Long-term frailty modeling using a non-proportional hazards model: Application with a melanoma dataset

Abstract

The semiparametric Cox regression model is often fitted in the modeling of survival data. One of its main advantages is the ease of interpretation, as long as the hazards rates for two individuals do not vary over time. In practice the proportionality assumption of the hazards may not be true in some situations. In addition, in several survival data is common a proportion of units not susceptible to the event of interest, even if, accompanied by a sufficiently large time, which is so-called immune, “cured,” or not susceptible to the event of interest. In this context, several cure rate models are available to deal with in the long term. Here, we consider the generalized time-dependent logistic (GTDL) model with a power variance function (PVF) frailty term introduced in the hazard function to control for unobservable heterogeneity in patient populations. It allows for non-proportional hazards, as well as survival data with long-term survivors. Parameter estimation was performed using the maximum likelihood method, and Monte Carlo simulation was conducted to evaluate the performance of the models. Its practice relevance is illustrated in a real medical dataset from a population-based study of incident cases of melanoma diagnosed in the state of São Paulo, Brazil.

Full text

Article
Long-term frailty modeling using a
non-proportional hazards model:
Application with a melanoma dataset
Vinicius F Calsavara
1
, Eder A Milani
2
, Eduardo Bertolli
3
and
Vera Tomazella
4
Abstract
The semiparametric Cox regression model is often fitted in the modeling of survival data. One of its main advantages is
the ease of interpretation, as long as the hazards rates for two individuals do not vary over time. In practice the
proportionality assumption of the hazards may not be true in some situations. In addition, in several survival data is
common a proportion of units not susceptible to the event of interest, even if, accompanied by a sufficiently large time,
which is so-called immune, “cured,” or not susceptible to the event of interest. In this context, several cure rate models
are available to deal with in the long term. Here, we consider the generalized time-dependent logistic (GTDL) model
with a power variance function (PVF) frailty term introduced in the hazard function to control for unobservable
heterogeneity in patient populations. It allows for non-proportional hazards, as well as survival data with long-term
survivors. Parameter estimation was performed using the maximum likelihood method, and Monte Carlo simulation was
conducted to evaluate the performance of the models. Its practice relevance is illustrated in a real medical dataset from a
population-based study of incident cases of melanoma diagnosed in the state of S~
ao Paulo, Brazil.
Keywords
Cure fraction, frailty model, generalized time-dependent logistic model, non-proportional hazard, melanoma, power
variance function distribution, survival model
1 Introduction
In survival analysis, the standard approach to the analysis of censored survival data is to consider the semi-
parametric proportional hazards model proposed by Cox,
1
which assumes that hazard ratios are constant over
time. However, in some situations the covariate effects may change over time and the Cox regression model may
not be adequate. In clinical study, prognostic factors such as treatment disappear with time. For example, some
types of cancer may respond well to chemotherapy initially, but the cancer cells may develop some tolerance to the
treatment through genetic mechanisms, resulting in loss of the treatment effect over time. Such situations clearly
represent non-proportional hazards scenarios. Schemper
2
noted that the Cox model has undoubtedly been used in
many cases in which proportionality assumptions are violated, with consequences for the results.
1
Department of Epidemiology and Statistics, A.C.Camargo Cancer Center, S~
ao Paulo, Brazil
2
Institute of Mathematics and Statistics, Federal University of Goia
´s, Goia
ˆnia, Brazil
3
Skin Cancer Department, A.C.Camargo Cancer Center, S~
ao Paulo, Brazil
4
Department of Statistics, Federal University of S~
ao Carlos, S~
ao Carlos, Brazil
Corresponding author:
Vinicius F Calsavara, Department of Epidemiology and Statistics, A.C.Camargo Cancer Center, Prof. Anto
ˆnio Prudente Street, 211, Liberdade, S~
ao
Paulo 01509-010, SP, Brazil.
Email: vinicius.calsavara@accamargo.org.br
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In practice, one usually fits a Cox proportional hazards model and assesses the proportionality assumption
based on the Schoenfeld residuals.
3–5
Another approach, suggested by Klein and Moeschberger
6
is to plot the log
of the cumulative hazard functions against time and check for parallelism. Several graphical methods for assess-
ment of the proportional hazards assumption have been proposed. Hess
7
considered eight graphical methods for
the detection of assumption violations using three real datasets with single binary covariates.
In the analysis of survival data, when departures from assumption are detected, several possible workarounds,
such as redefinition of covariates, model stratification by a covariate with a non-proportional hazard, use of time-
dependent covariate terms, use of separate models for disjunct time periods,
2
and fitting of a non-proportional
hazard model, can be applied.
Several techniques have been proposed to deal with non-proportional hazards; they include the non-parametric
accelerated failure time model proposed by Prentice
8
and Kalbfleisch and Prentice
9
; the hybrid hazard model
described by Etezadi-Amoli and Ciampi;
10
the extension of hybrid hazard models proposed by Louzada et al.,
11,12
and the generalized time-dependent logistic (GTDL) model proposed by Mackenzie.
13
Louzada-Neto et al.
14
presented a Bayesian approach to the GTDL model, and Milani et al.
15
extended the GTDL model by including
a gamma frailty term in the modeling. These models have been applied successfully to problems in which all units
are susceptible to the event of interest, that is, the existence of a cure fraction of the population is not possible.
However, some subjects will not failure (occurrence of the event of interest) because the units are “cured” in
several studies. The cure rate class of models considers such situations and has been studied by several authors in
recent years. The most popular cure rate model is the standard mixture model, proposed by Boag
16
and modified
by Berkson and Gage.
17
In this model, the population survival function is SðtÞ¼pþð1pÞS0ðtÞ, such that p2
ð0;1Þis referred to as the cured fraction, and S0ðtÞis a proper survival function for uncured patients. Common
choices for S0ðtÞare exponential, Weibull, log-logistic, log-normal distributions, among others.
Traditional cure rate models implicitly assume a homogeneous population for the susceptible units, but covar-
iate information can be included to explain the observable heterogeneity. However, a portion of unobserved
heterogeneity can be induced by several factors, such as environmental or genetic factors, or information that was
not considered in planning. Houggard
18
showed the advantages of considering two sources of heterogeneity
[observable (given by covariates) and unobservable] in a model. This random effect, called frailty, can be incor-
porated into the hazard function to control for unobservable heterogeneity of the units under study. In this
context, the frailty model is used widely.
19
This model is characterized by the inclusion of a random effect,
that is, an unobservable random variable that represents the information that cannot be or has not been observed.
The frailty term not only explains heterogeneity among individuals, but also enables assessment of covariate
effects that were not considered. The omission of an important covariate from a model will increase the amount of
unobservable heterogeneity affecting inferences about the model parameters. The inclusion of a frailty term can
help to alleviate this problem.
A frailty term can be included in an additive form in a model. However, a multiplicative effect of the frailty
term on the baseline hazard function is often used, as a generalization of the proportional hazards model intro-
duced by Cox.
1
This approach has been studied by several authors.
20–24
Other authors
25–31
have considered cure
rate models with frailty terms.
Another possibility for cure rate modeling is through defective models, which offer a strategy for the modeling
of survival data in the context of a cure fraction. Balka et al.
32,33
Rocha et al.
34–36
Scudilio et al.
37
and Calsavara
et al.
38,39
recently popularized the term “defective,” although the same idea had appeared in previous papers.
Instead of estimating the cure fraction pdirectly, as in a standard mixture model, the defective model provides an
alternative for the modeling of lifetime data with long-term survivors, once it has been made a cure rate model by
changing the usual domains of its parameters. When a probability distribution has this property, it is termed
“defective.” In a defective distribution, the integral of the density function is not 1, but a value in the range (0, 1),
which leads to a proportion of immunes in the population. The survival curve stabilizes at p2ð0;1Þ, meaning that
the survival function is improper. However, the impropriety of a survival function does not necessarily imply a
defective distribution.
In this paper, we consider another way to model survival datasets under non-proportional hazards and with the
possibility of a cure fraction of the population. Our strategy is to consider the GTDL model by including a power
variance function frailty term in the modeling, which is an extension of the model proposed by Milani et al.
15
We
illustrate the applicability of this model using a real medical dataset from a population-based study of incident
cases of melanoma diagnosed in the state of S~
ao Paulo, Brazil. Although melanoma is one of the best known by
the population, skin carcinomas are more incident than melanoma, but the survival of patients with melanoma is
worse due to its potential for metastatic dissemination. According to the Brazilian National Institute of Cancer,
2Statistical Methods in Medical Research 0(0)

