The Empirical Cressie-Read Test Statistics for Longitudinal

Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2014, Article ID 738953, 13 pages
http://dx.doi.org/10.1155/2014/738953
Research Article
The Empirical Cressie-Read Test Statistics for
Longitudinal Generalized Linear Models
Junhua Zhang,1 Ruiqin Tian,2 Suigen Yang,2,3 and Sanying Feng2
1
College of Mechanical Engineering, Beijing Information Science and Technology University, Beijing 100192, China
College of Applied Sciences, Beijing University of Technology, Beijing 100124, China
3
College of Sciences, Tianjin University of Commerce, Tianjin 300134, China
2
Correspondence should be addressed to Ruiqin Tian; [email protected]
Received 10 July 2013; Revised 19 December 2013; Accepted 24 January 2014; Published 18 March 2014
Academic Editor: Jen-Tzung Chien
Copyright © 2014 Junhua Zhang et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
For the marginal longitudinal generalized linear models (GLMs), we develop the empirical Cressie-Read (ECR) test statistic
approach which has been proposed for the independent identically distributed (i.i.d.) case. The ECR test statistic includes empirical
likelihood as a special case. By adopting this ECR test statistic approach and taking into account the within-subject correlation,
the efficiency theory results of estimation and testing based on ECR are established under some regularity conditions. Although a
working correlation matrix is assumed, there is no need to estimate the nuisance parameters in the working correlation matrix based
on the quadratic inference function (QIF). Therefore, the proposed ECR test statistic is asymptotically a standard 2 limit under the
null hypothesis. It is shown that the proposed method is more efficient even when the working correlation matrix is misspecified.
We also evaluate the finite sample performance of the proposed methods via simulation studies and a real data analysis.
1. Introduction
Longitudinal studies are increasingly common in many scientific research fields, including epidemiology, econometrics,
medicine, and life and social sciences. For example, longitudinal studies are often used in psychology to study developmental trends across the life span, in sociology to study life
events throughout lifetimes or generations, and in medical
area to take different treatments at the start of the study and to
see what kind of effects the assigned patients have with each
treatment by week or by year.
Longitudinal data modeling is a statistical method often
used in the experiments that are designed such that responses
on the same experimental units are observed at each repetition. However, generalized linear models (GLMs [1]) have
become a favored tool for the modelling of clustered and
longitudinal data by allowing for the non-Gaussian data and
nonlinear link functions, such as binomial or Poisson’s type
responses. The intrinsic complexity of longitudinal GLMs
makes them challenging. First, one characteristic of longitudinal data is the within-subject correlation among the
repeated measurements. Ignoring this within-subject correlation causes a loss of efficiency in general problems, and
the presence of correlation makes it hard to establish the
underlying asymptotic theory. Second, the full likelihood for
longitudinal data is often difficult to specify, particularly for
the correlated non-Gaussian data. Researchers had developed
appropriate methods and criteria for longitudinal GLMs,
where generalized estimating equations (GEE [2, 3]) or
a recently proposed related method based on quadratic
inference functions (QIF; see [4]) is an attractive option for
longitudinal regression, particularly when marginal relationships rather than within-subject correlation are of primary
interest.
In the present paper, we consider the popular marginal
longitudinal GLMs. Suppose that Y = (1 , . . . ,  ) is the
multivariate response for the  subject and X = (x1 , . . . ,
x ) is the  ×  matrix of the covariates for the th subject
( = 1, . . . , ). Observations from different subjects are independent, but those from the same subjects are correlated.
Assume that the marginal mean of  is
 ( | x ) = ℎ ( x ) ,
 = 1, . . . , ,  = 1, . . . ,  ,
(1)
2
where ℎ(⋅) is a known link function,  = (1 , . . . ,  ) is the
unknown parameter vector of interest, and x = (1 , . . . ,
 ) is a  × 1 vector for  = 1, . . . ,  . In order to simplify
notation, but without loss of generality, we assume  ≡ .
Recently, there has been considerable interest of investigating
the GLMs, such as Liang and Zeger [2], Zeger and Liang [3],
Fitzmaurice [5], Qu et al. [4], Pan [6], Balan and SchiopuKratina [7], Qu and Li [8], Wang and Qu [9], Wang [10], and
Li et al. [11].
The empirical likelihood method was introduced by
Owen [12–14] as a nonparametric method of inference based
on a data-driven likelihood ratio function in the i.i.d. case and
had been applied to various statistical models. For example,
Kolaczyk [15] and Chen and Cui [16] made an extension to the
generalized linear models. As for other studies, see Baggerly
[17], DiCiccio et al. [18], Chen and Qin [19], Qin and Lawless
[20], Zhu and Xue [21], and Li et al. [22] among others. Owen
[23] is one of the best references for this field. Although the
empirical likelihood method had been investigated by many
authors, those researches had mostly focused on independent
observations. For the analysis of longitudinal data, Xue and
Zhu [24] applied the empirical likelihood approach to the
varying coefficient models with longitudinal data, but they
did not consider the within-subject correlation structure of
the longitudinal data. Li et al. [25] and Li et al. [26] proposed
the generalized empirical likelihood method for longitudinal
data by introducing the known within-subject correlation
structure. Wang et al. [27] proposed two generalized empirical likelihood methods for the longitudinal linear model by
taking into consideration within-subject correlations. They
showed that, by plugging the estimator of the working
correlation matrix into the empirical log-likelihood ratio
(ELR), the resulting ELR is asymptotically a weighted sum
of independent 12 -variables with unknown weights. Thus,
the empirical likelihood method needs to be adjusted so that
it can be efficiently used to accommodate the correlation
inherent in longitudinal data. However, in many applications,
how to estimate the working correlation matrix effectively is
a challenging problem.
Instead of the empirical likelihood ratio statistic, Baggerly
[17] used the empirical Cressie-Read (ECR) test statistic and
showed that it is also asymptotically a Chi-squared distribution in the i.i.d. case. Bravo [28] extended the ECR approach
to inference for -mixing processes. The empirical CressieRead test statistic has user-specified parameter  ∈ (−∞, ∞)
and encompasses several commonly used testing statistics
as special cases, such as the empirical likelihood statistic
( = 0), the maximum entropy, minimum information or
the Kullback-Leibler statistic ( = −1), the Neyman-modified
2 statistic or the Euclidean likelihood statistic ( = −2), the
Freeman-Tukey statistic ( = −1/2), and Pearson’s 2 statistic
( = 1), the first two being defined in a limiting sense. Thus,
we can use these results to construct confidence regions of the
parameter  of interest.
In this paper we develop Baggerly’s results on the ECR
test statistic for the longitudinal data by combining the idea
of QIF in Qu et al. [4] and apply the proposed method to
the marginal longitudinal GLMs (1). Specifically, we derive
the asymptotic distributions of the maximum empirical
Journal of Applied Mathematics
Cressie-Read likelihood estimator (MECRLE) and the ECR
test statistic. Although the working correlation matrix is
assumed, the proposed method needs not to estimate the nuisance parameters associated with correlations. This is because
the inverse of the commonly used working correlation matrix
can be exactly represented or approximated by a linear combination of basis matrices such that the dimension of the
unbiased estimating functions is greater than the number
of unknown parameters. Thus, the efficiency results can be
obtained by combining the ECR method with the generalized
method of moments (GMM) proposed by Hansen [29].
Therefore, the proposed method inherits the advantages of
the empirical likelihood, QIF, and GMM methods, and the
proposed method represents a good alternative in this area
as well.
The paper is organized as follows. In Section 2, we
propose the more generalized method of the ECR test statistic
for marginal longitudinal GLMs by combining with the QIF
approach. The technical conditions and some asymptotic
properties are given in Section 3. In Section 4, we present the
results for simulation studies and a real dataset to illustrate the
proposed methods. The technical proofs of the main results
are presented in Section 5.
2. Model and ECR Test Statistic
Suppose that the population (X, Y) comes from the marginal
longitudinal generalized linear model (1). Then the marginal
mean of  is
 =  ( | x ) = ℎ ( x )
(2)
and the marginal variance of  is
Var ( | x ) = ] ( ) ,
(3)
where ](⋅) is a variance function and  is a scale parameter.
With assumptions on the first marginal moments, the
GEE [2] estimator of  is defined as the solution of

