REVSTAT – Statistical Journal Volume 16, Number 2, April 2018, 231–255 SEMI-PARAMETRIC LIKELIHOOD INFERENCE FOR BIRNBAUM–SAUNDERS FRAILTY MODEL Authors: N. Balakrishnan – Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada bala@mcmaster.ca Kai Liu – Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada liuk25@math.mcmaster.ca Received: April 2017 Revised: August 2017 Accepted: August 2017 Abstract: • Cluster failure time data are commonly encountered in survival analysis due to dif- ferent factors such as shared environmental conditions and genetic similarity. In such cases, careful attention needs to be paid to the correlation among subjects within same clusters. In this paper, we study a frailty model based on Birnbaum–Saunders frailty distribution. We approximate the intractable integrals in the likelihood function by the use of Monte Carlo simulations and then use the piecewise constant baseline haz- ard function within the proportional hazards model in frailty framework. Thereafter, the maximum likelihood estimates are numerically determined. A simulation study is conducted to evaluate the performance of the proposed model and the method of infer- ence. Finally, we apply this model to a real data set to analyze the effect of sublingual nitroglycerin and oral isosorbide dinitrate on angina pectoris of coronary heart disease patients and compare our results with those based on other frailty models considered earlier in the literature. Key-Words: • Birnbaum–Saunders distribution; censored data; cluster time data; frailty model; Monte Carlo simulation; piecewise constant hazards. AMS Subject Classification: • 62N02.
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REVSTAT – Statistical Journal
Volume 16, Number 2, April 2018, 231–255
SEMI-PARAMETRIC LIKELIHOOD INFERENCE
FOR BIRNBAUM–SAUNDERS FRAILTY MODEL
Authors: N. Balakrishnan
– Department of Mathematics and Statistics, McMaster University,Hamilton, Ontario, [email protected]
Kai Liu
– Department of Mathematics and Statistics, McMaster University,Hamilton, Ontario, [email protected]
Received: April 2017 Revised: August 2017 Accepted: August 2017
Abstract:
• Cluster failure time data are commonly encountered in survival analysis due to dif-ferent factors such as shared environmental conditions and genetic similarity. In suchcases, careful attention needs to be paid to the correlation among subjects within sameclusters. In this paper, we study a frailty model based on Birnbaum–Saunders frailtydistribution. We approximate the intractable integrals in the likelihood function bythe use of Monte Carlo simulations and then use the piecewise constant baseline haz-ard function within the proportional hazards model in frailty framework. Thereafter,the maximum likelihood estimates are numerically determined. A simulation study isconducted to evaluate the performance of the proposed model and the method of infer-ence. Finally, we apply this model to a real data set to analyze the effect of sublingualnitroglycerin and oral isosorbide dinitrate on angina pectoris of coronary heart diseasepatients and compare our results with those based on other frailty models consideredearlier in the literature.
Key-Words:
• Birnbaum–Saunders distribution; censored data; cluster time data; frailty model;
Monte Carlo simulation; piecewise constant hazards.
AMS Subject Classification:
• 62N02.
232 N. Balakrishnan and Kai Liu
Semi-Parametric Likelihood Inference for Birnbaum–Saunders Frailty Model 233
1. INTRODUCTION
It is of natural interest in medical or epidemiological studies to examine
the effects of treatments. Proportional hazards model, proposed by Cox [5], is
the most popular model for the analysis of such survival data which models the
hazard function as
h(t) = h0(t) exp(β′x),
where t, x and h0 are the time to certain event, set of covariates and baseline
hazard function, respectively. This model makes a critical assumption of inde-
pendent observations from the subjects. However, correlation commonly exists
in survival data due to shared environmental factors or genetic similarity. There-
fore, neglecting this correlation may lead to biased results. A convenient choice
for modeling these kinds of correlation in survival data is the frailty model. The
terminology frailty was first introduced by Vaupel et al. [20], while accounting
for the heterogeneity of individuals in distinct clusters. Generally speaking, the
more frail an individual is, the earlier the event of interest will be. A shared
frailty model introduces multiplicative random effects, which is referred to as the
frailty term, in the proportional hazards model, and is defined as follows. Let
(tij , δij , xij), i = 1, ..., n, j = 1, ..., mi, be the failure time, censoring indicator, and
the covariate vector of the jth individual in the ith cluster, where δij is 1 if tij is
not censored and 0 otherwise. Let yi be the frailty shared commonly by all the
subjects in the ith cluster. Then, given yi, tij are assumed to be independent
with hazard function
(1.1) h(tij |yi) = yih0(tij) exp(β′xij).
The frailties yi are assumed to be independent and identically distributed with
a distribution, called the frailty distribution. The baseline hazard h0(tij) is arbi-
trary. A common parametric choice of the baseline hazard is Weibull. Klein [13]
proposed a non-parametric estimate of the cumulative hazard function of baseline
distribution and then used a profile likelihood function.
The most prevailing choice of the frailty distribution is gamma distribution
due to its mathematical simplicity and the mathematical tractability of ensuing
inference [13]. It has a closed-form for the conditional likelihood function, given
the observed data, so that EM algorithm can be applied effectively to obtain
the maximum likelihood estimates. Another possibility is the positive stable
distribution proposed by Hougaard [10]. Furthermore, Hougaard [11] derived
power variance function from the positive stable distribution, which contains the
preceding frailty distributions as special cases. All these distributions have simple
Laplace transforms and therefore facilitates convenient computation of maximum
likelihood estimates. However, there is no real biological reason for their use.
Nevertheless, when the Laplace transform of the frailty distribution is unknown,
234 N. Balakrishnan and Kai Liu
the likelihood function becomes intractable. Lognormal distribution is one such
example. McGilchrist and Aisbett [16] developed a best linear unbiased prediction
(BLUP) estimation method in this case of lognormal frailty model. Balakrishnan
and Peng [2] proposed the generalized gamma frailty model since the generalized
gamma distribution contains the gamma, Weibull, lognormal and exponential
distributions all as special cases. Consequently, the generalized gamma frailty
model becomes more flexible and tend to provide good fit to data as displayed
by Balakrishnan and Peng [2].
The two-parameter Birnbaum–Saunders (BS) family of distributions was
originally derived as a fatigue model by Birnbaum and Saunders [3] for which
a more general derivation from a biological viewpoint was later provided by
Desmond [7]. This distribution possesses many interesting distributional prop-
erties and shape characteristics. In the present work, we use this BS model as
the frailty distribution along with a piecewise constant baseline hazard function
within the proportional hazards model to come up with a flexible frailty model.