about 6000 new cases of melanoma were expected to be identified in 2018
40
; according to the International Agency
for Research on Cancer, this number is about 7000.
41
An estimated 2000 deaths per year in Brazil are attributable
to melanoma.
40,41
In the study from which our dataset is drawn, patients diagnosed with melanoma were enrolled
from 2000 to 2014, with follow-up conducted until 2018; death due to cancer was the event of interest.
Our paper is organized as follows. In section 2, we present the GTDL and GTDL frailty distributions, prob-
ability density, survival and hazard functions, and their cure rate version. Inference methods based on the like-
lihood function are presented in section 3. In section 4, we consider a simulation study under different scenarios.
We numerically evaluate the asymptotic properties of the estimators, as well as the performance of the GTDL
frailty model in relation to several discrimination criteria. In addition, we evaluate performance using the like-
lihood ratio statistic to test the inclusion of the frailty term in the GTDL model considering several configurations
of sample size and degree of unobservable heterogeneity in the population. In section 5, we apply these procedures
to the real melanoma cancer dataset. We offer final remarks in section 6.
2 Background
In this section, we present the GTDL regression model and its frailty version, as well as the hazard, survival,
probability density functions, and the cases in which these models become a cure rate models. These models are
useful for the modeling data with non-proportional hazards and with the possibility of a long-term survivors in
the population.
2.1 GTDL regression model
Let T>0 be a random variable representing the failure time and h0ðtÞthe instantaneous failure rate or baseline
hazard function. According to Mackenzie,
13
the GTDL model with a hazard function is given by
h0ðt;x1Þ¼kexp atþx>
1b

1þexp atþx>
1b
 (1)
where k>0 is a scalar, ais a measure of the time effect, x>
1¼ðx1;...;xpÞ, and b>¼ðb1;...;bpÞare the sets of
covariates and their regression coefficients, respectively.
The corresponding probability density function f0ðt;x1Þand survival function S0ðt;x1Þare, respectively, as
follows
f0ðt;x1Þ¼ kexpðatþx>
1bÞ
1þexpðatþx>
1bÞ
"#
1þexpðatþx>
1bÞ
1þexpðx>
1bÞ
"#
k=a
and
S0ðt;x1Þ¼ 1þexpðatþx>
1bÞ
1þexpðx>
1bÞ
"#
k=a
(2)
The ratio of the hazard function for two individuals, iand j, with i6¼ jwhere i;j¼1;...;n, with different
covariate vectors is given by
sðt;x1i;x1jÞ¼h0ðt;x1iÞ
h0ðt;x1jÞ¼kexp atþx>
1ib

1þexp atþx>
1ib

1þexp atþx>
1jb

kexp atþx>
1jb

¼
1þexp atþx>
1jb

1þexp atþx>
1ib

exp x1ix1j
ðÞ
>b
hi
(3)
Calsavara et al. 3

Note that the time effect does not disappear in equation (3), and hence the non-proportionality becomes
evident. As mentioned by Mackenzie,
13
equation (1) is neither a proportional hazards model nor an accelerated
life model, but it will approach the proportional hazards model when ½1þexpðatþx>
1bÞ1 and, when this
condition holds, the estimates of the regression parameters bshould be similar in both models.
Model (1) is indicated for modeling lifetime data and for modeling phenomenon with monotone failure rates.
The shape of the hazard function takes several forms according to the value of parameter a.Whena>0, the
hazard function is increasing; when a<0 it is decreasing, and when a¼0 the hazard function is constant
over time.
The survival function is proper for a0, but when time effect ais negative, the GTDL model naturally
acquires an improper distribution, which is useful for the modeling of survival data in the presence of a surviving
fraction. The long-term survivors in the population is calculated as the limit of the survival function (2) when
a<0, given by
pðx1Þ¼lim
t!1 S0ðt;x1Þ¼ lim
t!1
1þexpðatþx>
1bÞ
1þexpðx>
1bÞ
"#
k=a
¼1þexpðx>
1bÞ

k=a2ð0;1Þ(4)
Figure 1 plots the baseline hazard and survival functions for different parameter values for the GTDL model
considering a group variable as the covariate.
The GTDL model has the advantage of allowing a cure rate without requiring extra parameters, as in tradi-
tional cure rate models. An additional advantage over cure rate models is that it does not make assumptions about
the existence of the cure rate. In the literature, models with this property have recently been termed “defective,”
when they accommodate a proportion of the cure fraction with dependence on a single parameter value.
32–37
2.2 GTDL frailty model
The concept of frailty provides a convenient way of introducing unobserved heterogeneity and associations into
models for survival data. The role of frailty in the univariate time scenario is to measure a possible heterogeneity
in order to identify the influence of covariates that were not incorporated into the modeling or cannot be mea-
sured. From the GTDL model given in equation (1), the hazard function of the ith individual with the frailty term
v
i
multiplicative is given by
15
hiðt;x1i;viÞ¼vih0ðt;x1i;viÞ¼vi
kexp atþx>
1ib

1þexp atþx>
1ib
 (5)
0 5 10 15 20 25 30
0.00 0.05 0.10 0.15 0.20
Time
Hazard function
Group10
Group11
Group20
Group21
Group30
Group31
010203040
0.0 0.2 0.4 0.6 0.8 1.0
Time
Survival function
Gruop10
Gruop11
Gruop20
Gruop21
Gruop30
Gruop31
Figure 1. Baseline hazard (left panel) and survival (right panel) functions from the GTDL model. The parameter values used are:
Group 1, a¼0:2;k¼0:2, and b¼1; Group 2, a¼0:5;k¼0:2, and b¼1; and Group 3, a¼0:001;k¼0:2, and b¼1. The
subscript numerals indicate the values of the fixed covariates. (For interpretation of the references to color in this figure legend, the
reader is referred to the online version of this article.)
4Statistical Methods in Medical Research 0(0)

interpreted as the conditional hazard function of the ith individual given the frailty term v
i
, which is characterized
as the frailty of ith subject. The conditional hazard function is greater than the baseline hazard function for vi>1
and smaller than baseline for vi<1; and in the special case of degenerated frailty v
i
¼1, the frailty model reduces
to the GTDL model (1). Its conditional survival function is easily obtained, given by
Siðt;x1i;viÞ¼S0ðt;x1iÞvi¼1þexp atþx>
1ib

