# Adjusted Empirical Likelihood for Varying Coefficient Partially

Authors: Peixin Zhao
Journal: Journal of Mathematics
 (2013)

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## Abstract

By constructing an adjusted auxiliary vector ingeniously, we propose an adjusted empirical likelihood ratio function for the parametric components of varying coefficient partially
linear models with censored data. It is shown that its limiting distribution is standard
central chi-squared. Then the confidence intervals for the parametric components are constructed. A simulation study and a real data analysis are undertaken to assess the finite
sample performance of the proposed method.


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## Body

## 1. Introduction

Outcome censoring often occurs in many disciplines such as econometrics, biostatistics, and bioinformatics. There have been many recent researches in the area of statistical inference for censored data. Wang and Zheng [1] and Wang and Li [2] considered the estimation problem for partly linear models based on different methods. Yang et al. [3] proposed an empirical likelihood method for a partially linear single-index model with right censored data. More results on the statistical inferences of censored data can be seen in [4–6]. In this paper, we consider the empirical likelihood inferences for a varying coefficient partially linear model with right censored data. Specifically, let {Yi,i=1,…,n} be the response variable and let {(Xi,Zi,Ui),i=1,…,n} be associated covariates. We consider the following varying coefficient partially linear model:
(1)Yi=XiTθ(Ui)+ZiTβ+ϵi,
where β is a q×1 vector of unknown parameters, θ(·) is a p×1 vector of unknown function, and ϵi is the model error such as E(ϵi∣Xi,Zi,Ui)=0. Due to the curse of dimensionality, as in [7], we assume that Ui is univariate. In this paper, we assume that the available data {(Yi*,δi,Xi,Zi,Ui), i=1,…,n}, are independent and identically distributed, where Yi*=min{Yi,Ci}, δi=I(Yi≤Ci) and Ci is a censoring variable. We also assume that {Ci,i=1,…,n} are i.i.d. variables with a distribution function G(·) and Ci is independent of the prognostic variables (Xi,Zi,Ui) and the response Yi.Model (1) has proved to be very useful as it combines the flexibility of nonparametric models and the interpretation of linear models. Recently, a variety of methods have been proposed for estimating the parameters and nonparametric functions in model (1) based on the uncensored data (see [7, 8]). In this paper, we recommend using the empirical likelihood method to make inference for β in the model (1) under right censored data. We define an empirical log-likelihood function based on the synthetic data and show that its asymptotic distribution is a mixture of central chi-squared distribution. Hence, the confidence regions for parametric components can be constructed if the unknown weights are estimated. In addition, by constructing an adjusted auxiliary vector ingeniously, we also propose an adjusted empirical likelihood ratio function for the parametric components and show that its limiting distribution is a standard central chi-squared distribution. Then the confidence intervals for parametric components are constructed and the estimation for the unknown weights is avoided. To demonstrate the performance of the proposed method, based on some numerical experiments such as the simulation study and the real data analysis, we conducted a comparison with the normal approximation method. Compared with the normal approximation method, the confidence intervals based on the empirical likelihood method perform fairly well.The confidence intervals obtained by the empirical likelihood method possess several attractive features compared to the conventional Wald-type confidence intervals, such as circumvention of asymptotic variance estimation to compute the standard error, flexible shapes of the confidence intervals determined by data and range-preserving property (see [9]). Wang and Li [2] considered the adjusted empirical likelihood inferences for a class of partially linear models with right censored data. Yang et al. [3] considered the adjusted empirical likelihood inferences for a partially linear single-index model with right censored data. This paper also contributes to the rapidly growing literature on the adjustment-based empirical likelihood method and provides additional positive results for the Wilks phenomena in the parametric components of varying coefficient partially linear models with censored data, which extends the application literature of the adjusted empirical likelihood method. The more works on empirical likelihood methods can be found in [10, 11].


