# Empirical Likelihood for Generalized Functional-Coefficient Regression Models with Multiple Smoothing Variables under Right Censoring Data

Authors: Hong-Xia Xu; Han-Sheng Zhong; Guo-Liang Fan
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

Empirical likelihood as a nonparametric approach has been demonstrated to have many desirable merits for constructing a confidence region. The purpose of this article is to apply the empirical likelihood method to study the generalized functional-coefficient regression models with multiple smoothing variables when the response is subject to random right censoring. The coefficient functions with multiple smoothing variables can accommodate various nonlinear interaction effects between covariates. The empirical log-likelihood ratio of an unknown parameter is constructed and shown to have a standard chi-squared limiting distribution at the true parameter. Based on this, the confidence region of the unknown parameter can be constructed. Simulation studies are carried out to indicate that the empirical likelihood method performs better than a normal approximation-based approach for constructing the confidence region.


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## Body

## 1. Introduction

In studying the relationship between a response and a set of predictor variables or regressors, the mean response variable is often assumed to be a linear regression function of the regressors. Recently, there has been increasing interest and activity in the general area of semiparametric regression modeling in statistics to analyse the high-dimensional data. Semiparametric models are often employed in regression analysis because they well balance between flexibility and fidelity. Among them, scholars have studied the semiparametric functional-coefficient regression model in recent years due to its direct connectivity to the classical linear model. The semiparametric functional-coefficient regression models with multiple smoothing variables have the following form:(1)Y=XTβ+ZTαU+ε,where Y is the response, X=X1,…,XpT, Z=Z1,…,ZdT, and U=U1,…,UdT are predictors, β=β1,…,βpT is a vector of p-dimensional unknown parameters, αU=α1U1,…,αdUdT is a d-dimensional vector of unspecified smooth coefficient functions, and ϵ is the model error.Model (1) belongs to the class of semiparametric functional-coefficient regression model and has many useful models as its special cases. Most of previous works are based on the assumption that all coefficient functions share the same smoothing variable. For instance, when all coefficient functions share the same smoothing variable and α⋅≡ constant, model (1) reduces to the usual linear regression model. When d=1 and Z≡1, model (1) reduces to the famous partially linear regression model which was first introduced by Engle et al. [1] to study the influence of weather on electricity demand and further studied by Chen [2], Härdle et al. [3], and Liang and Fan [4] among others. When all coefficient functions share the same smoothing variable, model (1) becomes the varying-coefficient partially linear model which has been widely studied in the literature as well, for instance, the work of Zhang et al. [5], Li et al. [6], Fan and Huang [7], and Huang and Zhang [8] among others. When the coefficient functions have different smoothing variables, Ip et al. [9] used a generalized likelihood ratio test to test coefficient functions in functional-coefficient regression models; Zhang and Li [10] discussed the functional-coefficient regression models with different smoothing variables in different coefficient functions and defined the integrated estimates of the coefficient functions by marginal integration; Zhang and Li [11] introduced averaged estimation for coefficient functions in functional-coefficient regression models with different smoothing variables; Zhang and Li [12] proposed a procedure for estimating the coefficient functions in the functional-coefficient regression models with different smoothing variables in different coefficient functions; Yang and Lee [13] discussed the semiparametric efficient estimation for generalized functional-coefficient regression models with multiple smoothing variables.However, the abovementioned related articles are based on the assumption that the data are fully observable. In practice, censored data are widely existing and often encountered. For example, in medical research, termination of studies or subjects leaving studies by various reasons occurs, which is referred to as the random right censoring. The techniques of complete data cannot be directly applied to censored data. Specifically, we consider the following generalized functional-coefficient regression models with multiple smoothing variables:(2)EϕYX,Z,U=XTβ+ZTαU,where Y is subject to random right censoring and ϕ⋅ is a unknown function, which is introduced to include various functions according to ones interest, not to deal with random right censoring. For example, it may be the identity function or I⋅≤t, which produces the conditional mean or the conditional probability function given covariates, respectively. When the response was random right censored, Buckley and James [14] gave a method of estimating parameters in the linear regression model; Bravo [15] considered the problem of estimation and inference in semiparametric varying-coefficients partially linear models; for semiparametric varying-coefficient models with different smoothing variables, Yang [16] applied a mean-preserving transformation method to construct estimators for unknown parameters and functions and established their asymptotic normalities.It is well known that the empirical likelihood (EL) method proposed by Owen [17] has many advantages over the normal approximation (NA) method and the bootstrap methods for constructing confidence intervals. The most appealing features are that the confidence regions constructed by the EL method do not require estimating the covariance of the estimator, and the shape and orientation of confidence regions are only determined by the data. In addition, the confidence region derived by the NA-based method is predetermined to be symmetric which may not be adequate when the underlying distribution is typically asymmetric. More discussions on advantages of the EL method over the existing methods can be found in the monograph of Owen [18]. In the past two decades, lots of scholars applied the EL approach to investigate various regression models, such as the partially linear model under right censored data [19], partially linear single-index model [20], semiparametric varying-coefficient partially linear regression models [21], longitudinal partially linear model with α-mixing errors [22], high-dimensional partially linear varying-coefficient model with measurement errors [23], linear quantile regression model with the covariates missing at random [24], among others.In this article, we shall use the EL method to study the generalized functional-coefficient regression models with multiple smoothing variables when the response is subject to random right censoring. The coefficient functions with different smoothing variables can accommodate various nonlinear interaction effects between covariates that are admitted in the model. The empirical log-likelihood ratio for the unknown parameter is constructed based on the smooth backfitting technique, and it is shown to possess a nonparametric version of Wilks’ theorem.The rest of this article is organized as follows. In Section2, we propose the empirical log-likelihood ratio statistic for the unknown parameter and give the main result. A simulation study is presented in Section 3. Assumption conditions and proofs of the main result are relegated to the Appendix.