1þexp x>
1ib

"#
kvi
a
(6)
According to the way in which the frailty term acts on the hazard function, natural frailty distribution
candidates are supposed to be non-negative, continuous, and time independent (i.e. gamma, log-normal,
inverse Gaussian, positive stable, and power variance function distributions, among others). In the literature,
the gamma distribution with mean 1 and variance hhas been widely used, as it permits easy algebraic
treatment.
In this paper, we consider the family of power variance function (PVF) distributions, as it presents as a
particular case the gamma, inverse Gaussian and positive stable distributions. The PVF distribution was suggested
by Tweedie
42
and derived independently by Hougaard.
43
Let Vbe a random variable following a PVF distribution
with parameters l,wand cwith density function written as
44
fðv;l;w;cÞ¼ewð1cÞv
l1
c
ðÞ
1
pX
1
k¼1
ð1Þkþ1½wð1cÞkð1cÞlkcCðkcþ1Þ
ckk!vkc1sinðkcpÞ
where l>0;w>0 and 0 <c1.
Following the historical definition of frailty originally introduced in the field of demography,
19
we use the
restriction E½V¼l¼1 such that V½V¼l2
w¼1
w:¼h, where his interpretable as a measure of unobserved
heterogeneity.
Note that if we build the likelihood function using the hazard and survival functions given in equations (5) and
(6), respectively, we would have more parameters than observations, considering univariate times. Thus, the
random effect can be integrated out to get a likelihood function not depending on unobserved quantities, con-
sequently the marginal survival function is given by
SðtÞ¼EV½Sðt;x1;vÞ ¼ Z1
0
Sðt;x1;vÞfvðvÞdv¼L
vlog S0ðt;x1Þ½
where fvðÞ is the probability density of the corresponding frailty distribution, S0ðÞ is the baseline survival func-
tion, and Lv½ denotes the Laplace transform of frailty distribution.
Considering as the baseline survival function from the GTDL model given in equation (2), the unconditional
survival, probability density, and hazard functions in the PVF frailty model are, respectively
Sðt;x1Þ¼exp 1c
ch 11þkh
að1cÞlog 1þexpðatþx>
1bÞ
1þexpðx>
1bÞ
"#()
c
0
@1
A
2
43
5(7)
fðt;x1Þ¼ kexpðatþx>
1bÞ
1þexpðatþx>
1bÞ1þkh
að1cÞlog 1þexpðatþx>
1bÞ
1þexpðx>
1bÞ
"#()
c1
exp 1c
ch 11þkh
að1cÞlog 1þexpðatþx>
1bÞ
1þexpðx>
1bÞ
"#()
c
0
@1
A
2
43
5
Calsavara et al. 5

and
hðt;x1Þ¼ kexpðatþx>
1bÞ
1þexpðatþx>
1bÞ1þkh
að1cÞlog 1þexpðatþx>
1bÞ
1þexpðx>
1bÞ
"#()
c1
¼h0ðt;x1Þ1þkh
að1cÞlog 1þexpðatþx>
1bÞ
1þexpðx>
1bÞ

c1(8)
where h0ð;x1Þis the hazard function from the GTDL model given in equation (1).
Henceforth, we will refer to the model in which the survival function is as shown in equation (7), as the GTDL
PVF frailty model. Note that the usual GTDL model (2) is obtained as h!0. In addition, the GTDL PVF frailty
model is a flexible model in the sense that it includes many other frailty models as special cases. For instance, the
GTDL gamma frailty model is obtained if c!0. In the case of c¼0:5, the GTDL inverse Gaussian frailty model
is derived. The GTDL positive stable frailty is a special case of the GTDL PVF frailty model in which some
asymptotic considerations are necessary to show this fact. We refer the interested readers to Wienke.
44
It is evident that the hazard function in equation (8) depends on the time; consequently, the GTDL PVF frailty
model is also of non-proportional hazard. As does the GTDL model, the GTDL PVF frailty model allows
negative values for the time effect (a<0). Thus, the corresponding long-term survivors is
pðx1Þ¼lim
t!1 Sðt;x1Þ
¼exp 1c
ch 11kh log 1 þexp x>
1b

að1cÞ
()
c
0
@1
A
2
43
52ð0;1Þ(9)
If parameter ais estimated to be negative, then the cure fractions for the GTDL and GTDL PVF frailty models
can be obtained from equations (4) and (9), respectively. If parameter ais estimated to be positive, then there is no
cure rate according to the two models, and functions (2) and (7) are proper survival functions.
As previously mentioned, the GTDL model (1) does not provide a reasonable parametric fit for modeling
phenomenon with non-monotone failure rates such as the bathtub-shaped and the unimodal failure rates which
are common in reliability and biological studies. In this sense, an advantage of the proposed model (8) over the
traditional GTDL model is the ability to accommodate various forms of the hazard function that can be used in a
variety of problems for modeling lifetime data. Figure 2 plots the baseline hazard and survival functions for
different parameter values in the GTDL PVF frailty model considering a group variable as a covariate.
0 1020304050
0.0 0.2 0.4 0.6 0.8
Time
Hazard function
Group11
Group10
Group21
Group20
Group31
Group30
Group41
Group40
0 1020304050
0.0 0.2 0.4 0.6 0.8 1.0
Time
Survival function
Group11
Group10
Group21
Group20
Group31
Group30
Group41
Group40
Figure 2. Hazard (left panel) and survival (right panel) functions from the GTDL PVF frailty model. The parameter values fixed are:
Group 1, a¼0:1;k¼0:6;b¼1;c¼0:01 and h¼1; Group 2, a¼0:2;k¼0:7;b¼1;c¼0:5 and h¼0:5; and Group 3, a¼
0:15;k¼0:3;b¼5;c¼0:9 and h¼0:3 and Group 4, a¼0:05, k¼1, b¼0:5;c¼0:9 and h¼1. The subscripted numerals
indicate the values of the fixed covariates. (For interpretation of the references to color in this figure legend, the reader is referred to
the online version of this article.)
6Statistical Methods in Medical Research 0(0)

