matrix S
e
for the measurement error model (9).
•replicate: the identification of the availability of repeated measurements in the con-
founders. replicate = “FALSE” represents no repeated measurements and
replicate = “TRUE” indicates that repeated measurements exist in the dataset. The
default is set as replicate = “FALSE”.
•RM: a pXdimensional user-specified vector with each entry representing the number of repe-
titions for the respective confounder. For example, RM = c(2,2,3) indicates that three
confounders in Xhave repeated measurements, where the first and second confounders
have two repetitions and the third one has three repetitions. The default of RM is set as the
pX-dimensional zero vector, i.e., RM = rep(0,px).
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In the argument data, the potential outcome can be continuous or binary and the treat-
ment in the second column is designed to be binary. For the columns of confounders, one
should place error-prone confounders from the third to the (px+2)-th column, and the
remaining columns record precisely-measured confounders in Z. The argument PS is used to
specify a link function g() that is used to characterize the propensity score in model (2), where
the logistic regression model (3) is taken as the default. Two arguments Psi and Kare used to
generate the working data X∗
iðk;cÞin Step 1 in Section 3.1. The default of Psi is given by
seq(0,1,length = 10), i.e., an interval [0, 1] with equal width divided into M= 10 sub-
intervals, and the default of Kis set as 200. px reflects the dimension of error-prone confound-
ers, with the default value set as the dimension of all confounders W
i
, revealing that all
confounders may be subject to measurement error. Setting px = 0 accommodates the situa-
tion where all confounders are precisely measured and there is no need to correct the measure-
ment error effects. On the contrary, if confounders do involve measurement error, specifying
px = 0 yields the naive estimate of τ
0
which ignores the measurement error effects. The argu-
ment extrapolate contains commonly used working functions for extrapolation listed in
Section 4.2. The default of the working extrapolation function is taken as the quadratic
function.
Finally, Sigma_e records the covariance matrix S
e
, which can be user-specified or esti-
mated by using auxiliary information, as described in Sections 3.2-3.4. Two arguments
replicate and RM are used to indicate the availability of repeated measurements and the
way of generating working data. Specifically, when replicate = “FALSE”, then (10) is
implemented to generate the working variables in the main dataset for Cases 1, 2, and 4,
respectively described in Sections 3.1, 3.2, and 3.4. On the other hand, the argument
replicate = “TRUE” reflects Case 3 described in Section 3.3, which uses (14) to replace
Step 1 in Section 3.1. The argument RM is in use to accompany with the argument
replicate. When replicate = “FALSE”, no repeated measurements are available in
the sample, and RM should be set as RM = rep(0,px). In contrast, if replicate =
“TRUE”, there are repeated measurements for the confounders, and in this case, users should
specify the number of repeated measurements for each confounder by setting a proper value
for RM. For example, setting RM = c(2,2,3) represents that three confounders have
repeated surrogate measurements, having 2, 2, and 3 replicates, respectively. When all argu-
ments are specified, the output of this function gives a vector ~
gas defined in Step 3.
VSE_PS
The function VSE_PS, reflecting variable selection and estimation of propensity scores, is used
to implement Step 4 in Section 3.1. The input function is given by
VSE_PS(V, y, method=“lasso”, cv=“TRUE“, alpha = 1),
with the following arguments:
•V: a (p+ 1) ×(p+ 1) matrix V
n
in (11).
•y: a (p+ 1)-dimensional vector ~
gin (11).
•method: it reflects the penalty function ρ
λ
() in (11) with choices presented in Section 4.3,
where “lasso”,“scad” and “mcp” are given by (20),(21) and (22), respectively.
•cv: the method for choosing the tuning parameter λ.cv=“TRUE” suggests the use of the
cross-validation method and cv=“FALSE” allows the use of the BIC, described in Section
4.4.
•alpha: a constant α2[0, 1] in (23).
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The argument Vis a user-specified matrix, with the default set as the identity matrix, and y
represents a vector derived by the output of SIMEX_EST. The argument method provides
the penalty functions in Section 4.3 that are implemented in the R package ncvreg. The argu-
ment cv gives two choices to determine the optimal tuning parameter, respectively deter-
mined by cross-validation and BIC. Finally, alpha reflects a user-specified value αin (23),
with the default value alpha = 1 that recovers the lasso method.
The output of this function gives a (p+ 1)-dimensional vector of the estimator of γ. In this
vector, components with zero values represent confounders that are unimportant and should
be excluded; components with nonzero values identify important confounders entering the
treatment model (12).