The precise specification of this model is detailed in Section 2. An estimation
method to obtain the maximum likelihood estimates of model parameters is pre-
sented in Section 3. A simulation study is conducted in Section 4 to assess the
performance of the proposed method and then the usefulness of the proposed
model and the method of inference is illustrated with a real data in Section 5.
Discussions and some concluding remarks are finally made in Section 6.
2. MODEL SPECIFICATION
2.1. BS distribution as frailty distribution
The BS distribution was originally derived to model fatigue failure caused
under cyclic loading [3]. The fatigue failure is due to the initiation, growth
and ultimate extension of a dominant crack. It is assumed that the total crack
extension Yj due to the jth cycle, for j = 1, ..., are independent and identically
distributed random variables with mean µ and variance σ2. Then, the distribution
of the failure time (i.e., time for the crack to exceed a certain threshold level) is
given by
(2.1) F (t; α, β) = Φ
[
1
α
{
( t
β
)1/2−
(β
t
)1/2}
]
, 0 < t < ∞, α, β > 0,
where Φ is the standard normal cumulative distribution function (CDF), and α
and β are the shape and scale parameters, respectively. We now assume that the
frailty random variable Yi in (1.1) follows the BS distribution defined in (2.1).
Semi-Parametric Likelihood Inference for Birnbaum–Saunders Frailty Model 235
Since 1α
{
(
Tβ
)1/2−
(
βT
)1/2}
is a standard normal random variable, the random
variable T is simply given by
(2.2) T = β
{
αZ
2+
[
(αZ
2
)2+ 1
]1/2}2
,
where Z ∼ N(0, 1). The probability density function (PDF) of T , derived from
(2.1), is given by
(2.3) f(t; α, β) =1
2√
2παβ
[
(β
t
)1/2+
(β
t
)3/2]
exp
[
− 1
2α2
( t
β+
β
t− 2
)
]
, t > 0.
The relation between T and Z in (2.2) enables us to obtain the mean and
variance of T easily as
E(T ) = β(
1 +1
2α2
)
,(2.4)
V (T ) = (αβ)2(
1 +5
4α2
)
.(2.5)
In the frailty model in (1.1), if the frailty term yi is assumed to follow the
BS distribution, for ensuring identifiability of model parameters, the mean of the
frailty distribution needs to be set as 1. More specifically, let Y1 be a BS random
variable with shape parameter α and scale parameter β with its mean as 1. Let
Y2 = cY1. Then, E(Y2) = cE(Y1) = c. Besides, we know that if Y1 ∼ BS(α, β),
then cY1 ∼ BS(α, cβ). Therefore, Y2 ∼ BS(α, cβ) with mean c. Then, given the
frailty term y2, the lifetime of the patients are modeled by the hazard function
h(t|y2) = y2h0(t) exp(β′x) = c y1h0(t) exp(β′x).
Let us define ch0(t) to be a new baseline hazard function h1(t), which is nothing
but rescaling the original baseline hazard function. Then, the model can be
rewritten as
h(t|y2) = y1h1(t) exp(β′x),
which is identical to a frailty model with frailty variable Y1 and baseline hazard
function h1(t) = ch0(t).
Thus, the scale parameter β can be written in terms of the shape parameter
α as
(2.6) β =2
2 + α2,
so that the variance of the frailty variable Yi becomes
(2.7) V (Yi) =4α2 + 5α4
α4 + 4α2 + 4,
which is constrained to be in the interval (0, 5).
Some important discussions on inferential issues for BS distribution can be
found in [1, 4, 8, 9, 15, 17, 18, 19].
236 N. Balakrishnan and Kai Liu
2.2. Piecewise constant hazard as baseline hazard function
The baseline hazard h0(t) in (1.1) is normally assumed in the parametric
setting to be that of exponential or Weibull distribution [11]. However, such a
strong parametric assumption is not always desirable as the resulting inference
may become non-robust. For this reason, we use a piecewise constant hazard func-
tion to approximate the baseline hazard so that it could capture inherent shape
and features of the hazard function better. Let J be the number of partitions
of the time interval, i.e., 0 = t(0) < t(1) < ··· < t(J), where t(J) > max(tij). The
points t(1), ..., t(J) are called cut-points. The piecewise constant hazard function
is then given by
h0(t) = γk for t(k−1) ≤ t < t(k) for k = 1, ..., J.
The corresponding cumulative hazard function is
(2.8) H0(t) =
k−1∑
q=1
γq
(
t(q) − t(q−1))
+ γk
(
t − t(k−1))
for t(k−1) ≤ t < t(k),
where γk is a constant hazard for interval[
t(k−1), t(k))
, k = 1, ..., J .
3. ESTIMATION METHOD
Let (tij , δij , xij), i = 1, ..., n, j = 1, ..., mi, be the failure time, censoring in-
dicator, and the covariate vector for the jth individual in the ith cluster and yi
be the frailty term. Then, the full likelihood function of the BS frailty model is
obtained from (1.1) as
L =n
∏
i=1
∫ ∞
0
( mi∏
j=1
h(tij |yi)δij S(tij |yi)
)
f(yi) dyi
=n
∏
i=1
∫ ∞
0
[ mi∏
j=1
(
yih0(tij) exp(β′xij))δij
× exp(
− yiH0(tij) exp(β′xij))
]
f(yi) dyi
=n
∏
i=1
[ mi∏
j=1
(
h0(tij) exp(β′xij))δij
(3.1)
×∫ ∞
0yδi·
i exp(
− yi
mi∑
j=1
H0(tij) exp(β′xij))
f(yi) dyi
]
=n
∏
i=1
[ mi∏
j=1
(
h0(tij) exp(β′xij))δij
Ii
]
,
Semi-Parametric Likelihood Inference for Birnbaum–Saunders Frailty Model 237
where δi· =∑mi
j=1 δij , H0 is the cumulative baseline hazard function with parame-
ter γ as given in (2.8), f is the PDF of the BS distribution with shape parameter
α and scale parameter β = 22+α2 as given in (2.3), and
Ii =
∫ ∞
0yδi·
i exp(
− yi
mi∑
j=1
H0(tij) exp(β′xij))
f(yi) dyi.
The above expression of Ii can be rewritten as
(3.2) Ii =
∫ ∞
−∞g(zi)
δi· exp(
− g(zi)
mi∑
j=1
H0(tij) exp(β′xij))
fZ(zi) dzi,
where fZ is the PDF of the standard normal distribution and
g(zi) =2
2 + α2
{
1 +α2z2
i
2+ α zi
(
1 +α2z2
i
4
)1/2}
.