∑̇  V−1
 (Y −  ) = 0,
(4)
=1
where  = (1 , . . . ,  ) , ̇  =  /, is an  ×  matrix
1/2
and V = A1/2
is a working covariance matrix with
 R()A
A being the  ×  diagonal matrix of marginal variances
Var(Y ) and R() as the working correlation matrix, where 
is a vector which fully characterizes R(). The main advantage
of the GEE method is that it yields a consistent estimator even
if the working correlation matrix is misspecified.
Qu et al. [4] introduced the quadratic inference function
(QIF) by assuming that the inverse of the working correlation
can be approximated by a linear combination of several basis
matrices; that is,
R−1 ≈ 1 1 + 2 2 + ⋅ ⋅ ⋅ +   ,
(5)
where 1 is the identity matrix, 2 , . . . ,  are symmetric
basis matrices which are determined by the structure of R(),
Journal of Applied Mathematics
3
and 1 , . . . ,  are constant coefficients. The advantage of this
approach is that it does not require estimation of linear
coefficients  ’s which can be viewed as nuisance parameters.
To implement the QIF approach, we need to choose the
basis for the inverse of the correlation matrix R(). Qu et
al. [4] and Qu and Song [30] found that, if R() is the
exchangeable working correlation matrix, R−1 = 1 1 + 2 2 ,
where 1 is the identity matrix and 2 is a matrix with 0 on
the diagonal and 1 off diagonal. If the working correlation
matrix is AR(1) with  = |−| , then R−1 can be written as
a linear combination of three basis matrices, that is, 1 1∗ +
2 2∗ + 3 3∗ , where 1∗ is the identity matrix, 2∗ can be 1
on the subdiagonal and 0 elsewhere, and 3 can be 1 on the
corners (1, 1) and (, ) and 0 elsewhere. However, 3∗ can
often be dropped out of the model, as removing 3∗ does not
affect the efficiency of the estimator too much, and this can
simplify the estimation procedure. More details can be found
in Qu et al. [4], Qu and Song [30], and Qu and Li [8]
among others. In addition, Qu and Lindsay [31] developed
an adaptive estimating equation approach to find a reliable
approximation to the inverse of the variance matrix.
Let
̇  A−1/2
1 A−1/2
(Y −  )


 () = (
̇  A−1/2
2 A−1/2
(Y −  )


).
..
.
(6)
0 can be based on the following constrained minimization
problem:

CR () = min {CR () | ∑ = 1,  ≥ 0,
=1

∑  () = 0} .
=1
As noted by Baggerly [17], a unique value for the righthand side of (9) exists, provided that zero is inside the
convex hull of the 1 (), . . . ,  () for given . Applying the
Lagrange multiplier method, we can obtain the solution of 
by minimizing (9). Let
=


2
−
∑ {( ) − 1} + 1 (∑ − 1)
 (1 + ) =1
=1

CR () =

2
−
∑ {( ) − 1} ,
 (1 + ) =1
=1
where 1 and 2 = (1 , . . . ,  ) are the Lagrange multipliers.
Let / = 0; we have
−1/(1+)
1
, ( ≠ − 1) ,
{ {1 +  +   ()}

 = {
(11)