EST_ATE
Upon the implementation of Steps 1-4, we then use the function EST_ATE to estimate ATE,
as discussed in Step 5. The implementation is given by
EST_ATE(data, PS = “logistic”, Psi = seq(0,1,length = 10),
K = 200, gamma, px = p, extrapolate=“quadratic”, Sigma_e,
replicate = “FALSE”, RM = 0, bootstrap = 100).
All the arguments in this function are the same as those in SIMEX_EST, except for the
argument gamma. The argument gamma records the estimate obtained from the implementa-
tion of Steps 1-4, and is used to estimate the propensity score bpiðk;cÞin Step 5. The function
EST_ATE provides the final estimate of ATE.
Furthermore, to provide a variance estimate and the resulting p-value for the estimated
ATE, we employ the bootstrap algorithm by repeatedly running the proposed procedure to a
sequence of bootstrap samples; then using the resulting estimates of ATE, we compute the
sample variance of those estimates; taking this as a bootstrap variance for the initially obtained
estimate of ATE, we calculate an associated p-value. The argument bootstrap is used to
specify the number of bootstrap samples the user wishes to consider; its default value is set as
100. Function EST_ATE outputs values with headings estimate,variance, and p-
value, which are a point estimate, the associated variance estimate, and the resulting p-value
of τ
0
, respectively.
6 Numerical studies
6.1 Implementation of AteMeVs
In this section, we implement the R package AteMeVs to the METABRIC data described in
Section 1. The dataset contains the information for 1422 patients, together with 331 gene
expressions. Following the notation in Section 2, we define Yand Tas “overall_survival” and
“hormone_therapy”, respectively. We let Wdenote those gene expressions. The following code
is used to prepare for the dataset.
read.table(“C://METABRIC_causal.csv”, sep = “,”, header = TRUE) ->
data_METABRIC
library(MASS)
library(ncvreg)
library(AteMeVs)
data = data_METABRIC[, 1:280]
gene = colnames(data_METABRIC)[3:280]
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set.seed(20651252)
n = dim(data)[1]
p = dim(data)[2] - 2
We now demonstrate Steps 1-3 by using the function SIMEX_EST and by setting K= 10
and Cto include the cutpoints equally dividing the interval [0, 2] into M= 10 subintervals.
Since there is no additional information to estimate S
e
, we follow Case 1 in Section 2.3 to spec-
ify S
e
as a diagonal matrix with common value s2 = 0.2. As noted in [32], measurements of
gene expressions are subject to measurement error, and therefore, we specify px = p, which
is 331. The implementation is given below:
Psi = seq(0, 2, length = 10)
K = 10
s2 = 0.2
y = as.vector(SIMEX_EST(data, Psi, K, px = p, Sigma_e = diag(s2, p)))
matrix(y,ncol=2)
[,1] [,2]
[1,] -0.132002775 -0.098403692
[2,] 0.192014131 -0.247497662
[3,] 0.333889793 0.153789525
[4,] -0.470314553 -0.486142020
[5,] -0.164594040 -0.040468911
[6,] -0.190031631 0.041317059
[7,] 0.443131214 0.108216814
[8,] -0.612154496 0.014734219
[9,] 0.201441389 -0.012663900
[10,] 0.611405760 -0.242673669
Due to the space constraint, we report partial results for the output yabove to show the esti-
mate ~
g.
Next, we use ~
gto demonstrate variable selection in Step 4. Since V
n
in (11) is user-specified,
we follow [23] and set V
n
as the identity matrix. To see the impact of variable selection by dif-
ferent methods, we examine three penalty functions (20),(21), and (22). Detailed demonstra-
tions with an application of the function VSE_PS are given below. We also display some
results in the command “VS” as follows.
