A combined gamma frailty and normal random-effects model for repeated, overdispersed time-to-event data

TL;DR: This paper presents, extends, and studies a model for repeated, overdispersed time-to-event outcomes, subject to censoring, and two estimation methods are presented.

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 Demetrio (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.

Summary

1 Introduction

• Time-to-event data are prominent in contemporary statistical analysis, not only for univariate outcomes but also in hierarchical settings.
• The timeto-event case is but one of the applications of their framework.
• Basic ingredients for their modeling framework, standard generalized linear models, extensions for overdispersion, and the generalized linear mixed model are the subject of Section 3.
• Avenues for parameter estimation and ensuing inferences are explored in Section 5, with particular emphasis on so-called partial marginalization and pseudo-likelihood estimation.

2.1 Recurrent asthma attacks in children

• These data have been studied in Duchateau and Janssen.
• 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.
• The different events are thus clustered within a patient and ordered in time.
• 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).
• 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.

3 Background

• The authors 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.
• Þ and cð , Þ. Often, and are termed ‘‘natural parameter’’ (or ‘‘canonical parameter’’) and ‘‘dispersion parameter’’, respectively.
• When the standard exponential-family models constrain the mean–variance relationship, so-called overdispersion is introduced.
• In their 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.

4.1 General model formulation

• The authors now need two different notations, ij and ij, to refer to the linear predictor and/or the natural parameter.
• It is convenient, but not strictly necessary, to assume that the two sets of random effects, hi and bi, are independent of each other.
• Obviously, parameterization (12) allows for random effects ij capturing overdispersion, and formulated directly at mean scale, whereas ij could be considered the GLMM component.
• They considered conjugacy, conditional upon the normally-distributed random effect bi.
• Fortunately, the Weibull and exponential cases satisfy this property, with gamma random effects.

4.2 Weibull- and exponential-type models for time-to-event data

• The general Weibull model for repeated measures, with both gamma and normal random effects can be expressed as f ð yijhi, biÞ ¼ First, it is implicit that the gamma random effects are independent.
• This need not be the case and, like in the Poisson case, extension via multivariate gamma distributions is possible.
• Third, it is evident that the classical gamma frailty model (i.e., no normal random effects) and the Weibull-based GLMM (i.e., no gamma random effects) follow as special cases.
• This is typically considered for the exponential model, but it holds for the Weibull model too, by observing that the Weibull model is nothing but an exponential model for the random variable Y ij.
• Fifth, the above expressions are derived for a two-parameter gamma density.

5 Estimation

• The key problem in maximizing (23) is the presence of N integrals over the random effects bi and hi.
• The authors consider so-called partial marginalization, in agreement with Molenberghs et al.
• 5 but, unlike these authors, also allowing for censorship.

5.1 Partial marginalization

• While closed-form expressions, can be used to implement maximum likelihood estimation, with numerical accuracy governed by the number of terms included in the Taylor series, one can also proceed by what Molenberghs et al.5 termed partial marginalization.
• By this the authors refer to integrating the conditional density over the gamma random effects only, leaving the normal random effects untouched.
• Now, in the survival case it is evidently very likely that censoring occurs.
• Focusing on rightcensored data, it is then necessary to integrate the marginal density over the survival time within the interval ½0,Ci .
• Recall that, while expressions of the type (16) appear to be for the univariate case, they extend without problem to the longitudinal setting as well.

5.2 Pseudo-likelihood

• Pseudo-likelihood,10,18 as generalized estimating equations19 are useful when the computational burden of full likelihood becomes burdensome and/or when robustness against misspecification of higher order moments is desirable.
• Essentially then, the joint distribution is replaced with a product of factors of marginal and/or conditional distributions of lower dimensions.
• Let us define pseudo-likelihood in general and formally, after which the authors turn to the special case of pairwise likelihood.
• Grouping the outcomes for subject i into a vector.

6 Marginal distributions and moments

• Along the lines of Molenberghs et al.5 and Molenberghs and Verbeke,29 the marginal density and moments are derived.
• Molenberghs and Verbeke29 showed that only a finite number of moments is finite.
• This holds not only for the combined model, but as soon as gamma random effects are combined with Weibull outcomes, i.e., it also applies to the conventional Weibullgamma model.
• Because it is possible that even the second and first moments may be infinite, it is wise to check the number of finite moments.