( = −1) ,
{ exp {  ()} ,
where  and the vector  = (1 , . . . ,  ) are determined by
−1/(1+)
1 
= 1,
∑{1 +  +   ()}
 =1
(7)
where  is a user-specified parameter. ECR (7) is a generalization of the empirical log-likelihood ratio statistic and can
also be defined as

2
−
{
{( ) − 1} ,
∑
{
{
{

+
)
(1
{
=1
{
{ 
{
{
CR () = {−2∑ log ( ) ,
{ =1
{
{
{

{
{
{
{2∑ log ( ) ,
{ =1
for  ≠ 0, −1,
for  = 0,
−1/(1+)
1 
 () = 0.
∑{1 +  +   ()}
 =1
(8)
for  = −1.
Let  be a multinomial likelihood supported on the observed
data, and suppose that we are interested in testing the null
hypothesis 0 :  = 0 , where 0 is the true parameter
vector. The empirical Cressie-Read family of test statistics for
(12)
In the present paper, we only are interested in the case of
 ≠ − 1. When  → −1, the estimator of  is viewed as the
empirical exponential tilting estimator (see [33, 34]). When
 ≠ − 1 and substituting (11) into (7), we obtain the following
empirical Cressie-Read test statistic:
CR () =
−∞ <  < ∞,
(10)
+ 2 ∑  () ,
−1/2
 −1/2
(̇  A  A (Y −  ))
It is easy to check that the GEE defined in (4) becomes a
linear combination of the above extended score vector of
∑=1  (). Note that the dimension of  () is  = , and it
is greater than the number of unknown parameters; thus the
GEE method is unavailable. Therefore, Qu et al. [4] proposed
the quadratic inference function by extending the method of
GMM proposed by Hansen [29] to obtain the estimator of .
In this paper, we consider an alternate method using the
following empirical Cressie-Read test statistics [17, 32]:
(9)

/(1+)
2
− 1] .
∑ [{1 +  +   ()}
 (1 + ) =1
(13)
Baggerly [17] had applied the empirical Cressie-Read test
statistic to the general parameter case and shown that all
members of the empirical Cressie-Read family have a Chisquared calibration and enjoy the advantages of the empirical
likelihood method. From the proof of Theorem 1 in this paper,
we can obtain that
̂ −1  () +  (1) ,
CR () =  () Σ

1
̂ −1  () +  (−1/2  ) ,
 =  (1 + )  () Σ

2
̂ −1  () +  ,  =  (−1/2 ) ,
 () = (1 + ) Σ
(14)

̂ = (1/) ∑  () ().
 () and Σ

=1
̂ satisfies CR ()
̂ = inf
̂ is
If 
CR
(),
the
estimator


∈B

called the maximum empirical Cressie-Read likelihood estimator (MECRLE) of the parameter .
where  () = (1/) ∑=1
4
Journal of Applied Mathematics
3. Asymptotic Properties
To establish the asymptotic properties, we need the following
regularity conditions.
̂ = (1/) ∑  () () converges a.s.
(C1) The matrix Σ

=1
to an invertible positive definite matrix Σ(). This
condition holds based on the weak law of large number when  tends to infinity and the cluster size is
fixed.
(C2) The domain B is a compact subset of R and the true
parameter value 0 is in its interior. There exists a
unique 0 ∈ B satisfying mean zero model assumption {1 (0 )} = 0.
(C3) Let  = (1 , . . . ,  ) where  = ( | x ); then
 is a.s. continuously differentiable in . Denote this
 ×  derivative matrix by ̇  ; then ̇  has full column
rank  a.s.
(C4) Assume that  ()/ is continuous in a neighborhood of the true value 0 and the rank of (1 (0 )/
̂ converges in prob) is . In addition, ( /)()
̂ converges in probability to [(1 /)(0 )] when 
ability to 0 .
Conditions (C1)–(C4) are actually quite mild and can be
easily satisfied, and these conditions are also found in Qin
and Lawless [20] and Qu et al. [4]. Conditions (C1) and (C4)
ensure that the asymptotic variance exists for the proposed
estimator. Condition (C2) ensures that there exists a √consistent solution in the compact subset B. (C3) is a common condition used in GLMs with longitudinal data.
Theorem 1. Assume that conditions (C1–C4) hold. Then, as
 → ∞, with probability tending to 1, CR () attains its minî in the interior of the ball ‖− ‖ ≤
mum value at some point 
0
−1/3
̂ ̂, and ̂ = ()
̂ satisfy

and ,
̂ ̂, ̂) = 0,
1 (,
̂ ̂, ̂) = 1,
2 (,
̂ ̂, ̂) = 0,
3 (,
(15)
Theorem 2. Suppose that the conditions of Theorem 1 hold,
and further assume that 2 ()/ is continuous about 
in a neighborhood of the true value 0 . Then, as  → ∞, one
has

̂ −  ) →
√ (
 (0, Ω) ,
0

 ” denotes the convergence in distribution and
where “→
−1
 ( ) 
 ( )
Ω = [( 1 0 ) Σ−1 (0 ) ( 1 0 )] .


1 (, , ) =
−1/(1+)
1 
 () ,
∑{1 +  +   ()}
 =1
Remark 4. The proposed method provides a way to find efficient estimates for marginal longitudinal GLMs when the
within-subject correlation is considered. It is known that we
can construct the confidence regions of 0 using Theorem 2.
̂ −  ) does not depend on
The asymptotic variance Ω of √(
0
any  and can be consistently estimated by

[{∑̂
[
̂
 ()
=1

× {∑̂
=1



1
1
 (1 + )2
̂
 ()

=1
−1
× ∑{1 +  +   ()}
=1

×(
 ()
) .