V = diag(1, length(y), length(y))
est_lasso_cv = VSE_PS(V, y, method = “lasso”, cv = “TRUE”)
est_scad_cv = VSE_PS(V, y, method = “scad”, cv = “TRUE”)
est_mcp_cv = VSE_PS(V, y, method = “mcp”, cv = “TRUE”)
cbind(est_lasso_cv,
est_scad_cv,
est_mcp_cv) -> VS
rownames(VS) = gene
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VS
est_lasso_cv est_scad_cv est_mcp_cv
brca1 0.0000000 0.0000000 0.0000000
brca2 0.0000000 0.0000000 0.0000000
palb2 0.0000000 0.0000000 0.0000000
pten 0.0000000 0.5182603 0.5230828
tp53 0.0000000 0.0000000 0.0000000
atm 0.0000000 0.0000000 0.0000000
cdh1 0.0000000 0.0000000 0.0000000
chek2 0.5457605 0.6601017 0.6649249
nbn 0.0000000 0.0000000 0.0000000
nf1 -0.5277423 -0.5634578 -0.5586342
stk11 0.0000000 0.0000000 0.0000000
bard1 0.0000000 0.0000000 0.0000000
mlh1 0.0000000 0.0000000 0.0000000
msh2 0.5342537 0.6485958 0.6534201
msh6 0.0000000 0.0000000 0.0000000
pms2 0.0000000 0.0000000 0.0000000
epcam 0.0000000 0.0000000 0.0000000
rad51c 0.0000000 0.0000000 0.0000000
rad51d 0.0000000 0.0000000 0.0000000
rad50 0.0000000 0.0000000 0.0000000
rb1 -0.8071897 -0.8429033 -0.8380785
rbl1 0.0000000 0.0000000 0.0000000
rbl2 0.0000000 0.0000000 0.0000000
ccna1 0.0000000 0.0000000 0.0000000
ccnb1 0.0000000 0.0000000 0.0000000
cdk1 0.0000000 0.0000000 0.0000000
ccne1 0.0000000 0.0000000 0.0000000
cdk2 0.0000000 0.0000000 0.0000000
cdc25a -0.5527873 -0.5885001 -0.5836741
ccnd1 0.5691736 0.6835180 0.6883442
cdk4 0.5165555 0.6309001 0.6357266
cdk6 -0.7121596 -0.7478720 -0.7430453
ccnd2 0.0000000 0.0000000 0.0000000
cdkn2a -0.5822496 -0.6179617 -0.6131343
cdkn2b 0.0000000 0.0000000 0.0000000
myc -0.6168412 -0.6525533 -0.6477256
cdkn1a 0.0000000 0.0000000 0.0000000
cdkn1b 0.0000000 0.0000000 0.0000000
e2f1 0.0000000 0.0000000 0.0000000
e2f2 0.0000000 -0.5323233 -0.5274951
Variable selection shows that zero values correspond to unimportant gene expressions and
nonzero ones suggest important gene expressions. The results show that informative gene
expressions are sparse regardless of variable selection methods. In the partial results displayed
here, we observe that some gene expressions, such as “e2f2” and “pten”, are selected by (21)
and (22) but not by (20). Moreover, selected gene expressions contain “cdk4”, “cdk6”, and
“ccnd1”, which is consistent with the findings of [38,39]; they found that “ccnd1” was
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associated with a good breast cancer prognosis and cdk6 has been shown to be regulated and
influenced by several mitogenic signaling pathways in breast cancer.
Finally, using the selected gene expressions, we estimate ATE by using the function EST_
ATE; the estimation result is shown as follows:
ate_lasso_cv = EST_ATE(data,
gamma = est_lasso_cv,
px = p,
Sigma_e = diag(s2, px))
ate_scad_cv = EST_ATE(data,
gamma = est_scad_cv,
px = p,
Sigma_e = diag(s2, px))
ate_mcp_cv = EST_ATE(data,
gamma = est_mcp_cv,
px = p,
Sigma_e = diag(s2, px))
> ate_mcp_cv
estimator variance p-value
[1,] 1.4773 0.04885097 2.326125e-11
> ate_scad_cv
estimator variance p-value
[1,] 1.631049 0.04049768 5.275382e-16
> ate_lasso_cv
estimator variance p-value
[1,] 1.116406 0.0357837 3.597e-09
An estimate of ATE is 1.4773, 1.631049, and 1.116406, respectively corresponding to the
estimates of γderived from (20),(21), and (22); and associated variance estimates derived by
the three methods are 0.04885097, 0.04049768, and 0.0357837, respectively. All the three
resulting p-values are smaller than the significance level 0.05, suggesting that the chemother-
apy treatment has a positive causal effect on increasing the survival of a patient. These results
are derived by accommodating the effects of informative gene expressions, including “ccnd1”,
“cdk4”, and “cdk6”, which are also identified to be informative by [9].
6.2 Comparisons of AteMeVs with other methods
To highlight the advantages of the package AteMeVs and underscore the importance of
addressing issues of measurement error and variable selection, we consider two additional sce-
narios: (i) using AteMeVs without implementing VSE_PS, and (ii) using existing packages
iWeigReg and CausalGAM, in comparison to the use of AteMeVs with different penalty func-
tions. In Scenario (i), we correct for measurement error in confounders but do not address the
exclusion of irrelevant confounders; in Scenario (ii), we aim to estimate ATE without taking
measurement error and variable selection into account. We summarize the numerical results
in Table 1, where ‘EST’ represents the estimate of ATE, ‘VAR’ is the variance associated with
the estimated ATE derived from the packages, and ‘p-value’ is the p-value derived from testing
the null hypothesis H
0
:τ
0
= 0.