7.1 Recurrent asthma attacks in children

• The authors will analyze the times-to-event, introduced in Section 2.1.
• The treatment effect 1 is stably identifiable in all four models.
• As a result, overdispersion now disappears, given that the standard error values are more trustworthy.
• Re-fitted results for all four models in this way are reported in Table 7.
• A related finding was reported in Geys et al.26 where excessive computational requirements could be avoided when using pseudo-likelihood.

7.2 Survival in cancer patients

• For the generalized Cauchy model, predictor function ’ is set equal to instead.
• Parameter estimates are presented in Table 9.
• The key scientific question is directed toward the difference in survival across cancer types.
• Thus, their analysis illustrates the occurrence, in real life, of distributions without finite moments, with all moments finite, and with a finite number of finite moments.
• The generalized Cauchy model has a finite mean and finite variance and provides a parsimonious description, unlike the generalized logistic model, in spite of its full series of finite moments.

8 Simulation study

• The authors aim to evaluate the performance of the combined model, Weibull model with gamma frailties and normal random effects, under full likelihood and pseudo-likelihood.
• The design of the simulation study was carried out under different settings, to investigate the impact of sample size, censoring percentage, and method of estimation.
• The authors used two sets of true parameters, similar in spirit to the ones in Table 7, without and with censoring (full likelihood).
• The Mahalanobis distance has the advance of taking the variance-covariance structure into account.
• While with increasing proportion of censored observations, within the same sample size, the relative distance seems to be stable when using pseudo-likelihood estimation method.

9 Concluding remarks

• Building upon work by Molenberghs et al.,5 the authors have studied the combination of normal and nonnormal random effects in the time-to-event case.
• The statistical loss of efficiency of pseudo-likelihood is relatively small, although the consistency behavior for the maximum-likelihood case is better.
• The gamma and normal random effects play distinct roles.
• The model can be extended further and/or adapted to specific cases.

Figures

Table 2. Advanced cancer data.

Table 3. Model elements for the exponential-gamma and Weibull-gamma models.

Table 10. Results of the simulation study, with 10% censored observation.

Table 9. Parameter estimates (standard errors) for generalized log-logistic, generalized logistic, and generalized Cauchy models, respectively, fitted to the survival data in cancer patients.

Table 7. Asthma data: combined model fitted with maximum likelihood and pseudo-likelihood, with and without censoring (model-based s.e.; empirically corrected s.e.).

Table 8. Asthma study: Wald test for treatment effect’s assessment in combined model.

Table 1. Asthma data for the first two patients.

Figure 1. Mahalanobis distance for different sample sizes and censored observation percentages. (a) 50 subjects, (b) 100 subjects, (c) 200 subjects, (d) 10 percent censoring, (e) 25 percent censoring and (f) 50 percent censoring.

Table 6. Asthma data: combined model fitted with maximum likelihood and pseudo-likelihood, with and without censoring (model-based s.e.; empirically corrected s.e.).

Table 4. Asthma study: full likelihood.

Table 5. Asthma study: Wald test results for the assessment of treatment effect.

Table 11. Results of the simulation study, with 25% censored observation.

Table 12. Results of the simulation study, with 50% censored observation.

Full text

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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
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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
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DOI: 10.1177/0962280214520730
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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)
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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
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A random variable Y follows an exponential family distribution if the density is of the form
f ðyÞf ðyj, Þ¼exp 
1
½y  ðÞ þ cðy, Þ

ð1Þ
for a specific set of unknown parameters  and , and for known functions ðÞ and cð, Þ. Often, 
and  are termed ‘‘natural parameter’’ (or ‘‘canonical parameter’’) and ‘‘dispersion parameter’’,
respectively. It is well known that
EðYÞ¼ ¼
0
ðÞð2Þ
VarðYÞ¼
2
¼ 
00
ðÞð3Þ
implying a mean–variance relationship: 
2
¼ 
00
½
0
1
ðÞ ¼ v ðÞ, with vðÞ 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 1Þð
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, ðÞ 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 VarðYÞ¼vðÞ. 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 ðyjÞ¼exp 
1
½yhðÞgðÞ þcðy, Þ

ð6Þ
f ðÞ¼exp ½ hðÞgðÞ þ c

ð, Þ

ð7Þ
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Q1. What are the contributions mentioned in the paper "A combined gamma frailty and normal random-effects model for repeated, overdispersed time-to-event data" ?

This paper presents, extends, and studies a model for repeated, overdispersed time-to-event outcomes, subject to censoring. A limited simulation study is conducted to examine the relative merits of these estimation methods.