## 2. Methodology

### 2.1. Introduction for Right Censored Semiparametric Model

LetW=XT,ZT,UTT and C be a set of covariates and a censoring variable, respectively. We assume that a random sample Wi,Yi,i=1,2,…,n of W,Y comes from model (2). In the presence of random right censoring, we observe a random sample Ti,δi of a pair T,δ instead of observing Yi. The observed Survival Time T is minY,C and the censoring indicator is δ=IY≤C, which equals 1 if Y≤C and 0 otherwise. Since the estimation of the model cannot be obtained directly, the following mean-preserving transformation is considered:(3)YG=δϕT1−GT−,where G is the distribution function of the censoring variable C. In (3), YG is zero when the response is censored and ϕY/1−GY− when the response is observed. On the other hand, under the assumptions given below, by noting that EδW,Y=1−GY−, it follows that(4)EYGW=w=EϕYW=w.AlthoughYG depends on the choice of ϕ, i.e, YG=YGϕ, we omit the dependence in the expression throughout the paper for notational convenience. It also can be seen that (4) holds for any ϕ.


### 2.2. Empirical Likelihood for Right Censored Data

In this section, we shall apply the EL method to construct confidence regions of the unknown parameter for generalized functional-coefficient regression models (2) with multiple smoothing variables. For the given β, α⋅ can be estimated by using some smoothing techniques with response YG,i−XiTβ. Here, we use the smoothing backfitting technique studied in Lee et al. [25] and Yang [16] and denote the resultant estimator as α^G⋅;β. Let α^G,jll⋅;β satisfy the following system of integral equations:(5)α^G,jlluj;β=α˜G,jlluj;β−∑k≠j∫Q^juj−1Q^jkuj,ukα^G,kuk;βduk,j=1,…,d,where Q^juj=n−1∑i=1nDijDijTKhjuj,Ui,jZi,j2, Q^jkuj,uk=n−1∑i=1nDijDikTKhjuj,Ui,jZi,jZi,k, and α˜G,juj;β=n−1Q^juj−1∑i=1nDijKhjuj,Ui,jZi,jYiG−XiTβ for Zi=Zi,1,…,Zi,dT, Ui=Ui,1,…,Ui,dT, Dij=1,Ui,j−uj/hjT, and Khu,v=Ku−v/h/∫Kw−v/hdw with a base kernel K. Then, α^Gll⋅;β=α^G,1ll⋅;βT,…,α^G,dll⋅;βTT is called the local linear smooth backfitting estimator with response YiG−XiTβ. Note that α^Gll⋅;β estimates not only the coefficient functions but also their derivatives. Thus, only the first components of α^G,jll⋅;βj=1,…,d constitute α^G⋅;β=α^G,1⋅;β,…,α^G,d⋅;βT.An auxiliary random variable is introduced as follows:(6)ηiβ=Xi−ZiTα^XUiYG,i−ZiTα^YGUi−Xi−ZiTα^XUiβ.Similar to Owen [26], an empirical log-likelihood ratio can be defined as(7)lnβ=−2max∑i=1nlognpi:∑i=1npiηiβ=0,pi≥0,∑i=1npi=1.For any givenβ, a unique value for lnβ exists, and we assume that 0 is inside the convex hull of the points η1β,…,ηnβ. By the Lagrange multiplier method, we can obtain pi=1+λτηiβ/n, and lnβ can be expressed as(8)lnβ=2∑i=1nlog1+λτηiβ,where λ is the solution of the following equation:(9)1n∑i=1nηiβ1+λτηiβ=0.Theorem 1.
Suppose that the conditions (C1)–(C11) in the Appendix hold. Ifβ0 is the true value of the parameter β, then(10)lnβ0⟶dχp2,asn⟶∞,where ⟶d stands for convergence in distribution, χp2 is a standard chi-square random variable with p degree of freedom.It follows from Theorem1 that an approximate 1−α level confidence region for β0 can be constructed by β:lnβ≤cα, where cα is chosen to satisfy Pχp2≤cα=1−α.