The results show that the package AteMeVs performs stably, irrespective to the degree of
measurement error and the choice of the penalty function; significant causal effects are
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revealed by the use of AteMeVs. In contrast, Scenario (i) shows insignificant causal effects
with p-values greater than 0.05, which might be caused by the involvement of irrelevant con-
founders, even though measurement error correction is taken into account. Moreover, with
the ignorance of measurement error effects and variable selection, it is interesting that the
package iWeigReg shows evidence for the significance of the causal effects, but the EST and
VAR are greater than those derived by the package AteMeVs. On the other hand, CausalGAM
does not provide evidence for suggesting ATE differs zero.
7 Discussion
The inverse-probability-weighting estimation method and its variants have proved to be useful
for estimating the average treatment effect within the causal inference framework. However,
their applications are hindered by two critical conditions. The validity of those methods relies
on the proper determination of propensity scores and the use of the precise measurements of
the covariates. When data lack these features [23], introduced a simulation-based method that
adapts the inverse-probability-weighting scheme to accommodate measurement error effects
as well as variable selection for calculating propensity scores. In this paper, we develop an R
package, called AteMeVs, to extend the method proposed by [23]. This package provides ana-
lysts a user-friendly tool for estimating the average treatment effect when working with error-
contaminated data and inconsequential confounders.
As the package AteMeVs is designed to handle classical measurement error model (9),
biased estimation is anticipated when model (9) is not feasible; the impact of the violation of
the model (9) was explored by [40] for survival analysis with covariate measurement error.
Further, the development of the package AteMeVs lies on the correct parametric modelling
for the propensity score. When such an assumption is untrue, estimation results obtained
from using AteMeVs may become invalid. However, in applications, the relationship between
the treatment and the confounders can be complex, making it difficult to have straightforward
representation through a convenient parametric model. It is useful to introduce semipara-
metric models to characterize propensity scores.
Table 1. Comparisons of estimation methods. LASSO(x) is the usage of the package AteMeVs with the LASSO penalty function, SCAD(x) is the usage of the package Ate-
MeVs with the SCAD penalty function, MCP(x) is the usage of the package AteMeVs with the MCP penalty function, Full(x) refers to Scenario (i), where x= 0.2, 0.5 or
0.7, representing identical diagonal elements in S
e
. The usage of iWeigReg and CausalGAM refer to Scenario (ii).
Method Estimator of ATE
EST VAR p-value
AteMeVs LASSO(0.2) 1.477 0.049 2.326e-11
LASSO(0.5) 1.739 0.043 6.83e-17
LASSO(0.7) 1.590 0.071 2.525e-09
SCAD(0.2) 1.631 0.040 5.275e-16
SCAD(0.5) 1.115 0.102 0.000
SCAD(0.7) 0.661 0.097 0.034
MCP(0.2) 1.116 0.036 3.597e-19
MCP(0.5) 1.274 0.066 7.084e-07
MCP(0.7) 1.440 0.099 4.599e-06
Scenario (i) Full(0.2) 0.027 0.031 0.878
Full(0.5) 0.300 0.079 0.285
Full(0.7) 0.005 0.075 0.986
Scenario (ii) iWeigReg 3.168 0.741 0.000
CausalGAM 0.121 0.038 0.534
https://doi.org/10.1371/journal.pone.0296951.t001
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While the package AteMeVs offers flexibility in handling measurement error and variable
selection, it has limitations. Currently, the package focuses on continuous error-prone random
variables, and it cannot handle error-contaminated confounders that are mixed with both con-
tinuous and discrete variables. Another notable issue concerns the size of data. The AteMeVs
is basically developed for settings where the number of confounders is smaller than the sample
size, which is driven by the setup considered in [23]. It is interesting to generalize the method
in [23] to handle high-dimensional error-prone data, where the dimension of confounders can
be diverging as the sample size approaches infinity. Creating R packages to conduct causal
inference about such data can be useful.
Finally, the package AteMeVs developed here can only handle outcomes with complete
observations. In the presence of incomplete responses with error-contaminated covariates,
such as survival data with covariate measurement error, it is important to address both the
censoring effects (e.g., [41]) and the measurement error effects when estimating causal effects.
It is interesting to devise causal inference methods to handle such data and then develop R
packages accordingly to extend the application scope of the package AteMeVs.
Acknowledgments
The authors thank two referees for their useful comments to significantly improve the presen-
tation of the initial manuscript.
Author Contributions
Conceptualization: Li-Pang Chen, Grace Y. Yi.
Methodology: Li-Pang Chen, Grace Y. Yi.
Software: Li-Pang Chen.
Supervision: Grace Y. Yi.
Visualization: Li-Pang Chen, Grace Y. Yi.
Writing – original draft: Li-Pang Chen, Grace Y. Yi.
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