In this paper, we also incorporate explanatory variables in the GTDL and GTDL PVF frailty models through
parameter a, which is a more reasonable approach because it can directly reflect the effect of a treatment. For
instance, for some treatment A, if the treatment effect is good, then some patients will be cured and the estimate
for awill be a<0; if the treatment is not sufficient, the estimate will be a>0. Given this capacity, the GTDL and
GTDL PVF frailty models are more flexible than are standard approaches.
13,15
In this sense, explanatory variables are incorporated in the model through the hazard function (1) and the scale
parameter awith a set of two-covariate vectors, x12Rpand x22Rqþ1, such that x>¼ðx>
1;x>
2Þ2Rwis a w-
dimensional covariate vector, where w¼pþqþ1. Importantly, parameter acan be estimated to be negative
(which leads to cure) or positive (indicating the absence of a cure rate). Thus, to guarantee a2R, we use an
identity link function, such as
aðx2iÞ¼x2>
ia
where x2>
i¼ð1;x2i1;x2i2;...;x2iqÞand a>¼ða0;a1;...;aqÞare the sets of covariates and their regression coeffi-
cients, respectively. In practice, the covariate vectors may be the same, i.e. x¼x1¼x2, but if the researcher has
prior knowledge about the variables that can be associated to cure rate, we suggest link this subset variables to the
aparameter.
An advantage of the GTDL and GTDL PVF frailty models over alternative models is the lack of assumption
about the existence of the cure rate; the time effect values lead to proper or improper distribution. Thus, these
models are flexible and can be applied in situations with and without a cure fraction.
3 Inference
In this section, we describe the inferential procedure, which is based on the maximum likelihood approach and the
asymptotic large sample theory. Let T>0 be a random variable representing the time until the occurrence of the
event of interest. Furthermore, let d
i
be the censoring indicator variable, that is, di¼0 if the observed time is
censored and di¼1 otherwise, i¼1;...;n. The observed dataset is D¼t;d;XÞð , where t¼t1;...;tn
ðÞ
>are the
observed lifetimes, d¼d1;...;dn
ðÞ
>are the censoring indicators, and Xis a matrix containing the covariate
information. Consider that T
i
s are independent and identically distributed random variables with survival and
hazard functions specified, respectively, by S;#; x1;x2
ðÞ
and h;#; x1;x2
ðÞ
, where #denotes a vector of unknown
parameters. We assume that Tis independent of the censoring time. Thus, the likelihood function of #under non-
informative censoring is expressed as
6
L#;D
ðÞ
/Y
n
i¼1
hðti;#; x1i;x2iÞdiSðti;#; x1i;x2iÞ
The corresponding log-likelihood function, ‘ð#Þ¼log L#;D
ðÞ
, is given by
‘ð#Þ/X
n
i¼1
dilog hðti;#; x1i;x2iÞþX
n
i¼1
log Sðti;#; x1i;x2iÞ
Thus, for the GTDL regression model the log-likelihood function for #¼a;b;kðÞ
>is
‘ð#Þ¼log kX
n
i¼1
diþX
n
i¼1
x>
2iaditiX
n
i¼1
dilog 1 þexp x>
2iatiþx>
1ib

kX
n
i¼1
1
x>
2ialog 1þexp x>
2iatiþx>
1ib

1þexpðx>
1ibÞ
"#()
þX
n
i¼1
dix>
1ib
(10)
Calsavara et al. 7

For the GTDL PVF frailty regression model the log-likelihood function for #¼a;b;k;h;c
ðÞ
>is
‘ð#Þ¼log kX
n
i¼1
diþX
n
i¼1
x>
2iaditiþX
n
i¼1
dix>
1ibX
n
i¼1
dilog 1 þexpðx>
2iaþx>
1ibÞ

þc1
ðÞ
X
n
i¼1
dilog 1 þhk
x>
2iað1cÞlog 1þexpðx>
2iatiþx>
1ibÞ
1þexpðx>
1ibÞ
"#() !
þX
n
i¼1
1c
ðÞ
ch 11þhk
x>
2iað1cÞlog 1þexpðx>
2iatiþx>
1ibÞ
1þexpðx>
1ibÞ

c
!
(11)
Maximum likelihood estimates (MLEs) for parameters from the GTDL and GTDL PVF frailty models are
obtained by numerically maximizing log-likelihood functions (10) and (11), respectively. Many routines are
available for numerical maximization. We used the optim routine in the R software
45
for numerical maximization.
The asymptotic properties of maximum likelihood estimators are needed to build confidence intervals and to
test hypotheses about the model parameters. Under certain regularity conditions, ^
#has asymptotic multivariate
normal distribution with mean #and variance and covariance matrix Rð^
#Þ, which is estimated by
^
Rð^
#Þ¼ @2‘ð#;DÞ
@#@#>#¼^
#
()
1
Thus, an approximate 100ð1aÞ% confidence interval for #iis ð^
#iza=2ffiffiffiffiffiffi
^
Rii
q;^
#iþza=2ffiffiffiffiffiffi
^
Rii
qÞ, where ^
Rii
denotes the ith diagonal element of the inverse of ^
Rand zadenotes the 100ð1aÞpercentile of the standard
normal random variable.
The asymptotic normality assumption of MLEs holds only under certain regularity conditions, which are not
easy to assess with our models. In the next section, we describe a simulation study performed to determine
whether the usual asymptotes of the MLEs hold. Many authors have performed simulations to assess the asymp-
totic behavior of MLEs, especially when the analytical investigation is not trivial.
35,38,39
We also conducted a
simulation study in order to assess the impact of unobservable heterogeneity on the cured fraction, as well as we
evaluate the performance of the models in estimating correctly the parameter a(negative) when in fact there is a
long-term survivors group. In addition, we investigate the performance of the GTDL frailty model in terms of the
same discrimination criteria when compared to the standard GTDL model, that is, without the frailty term.
4 Simulation study
In this section, we evaluate the performance of MLEs of the GTDL PVF frailty and GTDL model parameters
considering different sample sizes. To assess the covariate effects on the hazard function and time effect, we divide
the sample into two groups (X); control (group 0) and treatment (group 1). Subjects in the control and treatment
groups are assigned covariate values of 0 and 1, respectively. We introduce two regression parameters for a, that
is, aðxÞ¼a0þa1x, where a
0
is the intercept and a
1
is the associated group variable. The cure fractions from the
GTDL PVF frailty and GTDL models are functions of parameters as shown in equations (4) and (9), respectively.
To introduce random censoring, the distribution of censoring times is assumed to be exponential with rate s,
which is set to control the proportion of right-censored observations. Datasets ðti;di;xiÞfrom the GTDL PVF
frailty and GTDL models are generated using the steps shown in the Supplementary material.
4.1 Model fitting
We performed out an extensive Monte Carlo simulation considering sample sizes of n¼50, 100, 150, 200, 300,
500, 1000, 2000, 5000, and 10,000. For each scenario (combination of parameter values and sample size), we
computed average MLEs of the parameters, their standard deviations (SDs), bias, and root mean square errors
(RMSEs) of the MLEs of the parameters, and the empirical coverage probabilities (CPs) of 90% and 95%
confidence intervals. The standard error for the cure rate parameter was estimated using the delta method with
first-order Taylor’s approximation. All simulations were performed with the R software
45
and 1000 Monte Carlo
runs. Estimates were obtained using the BFGS algorithm of maximization, which is an option for the optim
8Statistical Methods in Medical Research 0(0)

