http://smm.sagepub.com/
Statistical Methods in Medical Research
http://smm.sagepub.com/content/early/2014/02/11/0962280214520730
The online version of this article can be found at:
DOI: 10.1177/0962280214520730
published online 12 February 2014Stat Methods Med Res
Geert Molenberghs, Geert Verbeke, Achmad Efendi, Roel Braekers and Clarice GB Demétrio
time-to-event data
A combined gamma frailty and normal random-effects model for repeated, overdispersed
Published by:
http://www.sagepublications.com
can be found at:Statistical Methods in Medical ResearchAdditional services and information for
http://smm.sagepub.com/cgi/alertsEmail Alerts:
http://smm.sagepub.com/subscriptionsSubscriptions:
http://www.sagepub.com/journalsReprints.navReprints:
http://www.sagepub.com/journalsPermissions.navPermissions:
What is This?
- Feb 12, 2014OnlineFirst Version of Record >>
at KU Leuven University Library on July 22, 2014smm.sagepub.comDownloaded from at KU Leuven University Library on July 22, 2014smm.sagepub.comDownloaded from
XML Template (2014) [6.2.2014–10:27am] [1–19]
//blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/SMMJ/Vol00000/140005/APPFile/SG-SMMJ140005.3d (SMM) [PREPRINTER stage]
Article
A combined gamma frailty and
normal random-effects model
for repeated, overdispersed
time-to-event data
Geert Molenberghs,
1,2
Geert Verbeke,
2,1
Achmad Efendi,
1,3
Roel Braekers
1,2
and Clarice GB Deme
´
trio
4
Abstract
This paper presents, extends, and studies a model for repeated, overdispersed time-to-event outcomes,
subject to censoring. Building upon work by Molenberghs, Verbeke, and Deme
´
trio (2007) and
Molenberghs et al. (2010), gamma and normal random effects are included in a Weibull model, to
account for overdispersion and between-subject effects, respectively. Unlike these authors, censoring is
allowed for, and two estimation methods are presented. The partial marginalization approach to full
maximum likelihood of Molenberghs et al. (2010) is contrasted with pseudo-likelihood estimation. A
limited simulation study is conducted to examine the relative merits of these estimation methods. The
modeling framework is employed to analyze data on recurrent asthma attacks in children on the one hand
and on survival in cancer patients on the other.
Keywords
exponential model, generalized Cauchy distribution, conjugacy, maximum likelihood, frailty model,
pseudo-likelihood, strong conjugacy, Weibull model
1 Introduction
Time-to-event data are prominent in contemporary statistical analysis, not only for univariate
outcomes but also in hierarchical settings. Apart from the need to accommodate such data
hierarchies for repeated survival outcomes, recurrent events, and the like,
1
it is possible that
overdispersion
2
is present in the data, relative to the standard generalized linear model
3,4
assumed, as well as censored observations.
1
I-BioStat, Universiteit Hasselt, Diepenbeek, Belgium
2
I-BioStat, Katholieke Universiteit Leuven, Leuven, Belgium
3
Study Program of Statistics, Universitas Brawijaya, Malang, Indonesia
4
ESALQ, Piracicaba, Sa
˜
o Paulo, Brazil
Corresponding author:
Geert Molenberghs, I-BioStat, Universiteit Hasselt, B-3590 Diepenbeek, Belgium.
Email: geert.molenberghs@uhasselt.be
Statistical Methods in Medical Research
0(0) 1–19
! The Author(s) 2014
Reprints and permissions:
sagepub.co.uk/journalsPermissions.nav
DOI: 10.1177/0962280214520730
smm.sagepub.com
at KU Leuven University Library on July 22, 2014smm.sagepub.comDownloaded from
XML Template (2014) [6.2.2014–10:27am] [1–19]
//blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/SMMJ/Vol00000/140005/APPFile/SG-SMMJ140005.3d (SMM) [PREPRINTER stage]
While each of these features has received attention, it is uncommon to treat all of them
simultaneously. Building upon their earlier work,
5
Molenberghset al.
6
presented a general
modeling framework for (non-)Gaussian overdispersed and hierarchical outcomes. The time-
to-event case is but one of the applications of their framework. They combine so-called conjugate
random effects for overdispersion with generalized linear mixed model ideas (GLMM
7–9
) for
between-subject effects. Here, we supplement their method with the possibility to accommodate
censorship.
Whereas Molenberghs et al.
5
focused on maximum likelihood, using so-called partial
marginalization, we supplement this inferential option with pairwise likelihood ideas.
10
A
simulation study is conducted to study the relative merits of these methods. The methodology is
applied to analyze data on recurrent asthma attacks in children on the one hand and on survival in
cancer patients on the other.
The paper is organized as follows. In Section 2, motivating case studies with a time-to-event
outcome are described, with analyses reported in Section 7. Basic ingredients for our modeling
framework, standard generalized linear models, extensions for overdispersion, and the generalized
linear mixed model are the subject of Section 3. The proposed, combined model is described and
further studied in Section 4. Avenues for parameter estimation and ensuing inferences are explored
in Section 5, with particular emphasis on so-called partial marginalization and pseudo-likelihood
estimation. Some cautionary remarks regarding the existence of the corresponding marginal
distributions’ moments are issues in Section 6. A simulation study is described and results
presented in Section 8.
2 Case studies
2.1 Recurrent asthma attacks in children
These data have been studied in Duchateau and Janssen.