The maximum likelihood estimates are hard to determine due to the in-
tractable integral in (3.2) present in the likelihood function in (3.1). A direct and
convenient way is to use Monte Carlo simulation to approximate the integral in
(3.2) as follows:
Ii = EZ
[
g(Z)δi· exp(
− g(Z)
mi∑
j=1
H0(tij) exp(β′xij))
]
=1
N
N∑
k=1
g(z(k))δi· exp
(
− g(z(k))
mi∑
j=1
H0(tij) exp(β′xij))
,
where z(k), k = 1, ..., N , are the realizations of standard normal random variable.
The log-likelihood function can then be approximated from (3.1) as
(3.3)
l =n
∑
i=1
[
mi∑
j=1
δij
(
log h0(tij) + β′xij
)
+ log1
N
N∑
k=1
g(z(k))δi· exp
(
− g(z(k))
mi∑
j=1
H0(tij) exp(β′xij))
]
.
Once the approximate log-likelihood function is obtained as in (3.3), Fisher’s
score function and the Hessian matrix with respect to the parameters α,β, γ can
be obtained readily upon taking partial derivatives of first- and second-order, and
pertinent details are presented in Appendix A. The MLEs of model parameters
can then be obtained by Newton–Raphson algorithm iteratively as
α(k)
β(k)
γ(k)
=
α(k−1)
β(k−1)
γ(k−1)
−
∂2l∂α2
∂2l∂α∂βT
∂2l∂α∂γT
∂2l∂α∂β
∂2l∂β∂βT
∂2l∂β∂γT
∂2l∂α∂γ
∂2l∂βT ∂γ
∂2l∂γ∂γT
−1
∂l∂α∂l∂β∂l∂γ
α=α(k−1),β=β(k−1)
,γ=γ(k−1)
.
238 N. Balakrishnan and Kai Liu
The iterations need to be continued until the desired tolerance level is achieved,
say, |θi+1 − θi| < 10−6. Finally, the standard errors of the estimates of α,β, γ can
be obtained from the inverse of the Hessian matrix evaluated at the determined
MLEs.
4. SIMULATION STUDY
An extensive simulation study is carried out here to assess the performance
of the proposed model and the method of estimation. We consider 4 scenarios:
(1) n = 100, m = 2, (2) n = 100, m = 4, (3) n = 100, m = 8 and (4) n = 400,
m = 2. Here, the clusters can be considered as hospitals and each subject as a
patient in these hospitals. The patients are randomly assigned to either a treat-
ment group or a control group with equal probability. The frailty term follows (1)
the BS distribution with shape parameter (2√
10−2)1/2
3 and scale parameter 98+
√10
,
(2) gamma distribution (GA) with shape parameter 2 and scale parameter 0.5,
(3) lognormal (LN) distribution with µ = − log(1.5)2 and σ2 = log(1.5). With these
choices of parameters, the mean and variance of the frailty distribution become 1
and 0.5, respectively, for all these frailty distributions. The standard exponential
distribution and the standard lognormal distribution are considered for baseline
distributions. We then set β = − log(2) = −0.6931 so that the hazard rate of
patients in the treatment group is half of those in the control group. Finally, the
censoring times are generated from the uniform distribution in [0, 4.5].
The simulation procedure is as follows:
(1) Generate n frailty values from frailty distributions, i.e., yi, i = 1, ..., n,
and assign each subject in the same cluster with same frailty value.
(2) Assign each patient to treatment group or control group with proba-
bility 0.5.
(3) Given the frailty term, the survival function is
S(tij |yi) = exp(
−yiH0(tij) exp(βxij))
and the cumulative distribution function is
F (tij |yi) = 1 − exp(
−yiH0(tij) exp(βxij))
,
which follows a uniform distribution (0,1). Therefore we generate uij
from Uniform(0,1) and set F (tij |yi) = uij .
(4) Calculate the baseline cumulative hazard function, which is
H0(tij) = − log(1 − uij)
yi exp(βxij).
(5) Solve for the lifetime according to the true baseline distribution, i.e.:
for standard exponential, tij = H0(tij); for standard lognormal, tij =
exp(
Φ−1(1 − exp(−H0(tij))))
since 1 − exp(−H0(tij)) = Φ(log(tij)).
Semi-Parametric Likelihood Inference for Birnbaum–Saunders Frailty Model 239
(6) Now, we generate censoring time cij from Uniform[0,4.5].
(7) Compare tij and cij . If tij <= cij , then set tij to be the observed time
and the censoring indicator δij = 1. If tij > cij , we set cij to be our
observed time and δij = 0.
We generated 1000 data sets under each setting and applied the proposed
semi-parametric BS frailty model to these data sets. For comparative purposes,
we fitted the simulated data sets with the parametric BS frailty model along with
gamma and lognormal frailty models. Thus, we fitted 6 models for each simu-
lated data with frailty distribution to be one of BS, gamma or lognormal, and
the baseline hazard function to be either piecewise constant hazard function or
Weibull hazard function. The primary parameters of interest are the treatment
effect and the frailty variance, and so our attention will focus on these parame-
ters. The estimates of the treatment effect are summarized in Figures 1 and 2,
while Figures 3 and 4 demonstrate how the estimates of the frailty variance dif-
fer under different models. The horizontal black lines are the true values of the
parameters of interest, while the vertical bars give 95% confidence intervals. The
three numbers on the top of each plot are the rejection rate and coverage proba-
bilities at confidence levels of 95% and 90%. The two numbers at the bottom of
each plot provide bias and mean square error for the different models considered.
Figures 1 and 2 clearly show that the choice of frailty distribution has little
impact on the estimate of treatment effect. When the true baseline distribution is
exponential, either Weibull baseline hazard or piecewise constant hazard function
will result in accurate estimation of the treatment effect. However, when the
true baseline distribution is lognormal, use of piecewise constant hazard baseline
distribution results in smaller bias and mean square error than when using the
Weibull distribution as baseline. This reveals that misspecification of the baseline
hazard function impacts the estimate of treatment effect and the semi-parametric
frailty models are therefore better than the parametric frailty models based on
robustness consideration.