## 2. Methodology and Results

As in [3], let YiG*=Yi*δi/(1-G(Yi*)); then we have E(YiG*∣Xi,Zi,Ui)=E(Yi∣Xi,Zi,Ui). This implies that
(2)YiG*=XiTθ(Ui)+ZiTβ+ϵi*,
where E(ϵi*∣Xi,Zi,Ui)=0. When G(·) is known as well, model (2) is a standard semiparametric varying coefficient partially linear model. For given β, using the same arguments as Fan and Huang [7], we can get the weighted local least-squares estimator of θ(u) by minimizing
(3)∑i=1n{YiG*-ZiTβ-∑k=1p[ak+bk(Ui-u)]Xik}2Kh(Ui-u),
where Kh(·)=h-1K(·/h), K(·) is a kernel function, h is a bandwidth, and Xik denotes the kth component of Xi.LetZ=(Z1,…,Zn)T, let YG*=(Y1G*,…,YnG*)T, let Ip be p×p identity matrix, let 0p be p×p zero matrix, let Ωu=diag(Kh(u-U1),…,letKh(u-Un)) be n×n diagonal matrix. Then, the solution to minimize (3) can be given by
(4)θ~(u)=(Ip,0p)(DuTΩuDu)-1DuTΩu(YG*-Zβ),
where Du=(X1⋯Xnh-1(U1-u)X1⋯h-1(Un-u)Xn)T.Let(Ip,0p)(DuTΩuDu)-1DuTΩu≡(S1(u),…,Sn(u)), let μ~(u)=∑k=1n‍Sk(u)ZkT, and let g~(u)=∑k=1n‍Sk(u)YkG*. Then we have θ~(u)=g~(u)-μ~(u)β. Substituting this into (2) and by a simple calculation we have
(5)Y˘iG*=Z˘iTβ+ϵi*,
where Z˘i=Zi-μ~(Ui)TXi, Y˘iG*=YiG*-g~(Ui)TXi. To construct the empirical likelihood ratio function for β, we introduce the auxiliary random vector
(6)η˘i(β)=Z˘i(Y˘iG*-Z˘iTβ).By (5) we have that β is true if and only if E{η˘i(β)}=o(1). Using such information, we can define a profile empirical log-likelihood ratio function for β. However, G(·) is usually unknown, and η˘i(β) cannot be used directly to make inference for β. To solve this problem, we replace G(·) in η˘i(β) by its estimator. In this paper, we employ the Kaplan-Meier estimator
(7)Gn(y)=1-∏i=1n‍(N+(Yi*)1+N+(Yi*))I(Yi*≤y,δi=0),
where N+(y)=∑j=1n‍I(Yj*>y). Hence, an estimator of η˘i(β) can be defined as
(8)η^i(β)=Z˘i(Y˘iGn*-Z˘iTβ),
where Y˘iGn*=YiGn*-g^(Ui)TXi and g^(u)=∑k=1n‍Sk(u)YkGn*. Then, we can define the profile empirical log-likelihood ratio function for β as
(9)R^(β)=-2max{∑i=1nlog(npi)∣pi≥0,∑i=1npi=1,∑i=1npiη^i(β)=0},
where pi=pi(β), i=1,…,n. A unique value for R^(β) exists, provided that 0 is inside the convex hull of the points (η^1(β),…,η^n(β)). Based on the method of Lagrange multipliers to find the optimal pi, then the empirical log-likelihood ratio function can be represented as
(10)R^(β)=2∑i=1n‍log(1+λTη^i(β)),
where λ=λ(β) is a q×1 vector, which satisfies
(11)∑i=1n‍η^i(β)(1+λTη^i(β))=0.
Let
(12)Ψ(u)=E(XXT∣U=u),Φ(u)=E(XZT∣U=u),H(s)=E{[Z-μ(U)TX]YG*I(s<Y*)}(1-G(s))(1-F(s-)),μ(u)=Ψ(u)-1Φ(u),ΛG(u)=∫-∞u‍11-G(s-)dG(s),Σ0(β)=E{[Z-μ(U)TX][Z-μ(U)TX]T×[YG*-ZTβ-XTθ(U)]2},Σ1=∫-∞∞‍H(s)H(s)T(1-F(s-))(1-ΔΛG(s))dG(s).
The following theorem gives the asymptotic distribution of R^(β).Theorem 1.
Suppose that conditions (C1)–(C5) in the Appendix hold. Ifβ0 is the true parameter, then
(13)R^(β0)→ℒw1χ1,12+w2χ1,22+⋯+wqχ1,q2,
where w1,…,wq are the eigenvalues of  Σ0-1(β0)Σ(β0), Σ(β0)=Σ0(β0)-Σ1, and χ1,12,…,χ1,q2 are independent standard chi-square random variables with 1 degree of freedom.From Theorem1, we can see that the asymptotic distribution of R^(β) is a mixture of central chi-square distributions. Hence, the confidence regions for the parametric components can be constructed if the unknown weights are estimated. Next, we give an adjusted empirical log-likelihood ratio function that has an asymptotic standard chi-square distribution. Then the confidence regions for the parametric components are constructed, and the estimation for the unknown weights is avoided. Let Fn(·) be the Kapaln-Meier estimator for the distribution function of Y, and
(14)Σ^0(β)=1n∑i=1n‍η^i(β)η^i(β)T,ΛnGn(u)=∫-∞u‍11-Gn(s-)dGn(s),Hn(s)=n-1∑i=1n[Zi-μ~(Ui)TXi]YiGn*I(s<Yi*)}(1-Gn(s))(1-Fn(s-)),Σ^1=1n∑i=1n‍(1-δi)Hn(Yi*)Hn(Yi*)T(1-ΔΛnGn(Yi*-))Σ^(β)=Σ^0(β)-Σ^1,Sn(β)=∑i=1n‍η^i(β)∑i=1n‍η^i(β)T.
Then, as in [3], an adjusted empirical log-likelihood ratio function can be defined by
(15)R^ad(β)=γn(β)R^(β),
where γn(β)=tr{Σ^-1(β)Sn(β)}/tr{Σ^0-1(β)Sn(β)}. The following theorem shows that the asymptotic distribution of R^ad(β) can be approximated by standard chi-square distribution.Theorem 2.
Suppose that conditions (C1)–(C5) in the Appendix hold. Ifβ0 is the true parameter, then
(16)R^ad(β0)→ℒχq2.Letχq2(1-α) be the 1-α quantile of χq2, (0<α<1). Theorem 2 implies that an approximate 1-α confidence region for β can be defined by
(17)Cα(β)={β∣R^ad(β)≤χq2(1-α)}.