−1/(1+)
(19)
}]
]
or by the same expression with the ̂ ’s, replaced by 1/.
Theorem 5. Suppose that the conditions of Theorem 2 hold;
then, under the hypothesis 0 :  = 0 , CR (0 ) is asymptotically a Chi-squared distribution with  degrees of freedom
as  → ∞.
Theorem 5 allows us to use the empirical Cressie-Read
test statistic for testing or obtaining the confidence regions for
the parameter 0 . For any 0 <  < 1, the confidence region
of 0 with asymptotic coverage 1 −  can be determined by
0 :  { ()} = 0.
(16)

−1
̂  ()}
̂
} {∑̂  ()

(20)
We can construct a goodness-of-fit statistic to test the
model assumption:
−1/(1+)
1 
2 (, , ) = ∑{1 +  +   ()}
,
 =1
3 (, , ) =
(18)
Remark 3. When  → 0, the empirical log-likelihood ratio
statistic is the special case of the empirical Cressie-Read test
statistic CR ().
 (CR (0 ) ≤ 2 ) = 1 − .
where
(17)
(21)
̂ the empirical Cressie-Read likeliWe call CR (0 ) − CR ()
̂ the model test statistic.
hood ratio statistic and call CR ()
Theorem 6. Suppose that the conditions of Theorem 2 hold
and  () has dimension  =  and  has dimension  with
̂ is
 < . Then, under the model assumption (21), CR ()
asymptotically a Chi-squared distribution with  −  degrees
of freedom as  → ∞; under the null hypothesis 0 :  = 0 ,
̂ is asymptotically a Chi-squared distribution
CR (0 )−CR ()
with  degrees of freedom as  → ∞.
Journal of Applied Mathematics
5
Table 1: The average coverage probabilities (ACP) and the average lengths (ALEN) of the confidence intervals for 1 , 2 , and 3 when the
nominal level is 0.95 and the true correlation structure is AR(1) structure.