The heterogeneity among clusters is explained by the frailty variance and
so it is important to investigate the frailty variance. The estimates of frailty
variance are shown in Figures 3 and 4. BS frailty model always has less mean
square error than the lognormal frailty model no matter what the true frailty
model is. Even though the gamma frailty model generally has smallest bias and
mean square error, its coverage probabilities are quite small and considerably
below the nominal level. Both parametric and semi-parametric BS frailty models
have coverage probabilities close to the nominal level, and so does the lognormal
frailty model. Furthermore, as the sample size gets larger, the estimates become
more precise. When the sample size is small, the rejection rate is small for BS and
lognormal frailty models, but they become larger when the sample size increases.
240 N. Balakrishnan and Kai Liu
● ● ● ● ● ●
0.951 0.95 0.954 0.955 0.95 0.95
0.943 0.944 0.944 0.946 0.945 0.946
0.89 0.888 0.896 0.888 0.888 0.886
−0.0144 −0.0113 −0.0167 −0.0184 −0.0137 −0.0112
0.0535 0.0515 0.0512 0.0514 0.0535 0.0516
● ● ● ● ● ●
1 1 1 1 1 1
0.956 0.953 0.955 0.951 0.955 0.953
0.901 0.889 0.909 0.892 0.9 0.889
−0.0055 −0.0049 −0.0026 −0.0061 −0.005 −0.0046
0.0227 0.0226 0.0214 0.0222 0.0225 0.0225
● ● ● ● ● ●
1 1 1 1 1 1
0.941 0.942 0.949 0.942 0.941 0.944
0.891 0.887 0.895 0.888 0.892 0.889
−0.0057 −0.0058 0.0026 −0.0018 −0.0052 −0.0054
0.0104 0.0104 0.0103 0.0105 0.0104 0.0104
● ● ● ● ● ●
0.998 0.998 0.998 0.998 0.998 0.998
0.944 0.951 0.937 0.941 0.942 0.95
0.888 0.885 0.881 0.884 0.889 0.886
−0.0105 −0.0098 −0.0121 −0.0171 −0.0101 −0.01
0.0261 0.0257 0.0248 0.0251 0.0261 0.0258
● ● ● ● ● ●
1 1 1 1 1 1
0.952 0.953 0.952 0.947 0.95 0.953
0.904 0.908 0.893 0.895 0.903 0.909
−0.0034 −0.0025 −0.0055 −0.009 −0.003 −0.0027
0.0123 0.0119 0.0121 0.0117 0.0123 0.0119
n=100, m=2 n=100, m=4 n=100, m=8
n=200, m=2 n=400, m=2
−1.5
−1.0
−0.5
0.0
0.5
−1.5
−1.0
−0.5
0.0
0.5
bp bw gp gw lp lw bp bw gp gw lp lw
fitted models
treatm
ent
effect
fitted models
●aaaaa
●aaaaa
●aaaaa
●aaaaa
●aaaaa
●aaaaa
BS frailty, piecewise constant hazards
BS frailty, Weibull baseline
gamma frailty, piecewise constant hazards
gamma frailty, Weibull baseline
LN frailty, piecewise constant hazards
LN frailty, Weibull baseline
True frailty dist. is BS and true baseline dist. is exp
● ● ● ● ● ●
0.947 0.949 0.951 0.947 0.946 0.95
0.946 0.942 0.943 0.943 0.945 0.942
0.905 0.908 0.91 0.907 0.904 0.906
−0.0211 −0.0195 −0.0106 −0.0167 −0.0211 −0.0207
0.0571 0.0551 0.0505 0.0531 0.0571 0.0557
● ● ● ● ● ●
0.999 0.999 0.999 0.999 0.999 0.999
0.956 0.95 0.954 0.943 0.957 0.949
0.9 0.902 0.904 0.895 0.902 0.903
−0.0056 −0.0058 0.0032 −0.0012 −0.005 −0.0056
0.0233 0.0236 0.0221 0.0231 0.0232 0.0235
● ● ● ● ● ●
1 1 1 1 1 1
0.944 0.943 0.945 0.946 0.944 0.945
0.896 0.894 0.894 0.892 0.898 0.894
−0.0058 −0.006 0.0022 −0.0026 −0.0056 −0.006
0.011 0.0108 0.0105 0.0107 0.0109 0.0108
● ● ● ● ● ●
0.997 0.998 0.998 0.998 0.997 0.998
0.941 0.947 0.947 0.948 0.942 0.949
0.905 0.902 0.902 0.901 0.903 0.9
−0.0024 −0.0044 0.0055 −0.0011 −0.0016 −0.0047
0.0262 0.0258 0.0239 0.0248 0.0262 0.0258
● ● ● ● ● ●
1 1 1 1 1 1
0.954 0.947 0.955 0.939 0.954 0.945
0.893 0.894 0.888 0.889 0.894 0.896
−0.0062 −0.0071 −3e−04 −0.0061 −0.0054 −0.0074
0.0132 0.0133 0.0122 0.0128 0.0132 0.0133
n=100, m=2 n=100, m=4 n=100, m=8
n=200, m=2 n=400, m=2
−1.5
−1.0
−0.5
0.0
0.5
−1.5
−1.0
−0.5
0.0
0.5
bp bw gp gw lp lw bp bw gp gw lp lw
fitted models
treatm
ent effect
fitted models
●aaaaa
●aaaaa
●aaaaa
●aaaaa
●aaaaa
●aaaaa
BS frailty, piecewise constant hazards
BS frailty, Weibull baseline
gamma frailty, piecewise constant hazards
gamma frailty, Weibull baseline
LN frailty, piecewise constant hazards
LN frailty, Weibull baseline
True frailty dist. is gamma and true baseline dist. is exp
● ● ● ● ● ●
0.954 0.954 0.96 0.956 0.954 0.953
0.943 0.949 0.941 0.945 0.945 0.948
0.893 0.887 0.891 0.888 0.892 0.888
−0.0138 −0.0103 −0.0203 −0.0228 −0.0133 −0.0103
0.0521 0.0503 0.0505 0.0514 0.0522 0.0505
● ● ● ● ● ●
1 1 1 1 1 1
0.951 0.949 0.952 0.943 0.95 0.949
0.893 0.887 0.901 0.887 0.89 0.887
−0.0057 −0.0048 −0.0046 −0.0074 −0.0053 −0.0047
0.0226 0.0224 0.0219 0.0227 0.0225 0.0223
● ● ● ● ● ●
1 1 1 1 1 1
0.939 0.937 0.936 0.942 0.938 0.937
0.89 0.892 0.89 0.887 0.891 0.892
−0.0055 −0.0056 0.0012 −0.0025 −0.0053 −0.0055
0.0104 0.0103 0.0104 0.0105 0.0104 0.0103
● ● ● ● ● ●
0.998 0.998 0.998 0.998 0.998 0.998
0.943 0.952 0.94 0.943 0.944 0.952
0.887 0.878 0.885 0.88 0.887 0.878
−0.01 −0.0093 −0.0153 −0.0187 −0.0097 −0.0096
0.0256 0.025 0.0248 0.0254 0.0255 0.025
● ● ● ● ● ●
1 1 0.999 1 1 1
0.954 0.954 0.942 0.948 0.953 0.952
0.896 0.913 0.896 0.912 0.896 0.912
−0.0035 −0.0024 −0.0085 −0.0116 −0.0033 −0.0027
0.0121 0.0117 0.0121 0.0114 0.0121 0.0117
n=100, m=2 n=100, m=4 n=100, m=8
n=200, m=2 n=400, m=2
−1.5
−1.0
−0.5
0.0
0.5
−1.5
−1.0
−0.5
0.0
0.5
bp bw gp gw lp lw bp bw gp gw lp lw
fitted models
treatm
ent effect
fitted models
●aaaaa
●aaaaa
●aaaaa
●aaaaa
●aaaaa
●aaaaa
BS frailty, piecewise constant hazards
BS frailty, Weibull baseline
gamma frailty, piecewise constant hazards
gamma frailty, Weibull baseline
LN frailty, piecewise constant hazards
LN frailty, Weibull baseline
True frailty dist. is LN and true baseline dist. is exp
Figure 1: Estimate of treatment effect when the true baseline distributionis exponential.
