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 T denote the observed binary treatment (e.g., chemotherapy) with T = 1 if treated and T = 0 if untreated. Accordingly, we consider counterfactual responses corresponding to the treatment status. For t ∈ {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,

To facilitate an individual’s characteristics, we let W denote 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 T on W, we consider the conditional probability 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 consistently estimate τ₀ (e.g., [25]), which basically is due to the property (e.g., [23]) (1) 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 outcomes 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, π is frequently characterized by a parametric model: (2) where γ = (γ₀, γ₁, ⋯, γ_p)^⊤ is the vector of regression parameters of dimension p + 1, with γ₀ representing the intercept; and g(⋅) is a link function. For ease of exposition and the inclusion of the intercept, we slightly abuse the notation W in (2) by including 1 to the original p-dimensional 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: (3)
• the probit regression model: (4) with Φ(⋅) representing the cumulative distribution function of the standard normal distribution, and
• the complementary log-log regression model: (5)

2.2 The IPW estimator

With the setup in Section 2.1, the estimation of τ₀ 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 i with i = 1, …, n.

The estimation of τ₀ 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, …, n based on estimating parameter γ in model (2). Let S_i(γ;W_i) denote the likelihood score function obtained from subject i that is derived from fitting model (2). If the true value of W_i is available, one may solve (6) for γ to obtain a consistent estimate of γ, denoted , provided usual regularity conditions. Then we calculate π_i with γ in (2) replaced by the estimate , and let denote the resulting value of π_i.

In the second step, utilizing the property (1) yields a consistent estimate of the ATE τ₀ by the following IPW estimator, as initiated by [25]: (7) To mitigate unstable numerical results caused by extreme values of that may be close to 0 or 1, [12] proposed a stable version of (7), which also offers a consistent estimator of τ₀: (8) 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 so that all error-prone variables are included in X_i and all precisely measured variables go to Z_i. Let denote the observed surrogate measurement of X_i.

To characterize the relationship between and X_i, we consider the classical additive error model (e.g., [26, 27]) (9) where the error term e_i is independent of {T_i, X_i, Z_i, Y_i} and follows N(0, Σ_e) with covariance matrix Σ_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 measurement error in is reflected by the value of Σ_e. In this paper, we consider the following four cases for Σ_e:

1. Case 1: Σ_e is known;
2. Case 2: Σ_e is unknown and estimated from repeated surrogate measurements for a subset, say , of {1, 2, ⋯, n} with and m < n;
3. Case 3: Σ_e is unknown and estimated from repeated surrogate measurements of X_i for i = 1, ⋯, n, where n_i ≥ 2 that may or may not depend on i;
4. Case 4: Σ_e is unknown and estimated from an external validation sample , where is index set for the subjects in the validation sample.

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 Σ_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 for i = 1, ⋯, n, where includes the informative confounders associated with T_i and Y_i, and contains noninformative confounders. We further write and so that represents the subvector of error-free confounders in W_i and is the subvector of error-prone confounders in W_i. Let and denote the dimension of Z_i and X_i, respectively. We let denote the observed version of X_i, where and are the observed measurements of and , respectively.

3.1 Implementation steps for Case 1

First, we describe the algorithm for Case 1 where Σ_e is user-specified. The algorithm of estimating τ₀ 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 , where K is a user-specified positive integer, is a sequence of M non-negative values taken from with a given and ψ₁ = 0, and (10)

Step 2. Estimation of Treatment Model Parameters:

Parameter γ in model (2) is estimated by solving (6) with X_i replaced by , and let denote the resulting estimate. Calculate for .

Step 3. Extrapolation:

For j = 0, 1, 2, …, p, let denote the jth element of ; fit a regression model to and extrapolate it to ψ = −1; and let denote the resulting extrapolated value of γ_j, the jth element of γ. Write .

Step 4. Variable Selection:

Minimize the penalized quadratic loss function (11) 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 with and corresponding to the non-zero and zero components in , respectively. With unimportant variables and excluded from the initial model (2), the final treatment model is taken as (12) where is the vector of model parameters associated with important covariates .

For k = 1, ⋯, K and , calculate an estimate, say, , of τ₀ using (8) with replaced by the propensity score for subject i, determined by the selected treatment model (12) with replaced by , the subvector of that corresponds to . Then calculate Finally, fit a regression model to and extrapolate it to ψ = −1. The resulting value, denoted as , is taken an estimate of τ₀.

3.2 Implementation steps for Case 2

Consider Case 2 where repeated measurements of X_i are available for subjects, and surrogates and X_i are linked via the measurement error model where for ; e_ij follows N(0, Σ_e) with unknown covariance matrix Σ_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 Σ_e by (13) where .

To estimate τ₀, we repeat the five steps described in Section 3.1 with Σ_e in (10) replaced by (13).