function in R. In our simulation studies, we fixed the parameter c!0 (GTDL gamma frailty model) for all fitted
models in order to corroborate with the results obtained in the application section.
Results for the GTDL and GTDL gamma frailty models are summarized in Table 1. The estimation method
worked very well, as the sample size increased the bias gets to 0 for all parameters. The RMSEs and SDs decreased
to 0 as the sample size increased, and besides they are closer (RMSEs and SDs) when the sample size was n150.
Empirical CPs for all parameters appeared to be reasonably close to the nominal level with increasing sample size,
regardless of model. Under the GTDL gamma frailty model, the empirical CPs for kand hwere below than the
nominal level for n100. Considering the scenario of n¼2000, the empirical distributions of parameter estimates
for both models are shown in the Supplementary material. The plots indicate that the normal distribution
provides reasonable approximations for estimator distributions (see Figure 1 in the supplementary material).
In the supplementary material, we also illustrated the behavior of MLEs of the parameters from GTDL
gamma frailty model when a continuous covariate is considered in the data generation process, as well as
when two covariates (one binary and a continuous covariates) are considered in the GTDL gamma frailty
model. The results are similar with previously obtained suggesting that the models can separate the effects on
the two components (aand bparameters).
4.2 Impact of frailty on the cured fraction
To evaluate the impact of the frailty term in the estimation of the cure rates, we conducted a simulation consid-
ering several degrees of unobservable heterogeneity in the population. For each dataset simulated, the GTDL
gamma frailty and GTDL models were fitted and then the cure rates were compared with the fixed true values. We
generated datasets from the GTDL gamma frailty model considering different sample sizes and parameter values.
We fixed a0¼0:36;a1¼0:24;k¼1:4, b1¼2:7, and h¼f0;0:05;0:1;0:3;0:7;1:1;1:5g. The corresponding
cure rates are for groups 0 and 1 are p0¼f0:068;0:080;0:092;0:139;0:220;0:286;0:340gand p1¼f0:859;0:860;
0:860;0:862;0:866;0:869;0:872g, respectively. Based on 1000 datasets, we calculated the average RMSEs of MLEs
to cure rates.
Figure 3 shows the RMSEs of the MLEs of the cure fractions obtained from the GTDL gamma frailty and
GTDL models, considering several degrees of heterogeneity in the sample.
As expected, RMSE decreased to 0 as the sample size increased, regardless of group. However, RMSEs
increased with the degree of unobservable heterogeneity, mainly in group 0. For both models, the estimated
cure fractions are close to the true values, as indicated by the RMSEs of the cure rates.
4.3 Impact of cure rate and sample size in the estimating of aparameter
As previously mentioned, if parameter ais estimated to be negative, then the cure fraction is computed as function
of model parameters. In this sense, it is important to evaluate the sensitivity of the model in to identify correctly an
aless than zero when in fact there is an immune group. To evaluate the impact of small cure probabilities, we
conducted a simulation considering several values of sample size and cure rates p¼f0:01;0:03;0:045;0:11;0:20g.
For each dataset simulated, we fitted the models and then based on 1000 datasets we calculated the percentage of
cases in which estimates of awere less than zero. The results are shown in Table 2. As expected, the percentages
(model correctly identifies the cure fraction, i.e., a<0) increased with the sample size and cured rate fixed. As cure
probabilities decreased, the GTDL gamma frailty model had difficulty in identifying an immune group. However,
the percentages increased with sample size, regardless of the model. In the scenario of cure rate fixed at 0.01, the
GTDL model correctly identified the avalue in 100% when n¼1000, but in the GTDL gamma frailty model was
87.8%. For sample size n¼2000 the rate was higher than 95% (it was omitted in the table). When the cure rate is
greater than or equal to 0.11 and small sample size, both models correctly identified the cure fraction in more than
92% of the cases.
4.4 Model discrimination
In the fourth simulation, we investigated the performance of various discrimination criteria in the selection of the
correct model from the GTDL gamma frailty and GTDL models. We considered the Akaike information criterion
(AIC), corrected Akaike information criterion (AICc), Bayesian information criterion (BIC), Hannan–Quinn
information criterion (HQIC), and consistent Akaike information criterion (CAIC). These functions were com-
puted as follows: AIC ¼2‘þ2k;AICc ¼AIC þ2kðkþ1Þ=ðnk1Þ, BIC ¼2‘þklog n,HQIC¼2‘þ
2klogðlog nÞ, and CAIC ¼2‘þkðlog nþ1Þ, where ‘is the maximized log-likelihood function value, kis the
Calsavara et al. 9

Table 1. Bias, square roots of the mean squared errors (RMSEs), and standard deviations (SDs) of the maximum likelihood estimates,
and empirical coverage probabilities (CPs) of 90% and 95% confidence intervals for the simulated data.
GTDL model GTDL frailty model
a
0
a
1
kb
1
a
0
a
1
kb
1
h
n0.6 0.4 1 2.4 0.36 0.24 1.4 2.7 1.1
50 Bias 0.025 0.071 0.047 0.063 0.071 0.390 0.134 0.015 0.139
RMSE 0.206 1.713 0.340 0.727 0.235 1.277 0.818 1.171 0.985
SD 0.204 1.712 0.337 0.725 0.224 1.216 0.807 1.172 0.975
CP(90%) 0.902 0.919 0.894 0.914 0.915 0.953 0.868 0.939 0.805
CP(95%) 0.941 0.962 0.925 0.965 0.956 0.980 0.912 0.980 0.837
100 Bias 0.023 0.005 0.037 0.004 0.060 0.114 0.061 0.031 0.139
RMSE 0.130 0.147 0.234 0.455 0.163 0.551 0.522 0.743 0.765
SD 0.128 0.147 0.231 0.456 0.152 0.540 0.519 0.743 0.752
CP(90%) 0.906 0.908 0.900 0.902 0.888 0.942 0.871 0.915 0.816
CP(95%) 0.955 0.953 0.947 0.948 0.940 0.972 0.902 0.968 0.856
150 Bias 0.005 0.003 0.020 0.007 0.032 0.054 0.035 0.007 0.090
RMSE 0.100 0.113 0.184 0.359 0.121 0.326 0.399 0.588 0.634
SD 0.100 0.113 0.183 0.359 0.117 0.322 0.398 0.588 0.628
CP(90%) 0.904 0.908 0.904 0.898 0.910 0.929 0.893 0.901 0.885
CP(95%) 0.951 0.954 0.946 0.958 0.947 0.964 0.931 0.954 0.919
200 Bias 0.011 0.003 0.022 0.007 0.029 0.031 0.024 0.016 0.095
RMSE 0.090 0.101 0.162 0.315 0.107 0.256 0.363 0.501 0.560
SD 0.089 0.101 0.161 0.315 0.103 0.255 0.362 0.501 0.553
CP(90%) 0.888 0.894 0.908 0.898 0.911 0.918 0.885 0.905 0.888
CP(95%) 0.945 0.945 0.942 0.952 0.949 0.968 0.927 0.950 0.924
300 Bias 0.006 0.001 0.011 0.001 0.020 0.020 0.010 0.020 0.063
RMSE 0.070 0.080 0.125 0.254 0.088 0.193 0.284 0.409 0.471
SD 0.069 0.080 0.124 0.254 0.086 0.192 0.284 0.408 0.467
CP(90%) 0.917 0.892 0.915 0.891 0.905 0.911 0.907 0.909 0.910
CP(95%) 0.956 0.942 0.959 0.948 0.945 0.959 0.950 0.955 0.945
500 Bias 0.003 0.001 0.011 0.000 0.011 0.008 0.020 0.005 0.034
RMSE 0.054 0.060 0.100 0.190 0.069 0.134 0.234 0.311 0.392
SD 0.053 0.060 0.100 0.190 0.068 0.134 0.233 0.311 0.391
CP(90%) 0.896 0.891 0.893 0.918 0.894 0.910 0.898 0.903 0.904
CP(95%) 0.949 0.947 0.949 0.959 0.949 0.965 0.942 0.953 0.950
1000 Bias 0.002 0.002 0.007 0.004 0.007 0.004 0.007 0.001 0.026
RMSE 0.037 0.042 0.069 0.131 0.049 0.095 0.159 0.218 0.275
SD 0.037 0.042 0.068 0.131 0.048 0.095 0.159 0.218 0.274
CP(90%) 0.908 0.900 0.905 0.917 0.906 0.884 0.902 0.904 0.915
CP(95%) 0.953 0.954 0.953 0.962 0.951 0.941 0.960 0.953 0.958
2000 Bias 0.001 0.001 0.004 0.006 0.005 0.000 0.000 0.001 0.020
RMSE 0.025 0.028 0.049 0.096 0.035 0.066 0.109 0.151 0.194
SD 0.025 0.028 0.048 0.096 0.035 0.066 0.109 0.151 0.193
CP(90%) 0.910 0.932 0.914 0.889 0.899 0.892 0.916 0.915 0.899
CP(95%) 0.960 0.968 0.959 0.948 0.942 0.953 0.958 0.960 0.960
5000 Bias 0.001 0.001 0.001 0.001 0.002 0.000 0.001 0.003 0.006
RMSE 0.017 0.018 0.032 0.061 0.021 0.039 0.073 0.099 0.122
SD 0.017 0.018 0.032 0.061 0.02 0.039 0.074 0.099 0.122
CP(90%) 0.906 0.904 0.903 0.908 0.904 0.917 0.896 0.902 0.901
CP(95%) 0.960 0.949 0.951 0.953 0.958 0.962 0.944 0.952 0.95
(continued)
10 Statistical Methods in Medical Research 0(0)