Semi-Parametric Likelihood Inference for Birnbaum–Saunders Frailty Model 241
●
●●
●●
●
0.918 0.931 0.919 0.93 0.918 0.93
0.944 0.913 0.941 0.917 0.943 0.916
0.903 0.849 0.89 0.859 0.901 0.849
3e−04 −0.1071 −0.0179 −0.0776 0.0031 −0.103
0.0582 0.0902 0.0588 0.0735 0.0578 0.0888
●●
●●
●●
0.996 0.998 0.997 0.997 0.996 0.998
0.954 0.915 0.949 0.915 0.955 0.915
0.893 0.848 0.893 0.838 0.896 0.845
9e−04 −0.07 −0.0095 −0.0676 0.002 −0.0683
0.0262 0.0372 0.0255 0.036 0.026 0.0367
●●
●●
●●
1 1 1 1 1 1
0.953 0.902 0.958 0.873 0.953 0.902
0.902 0.824 0.903 0.794 0.901 0.826
−0.0017 −0.0592 −0.0085 −0.0685 −0.0011 −0.0584
0.0118 0.0176 0.0111 0.0198 0.0118 0.0174
●
●●
●●
●
0.994 0.996 0.996 0.994 0.994 0.996
0.949 0.887 0.945 0.914 0.951 0.891
0.907 0.815 0.898 0.846 0.908 0.82
0.004 −0.105 −0.0143 −0.073 0.0064 −0.1014
0.0283 0.0499 0.0283 0.037 0.028 0.0489
●
●●
●●
●
1 1 1 1 1 1
0.954 0.885 0.956 0.914 0.95 0.891
0.892 0.805 0.903 0.859 0.895 0.818
0.0114 −0.0971 −0.0061 −0.0665 0.0136 −0.0934
0.0133 0.0273 0.0132 0.0188 0.0133 0.0264
n=100, m=2 n=100, m=4 n=100, m=8
n=200, m=2 n=400, m=2
−1.5
−1.0
−0.5
0.0
0.5
−1.5
−1.0
−0.5
0.0
0.5
bp bw gp gw lp lw bp bw gp gw lp lw
fitted models
treatm
ent
effect
fitted models
●aaaaa
●aaaaa
●aaaaa
●aaaaa
●aaaaa
●aaaaa
BS frailty, piecewise constant hazards
BS frailty, Weibull baseline
gamma frailty, piecewise constant hazards
gamma frailty, Weibull baseline
LN frailty, piecewise constant hazards
LN frailty, Weibull baseline
True frailty dist. is BS and true baseline dist. is LN
●
●●
●
●
●
0.917 0.931 0.924 0.932 0.918 0.934
0.953 0.922 0.955 0.919 0.951 0.923
0.907 0.858 0.912 0.84 0.91 0.858
−0.006 −0.1157 −0.0163 −0.1206 −0.0028 −0.1124
0.0595 0.093 0.0567 0.0955 0.0587 0.0917
●●
●●
●●
0.996 0.996 0.997 0.997 0.996 0.996
0.952 0.921 0.96 0.923 0.954 0.924
0.901 0.844 0.903 0.852 0.902 0.846
0.0051 −0.0665 −8e−04 −0.0541 0.0066 −0.0647
0.0264 0.0369 0.025 0.0345 0.0262 0.0365
●●
●●
●●
1 1 1 1 1 1
0.939 0.902 0.939 0.878 0.938 0.903
0.881 0.82 0.884 0.802 0.884 0.825
0.003 −0.0561 −0.0025 −0.0631 0.0036 −0.0552
0.0129 0.0187 0.0126 0.0202 0.0129 0.0186
●
●●
●●
●
0.996 0.997 0.998 0.996 0.996 0.997
0.953 0.914 0.951 0.931 0.954 0.915
0.902 0.848 0.905 0.881 0.898 0.852
0.0142 −0.0947 0.0038 −0.0561 0.0175 −0.0902
0.0275 0.0463 0.0263 0.0325 0.0272 0.0452
●
●●
●●
●
1 1 1 1 1 1
0.955 0.861 0.957 0.915 0.952 0.866
0.903 0.787 0.904 0.859 0.903 0.793
0.0089 −0.1015 −0.0023 −0.0604 0.0121 −0.0969
0.0141 0.0294 0.0136 0.0179 0.014 0.0282
n=100, m=2 n=100, m=4 n=100, m=8
n=200, m=2 n=400, m=2
−1.5
−1.0
−0.5
0.0
0.5
−1.5
−1.0
−0.5
0.0
0.5
bp bw gp gw lp lw bp bw gp gw lp lw
fitted models
treatm
ent effect
fitted models
●aaaaa
●aaaaa
●aaaaa
●aaaaa
●aaaaa
●aaaaa
BS frailty, piecewise constant hazards
BS frailty, Weibull baseline
gamma frailty, piecewise constant hazards
gamma frailty, Weibull baseline
LN frailty, piecewise constant hazards
LN frailty, Weibull baseline
True frailty dist. is gamma and true baseline dist. is LN
●
●●
●●
●
0.926 0.93 0.932 0.938 0.926 0.932
0.949 0.913 0.94 0.907 0.948 0.91
0.898 0.855 0.9 0.854 0.9 0.853
−0.0014 −0.1082 −0.0225 −0.0848 0.0011 −0.1046
0.0562 0.0881 0.0576 0.0744 0.0558 0.0869
●●
●●
●●
0.997 0.998 0.998 0.998 0.997 0.998
0.954 0.914 0.952 0.908 0.954 0.917
0.897 0.845 0.886 0.829 0.896 0.847
−1e−04 −0.0708 −0.012 −0.071 8e−04 −0.0695
0.0258 0.0369 0.0256 0.0367 0.0257 0.0365
●●
●●
●●
1 1 1 1 1 1
0.952 0.899 0.951 0.875 0.952 0.9
0.893 0.826 0.896 0.79 0.897 0.825
−0.0018 −0.0593 −0.009 −0.0697 −0.0014 −0.0586
0.0118 0.0175 0.0115 0.0198 0.0118 0.0174
●
●●
●●
●
0.994 0.996 0.996 0.997 0.994 0.997
0.945 0.887 0.941 0.906 0.946 0.891
0.902 0.811 0.898 0.839 0.902 0.814
0.0037 −0.105 −0.0177 −0.0771 0.0057 −0.1018
0.0279 0.0493 0.0282 0.0382 0.0276 0.0483
●
●●
●●
●
1 1 1 1 1 1
0.954 0.879 0.955 0.912 0.953 0.881
0.9 0.797 0.901 0.854 0.901 0.807