1
2
3
1
2
3
1
2
3
1
2
3
60
CS
100
60
AR(1)
100
EL
ACP
0.9120
0.9140
0.9280
0.9160
0.9180
0.9240
0.9220
0.9240
0.9320
0.9300
0.9360
0.9420
ET
ALEN
0.4647
0.6375
0.4520
0.4646
0.6396
0.4518
0.3608
0.4959
0.3513
0.3613
0.4964
0.3514
ACP
0.9060
0.8820
0.9150
0.9120
0.9080
0.9180
0.9180
0.9120
0.9280
0.9200
0.9320
0.9380
4. Numerical Studies
4.1. Simulation Studies. In this subsection, we report the simulation study to illustrate the finite sample properties of the
proposed ECR test statistic. For simplicity, we only compute
the empirical likelihood (EL,  = 0) and the empirical
exponential tilting (ET,  → −1) and compare the proposed
methods with the GEE and QIF methods. Throughout the
simulation study, each dataset comprised  = 60 and 100
subjects and  = 5 observations per subject over time. For
each case, we repeat the experiment 500 times.
Consider the following logistic regression model. The
response variable  is binary and its marginal expectation
given x is
3
logit ( ) = ∑ , 0 ,
 = 1, . . . , ,  = 1, . . . , 5,
(22)
=1
where 0 = [1, 2, −0.8] and the covariate x = (,1 , . . . ,
,3 ) has a multivariate normal distribution with mean zero,
marginal variance 1, and an AR(1) correlation matrix with
autocorrelation coefficient 0.7. The binary response vector
for each cluster has mean specified by (22) and an AR(1)
correlation structure with an autocorrelation coefficient  =
0.7. Table 1 reports the simulation results for the average coverage probabilities and the average lengths of the confidence
intervals for 1 , 2 , and 3 when the nominal level is 0.95 and
the true correlation structure is AR(1) structure.
From Table 1, it is easy to see that the empirical likelihood method performs much better in terms of coverage
accuracies of the confidence intervals even when the working
correlation structure is misspecified. And when the working
correlation structure is correctly specified, the performances
of all the methods are usually slightly better. Table 1 also
shows that the average coverage probabilities obtained by
all methods tend to the nominal level 0.95 and the average
lengths decrease as  increases.
When sample size is 60 and the true correlation structure
is AR(1), the histograms and the QQ plots of the 500 maximum empirical likelihood estimators ̂1 , ̂2 , and ̂3 based
GEE
ALEN
0.4741
0.6539
0.4600
0.4713
0.6512
0.4577
0.3650
0.5033
0.3550
0.3641
0.5016
0.3541
ACP
0.9000
0.8400
0.9100
0.9100
0.8940
0.9100
0.9200
0.8760
0.9240
0.9220
0.9160
0.9240
QIF
ALEN
0.4658
0.5561
0.4565
0.4625
0.6152
0.4432
0.3663
0.4288
0.3599
0.3541
0.4857
0.3597
ACP
0.9120
0.8700
0.8960
0.9120
0.8740
0.9040
0.9480
0.9020
0.9280
0.9460
0.9140
0.9340
ALEN
0.4879
0.6642
0.4378
0.4742
0.6569
0.4432
0.3848
0.5149
0.3585
0.3716
0.5128
0.3537
on the empirical likelihood method are plotted in Figure 1
and the histograms and the QQ plots of the 500 maximum
empirical exponential tilting estimators ̂1 , ̂2 , and ̂3 based
on the empirical exponential tilting method are plotted
in Figure 2 under the misspecified and correct correlation
structures, respectively. Figures 1 and 2 show that empirically
these estimators are asymptotically normal even when the
working correlation structure is misspecified.
4.2. Application to Real Data. To examine the performance of
the proposed method, we analyze real data [35, 36] from a sixweek frequent magnetic resonance imaging (MRI) substudy
of the Betaseron clinical trial conducted at the University
of British Columbia in relapsing-remitting multiple sclerosis
involving 52 patients. The real data concerns a longitudinal
clinical trial to assess the effects of neutralizing antibodies
on interferon beta-1b (IFNB) in relapsing-remitting multiple
sclerosis (MS), which is a disease that destroys the myelin
sheath that surrounds the nerves. All patients were randomized into three treatment groups with allocation of 17 patients
being treated by placebo, 17 by low dose, and 16 by high dose.
This dataset has been studied by Song [37] and Li et al. [11].
There exist the missing values in this dataset; for convenience,
we only analyze the balanced longitudinal data which contain
39 patients.
For the analysis of this data, the binary response variable
is exacerbation, which refers to whether an exacerbation
began since the previous MRI scan, and is 1 for yes and 0
for no. Several baseline covariates are included in the model.
They are treatment (trt), time () in weeks, squared time (2 ),
and duration of disease (dur) in years. Here trt is treated as an
ordinal covariate with scale 0, 1, 2 representing zero (placebo),
low, and high dosage of the drug treatment. We consider the
following marginal logistic model for the data:
logit ( ) = 0 + 1 trt + 2  + 3 2 + 4 dur ,
(23)
where  is the probability of exacerbation at visit  for subject
. Two correlation structures (exchangeable (CS) and AR(1))
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(A)
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−2
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(C)
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(A)
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100
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(D)
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−5
0
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Theoretical quantiles
(F)
(b)
Figure 1: The histograms and the QQ plots for the maximum empirical likelihood estimators based on the empirical likelihood method for
 = 60. When the true correlation structure is AR(1), the left plots are obtained by using the misspecified CS working correlation structure
and the right plots are obtained by using the true AR(1) working correlation structure.
7
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2.5
0
3
(B)
Sample quantiles
Sample quantiles
0
−2
4
1
0
−5
100
0
5
−0.5
−1
−1.5
−2
−5
0
5
Theoretical quantiles
Theoretical quantiles
(E)
(D)
(F)
200
200
150
150
150
100
100
50
0
−1
Density
200
Density
Density
(a)
50
50
0
1
1
2
0
3
0
2
0
4
2
(A)
(B)
−1
3
0
0.5
2
1.5
1
0.5
Sample quantiles
4
2.5
Sample quantiles
Sample quantiles
−2
(C)
3
0
−5
100
3
2
1
0
Theoretical quantiles
5
−5
0
5
0
−0.5
−1
−1.5
−5
0
Theoretical quantiles
Theoretical quantiles
(E)
(F)
(D)
5
(b)
Figure 2: The histograms and the QQ plots for the maximum empirical exponential tilting estimators based on the empirical exponential
tilting method for  = 60. When the true correlation structure is AR(1), the left plots are obtained by using the misspecified CS working
correlation structure and the right plots are obtained by using the true AR(1) working correlation structure.
8
Journal of Applied Mathematics
Table 2: The parameter estimators (the corresponding standard errors) for CS and AR(1) correlation structures.
EL
−0.3834 (0.3820)
−0.0321 (0.1239)
−0.0344 (0.0141)
0.00031 (0.00013)
−0.0574 (0.0189)
−0.4212 (0.4016)
−0.0590 (0.1236)
−0.0295 (0.0146)
0.00026 (0.00013)
−0.0620 (0.0206)
̂0
̂1
̂2
̂3
̂4
̂0
̂1
̂2
̂3
̂
CS
AR(1)
4
ET
−0.3051 (0.3872)
−0.0454 (0.1259)
−0.0349 (0.0143)
0.00031 (0.00013)
−0.0700 (0.0200)
−0.4382 (0.4041)
−0.0599 (0.1242)
−0.0278 (0.0147)
0.00025 (0.00013)
−0.0645 (0.0208)
GEE
−0.4030 (0.4096)
−0.0809 (0.1510)
−0.0301 (0.0146)
0.00027 (0.00013)
−0.0617 (0.0227)
−0.4009 (0.3682)
−0.0767 (0.1195)
−0.0309 (0.0138)
0.00028 (0.00012)
−0.0606 (0.0179)
QIF
−0.5039 (0.4005)
−0.0667 (0.1206)
−0.0296 (0.0146)
0.00027 (0.00010)
−0.0530 (0.0255)
−0.5043 (0.4145)
−0.0711 (0.1506)
−0.0290 (0.0089)
0.00027 (0.00008)
−0.0558 (0.0249)
Table 3: 95% confidence intervals (CI) and the lengths (LEN) of the confidence intervals for CS and AR(1) correlation structures.
0
1
CS
2
3
4
0
1
AR(1)
2
3
4
CI
LEN
CI
LEN
CI
LEN
CI
LEN
CI
LEN
CI
LEN
CI
LEN
CI
LEN
CI
LEN
CI
LEN
EL
[−1.1321, 0.3654]
1.4975
[−0.2749, 0.2107]
0.4856
[−0.0620, −0.0068]
0.0552
[0.0001, 0.0006]
0.0005
[−0.0945, −0.0203]
0.742
[−1.2084, 0.3659]
1.5743
[−0.3012, 0.1832]
0.4844
[−0.0580, −0.0009]
0.0571
[0, 0.0005]
0.0005
[−0.1023, −0.0217]
0.0806
ET
[−1.0640, 0.4539]
1.5179
[−0.2922, 0.2014]
0.4936
[−0.0629, −0.0069]
0.0560
[0.0001, 0.0006]
0.0005
[−0.1092, −0.0308]
0.0784
[−1.2302, 0.3537]
1.5839
[−0.3033, 0.1834]
0.4867
[−0.0566, 0.0010]
0.0576
[0, 0.0005]
0.0005
[−0.1053, −0.0238]
0.0815
are considered in this analysis. Table 2 reports the coefficient
estimators and the corresponding standard errors for CS and
AR(1) correlation structures, respectively. Table 3 reports the
95% confidence intervals and the lengths of the confidence
intervals for CS and AR(1) correlation structures.
From Tables 2 and 3, we can see that the parameter
estimators are very similar for all four methods under the CS
and AR(1) working correlation structures. For the CS working correlation structure, the empirical likelihood and the
empirical exponential tilting methods have smaller standard
errors and smaller interval lengths than the GEE and QIF
methods. However, the GEE method has the smaller standard
errors and the smaller interval lengths under the AR(1)
working correlation structure. Similar to the results in Song
[37], we also find that the baseline disease severity measured
as the duration of disease before the trial is an important
explanatory variable associated with the risk of exacerbation,
GEE
[−1.2058, 0.3998]
1.6056
[−0.3768, 0.2150]
0.5918
[−0.0587, −0.0016]
0.0571
[0.00002, 0.00052]
0.0005
[−0.1061, −0.0172]
0.0889
[−1.1226, 0.3208]
1.4434
[−0.3109, 0.1576]
0.4685
[−0.0580, −0.0037]
0.0543
[0.00004, 0.00052]
0.00048
[−0.0957, −0.0255]
0.0702
QIF
[−1.1504, 0.3596]
1.5100
[−0.3680, 0.2065]
0.5745
[−0.0604, −0.0039]
0.0565
[0.00008, 0.00056]
0.00048
[−0.1030, −0.0030]
0.09995
[−1.2268, 0.2760]
1.5028
[−0.3663, 0.2240]
0.5903
[−0.0465, −0.0115]
0.0350
[0.00011, 0.00052]
0.00041
[−0.1047, −0.0070]
0.0977
and we also do not find strong evidence that the drug
treatment is effective in reducing the risk of exacerbation.
Therefore, all the methods have comparable performance,
and these findings are close to the existing analysis in Song
[37].
5. Proofs of the Theorems
In this section, we present rigorous proofs of our results stated
in Section 3.
Proof of Theorem 1. Assume that  ≠ − 1;  defined by (10)
takes derivatives with respect to  ; we have