## 3. Numerical Results

In this section, we conduct several simulation experiments to illustrate the finite sample performances of the proposed method and consider a real data set analysis for further illustration.


### 3.1. Simulation Studies

To evaluate the performance of the proposed adjustment-based empirical likelihood (AEL) method, we present some simulation results. In the simulation, we simulated data from the following model:(18)Y=(2U2-1)X+Z+ϵ,
where β=1, θ(u)=2u2-1. We generated n=50,100 and 150 subjects, respectively, and the covariates X~N(0,1), Z~N(1,1.5), and U~U(-1,1). Y is generated according to the model with ϵ~N(0,0.5). The censoring variable C~N(μ,1), where μ=3.5, 2.1 and 1.3, respectively, such that the corresponding censoring rates (CR) are about 0.1,0.3, and 0.45. We use the Epanechnikov kernel function K(u)=0.75(1-u2)+, and the cross-validation bandwidth hCV is obtained by minimizing
(19)CV(h)=∑i=1n‍(YiGn*-XiTθ^[i](Ui)-ZiTβ^[i])2,
where β^[i] and θ^[i](·) are estimators of β and θ(·) which are computed with all of the measurements, but the ith subject is deleted. β^[i] can be obtained by minimizing (10), θ^[i](·) can be obtained by replacing G(·), and β in (4) by Gn(·) and β^[i], respectively.For comparison, we consider two approaches for constructing the confidence intervals: the AEL method proposed in this paper and normality approximation (NA) method proposed by [12]. To construct confidence intervals based on the NA method, we need to estimate the asymptotic variance. However, note that the asymptotic variance of the estimator has a complicated structure; then we estimate the asymptotic variance by the bootstrap method. The average lengths and coverage probabilities of the confidence intervals are shown in Table 1. Here the nominal level is taken as 1-α=95%, and the simulation is computed with 1000 simulation runs. From Table 1, we can see the following observations.(i)
For any given level of CR and the size of sample, although the coverage probabilities of the confidence intervals obtained by the AEL method and the NA method are similar, the lengths of the confidence intervals obtained by the AEL method are shorter than those obtained by the NA method. This implies that the confidence intervals obtained by the AEL method outperform those obtained by the NA method.(ii)
For the given size of samplen, the performances of the confidence intervals, obtained by the AEL method under the moderate censoring rate, are all close to the nominal level 95%. This implies that the adjustment scheme is workable for the moderate censoring rate. However, the simulation results also suggest that larger samples would be needed when the censoring rate is relatively high such as CR=0.45.Table 1
The average lengths and coverage probabilities of the 95% confidence intervals forβ based on the AEL method and the NA method.
n
CR
Confidence interval
Length
Coverage
AEL
NA
AEL
NA
AEL
NA
0.1
(0.924, 1.247)
(0.828, 1.344)
0.323
0.516
0.914
0.916
50
0.3
(0.915, 1.259)
(0.825, 1.347)
0.344
0.522
0.896
0.894
0.45
(0.906, 1.264)
(0.824, 1.350)
0.358
0.526
0.863
0.864
0.1
(0.927, 1.220)
(0.913, 1.270)
0.293
0.357
0.935
0.937
100
0.3
(0.922, 1.223)
(0.907, 1.273)
0.301
0.366
0.931
0.930
0.45
(0.916, 1.227)
(0.906, 1.276)
0.311
0.370
0.928
0.928
0.1
(0.960, 1.161)
(0.942, 1.239)
0.201
0.297
0.948
0.948
150
0.3
(0.952, 1.199)
(0.939, 1.237)
0.247
0.298
0.945
0.943
0.45
(0.949, 1.202)
(0.933, 1.236)
0.253
0.303
0.943
0.942