1
Asthma is occurring more and more
frequently in very young children (between 6 and 24 months). Therefore, a new application of an
existing antiallergic drug is administered to children who are at higher risk to develop asthma in
order to prevent it. A prevention trial is set up with such children randomized to placebo or drug,
and the asthma events that developed over time are recorded in a diary. Typically, a patient has
more than one asthma event. The different events are thus clustered within a patient and ordered in
time. This ordering can be taken into account in the model. The data are presented in calendar time
format, where the time at risk for a particular event is the time from the end of the previous event
(asthma attack) to the start of the next event (start of the next asthma attack). A particular patient
has different periods at risk during the total observation period which are separated either by an
asthmatic event that lasts one or more days or by a period in which the patient was not under
observation. The start and end of each such risk period is required, together with the status indicator
to denote whether the end of the risk period corresponds to an asthma attack or not. Data for the
first two patients are listed in Table 1.
2.2 Survival in cancer patients
Hand et al.
11
presented data on patients with advanced cancer of the stomach, bronchus, colon,
ovary, or breast, who were treated, in addition to standard treatment, with ascorbate. The outcome
of interest, survival time in days, is recorded to address the question as to whether survival times
differ with the organ being affected. Individual-patient data are listed in Table 2. There are no
censored observations in this case.
2 Statistical Methods in Medical Research 0(0)
at KU Leuven University Library on July 22, 2014smm.sagepub.comDownloaded from
XML Template (2014) [6.2.2014–10:27am] [1–19]
//blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/SMMJ/Vol00000/140005/APPFile/SG-SMMJ140005.3d (SMM) [PREPRINTER stage]
3 Background
Our model is based upon the generalized linear model and two of its extensions, the first one to
accommodate overdispersion, and the second one to account for data hierarchies, such as in
longitudinal data. We briefly review these building blocks.
Table 2. Advanced cancer data.
Stomach Bronchus Colon Ovary Breast
124 81 248 1234 1235
42 461 377 89 24
25 20 189 201 1581
45 450 1843 356 1166
412 246 180 2970 40
51 166 537 456 727
1112 63 519 3808
46 64 455 791
103 155 406 1804
876 859 365 3460
146 151 942 719
340 166 776
396 37 372
223 163
138 101
72 20
245 283
Average
286.0 211.6 457.4 884.3 1395.9
Survival time in days per patient and per organ affected.
Table 1. Asthma data for the first two patients.
Patient ID Drug Begin End Status
100151
1 0 22 90 1
1 0 96 325 1
1 0 329 332 1
1 0 338 369 1
1 0 370 412 1
1 0 418 422 1
1 0 426 474 1
1 0 477 526 1
1 0 530 600 0
2 1 0 180 1
2 1 189 267 1
2 1 273 581 1
2 1 582 600 0
The column labeled ‘‘Status’’ referred to whether (1) or not (0) censoring has occurred.
Molenberghs et al. 3
at KU Leuven University Library on July 22, 2014smm.sagepub.comDownloaded from
XML Template (2014) [6.2.2014–10:27am] [1–19]
//blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/SMMJ/Vol00000/140005/APPFile/SG-SMMJ140005.3d (SMM) [PREPRINTER stage]
A random variable Y follows an exponential family distribution if the density is of the form
f ðyÞf ðyj, Þ¼exp
1
½y ðÞ þ cðy, Þ
ð1Þ
for a specific set of unknown parameters and , and for known functions ðÞ and cð, Þ. Often,
and are termed ‘‘natural parameter’’ (or ‘‘canonical parameter’’) and ‘‘dispersion parameter’’,
respectively. It is well known that
EðYÞ¼ ¼
0
ðÞð2Þ
VarðYÞ¼
2
¼
00
ðÞð3Þ
implying a mean–variance relationship:
2
¼
00
½
0
1
ðÞ ¼ v ðÞ, with vðÞ the so-called variance
function. In the exponential case, one assumes
f ðyÞ¼’e
’y
ð4Þ
with mean ’
1
and variance ’
2
. This extends in the Weibull case to
f ðyÞ¼’y
1
e
’y
EðYÞ¼’
1=
ð
1
þ 1Þ
VarðYÞ¼’
2=
ð2
1
þ 1Þð
1
þ 1Þ
2
Note that the Weibull model does not belong to the exponential family in a conventional sense,
unless in a somewhat contrived fashion where Y is replaced by Y
. In the mean and variance
expressions for the Weibull, ðÞ represents the gamma function.
When the standard exponential-family models constrain the mean–variance relationship,
so-called overdispersion is introduced. Early reviews are provided by Hinde and Deme
´
trio
2
provide general treatments of overdispersion. The Poisson case received particular attention by
Breslow
12
and Lawless.
13
A natural step is to allow the overdispersion parameter 6¼ 1, so that
(3) produces VarðYÞ¼vðÞ. A convenient route is through a two-stage approach. Generally, the
two-stage approach is made up of considering a distribution for the outcome, given a random effect
f ðy
i
j
i
Þ which, combined with a model for the random effect, f ð
i
Þ, produces the marginal model
f ðy
i
Þ¼
Z
f ðy
i
j
i
Þf ð
i
Þd
i
ð5Þ
In our exponential and Weibull cases, it is in line with the data range to assume such a random effect
to follow a gamma distribution, giving rise to the exponential-gamma and Weibull-gamma models.
The model elements are listed in Table 3.
The choice of the gamma distribution can also be motivated through the concept of
conjugacy.
14,15
To simplify notation, drop the indices for the purpose of the definition. The
hierarchical and random-effects densities are said to be conjugate if and only if they can be
written in the generic forms
f ðyjÞ¼exp
1
½yhðÞgðÞ þcðy, Þ
ð6Þ
f ðÞ¼exp ½ hðÞgðÞ þ c
ð, Þ
ð7Þ
4 Statistical Methods in Medical Research 0(0)
at KU Leuven University Library on July 22, 2014smm.sagepub.comDownloaded from