3.3 Implementation steps for Case 3

Now we consider Case 3 described in Section 2.3, where Σ_e is unknown but repeated surrogate measurements are 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 and i = 1, ⋯, n, we generate n_i variates independently from the standard normal distribution, and let {d_ij(ψ): j = 1, ⋯, n_i} denote them. Calculate and

Then for k = 1, ⋯, K, we define (14) where . Estimation of τ₀ can then be proceeded following the five steps in Section 3.1, with in (10) replaced by in (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 with , and an external validation sample with size , where the index sets and do not overlap. We assume that and X_k are related via (9) for . Further, we make the transportability assumption, considered in [30].

With the availability of X_k and in , we can empirically estimate Σ_X = var(X_k) and , and denote the resulting estimators by and respectively, where and .

Consequently, by the additivity of covariance matrices in (9), we estimate Σ_e by (15) Then estimation of τ₀ is carried out following the five steps in Section 3.1, with Σ_e in (10) replaced by the estimator in (15).

4.1 Choice of K and in step 1

Integer K determines the repetition of simulated data for each given ψ. To reduce Monte Carlo errors, a larger value of K is expected to produce a more stable result of . On the other hand, a larger value of K requires a substantially longer computational time. Therefore, a suitable choice of K is driven by the trade-off between the computation time and the accuracy of the results. Empirical experience (e.g., [23, 31–33]) suggests setting K to be 50, 100, 200, or 500 may be reasonable for many applications.

Regarding the choice of , one may take to be 1 or 2, and divide the interval equally into M sub-intervals, where M may be taken as 5, 10, or other positive integers. Then 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 β₀, β₁, β₂, and β₃, to approximate the true extrapolation functions in implementing Steps 3 and 5:

1. the quadratic function (16)
2. the linear function (17)
3. the rational linear function (18)
   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 (19)

4.3 Choices of penalty function in step 4

In implementing Step 4 in Section 3.1, we consider the following commonly used penalty functions ρ_λ(u) that are included in the R package ncvreg:

1. the least absolute shrinkage and selection operator (LASSO) penalty [34]: (20)
2. the smoothly clipped absolute deviation (SCAD) penalty [35]: (21) where I(⋅) is the indicator function, u₊ = max{u, 0}, and a = 3.7. Here is 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]: (22) with a = 3.
4. the Elastic Net [37]: (23) with α ∈ [0, 1]. If α = 1, then (23) reduces to (20); when α = 0, then (23) gives the L₂-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 Λ of possible values for the tuning parameter λ, and for λ ∈ Λ, let and let df_λ denote the number of non-zero elements of . We define Then the optimal tuning parameter λ* is chosen as the minimizer of BIC(λ):
2. V-fold cross validation (CV)
   The original dataset is first divided into V subsamples with an equal size, where V is 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 evaluate 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 The optimal tuning parameter λ* is then determined by the minimizer of CV(λ): This approach can be realized by employing the function cv.ncvreg in the R package ncvreg.

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 in Step 1.
• px: the dimension of X.
• K: a positive integer K in 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 rational linear function (18), and cubic refers to the cubic polynomial function (19).
• Sigma_e: the covariance matrix Σ_e for the measurement error model (9).
• replicate: the identification of the availability of repeated measurements in the confounders. 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 dimensional user-specified vector with each entry representing the number of repetitions for the respective confounder. For example, RM = c(2,2,3) indicates that three confounders in X have 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 -dimensional zero vector, i.e., RM = rep(0,px).

In the argument data, the potential outcome can be continuous or binary and the treatment 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 K are used to generate the working data in 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 subintervals, and the default of K is set as 200. px reflects the dimension of error-prone confounders, 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 situation where all confounders are precisely measured and there is no need to correct the measurement error effects. On the contrary, if confounders do involve measurement error, specifying px = 0 yields the naive estimate of τ₀ which ignores the measurement error effects. The argument 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 Σ_e, which can be user-specified or estimated 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 arguments are specified, the output of this function gives a vector as 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 in (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 α ∈ [0, 1] in (23).

The argument V is 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 argument cv gives two choices to determine the optimal tuning parameter, respectively determined by cross-validation and BIC. Finally, alpha reflects a user-specified value α in (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 implementation of Steps 1-4, and is used to estimate the propensity score in 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 τ₀, respectively.

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 Y and T as “overall_survival” and “hormone_therapy”, respectively. We let W denote 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]
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 to include the cutpoints equally dividing the interval [0, 2] into M = 10 subintervals. Since there is no additional information to estimate Σ_e, we follow Case 1 in Section 2.3 to specify Σ_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 y above to show the estimate .

Next, we use to 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 different methods, we examine three penalty functions (20), (21), and (22). Detailed demonstrations 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
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 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 γ derived 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 chemotherapy 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 scenarios: (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 functions. 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 H₀: τ₀ = 0.

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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 AteMeVs 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 Σ_e. The usage of iWeigReg and CausalGAM refer to Scenario (ii).

https://doi.org/10.1371/journal.pone.0296951.t001

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 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 confounders, 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.