0.0105 −0.0981 −0.0096 −0.0706 0.0122 −0.0948
0.0131 0.0272 0.0133 0.0194 0.013 0.0264
n=100, m=2 n=100, m=4 n=100, m=8
n=200, m=2 n=400, m=2
−1.5
−1.0
−0.5
0.0
0.5
−1.5
−1.0
−0.5
0.0
0.5
bp bw gp gw lp lw bp bw gp gw lp lw
fitted models
treatm
ent effect
fitted models
●aaaaa
●aaaaa
●aaaaa
●aaaaa
●aaaaa
●aaaaa
BS frailty, piecewise constant hazards
BS frailty, Weibull baseline
gamma frailty, piecewise constant hazards
gamma frailty, Weibull baseline
LN frailty, piecewise constant hazards
LN frailty, Weibull baseline
True frailty dist. is LN and true baseline dist. is LN
Figure 2: Estimate of treatment effect when the true baseline distributionis lognormal.
242 N. Balakrishnan and Kai Liu
● ●● ●
● ●
0.353 0.402 0.993 0.993 0.021 0.042
0.908 0.915 0.433 0.477 0.919 0.928
0.861 0.868 0.365 0.416 0.885 0.896
−0.0021 −0.0123 −0.0777 −0.0653 0.1103 0.0941
0.1044 0.0911 0.0192 0.0168 0.2734 0.248
● ●● ●
● ●
0.976 0.979 0.999 0.999 0.972 0.975
0.905 0.903 0.263 0.336 0.925 0.927
0.857 0.862 0.218 0.305 0.898 0.896
−0.0326 −0.0335 −0.0965 −0.0819 0.0204 0.0199
0.0294 0.0286 0.0156 0.0135 0.0484 0.0473
● ● ● ●● ●
1 1 1 0.998 1 1
0.914 0.917 0.164 0.347 0.95 0.952
0.865 0.861 0.15 0.332 0.916 0.916
−0.035 −0.0349 −0.084 −0.0694 0.0084 0.0089
0.0139 0.0137 0.0112 0.0102 0.0204 0.0201
● ●● ●
● ●
0.788 0.819 0.993 0.995 0.678 0.744
0.92 0.91 0.39 0.433 0.932 0.93
0.857 0.865 0.338 0.388 0.902 0.905
−0.0047 −0.0089 −0.0816 −0.0659 0.0751 0.0705
0.0558 0.051 0.0148 0.0131 0.1187 0.1104
● ● ● ●● ●
0.983 0.991 0.998 1 0.98 0.987
0.933 0.924 0.26 0.36 0.958 0.949
0.877 0.876 0.21 0.319 0.918 0.912
−0.0286 −0.0324 −0.0872 −0.0769 0.0271 0.0235
0.0252 0.0221 0.0122 0.0108 0.043 0.0377
n=100, m=2 n=100, m=4 n=100, m=8
n=200, m=2 n=400, m=2
−1
0
1
2
−1
0
1
2
bp bw gp gw lp lw bp bw gp gw lp lw
fitted models
frailt
y v
ari
ance
fitted models
●aaaaa
●aaaaa
●aaaaa
●aaaaa
●aaaaa
●aaaaa
BS frailty, piecewise constant hazards
BS frailty, Weibull baseline
gamma frailty, piecewise constant hazards
gamma frailty, Weibull baseline
LN frailty, piecewise constant hazards
LN frailty, Weibull baseline
True frailty dist. is BS and true baseline dist. is exp
● ●
● ●
● ●
0.544 0.606 0.99 0.989 0.018 0.056
0.954 0.952 0.558 0.584 0.974 0.974
0.908 0.894 0.495 0.528 0.961 0.964
0.1638 0.1578 −0.0199 −0.0012 0.3741 0.3703
0.1586 0.1469 0.0175 0.0187 0.6082 0.6097
● ●● ●
● ●
0.999 0.998 0.993 0.997 0.999 0.998
0.942 0.942 0.518 0.528 0.991 0.991
0.871 0.875 0.472 0.483 0.953 0.952
0.1281 0.1285 −0.0272 −0.0106 0.2427 0.2449
0.0587 0.0578 0.0089 0.0092 0.1538 0.154
● ●● ●
● ●
1 1 0.998 1 1 1
0.921 0.92 0.52 0.594 0.963 0.962
0.851 0.852 0.474 0.557 0.872 0.87
0.1115 0.1128 −0.0298 −0.0141 0.208 0.2104
0.0332 0.0332 0.0044 0.0044 0.0862 0.0869
● ●
● ●
● ●
0.929 0.953 0.992 0.992 0.88 0.928
0.956 0.945 0.58 0.59 0.987 0.988
0.903 0.883 0.538 0.53 0.974 0.968
0.1437 0.155 −0.0204 −0.0023 0.2813 0.3036
0.0809 0.0819 0.0089 0.0096 0.2337 0.2465
● ●● ●
● ●
1 1 0.996 0.995 1 1
0.929 0.919 0.613 0.602 0.981 0.972
0.859 0.843 0.56 0.547 0.895 0.874
0.1351 0.1402 −0.0233 −0.0068 0.2499 0.262
0.0482 0.0482 0.0053 0.0057 0.1298 0.1353
n=100, m=2 n=100, m=4 n=100, m=8
n=200, m=2 n=400, m=2
−1
0
1
2
3
−1
0
1
2
3
bp bw gp gw lp lw bp bw gp gw lp lw
fitted models
frailt
y v
ari
ance
fitted models
●aaaaa
●aaaaa
●aaaaa
●aaaaa
●aaaaa
●aaaaa
BS frailty, piecewise constant hazards
BS frailty, Weibull baseline
gamma frailty, piecewise constant hazards
gamma frailty, Weibull baseline
LN frailty, piecewise constant hazards
LN frailty, Weibull baseline
True frailty dist. is gamma and true baseline dist. is exp
● ● ● ●● ●
0.316 0.355 0.994 0.994 0.032 0.042
0.882 0.881 0.399 0.429 0.895 0.894
0.837 0.833 0.317 0.384 0.858 0.857
−0.0447 −0.0569 −0.0903 −0.0742 0.0501 0.0297
0.0991 0.0869 0.0195 0.0176 0.2313 0.194
● ● ● ●● ●
0.964 0.967 0.998 1 0.958 0.957