## 2.1. Introduction for Right Censored Semiparametric Model

LetW=XT,ZT,UTT and C be a set of covariates and a censoring variable, respectively. We assume that a random sample Wi,Yi,i=1,2,…,n of W,Y comes from model (2). In the presence of random right censoring, we observe a random sample Ti,δi of a pair T,δ instead of observing Yi. The observed Survival Time T is minY,C and the censoring indicator is δ=IY≤C, which equals 1 if Y≤C and 0 otherwise. Since the estimation of the model cannot be obtained directly, the following mean-preserving transformation is considered:(3)YG=δϕT1−GT−,where G is the distribution function of the censoring variable C. In (3), YG is zero when the response is censored and ϕY/1−GY− when the response is observed. On the other hand, under the assumptions given below, by noting that EδW,Y=1−GY−, it follows that(4)EYGW=w=EϕYW=w.AlthoughYG depends on the choice of ϕ, i.e, YG=YGϕ, we omit the dependence in the expression throughout the paper for notational convenience. It also can be seen that (4) holds for any ϕ.


## 2.2. Empirical Likelihood for Right Censored Data

In this section, we shall apply the EL method to construct confidence regions of the unknown parameter for generalized functional-coefficient regression models (2) with multiple smoothing variables. For the given β, α⋅ can be estimated by using some smoothing techniques with response YG,i−XiTβ. Here, we use the smoothing backfitting technique studied in Lee et al. [25] and Yang [16] and denote the resultant estimator as α^G⋅;β. Let α^G,jll⋅;β satisfy the following system of integral equations:(5)α^G,jlluj;β=α˜G,jlluj;β−∑k≠j∫Q^juj−1Q^jkuj,ukα^G,kuk;βduk,j=1,…,d,where Q^juj=n−1∑i=1nDijDijTKhjuj,Ui,jZi,j2, Q^jkuj,uk=n−1∑i=1nDijDikTKhjuj,Ui,jZi,jZi,k, and α˜G,juj;β=n−1Q^juj−1∑i=1nDijKhjuj,Ui,jZi,jYiG−XiTβ for Zi=Zi,1,…,Zi,dT, Ui=Ui,1,…,Ui,dT, Dij=1,Ui,j−uj/hjT, and Khu,v=Ku−v/h/∫Kw−v/hdw with a base kernel K. Then, α^Gll⋅;β=α^G,1ll⋅;βT,…,α^G,dll⋅;βTT is called the local linear smooth backfitting estimator with response YiG−XiTβ. Note that α^Gll⋅;β estimates not only the coefficient functions but also their derivatives. Thus, only the first components of α^G,jll⋅;βj=1,…,d constitute α^G⋅;β=α^G,1⋅;β,…,α^G,d⋅;βT.An auxiliary random variable is introduced as follows:(6)ηiβ=Xi−ZiTα^XUiYG,i−ZiTα^YGUi−Xi−ZiTα^XUiβ.Similar to Owen [26], an empirical log-likelihood ratio can be defined as(7)lnβ=−2max∑i=1nlognpi:∑i=1npiηiβ=0,pi≥0,∑i=1npi=1.For any givenβ, a unique value for lnβ exists, and we assume that 0 is inside the convex hull of the points η1β,…,ηnβ. By the Lagrange multiplier method, we can obtain pi=1+λτηiβ/n, and lnβ can be expressed as(8)lnβ=2∑i=1nlog1+λτηiβ,where λ is the solution of the following equation:(9)1n∑i=1nηiβ1+λτηiβ=0.Theorem 1.
Suppose that the conditions (C1)–(C11) in the Appendix hold. Ifβ0 is the true value of the parameter β, then(10)lnβ0⟶dχp2,asn⟶∞,where ⟶d stands for convergence in distribution, χp2 is a standard chi-square random variable with p degree of freedom.It follows from Theorem1 that an approximate 1−α level confidence region for β0 can be constructed by β:lnβ≤cα, where cα is chosen to satisfy Pχp2≤cα=1−α.