Table 1. Continued
GTDL model GTDL frailty model
a
0
a
1
kb
1
a
0
a
1
kb
1
h
n0.6 0.4 1 2.4 0.36 0.24 1.4 2.7 1.1
10,000 Bias 0.000 0.000 0.000 0.002 0.000 0.001 0.001 0.001 0.000
RMSE 0.012 0.014 0.023 0.043 0.015 0.029 0.052 0.069 0.087
SD 0.012 0.014 0.023 0.043 0.015 0.029 0.052 0.069 0.087
CP(90%) 0.886 0.894 0.889 0.900 0.900 0.890 0.895 0.903 0.901
CP(95%) 0.942 0.944 0.936 0.952 0.952 0.950 0.954 0.950 0.958
Sample size
RMSE of cured fraction
50 100 300 500 1000 2000 5000 10000
0.00 0.02 0.04 0.06 0.08 0.10
Group 0
GTDL frailty model
Sample size
RMSE of cure fraction
50 100 300 500 1000 2000 5000 10000
0.00 0.02 0.04 0.06 0.08
Group 1
GTDL frailty model
Sample size
RMSE of cure fraction
50 100 300 500 1000 2000 5000 10000
0.00 0.02 0.04 0.06 0.08 0.10
Group 0
GTDL model
Sample size
RMSE of cure fraction
50 100 300 500 1000 2000 5000 10000
0.00 0.02 0.04 0.06 0.08
Group 1
GTDL model
Figure 3. RMSEs of MLEs of the cure fraction obtained from the GTDL gamma frailty and GTDL models. (For interpretation of the
references to color in this figure legend, the reader is referred to the online version of this article.)
Table 2. Percentage of cases in which estimates of aare less than zero when in fact there is a long-term survivors group.
Sample size
Model Cure rate fixed 50 100 150 200 300 400 500 1000
GTDL 0.20 99.7 100 100 100 100 100 100 100
0.11 92.9 98.8 100 100 100 100 100 100
0.045 83.3 93.0 96.9 99.6 99.8 100 100 100
0.03 79.5 89.2 96.3 98.8 99.5 99 100 100
0.01 71.2 83.9 91.4 96.2 98.6 99.1 99.5 100
GTDL frailty 0.20 99.2 99.9 100 100 100 100 100 100
0.11 93.2 99.3 100 100 100 100 100 100
0.045 62.9 83.9 93.2 96.7 99.4 99.7 100 100
0.03 52.6 73.4 84.2 90.7 96.2 98.0 99.4 100
0.01 30.8 40.3 49.5 56.4 62.2 70.1 73.8 87.8
Calsavara et al. 11

number of parameters in the fitted model, and nis the sample size. Given a set of candidate models, the preferred
model will provide the minimum values.
To evaluate the performance of these information criteria, we generated datasets from the GTDL gamma
frailty and GTDL models (h¼0) using a binary explanatory variable, and fixing the same parameter values and
n¼50;100;300;500 and 1000.
For each simulated dataset, we calculated discrimination criteria from fits of the GTDL gamma frailty model
and the corresponding GTDL model without a frailty term. Based on 1000 datasets for each situation, we
calculated the mean differences in information criterion values between the GTDL and GTDL gamma frailty
models, as well as the observed selection proportions (with the GTDL gamma frailty model preferred) for each of
the five criteria. Figure 4 shows the mean differences and selection proportions for the discrimination criteria. A
positive mean difference means that, on average, the information criterion value from the fitted GTDL gamma
frailty model is smaller than that from the GTDL model, indicating an advantage of the GTDL gamma frailty
model. Note that the mean difference is always positive for n1000 and h0:7, regardless of information cri-
terion. These results show that the information criteria can distinguish between the models in the presence of high
heterogeneity, as the mean differences for the correct model were always greater than those for the incorrect
model. Selection proportions for the correct model were always high, especially when the AIC was used. We will
use the AIC in Section 5 because this criterion presented the greatest mean difference (Figure 4).
4.5 Hypothesis testing
As we are interested in estimating the degree of unobservable heterogeneity in the model including the frailty term,
we assessed whether the inclusion of this term in the GTDL model is necessary using the null hypothesis
H0:h¼0. The statistic used most commonly for this purpose is the likelihood ratio. Asymptotically, this statistic
has the distribution v2
1, but under H
0
the parameter value is on the boundary of the parametric space and
problems can occur when testing the null hypothesis. The likelihood ratio test (LRT) is given by
K¼2f‘ð^
#Þ‘ð^
#0Þg, where ^
#0is the maximum likelihood estimator of ^
#under H
0
. Under certain regularity
conditions, Maller and Zhou
46
showed that the statistical distribution Kis a mixture in proportions 50%=50% of
a chi-squared distribution with one degree of freedom and a point mass at 0, that is P½Kn¼0:5þ0:5P½v2
1n.
To evaluate the performance of the LRT in testing the null hypothesis (equivalent absence of heterogeneity),
datasets were simulated considering different values of h¼f0;0:05;0:1;0:3;0:7;1:1;1:5g;a0¼0:36;
a1¼0:24;k¼1:4;b1¼2:7, and several sample sizes. For each configuration, we calculated the rate of rejec-
tion of the null hypothesis. The size and power of the tests are presented in Table 3. As expected, the H
0
rejection
rate decreases as happroaches 0. This rate increases with sample size and h. If the null hypothesis is false, the
rejection rate increases with the sample size. If the null hypothesis is true, the rejection rate is below 5% signif-
icance for small samples, but converges to the expected rate size as the sample size increases.
5 Application
To illustrate applicability of the proposed model, we consider a real cancer dataset. We fitted the GTDL and
GTDL PVF frailty models and their special models to the dataset and compared them with survival curve
estimates obtained using the Kaplan–Meier estimator.
47
For each fitted model, we provide the MLEs and stan-
dard error estimate, 95% confidence interval estimates for the parameters, and AIC value. Estimates of the
standard error for the cure fraction parameter were obtained using the delta method with first-order Taylor’s
approximation.
5.1 Melanoma cancer data
The melanoma dataset is from a retrospective survey of 7166 records of patients diagnosed with melanoma in the
state of S~
ao Paulo, Brazil, between 2000 and 2014, with follow-up conducted until 2018. It was provided by the
Fundac¸~
ao Oncocentro de S~
ao Paulo (FOSP), which is responsible for coordinating the Hospital Cancer Registry of
the State of S~
ao Paulo. The FOSP is a public institution connected to the State Health Secretariat, which assists in
the preparation and implementation of healthcare policies in the field of Oncology, and serves as an instrument
so that oncology hospitals can prepare their own protocols and improve their care practices.
48
Death due to cancer was defined as the event of interest. The main goal was to assess the impact of surgery on
specific survival. Of the 7166 patients, 6307 underwent surgery and 859 did not. A total of 2067 events occurred
during follow-up period: 1561 (24.75%) occurred among patients who underwent surgery and 506 (58.9%)
12 Statistical Methods in Medical Research 0(0)