0.861 0.863 0.207 0.297 0.89 0.892
0.812 0.808 0.184 0.265 0.851 0.846
−0.073 −0.0745 −0.1107 −0.0976 −0.029 −0.0304
0.0315 0.0308 0.0186 0.0168 0.043 0.0419
● ● ● ● ● ●
1 1 1 0.998 1 1
0.831 0.823 0.119 0.286 0.895 0.897
0.764 0.76 0.109 0.277 0.846 0.838
−0.0734 −0.0732 −0.0988 −0.0813 −0.0379 −0.0375
0.0169 0.0168 0.015 0.0123 0.0191 0.0189
● ● ● ●● ●
0.719 0.771 0.993 0.997 0.617 0.681
0.883 0.875 0.298 0.409 0.904 0.904
0.837 0.838 0.26 0.371 0.869 0.867
−0.048 −0.0516 −0.0982 −0.0803 0.0172 0.0132
0.0537 0.0496 0.0176 0.0146 0.096 0.0903
● ● ● ●● ●
0.973 0.976 0.999 0.997 0.965 0.97
0.89 0.884 0.193 0.265 0.921 0.914
0.822 0.825 0.163 0.225 0.871 0.872
−0.0693 −0.0736 −0.1046 −0.0944 −0.024 −0.0285
0.0271 0.0248 0.0159 0.0142 0.0369 0.0329
n=100, m=2 n=100, m=4 n=100, m=8
n=200, m=2 n=400, m=2
−1
0
1
2
−1
0
1
2
bp bw gp gw lp lw bp bw gp gw lp lw
fitted models
frailt
y v
ari
ance
fitted models
●aaaaa
●aaaaa
●aaaaa
●aaaaa
●aaaaa
●aaaaa
BS frailty, piecewise constant hazards
BS frailty, Weibull baseline
gamma frailty, piecewise constant hazards
gamma frailty, Weibull baseline
LN frailty, piecewise constant hazards
LN frailty, Weibull baseline
True frailty dist. is LN and true baseline dist. is exp
Figure 3: Estimate of frailty variance when the true baseline distributionis exponential.
Semi-Parametric Likelihood Inference for Birnbaum–Saunders Frailty Model 243
Semi-Parametric Likelihood Inference for Birnbaum–Saunders Frailty Model 251
−3 −2 −1 0 1 2 3
−2
−1
01
23
BS, Weibull
norm quantiles
Devia
nce r
esid
ual
−3 −2 −1 0 1 2 3
−2
−1
01
2
BS, piecewise
norm quantiles
Devia
nce r
esid
ual
−3 −2 −1 0 1 2 3
−2
−1
01
23
Gamma, Weibull
norm quantiles
Devia
nce r
esid
ual
−3 −2 −1 0 1 2 3
−2
−1
01
2
Gamma, piecewise
norm quantiles
Devia
nce r
esid
ual
−3 −2 −1 0 1 2 3
−2
−1
01
23
LN, Weibull
norm quantiles
Devia
nce r
esid
ual
−3 −2 −1 0 1 2 3
−2
−1
01
23
LN, piecewise
norm quantiles
Devia
nce r
esid
ual
−3 −2 −1 0 1 2 3
−2
01
23
IG, Weibull
norm quantiles
Devia
nce r
esid
ual
−3 −2 −1 0 1 2 3
−2
−1
01
2
IG, piecewise
norm quantiles
Devia
nce r
esid
ual
Figure 9: QQ plots for deviance residuals.
6. DISCUSSION AND CONCLUDING REMARKS
In this work, we have proposed a semi-parametric frailty model with BS frailty
distribution. The non-parametric choice of baseline hazard function provides a ro-
bust and flexible way to model the data. The determination of MLEs becomes
very difficult due to the intractable integrals present in the likelihood function.
For this reason, Monte Carlo simulations are used to approximate the likelihood
function upon exploiting the relationship between BS and standard normal distri-
butions and then expressing those integrals as expectations of some functions of
standard normal variables. From the simulation study carried out and the illus-
trative example analyzed, the semi-parametric BS frailty model is seen to be quite
robust in estimating the covariate effects as well as the frailty variance. Interest-
ingly, it is seen to be even better than the three-parameter generalized gamma
frailty model though the latter has an extra shape parameter. It is of interest to
mention that the work carried out here can be generalized in two different direc-
tions. The BS distribution can be generalized by assuming that the variable Z in
(2.2) follows a standard elliptically symmetric distribution, including power expo-
nential, Laplace, Student t and logistic distributions. SuchageneralizedBirnbaum–
Saunders (GBS) distribution (see [14]) could be assumed for the frailty term yi
in (1.1) and then the resulting GBS frailty model could be studied in detail.
Next, we could allow for the possibility of a cure of patients within the context of
BS frailty model and develop the corresponding analysis. Work is currently under
progress on these problems and we hope to report these findings in a future paper.