2
=−
−(1+) + 1 + 2  () = 0,

(1 + )  
(24)
Journal of Applied Mathematics
9
Δ
which implies that
−(1+) =
(1 + )  (1 + 2  ())
2
(25)
.
Multiplying  at both sides of (25) and taking sum and noting
that ∑=1  = 1 and ∑=1   () = 0, we can derive that

∑− =
=1
(1 + )  1
.
2
(26)
For convenience, let  () = (1/) ∑=1  (). We now
find an approximation for  in terms of 0  () =  (1). By
∑=1  = 1 and applying the Taylor expansion, we have
(2 + ) 2

1 
+  (−1/2  )] = 1,
∑ [1 −  +
 =1
1 +  2(1 + )2
where  = +0  () and  is defined by (33). Rearranging
(34) and ignoring terms of order  (2 ), we have
Substituting (26) into (25), we have

−(1+) = ∑− +
(1 +
) 1+ 2 
()
2
=1
 + 0  ()
.
(27)
=
By Theorem 1 in Baggerly [17], there exists a constant  > 0
such that

2
−
∑ [( ) − 1] ≤ ,
 (1 + ) =1
−(1+) = 1+ +  +
(1 +
()
2
.
(29)
Therefore, we can obtain that

1
+ 2 0 ∑ ()  () 0 ]

=1
+  (−1/2  ) .
That is,
=
2 +  2 ̂
  Σ − 0  () +  (−1/2  ) .
2 (1 + ) 0 0
(30)
=
−1
=  ( ) and  = ((1 + )/2)2 . The next
where  = /
step is to bound the convergence rate of . Let  = 0 , where
‖0 ‖ = 1. Invoking Lemma 3.2 in K¨unsch [38] and ( ()) =
0, it is easy to show that 0  () =  (1/2 ). Applying the
Taylor expansion to ∑=1   () = 0, we have
1 
1
( () +  ()  () 0 ) +  ]
∑ [ () −
 =1
1+
CR () =
=
(31)
where  =  ( (), , ). Assume that  =  (−] ), ] >
0. By the central limit theorem, we have (1/) ∑=1  () =
 (−1/2 ) and (1/) ∑=1  () =  (−3/2 ). Let Σ() =
Var{ ()}; it is easy to show that
a.s.
(32)
If ] > 1/2 (or ] < 1/2), then the dominant term in the right̂0=
hand side of (31) is (1/) ∑=1  () =  (−1/2 ), while Σ
 (−] ), which implies that the sum cannot be zero unless
] = 1/2. Therefore, we obtain that  =  (−1/2 ). By (31), we
have

 =  (−1/2 ) .
(33)
 
  () +  (−1/2  ) .
2 0 
(37)

/(1+)
2
− 1]
∑ [(1 +  + 0  ())
 (1 + ) =1

2
1
∑ [ + 0  () −
2
2
+ )
(1
+
)
(1
=1
×2 0  ()  () 0 +  (−1 ) ]
=
1 2 ̂

[2 + 20  () −
  Σ
2
1
+
 0 0
(1 + )
+ (−1 ) ]

̂ −1 ( 1 ∑ ()) +  ,
0 = (1 + ) Σ
 =1
(36)
By (33) and (37) and applying the Taylor expansion for
CR (), it is easy to show that
= 0,
Δ 1
̂=
Σ
∑ ()  () → Σ () ,
 =1 
(35)
From (33) and (36), we have
−1/(1+)
1
,
 = [1 +  +   ()]

1+
2+
[20  ()
2 (1 + )
(28)
which implies that ∑=1 − = +1 + , where  =  ( ).
Substituting this expression into (27), we have
) 1+ 2 
(34)
=