Sample size
Mean difference
50 100 300 500 1000
−10 −5 051015
20
θ=0
Sample size
Observed selection proportions based on information criteria (%)
50 100 300 500 1000
0 20 40 60 80 100
θ=0
Sample size
Mean difference
50 100 300 500 1000
−10 −5 0 5 10 15 20
θ=0.3
Sample size
Observed selection proportions based on information criteria (%)
50 100 300 500 1000
0 20 40 60 80 100
θ=0.3
Sample size
Mean difference
50 100 300 500 1000
−10 −5 0 5 10 15 20
θ=0.7
Sample size
Observed selection proportions based on information criteria (%)
50 100 300 500 1000
0 20 40 60 80 100
θ=0.7
Sample size
Mean difference
50 100 300 500 1000
−10 −5 0 5 10 15 20
θ=1.1
Sample size
Observed selection proportions based on information criteria (%)
50 100 300 500 1000
02040
60 80 100
θ=1.1
Figure 4. Top row: Mean differences in information criteria obtained from fitted GTDL gamma frailty and GTDL models, when data were generated from the GTDL gamma frailty
model. Bottom row: Observed selection proportions (correct model) based on information criteria. (For interpretation of the references to color in this figure legend, the reader is
referred to the version of this article.)
Calsavara et al. 13

occurred among those who did not undergo surgery. The maximum observation time was approximately
18.54 years and the median follow-up time was 5.24 years.
The staging system proposed by the American Joint Committee on Cancer (AJCC) is commonly used world-
wide for several solid tumors, including melanoma. According to the latest edition,
49
clinical stages I and II
correspond to the melanoma limited to the skin, which is associated to a better prognosis. These patients are
normally treated with surgery and the great majority will be alive after 10 years of follow-up. Clinical stage III
corresponds to nodal spreading of the melanoma, and in this scenario surgery is routinely associated to radio-
therapy and/or some modality of systemic treatment such as immunotherapy or targeted therapies.
50
In the
literature, the melanoma specific survival after 10 years in these patients may vary from 24% to 88%.
50
Clinical stage IV corresponds to metastatic disease, which carries the worst prognostic.
49
Even though several
new modalities of treatment have been reported in the latest years, treating these patients is still challenging.
51
Some of these patients may undergo surgery at some time, but it is more likely that they will need systemic
treatment.
52
In our study, from available information about stage clinical and surgery, around 75% of the patients
who underwent surgery were in stage clinical I and II, while 65% of the patients who did not undergo surgery were
in stage clinical III and IV.
Figure 5 shows a plot of log cumulative baseline hazard rates against time (follow-up period) for the “treatment
received” variable. According to Klein and Moeschberger,
6
if the proportionality assumption holds, then these
curves should be approximately parallel, with constant vertical separation between them. This plot suggests that
the hazard are non-proportional. In particular the proportional hazards model is questionable before five years.
Table 3. Rates of rejection of the null hypothesis (absence of unobservable heterogeneity) at 5% nominal significance level for several
sample sizes and degrees of unobservable heterogeneity.
Sample size
h50 100 300 500 1000 2000 5000 10000
0 0.033 0.049 0.052 0.041 0.042 0.038 0.049 0.049
0.05 0.022 0.036 0.015 0.033 0.016 0.038 0.101 0.233
0.1 0.029 0.056 0.057 0.085 0.098 0.173 0.404 0.608
0.3 0.051 0.109 0.168 0.294 0.483 0.732 0.973 0.998
0.7 0.092 0.200 0.453 0.665 0.899 0.995 1 1
1.1 0.159 0.257 0.666 0.852 0.992 1 1 1
1.5 0.194 0.322 0.733 0.924 0.999 1 1 1
0 5 10 15 20
−1.0 −0.8 −0.6 −0.4 −0.2
Time (years)
log(Cumulative hazard function)
Treatment received
No surgery
Surgery
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−6
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0
0.24 0.69 1.3 1.9 2.9 4.3 6.5 10
Time (years)
Beta(t) for x
Figure 5. Left panel: Plot of log cumulative baseline hazard rates versus time on study for the treatment received variable. Right
panel: Standardized Schoenfeld residuals þ^
bfor the covariate Treatment received plotted from Cox model fitted.
14 Statistical Methods in Medical Research 0(0)

Figure 5 also shows a plot of standardized Schoenfeld residuals against time for this covariate. The results of
proportional hazards assumption testing for a Cox regression model fit
5
are displayed in Table 4; they provided
strong evidence that this variable had a non-constant effect over time.
To evaluate the effect of the surgery in the lifetime, we fitted the GTDL and GTDL PVF frailty models to
the dataset. For illustrative purposes, we link parameter ato treatment received through an identity link
function. Thus
aðxiÞ¼a0þxia1
where x
i
is a group variable, where x
i
¼1 and 0 indicate patients undergoing and not undergoing surgery, respec-
tively for i¼1;...;7166; and a>¼ða0;a1Þrepresents the regression coefficients. The results of the fitted GTDL
and GTDL PVF frailty models are given in Table 5. Notice that the estimate of cis close to zero indicating that a
GTDL gamma frailty model can be considered. In this sense, we also fitted the main special cases, GTDL inverse
Gaussian (c¼0:5) and gamma (c!0) frailty models. According to the AIC value, the GTDL gamma frailty
model seems to be the better choice among the four models.
The results suggest a significant effect of treatment in the lifetime, regardless of model, as the 95% confidence
interval the b
1
does not include 0. In addition, the measure of the time effect differs between groups (a
0
and a
1
are
significant). Note that ^
a0<0 and ^
a0þ^
a1<0 in the four models, which means that the distributions were
Table 4. Test of proportional hazards assumption.
Variable qv
2p-value
Treatment received 0.282 156 <0.0001
Table 5. Maximum likelihood estimates (MLEs), standard errors (SEs), 95% asymptotic confidence intervals (CIs), maximum of the
log-likelihood function ½max ‘ð#Þ, and AIC values obtained by fitting the GTDL and GTDL frailty models to the melanoma dataset.
Model GTDL model GTDL PVF frailty model
CI (95%) CI (95%)
Parameter MLE SE Lower Upper MLE SE Lower Upper
h– – – – 1.033 0.284 0.476 1.590
c– – – – 0.020 0.466 0.001 0.933
k1.058 0.061 0.939 1.178 1.345 0.139 1.072 1.618
a
0
0.607 0.037 0.679 0.535 0.380 0.106 0.588 0.173
a
1
0.430 0.039 0.354 0.505 0.252 0.092 0.072 0.433
b
1
2.432 0.074 2.576 2.287 2.644 0.113 2.865 2.423
p
0
0.299 0.018 0.263 0.334 0.293 0.020 0.254 0.333
p
1
0.605 0.013 0.579 0.630 0.583 0.019 0.547 0.620
max ‘ð#Þ7161.299 7150.338
AIC 14330.598 14312.676
Model GTDL Inv. Gaussian model GTDL Gamma frailty model
h0.823 0.286 0.268 1.388 1.153 0.244 0.675 1.632
k1.260 0.104 1.055 1.465 1.418 0.126 1.171 1.666
a
0
0.476 0.047 0.567 0.384 0.358 0.061 0.477 0.239
a
1
0.333 0.043 0.249 0.416 0.237 0.054 0.132 0.341
b
1
2.583 0.091 2.762 2.404 2.701 0.099 2.895 2.507
p
0
0.303 0.019 0.266 0.340 0.290 0.020 0.251 0.329
p
1
0.595 0.016 0.563 0.627 0.579 0.018 0.543 0.616
max ‘ð#Þ7154.150 7150.065
AIC 14318.300 14310.130
Calsavara et al. 15