252 N. Balakrishnan and Kai Liu
APPENDIX A — FIRST- AND SECOND-ORDER DERIVATIVES
OF THE LOG-LIKELIHOOD FUNCTION
The first- and second-order derivatives of the log-likelihood function with
respect to α,β and γ are as follows:
∂l
∂α=
n∑
i=1
1
Ii
∂Ii
∂α,
∂l
∂β=
n∑
i=1
[
mi∑
j=1
δij xij +1
Ii
∂Ii
∂β
]
,
∂l
∂γ=
n∑
i=1
[ mi∑
j=1
δij
h0(tij)
dh0(tij)
dγ+
1
Ii
∂Ii
∂γ
]
;
∂2l
∂α2=
n∑
i=1
[
− 1
I2i
(
∂Ii
∂α
)2
+1
Ii
∂2Ii
∂α2
]
,
∂2l
∂α∂βT=
n∑
i=1
[
− 1
I2i
∂Ii
∂α
(
∂Ii
∂β
)T
+1
Ii
∂2Ii
∂α∂βT
]
,
∂2l
∂α∂γT=
n∑
i=1
[
− 1
I2i
∂Ii
∂α
(
∂Ii
∂γ
)T
+1
Ii
∂2Ii
∂α∂γT
]
,
∂2l
∂β∂βT=
n∑
i=1
[
− 1
I2i
∂Ii
∂β
(
∂Ii
∂β
)T
+1
Ii
∂2Ii
∂β∂βT
]
,
∂2l
∂β∂γT=
n∑
i=1
[
− 1
I2i
∂Ii
∂β
(
∂Ii
∂γ
)T
+1
Ii
∂2Ii
∂β∂γT
]
,
∂2l
∂γ∂γT=
n∑
i=1
mi∑
j=1
− δij
h0(tij)2dh0(tij)
dγ
(
dh0(tij)
dγ
)T
+n
∑
i=1
mi∑
j=1
δij
h0(tij)
d2h0(tij)
dγdγT
+n
∑
i=1
[
− 1
I2i
∂Ii
∂γ
(
∂Ii
∂γ
)T
+1
Ii
∂2Ii
∂γ∂γT
]
,
where
∂Ii
∂α= δi·E1,i −
[ mi∑
j=1
H0(tij) exp(β′xij)
]
E2,i,
∂2Ii
∂α2= δi·(δi· − 1)E3,i − 2δi·
[ mi∑
j=1
H0(tij) exp(β′xij)
]
E4,i + δi·E5,i
+
[ mi∑
j=1
H0(tij) exp(β′xij)
]2
E6,i −[ mi
∑
j=1
H0(tij) exp(β′xij)
]
E7,i,
Semi-Parametric Likelihood Inference for Birnbaum–Saunders Frailty Model 253
∂2Ii
∂α∂βT=
{
[ mi∑
j=1
H0(tij) exp(β′xij)
]
E8,i − (δi· + 1)E2,i
}
×[ mi
∑
j=1
H0(tij) exp(β′xij)xij
]
,
∂Ii
∂β= − E9,i
[ mi∑
j=1
H0(tij) exp(β′xij)xij
]
,
∂Ii
∂γ= − E9,i
[ mi∑
j=1
dH0(tij)
dγexp(β′xij)
]
,
∂2Ii
∂β∂βT= E10,i
[ mi∑
j=1
H0(tij) exp(β′xij)xij
][ mi∑
j=1
H0(tij) exp(β′xij)xij
]T
− E9,i
[ mi∑
j=1
H0(tij) exp(β′xij)xij xijT
]
,
∂2Ii
∂β∂γT= E10,i
[
mi∑
j=1
H0(tij) exp(β′xij)xij
][
mi∑
j=1
exp(β′xij)dH0(tij)
dγ
]T
− E9,i
[
mi∑
j=1
exp(β′xij)xij
(
dH0(tij)
dγ
)T]
,
∂2Ii
∂γ∂γT= E10,i
[
mi∑
j=1
exp(β′xij)dH0(tij)
dγ
][
mi∑
j=1
exp(β′xij)dH0(tij)
dγ
]T
− E9,i
[
mi∑
j=1
exp(β′xij)d2H0(tij ; γ)
dγ dγT
]
;
in the above expressions, the quantities El,i (l = 1, ..., 10) are given by
E1,i =1
N
N∑
k=1
g(z(k))δi·−1 exp
(
−g(z(k))
mi∑
j=1
H0(tij) exp(β′xij))dg(z(k))
dα,
E2,i =1
N
N∑
k=1
g(z(k))δi· exp
(
−g(z(k))
mi∑
j=1
H0(tij) exp(β′xij))dg(z(k))
dα,
E3,i =1
N
N∑
k=1
g(z(k))δi·−2 exp
(
−g(z(k))
mi∑
j=1
H0(tij) exp(β′xij))
(
dg(z(k))
dα
)2
,
E4,i =1
N
N∑
k=1
g(z(k))δi·−1 exp
(
−g(z(k))
mi∑
j=1
H0(tij) exp(β′xij))
(
dg(z(k))
dα
)2
,
E5,i =1
N
N∑
k=1
g(z(k))δi·−1 exp
(
−g(z(k))
mi∑
j=1
H0(tij) exp(β′xij))d2g(z(k))
dα2,
254 N. Balakrishnan and Kai Liu
E6,i =1
N
N∑
k=1
g(z(k))δi· exp
(
−g(z(k))
mi∑
j=1
H0(tij) exp(β′xij))
(
dg(z(k))
dα
)2
,
E7,i =1
N
N∑
k=1
g(z(k))δi· exp
(
−g(z(k))
mi∑
j=1
H0(tij) exp(β′xij))d2g(z(k))
dα2,
E8,i =1
N
N∑
k=1
g(z(k))δi·+1 exp
(
−g(z(k))
mi∑
j=1
H0(tij) exp(β′xij))dg(z(k))
dα,
E9,i =1
N
N∑
k=1
g(z(k))δi·+1 exp
(
−g(z(k))
mi∑
j=1
H0(tij) exp(β′xij))
,
E10,i =1
N
N∑
k=1
g(z(k))δi·+2 exp
(
−g(z(k))
mi∑
j=1
H0(tij) exp(β′xij))
.
ACKNOWLEDGMENTS
The authors express their sincere thanks to the guest editor, Professor Sat
Gupta, for extending an invitation and to the anonymous reviewers for their
useful comments and suggestions on an earlier version of this manuscript which
led to this improved version.
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