## 3. Simulation Study

In this section, we conduct some simulations to assess the performance of the proposed empirical likelihood inference procedure. Throughout this section, the kernel function is taken to beKu=0.751−u2, u≤1, and the bandwidth h is selected by the cross-validation procedure. We compare our proposed EL method with the normal approximation (NA) method through simulated examples and a real data example.


### 3.1. Simulated Examples

In this subsection, consider the following model:(11)Y=X1β1+X2β2+Z1α1U1+Z2α2U2+ϵ,where β1=1.5, β2=2, α1U=3U2, and α2U=sinπU. The random smoothing variables U1 and U2 are independently generated from U0,1. The covariates X=X1,X2T and Z=Z1,Z2T are drawn independently from multivariate normal distribution with zero mean and covariance matrix 0.5k−l. The model error ϵ is drawn from normal distribution N0,1 and the uniform distribution U−1,1, respectively.For random right censoring, we consideredC∼U−6,6+c0, where c0 is a constant to control the percentage of censoring. In our simulations, we take c0 as 2.5, 4.0, and 6.6 to yield approximately 30%, 20%, and 10% censoring rate, respectively. We also take ϕt=tIt≤t0 for the transformation of the response, where we set t0 to the largest uncensored observation.The sample sizen is chosen to be 100, 200, and 300, respectively. The coverage probabilities (CPs) and the average lengths (ALs) are calculated based on the EL and NA methods for the normal error and uniform distribution error with 1000 replicates. The simulation results of CPs and ALs with the nominal level 1−α=0.95 are reported in Table 1. Further, we give the confidence regions of β1,β2 with the nominal level 1−α=0.95 in Figures 1 and 2 for the normal error and uniform distribution error, respectively.Table 1
The coverage probabilities (CPs) and average lengths (ALs) of the confidence intervals based on the EL and NA methods with nominal level1−α = 0.95.
ϵ
n
CR (%)
CP
AL
EL
NA
ELβ1
ELβ2
NAβ1
NAβ2
N
0,1
100
10
0.9330
0.9250
0.0249
0.0244
0.0257
0.0255
20
0.9180
0.9110
0.0270
0.0271
0.0278
0.0283
30
0.9000
0.8980
0.0281
0.0282
0.0290
0.0291
200
10
0.9390
0.9330
0.0169
0.0170
0.0172
0.0181
20
0.9240
0.9210
0.0188
0.0187
0.0210
0.0213
30
0.9170
0.9150
0.0209
0.0211
0.0228
0.0231
300
10
0.9440
0.9360
0.0080
0.0090
0.0103
0.0108
20
0.9380
0.9320
0.0115
0.0123
0.0131
0.0136
30
0.9260
0.9200
0.0140
0.0133
0.0150
0.0151
U
−
1,1
100
10
0.9290
0.9210
0.0250
0.0248
0.0261
0.0262
20
0.9070
0.9050
0.0277
0.0276
0.0283
0.0285
30
0.8980
0.8950
0.0288
0.0286
0.0294
0.0297
200
10
0.9370
0.9310
0.0171
0.0174
0.0182
0.0180
20
0.9220
0.9190
0.0190
0.0191
0.0217
0.0216
30
0.9150
0.9110
0.0214
0.0212
0.0233
0.0239
300
10
0.9410
0.9350
0.0118
0.0117
0.0123
0.0129
20
0.9360
0.9300
0.0132
0.0134
0.0140
0.0142
30
0.9230
0.9140
0.0144
0.0140
0.0152
0.0163Figure 1
95% confidence regions for (β1, β2) with CR = 10% (a), CR = 20% (b), and CR = 30% (c) based on ϵ∼N0,1.
(a)
(b)
(c)Figure 2
95% confidence regions for (β1, β2) with CR = 10% (a), CR = 20% (b), and CR = 30% (c) based on ϵ∼U−1,1.
(a)
(b)
(c)Table1 indicates that the EL method outperforms the NA method, since the EL method gives more accurate coverage probability and shorter average length for almost all settings. We also find that the EL and NA methods perform worse as the censoring rate increases or the sample size decreases. Furthermore, we can see from Figures 1 and 2 that, the EL method performs better than the NA method for different settings. When the censoring rate increases, the confidence region gets larger for two different model error cases. In sum, we strongly recommend our proposed EL method to construct the confidence regions for the unknown parameter in generalized functional-coefficient regression models with multiple smoothing variables under right censoring data.