### 3.2. Application to CGD Data

We illustrate the applicability of our proposed method using the chronic granulomatous disease (CGD) data set from the International CGD Cooperative Study Group. This data set was designed to have a single interim analysis when the follow-up data as of July 15, 1989 were complete. The monitoring committee for the trial terminated the trial at a meeting on September 22, 1989. The treatment given each patient was unblinded at the first scheduled visit for the patient following the decision of the monitoring committee. More details for this data description and analysis can be seen in [13].The variables contained here areX: treatment code, 1= rIFN, 2= placebo; Z1: pattern of inheritance, 1=X-linked, 2= autosomal recessive; Z2: age, in years; Z3: height, in cm; Z4: weight, in kg; Z5: using corticosteroids at time of study entry, 1= yes, 2= no; Z6: using prophylactic antibiotics at time of study entry, 1= yes, 2= no; Z7: 1= male, 2= female; Z8: hospital category, 1= US-NIH, 2= US-other, 3= Europe, Amsterdam, 4= Europe, other; T: elapsed time (in days) from randomization to diagnosis of a serious infection or if a censored observation: elapsed time from randomization to censoring date; δ: censoring indicator, 1= Noncensored observation, 2= censored observation. S: sequence number. For each patient, the infection records are in sequence number order.Based on the generalized likelihood ratio testing, Jiang and Qian [14] show that this data set can be fitted by the following model:
(20)Y=Xθ(U)+∑k=08‍Zkβk+ε,
where Y=ln(T), U=S/20, and Z0=1 as the intercept term. The estimators and 95% confidence intervals are shown in Table 2. From Table 2, we see that the covariates Z1, Z5, and Z7 have somewhat stronger associations with the response. In addition, we also can see that the confidence intervals obtained by the AEL method are workable.Table 2
Application to CGD data. The maximum empirical likelihood estimators (MELE) and 95% confidence intervals (CI) for the parametric components based on the AEL method.
β
0
β
1
β
2
β
3
β
4
MELE
4.470
−0.288
0.013
0.004
−0.007
CI
(4.210, 4.680)
(−0.473, −0.158)
(0.002, 0.021)
(0.002, 0.005)
(−0.012, −0.004)
β
5
β
6
β
7
β
8
MELE
0.398
−0.097
0.305
−0.021
CI
(0.268, 0.501)
(−0.267, 0.033)
(0.178, 0.412)
(−0.128, 0.045)