[0  ()
2
(1 + )
̂ −1  ()
+ 2 (1 + )  () Σ

+ 2 (−1/2 ) 
̂ −1  ()
− (1 + )  () Σ

̂ −1 
̂ −1  −  Σ
− 2 () Σ

+ (−1 )]
10
Journal of Applied Mathematics
=

̂ −1  ()
[ (1 + )  () Σ

(1 + )2
where  −  > 0 and  is the smallest eigenvalue of

̂ −1  ()
+ 2 (1 + )  () Σ

− (1 +
̂ −1 
)  () Σ

(
()] +  (1)
(38)
Similar to the proof of Lemma 1 in Qin and Lawless [20],
let  = 0 + −1/3 , for  ∈ { | ‖ − 0 ‖ = −1/3 }, where
‖‖ = 1. We first give a lower bound for CR () on the surface
of the ball. When ‖ ()‖3 < ∞ and ‖ − 0 ‖ ≤ −1/3 , we
can obtain that
1 
 () = (1 + ) [ ∑ ()  ()]
 =1
−1
1
× [ ∑ ()] +  (−1/3 )
 =1
(39)
a.s.
holds uniformly for  ∈ { | ‖ − 0 ‖ = −1/3 }. From (32),
(38), and (39), we have

−1
1 
1 
CR () = [ ∑ ()] [ ∑ ()  ()]
 =1
 =1

−1
1 
1   ( )
× [ ∑ (0 ) + ∑  0 −1/3 ]
 =1
 =1 
=  [ (
)
(log log )
)
1 (0 )
) −1/3 ] Σ−1 (0 )

1/2
× [ (−1/2 (log log )
+ (
)
1 (0 )
) −1/3 ] +  (−1/3 )

≥ ( − ) 1/3 ,
a.s.,
a.s.
It is known that, when  ∈ { | ‖ − 0 ‖ ≤ −1/3 }, CR () is
a continuous function about  and has a minimum value in
̂ satisfies
the interior of this ball. In addition, 

CR () 
−1/(1+)
2

=
[1 +  +   ()]
∑
2

 =̂ (1 + ) =1
 ()
×( 
)





= 0.
 ̂
=

1 (, 0, 0)
1
=−
∑ () ,

 (1 + ) =1
(40)

1 (, 0, 0)
1
=
−
 ()  () ;
∑
 (1 + ) =1 

2 (, 0, 0)
= 0,


+ (
(42)
1 (, 0, 0) 1   ()
= ∑
,

 =1 
1 
× [ ∑ ()  ()]
 =1
1/2
1 
× [ ∑ (0 )] +  (1)
 =1
Proof of Theorem 2. We take derivatives for 1 , 2 , and 3
with respect to , , and  ; we have
1 
1   ( )
= [ ∑ (0 ) + ∑  0 −1/3 ]
 =1
 =1 
−1/2
−1
(43)
1 
× [ ∑ ()] +  (−1/3 )
 =1
+  (

1 
1 
CR (0 ) = [ ∑ (0 )] [ ∑ (0 )  (0 )]
 =1
 =1
=  (log log ) ,

−1/3
(41)
Similarly,
̂ −1  () +  (1) .
=  () Σ

=  (−1/3 ) ,
 ( )
1 (0 )
) Σ−1 (0 )  ( 1 0 ) .


(44)
2 (, 0, 0)
1
=−
,

1+

2 (, 0, 0)
1
=
−
 () ;
∑
 (1 + ) =1 

3 (, 0, 0)
= 0,

3 (, 0, 0)
= 0,



 ()
3 (, 0, 0)
1
=
( 
) .
∑
2


(1 + ) =1
Journal of Applied Mathematics
11
By the conditions and Theorem 1 and the Taylor expanding of
̂ ̂, ̂),  (,
̂ ̂, ̂), and  (,
̂ ̂, ̂) at ( , 0, 0), we have
1 (,
2
3
0
̂ ̂, ̂)
0 = 1 (,
= 1 (0 , 0, 0) +
+
1 (0 , 0, 0) ̂
( − 0 )

1 (0 , 0, 0)
(̂ − 0)

11 12 13
31
(
(
=(
(
2 (0 , 0, 0)
(̂ − 0)

1
Σ (0 )
1+
1
 (1 (0 ))
−
1+
2
((1 + )
−
(
1 (0 )
)

1 (0 )
1
 (1 (0 ))  (
)
1+

)
1
)
−
0
).
1+
)

0
0
)
(48)
By the above expression and 1 (0 , 0, 0) = (1/)∑=1  (0 )
=  (−1/2 ), we can derive that  =  (−1/2 ). Note that
( (0 )) = 0, and Σ(0 ) is the positive definite matrix; it is
easy to show that
̂ ̂, ̂)
0 = 3 (,
3 (0 , 0, 0) ̂
( − 0 )

−1
−1
−1
−1
11
+ 11
[13 33.1
31 ] 11
3 (0 , 0, 0)
(̂ − 0)

−1 = (
0
−1
−1
−33.1
31 11
3 (0 , 0, 0)
+
(̂ − 0) +  ( ) ,

−1
−1
0 −11
13 33.1
−1
22
0
),
0
−1
33.1
(49)
(45)
̂ −  ‖ + ‖̂‖ + ‖̂‖. We further have
where  = ‖
0
−1
where 33.1 = −31 11
13 . Substituting (49) into (47), we have
̂ −  ) = −1  −1 √ ( , 0, 0) +  (1) ,
√ (
1

0
0
33.1 31 11
−1 (0 , 0, 0) +  ( )
(1 − 2 (0 , 0, 0) +  ( ))
−3 (0 , 0, 0) +  ( )
−1
−1
−1
−1
√ (̂ − 0) = − {11
+ 11
[13 33.1
31 ] 11
}
(50)
× √1 (0 , 0, 0) +  (1) .
1 (0 , 0, 0) 1 (0 , 0, 0) 1 (0 , 0, 0)