### 3.2. Real Data Analysis

In this subsection, we apply our proposed empirical likelihood method to analyse the lung cancer data from the Veteran’s Administration lung cancer trial [27]. In this trial, a standard or testing chemotherapy was randomly assigned to a male suffering from inoperable lung cancer, and 137 males entered into the follow-up at the start of this trial. There are 8 variables for all the patients which were also recorded: Survival in Days (denoted as Y), Status (1 = dead and 0 = censored) (denoted as δ), Treatment Indicator (1 = standard and 2 = test) (denoted as X1), Cell Type (1 = squamous, 2 = small cell, 3 = adeno, and 4 = large) (denoted as X2), Age (denoted as U1), Karnofsky Score (denoted as U2), Months from Diagnosis (denoted as Z1), and Prior Therapy (0 = no and 10 = yes) (denoted as Z2). Now, we apply model (2) to fit the data and use our proposed EL method and NA method to construct the confidence regions of β1,β2. The censoring rate of the data is about 6.57%. The profile least squares estimator of β is estimated as β^=0.1587,0.6003T. The confidence regions based on the EL method and NA method of β1,β2 with nominal level 1−α=0.95 and 0.90 are given in Figure 3. From Figure 3, we can see that the confidence regions contain the profile least squares estimate for EL and NA approaches. Furthermore, the confidence regions based on the EL method are smaller than those based on the NA method for different scenarios. Besides, the confidence regions with nominal level 1−α=0.95 are larger than those with nominal level 1−α=0.90. Also, from the confidence regions of the EL method, we can see that the Cell Type is an important factor in the interpretation of Survival in Days. The confidence regions of the Treatment Indicator include zero which indicates that the Treatment Indicator is not a significant factor, that is, there is no difference between the two chemotherapies in the lung cancer treatment.Figure 3
The confidence regions ofθ1,θ2 based on EL and NA methods with nominal level 1−α=0.95 (a) and 0.90 (b), respectively.
(a)
(b)