## 3.1. Simulation Studies

To evaluate the performance of the proposed adjustment-based empirical likelihood (AEL) method, we present some simulation results. In the simulation, we simulated data from the following model:(18)Y=(2U2-1)X+Z+ϵ,
where β=1, θ(u)=2u2-1. We generated n=50,100 and 150 subjects, respectively, and the covariates X~N(0,1), Z~N(1,1.5), and U~U(-1,1). Y is generated according to the model with ϵ~N(0,0.5). The censoring variable C~N(μ,1), where μ=3.5, 2.1 and 1.3, respectively, such that the corresponding censoring rates (CR) are about 0.1,0.3, and 0.45. We use the Epanechnikov kernel function K(u)=0.75(1-u2)+, and the cross-validation bandwidth hCV is obtained by minimizing
(19)CV(h)=∑i=1n‍(YiGn*-XiTθ^[i](Ui)-ZiTβ^[i])2,
where β^[i] and θ^[i](·) are estimators of β and θ(·) which are computed with all of the measurements, but the ith subject is deleted. β^[i] can be obtained by minimizing (10), θ^[i](·) can be obtained by replacing G(·), and β in (4) by Gn(·) and β^[i], respectively.For comparison, we consider two approaches for constructing the confidence intervals: the AEL method proposed in this paper and normality approximation (NA) method proposed by [12]. To construct confidence intervals based on the NA method, we need to estimate the asymptotic variance. However, note that the asymptotic variance of the estimator has a complicated structure; then we estimate the asymptotic variance by the bootstrap method. The average lengths and coverage probabilities of the confidence intervals are shown in Table 1. Here the nominal level is taken as 1-α=95%, and the simulation is computed with 1000 simulation runs. From Table 1, we can see the following observations.(i)
For any given level of CR and the size of sample, although the coverage probabilities of the confidence intervals obtained by the AEL method and the NA method are similar, the lengths of the confidence intervals obtained by the AEL method are shorter than those obtained by the NA method. This implies that the confidence intervals obtained by the AEL method outperform those obtained by the NA method.(ii)
For the given size of samplen, the performances of the confidence intervals, obtained by the AEL method under the moderate censoring rate, are all close to the nominal level 95%. This implies that the adjustment scheme is workable for the moderate censoring rate. However, the simulation results also suggest that larger samples would be needed when the censoring rate is relatively high such as CR=0.45.Table 1
The average lengths and coverage probabilities of the 95% confidence intervals forβ based on the AEL method and the NA method.
n
CR
Confidence interval
Length
Coverage
AEL
NA
AEL
NA
AEL
NA
0.1
(0.924, 1.247)
(0.828, 1.344)
0.323
0.516
0.914
0.916
50
0.3
(0.915, 1.259)
(0.825, 1.347)
0.344
0.522
0.896
0.894
0.45
(0.906, 1.264)
(0.824, 1.350)
0.358
0.526
0.863
0.864
0.1
(0.927, 1.220)
(0.913, 1.270)
0.293
0.357
0.935
0.937
100
0.3
(0.922, 1.223)
(0.907, 1.273)
0.301
0.366
0.931
0.930
0.45
(0.916, 1.227)
(0.906, 1.276)
0.311
0.370
0.928
0.928
0.1
(0.960, 1.161)
(0.942, 1.239)
0.201
0.297
0.948
0.948
150
0.3
(0.952, 1.199)
(0.939, 1.237)
0.247
0.298
0.945
0.943
0.45
(0.949, 1.202)
(0.933, 1.236)
0.253
0.303
0.943
0.942


## 3.2. Application to CGD Data

We illustrate the applicability of our proposed method using the chronic granulomatous disease (CGD) data set from the International CGD Cooperative Study Group. This data set was designed to have a single interim analysis when the follow-up data as of July 15, 1989 were complete. The monitoring committee for the trial terminated the trial at a meeting on September 22, 1989. The treatment given each patient was unblinded at the first scheduled visit for the patient following the decision of the monitoring committee. More details for this data description and analysis can be seen in [13].The variables contained here areX: treatment code, 1= rIFN, 2= placebo; Z1: pattern of inheritance, 1=X-linked, 2= autosomal recessive; Z2: age, in years; Z3: height, in cm; Z4: weight, in kg; Z5: using corticosteroids at time of study entry, 1= yes, 2= no; Z6: using prophylactic antibiotics at time of study entry, 1= yes, 2= no; Z7: 1= male, 2= female; Z8: hospital category, 1= US-NIH, 2= US-other, 3= Europe, Amsterdam, 4= Europe, other; T: elapsed time (in days) from randomization to diagnosis of a serious infection or if a censored observation: elapsed time from randomization to censoring date; δ: censoring indicator, 1= Noncensored observation, 2= censored observation. S: sequence number. For each patient, the infection records are in sequence number order.Based on the generalized likelihood ratio testing, Jiang and Qian [14] show that this data set can be fitted by the following model:
(20)Y=Xθ(U)+∑k=08‍Zkβk+ε,
where Y=ln(T), U=S/20, and Z0=1 as the intercept term. The estimators and 95% confidence intervals are shown in Table 2. From Table 2, we see that the covariates Z1, Z5, and Z7 have somewhat stronger associations with the response. In addition, we also can see that the confidence intervals obtained by the AEL method are workable.Table 2
Application to CGD data. The maximum empirical likelihood estimators (MELE) and 95% confidence intervals (CI) for the parametric components based on the AEL method.
β
0
β
1
β
2
β
3
β
4
MELE
4.470
−0.288
0.013
0.004
−0.007
CI
(4.210, 4.680)
(−0.473, −0.158)
(0.002, 0.021)
(0.002, 0.005)
(−0.012, −0.004)
β
5
β
6
β
7
β
8
MELE
0.398
−0.097
0.305
−0.021
CI
(0.268, 0.501)
(−0.267, 0.033)
(0.178, 0.412)
(−0.128, 0.045)



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*Source: 204363-2013-01-31.xml*