improper, leading to cure rates in the two groups. The results also show that the estimated long-term survivors in
the four models are similar, as seen in the simulation study.
As mentioned previously, of the four fitted models, the GTDL gamma frailty model gave the best fit according
to the AIC value. Although the GTDL PVF frailty model can be also considered, the difference between AIC
values is small and the parameter estimates are similar.
Taking into account the AIC criterion, max ‘ð) values and number of parameters in the model, we select the
GTDL gamma frailty model as our working model. Note that ^
h¼1:153, which indicates a reasonable degree of
unobserved heterogeneity in the sample. In addition, the estimated time effects from GTDL gamma frailty model
were ^
a0¼0:358; CI(95%Þ¼½0:477;0:239in the no surgery group and ^
a0þ^
a1¼0:121; CI
(95%Þ¼½0:153;0:089in the surgery group. These estimates evidence that the time effect is not the same in
both groups. As the time effects are negative, the model suggests that there are long-term survivors, as can be seen
in the estimated proportions, ^
p0¼0:290 with standard error 0.02 (no surgery) and ^
p1¼0:579 with standard error
0.018 (surgery).
0 5 10 15 20
0.0 0.2 0.4 0.6 0.8 1.0
Time (years)
Survival Function
Treatment received
No surgery
Surgery
Model
GTDL
GTDL frailty
0 5 10 15 20
0.0 0.2 0.4 0.6 0.8
Time (years)
Hazard function
Treatment received
No surgery
Surgery
Model
GTDL
GTDL frailty
Figure 6. Left panel: Estimated survival curve obtained via Kaplan–Meier (black line) for melanoma dataset, and estimated survival
function according to GTDL (red line) and GTDL gamma frailty models (green line). Right panel: Estimated hazard function according
to GTDL model and GTDL gamma frailty model. (For interpretation of the references to color in this figure legend, the reader is
referred to the web version of this article.)
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0.6 0.7 0.8 0.9 1.0
0.6 0.7 0.8 0.9 1.0
Predict values
Kaplan−Meier estimates
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Surgery
●
●
Model
GTDL
GTDL frailty
Figure 7. Plots of the Kaplan–Meier estimates for the survival function versus the respective predicted values obtained from the
GTDL (red points) and GTDL gamma frailty models (green points) stratified by treatment received. (For interpretation of the
references to color in this figure legend, the reader is referred to the web version of this article.)
16 Statistical Methods in Medical Research 0(0)

Overall, the models reasonably fit Kaplan–Meier curves. However, the GTDL frailty model enables the quan-
tification of unobserved heterogeneity, which is of great importance in clinical practice. We thus tested the
suitability of the frailty term in the GTDL model using LRT, as described for the simulation study. We obtained
K¼40:936 (p-value <0.0001), which provides evidence in favor of the inclusion of the frailty term.
Figure 6 shows the estimated survival and hazard functions from the GTDL and GTDL frailty models. In both
models, but more so in the GTDL frailty model, the survival function estimates are close to the Kaplan–Meier
curves. In addition, the hazard function curves are higher for patients who did not undergo surgery, mainly in the
first five years of follow-up, regardless of models. In both models, the fitted hazard functions decrease over time;
the curves also cross over time. Such crossing does not occur in the traditional GTDL model (1), which is a
disadvantage.
Finally, it is compared in Figure 7 the empirical estimated based on the Kaplan–Meier versus the correspond-
ing predicted values obtained from the GTDL and GTDL gamma frailty models. Through this approach, it is
evident that GTDL gamma frailty model provided predicted values closer than GTDL model, regardless of the
type of treatment patient received, which indicates a satisfactory fit of the proposed model. As can be seen in the
plots, some deviations of predictions occurred in relation the Kaplan–Meier estimates (overestimated or under-
estimated), but these differences are acceptable, once the higher difference between predicted values and Kaplan–
Meier estimates was 0.049 in the no surgery group, while in the surgery group it was 0.012. In the medical clinic,
this small difference is acceptable.
6 Concluding remarks
In this paper, we considered the GTDL model with a PVF frailty term for right-censored data with the potential
existence of long-term survivors in the population. An advantage of the studied model over alternatives is that it
does not make assumptions about the existence of the cure rate, once the parameter avalue has led to proper
ða>0Þor improper ða<0Þdistribution; this makes the model flexible and applicable to situations with and
without cure fractions. If parameter ais estimated to be negative, then the cure fraction is computed as a function
of the GTDL model parameters. In addition, the inclusion of a frailty term in the hazard function enables the
quantification of unobserved heterogeneity by means of the parameter h. In our simulation study, conducted to
illustrate the frequent properties of the MLEs of the parameters, the bias and RMSEs appeared to trend reason-
ably close to 0 as the sample size increased. The simulation study showed that the GTDL frailty model is not
indicated for small (n150) samples. In practice, the model is often chosen based on a selection criterion.
Therefore, we evaluated the performance of the GTDL frailty model against that of the GTDL model using
several such criteria. A simulation revealed that, on average, the AIC value from the fitted GTDL model was
smallest and, consequently, that this model performed best. The practical relevance and applicability of the
studied models were demonstrated using a real dataset. Although further research on this approach must be
conducted, our initial results suggest that this model enhances the analysis of non-proportional hazards in the
presence or absence of long-term survivors.
Acknowledgements
The authors thank the Fundac¸ ~
ao Oncocentro de S~
ao Paulo for providing the melanoma dataset. They also thank the two
referees for their comments which greatly improved this paper.
Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this
article.
Funding
The author(s) received no financial support for the research, authorship, and/or publication of this article.
ORCID iD
Vinicius F Calsavara https://orcid.org/0000-0003-2332-5863
Supplemental material
Supplemental material for this article is available online.
Calsavara et al. 17

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