( 2 (0 , 0, 0) 2 (0 , 0, 0) 2 (0 , 0, 0) )
=(
)



3 (0 , 0, 0) 3 (0 , 0, 0) 3 (0 , 0, 0)



)
(

Note that √1 (0 , 0, 0) →
 (0, Σ(0 )); we can derive that

̂ −  ) →
√ (
 (0, Ω) ,
0

√ (̂ − 0) →  (0, Φ) ,
(51)
where
−1

 ( )
 ( )
Ω = [( 1 0 ) Σ−1 (0 )  ( 1 0 )] ,


(46)
By the above expression, we can obtain that
−1 (0 , 0, 0) +  ( )
̂
 ( )
( ̂ ) = −1 (
),
̂−

 ( )
0
0
1
2 (0 , 0, 0)
+
(̂ − 0) +  ( ) ,

̂
× ( ̂ ) .
̂−

0
0
−
2 (0 , 0, 0) ̂
= 2 (0 , 0, 0) +
( − 0 )

+
a.s.,
 = (21 22 0 )
̂ ̂, ̂)
1 = 2 (,
= 3 (0 , 0, 0) +
1 (0 , 0, 0) 1 (0 , 0, 0) 1 (0 , 0, 0)



Δ ( 2 (0 , 0, 0) 2 (0 , 0, 0) 2 (0 , 0, 0) )
 = (
)



3 (0 , 0, 0) 3 (0 , 0, 0) 3 (0 , 0, 0)



(
)
→ ,
1 (0 , 0, 0)
+
(̂ − 0) +  ( ) ,

+
where
Φ = (1 + ) Σ−1 (0 )

× { −  (
(47)
1 (0 )
 ( )
) Ω( 1 0 ) Σ−1 (0 )} .


(52)
12
Journal of Applied Mathematics

Proof of Theorem 5. Since √ (0 ) →
 (0, Σ(0 )) and
Δ


̂ ) = (1/) ∑  ( ) ( ) → Σ( ), a.s., by (38), it
Σ(
0
0 
0
0
=1
is easy to show that
On the other hand, invoking the same argument, we have
̂
CR (0 ) − CR ()
̂ −1 ( )  ( )
=  (0 ) Σ
0
0

 −1
̂ ( )
CR (0 ) = [√ (0 )] Σ
0
̂ Σ
̂  ()
̂ +  (1)
̂ −1 ()
−  ()



× [√ (0 )] +  (1) → 2 .
(53)

= 1
(0 , 0, 0) Σ−1 (0 ) 1 (0 , 0, 0)
(56)

Proof of Theorem 6. From (33), (38), and (50) and by the
Taylor expansion and some simple calculations, we have
̂ =  ()
̂ Σ
̂  ()
̂ +  (1)
̂ −1 ()
CR ()





+ (
=
× 2 [√Σ−1/2 (0 ) 1 (0 , 0, 0)] +  (1) ,
 ( ) ̂

[ ( ) +  0 (
− 0 )
1+  0

−1/2
Δ
since 22 = 2 , where 2 = Σ−1/2 (0 )(1 (0 )/)Ω
(1 (0 )/) Σ−1/2 (0 ). Further,

) ] ̂ +  (1)
tr {Σ−1/2 (0 )  (

[ ( , 0, 0)
1 +  1 0
+
+ (
(0 , 0, 0)
×Σ
)]
(0 ) } = .

̂ →
Thus, we can obtain that CR (0 ) − CR ()
 2 .
× (0 , 0, 0) +  (−1/2 )] +  (1)
= [√Σ−1/2 (0 ) 1 (0 , 0, 0)]
Conflict of Interests
The authors declare that there is no conflict of interests
regarding the publication of this paper.

× 1 [√Σ−1/2 (0 ) 1 (0 , 0, 0)] +  (1) ,
(54)
Δ
since 12 = 1 , where 1 = [ − Σ−1/2 (0 )(1 (0 )/)
Ω(1 (0 )/) Σ−1/2 (0 )]. From √Σ−1/2 (0 )1 (0 , 0, 0)

→
 (0, ) and
tr (1 ) = tr { − Σ−1/2 (0 )  (
1 (0 )
) Ω


1 (0 )
) Σ−1/2 (0 )}

=  − tr {Σ−1/2 (0 )  (
1 (0 )
) Ω


 ( )
×( 1 0 ) Σ−1/2 (0 )}

=  − ,

−1/2

−1
−1
−1
−1
× [− {11
+ 11
[13 33.1
31 ] 11
} 1
×(

1 (0 )
 ( )
) Ω( 1 0 )


(57)
−1
−1
13 33.1
31 11
1
−1/2
× 1 [√Σ−1/2 (0 ) 1 (0 , 0, 0)] +  (1)
= [√Σ−1/2 (0 ) 1 (0 , 0, 0)]
  ̂ ̂
=
 ()  +  (1)
1+ 
=
− [√Σ−1/2 (0 ) 1 (0 , 0, 0)]
2
̂ →
 −
.
it is easy to show that CR ()
(55)
Acknowledgments
Junhua Zhang’s research was supported by the National
Nature Science Foundation of China (11002005) and the
Training Programme Foundation for the Beijing Municipal
Excellent Talents (2013D005007000005). Ruiqin Tian, Suigen
Yang, and Sanying Feng’s researches were supported by the
National Nature Science Foundation of China (11101014), the
Natural Science Foundation of Beijing (1142002), the Science
and Technology Project of Beijing Municipal Education
Commission (KM201410005010), the Specialized Research
Fund for the Doctoral Program of Higher Education of China
(20101103120016), and the Doctoral Fund of Innovation of
Beijing University of Technology.
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