## 3.1. Simulated Examples

In this subsection, consider the following model:(11)Y=X1β1+X2β2+Z1α1U1+Z2α2U2+ϵ,where β1=1.5, β2=2, α1U=3U2, and α2U=sinπU. The random smoothing variables U1 and U2 are independently generated from U0,1. The covariates X=X1,X2T and Z=Z1,Z2T are drawn independently from multivariate normal distribution with zero mean and covariance matrix 0.5k−l. The model error ϵ is drawn from normal distribution N0,1 and the uniform distribution U−1,1, respectively.For random right censoring, we consideredC∼U−6,6+c0, where c0 is a constant to control the percentage of censoring. In our simulations, we take c0 as 2.5, 4.0, and 6.6 to yield approximately 30%, 20%, and 10% censoring rate, respectively. We also take ϕt=tIt≤t0 for the transformation of the response, where we set t0 to the largest uncensored observation.The sample sizen is chosen to be 100, 200, and 300, respectively. The coverage probabilities (CPs) and the average lengths (ALs) are calculated based on the EL and NA methods for the normal error and uniform distribution error with 1000 replicates. The simulation results of CPs and ALs with the nominal level 1−α=0.95 are reported in Table 1. Further, we give the confidence regions of β1,β2 with the nominal level 1−α=0.95 in Figures 1 and 2 for the normal error and uniform distribution error, respectively.Table 1
The coverage probabilities (CPs) and average lengths (ALs) of the confidence intervals based on the EL and NA methods with nominal level1−α = 0.95.
ϵ
n
CR (%)
CP
AL
EL
NA
ELβ1
ELβ2
NAβ1
NAβ2
N
0,1
100
10
0.9330
0.9250
0.0249
0.0244
0.0257
0.0255
20
0.9180
0.9110
0.0270
0.0271
0.0278
0.0283
30
0.9000
0.8980
0.0281
0.0282
0.0290
0.0291
200
10
0.9390
0.9330
0.0169
0.0170
0.0172
0.0181
20
0.9240
0.9210
0.0188
0.0187
0.0210
0.0213
30
0.9170
0.9150
0.0209
0.0211
0.0228
0.0231
300
10
0.9440
0.9360
0.0080
0.0090
0.0103
0.0108
20
0.9380
0.9320
0.0115
0.0123
0.0131
0.0136
30
0.9260
0.9200
0.0140
0.0133
0.0150
0.0151
U
−
1,1
100
10
0.9290
0.9210
0.0250
0.0248
0.0261
0.0262
20
0.9070
0.9050
0.0277
0.0276
0.0283
0.0285
30
0.8980
0.8950
0.0288
0.0286
0.0294
0.0297
200
10
0.9370
0.9310
0.0171
0.0174
0.0182
0.0180
20
0.9220
0.9190
0.0190
0.0191
0.0217
0.0216
30
0.9150
0.9110
0.0214
0.0212
0.0233
0.0239
300
10
0.9410
0.9350
0.0118
0.0117
0.0123
0.0129
20
0.9360
0.9300
0.0132
0.0134
0.0140
0.0142
30
0.9230
0.9140
0.0144
0.0140
0.0152
0.0163Figure 1
95% confidence regions for (β1, β2) with CR = 10% (a), CR = 20% (b), and CR = 30% (c) based on ϵ∼N0,1.
(a)
(b)
(c)Figure 2
95% confidence regions for (β1, β2) with CR = 10% (a), CR = 20% (b), and CR = 30% (c) based on ϵ∼U−1,1.
(a)
(b)
(c)Table1 indicates that the EL method outperforms the NA method, since the EL method gives more accurate coverage probability and shorter average length for almost all settings. We also find that the EL and NA methods perform worse as the censoring rate increases or the sample size decreases. Furthermore, we can see from Figures 1 and 2 that, the EL method performs better than the NA method for different settings. When the censoring rate increases, the confidence region gets larger for two different model error cases. In sum, we strongly recommend our proposed EL method to construct the confidence regions for the unknown parameter in generalized functional-coefficient regression models with multiple smoothing variables under right censoring data.


## 3.2. Real Data Analysis

In this subsection, we apply our proposed empirical likelihood method to analyse the lung cancer data from the Veteran’s Administration lung cancer trial [27]. In this trial, a standard or testing chemotherapy was randomly assigned to a male suffering from inoperable lung cancer, and 137 males entered into the follow-up at the start of this trial. There are 8 variables for all the patients which were also recorded: Survival in Days (denoted as Y), Status (1 = dead and 0 = censored) (denoted as δ), Treatment Indicator (1 = standard and 2 = test) (denoted as X1), Cell Type (1 = squamous, 2 = small cell, 3 = adeno, and 4 = large) (denoted as X2), Age (denoted as U1), Karnofsky Score (denoted as U2), Months from Diagnosis (denoted as Z1), and Prior Therapy (0 = no and 10 = yes) (denoted as Z2). Now, we apply model (2) to fit the data and use our proposed EL method and NA method to construct the confidence regions of β1,β2. The censoring rate of the data is about 6.57%. The profile least squares estimator of β is estimated as β^=0.1587,0.6003T. The confidence regions based on the EL method and NA method of β1,β2 with nominal level 1−α=0.95 and 0.90 are given in Figure 3. From Figure 3, we can see that the confidence regions contain the profile least squares estimate for EL and NA approaches. Furthermore, the confidence regions based on the EL method are smaller than those based on the NA method for different scenarios. Besides, the confidence regions with nominal level 1−α=0.95 are larger than those with nominal level 1−α=0.90. Also, from the confidence regions of the EL method, we can see that the Cell Type is an important factor in the interpretation of Survival in Days. The confidence regions of the Treatment Indicator include zero which indicates that the Treatment Indicator is not a significant factor, that is, there is no difference between the two chemotherapies in the lung cancer treatment.Figure 3
The confidence regions ofθ1,θ2 based on EL and NA methods with nominal level 1−α=0.95 (a) and 0.90 (b), respectively.
(a)
(b)



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*Source: 1261426-2020-04-01.xml*