Abstract
Cure models have been widely used in biomedical research and clinical trials to analyze survival
data, when the study population is a mixture of susceptible and non-susceptible individuals.
These models assume that individuals experiencing the event of interest are homogeneous in
risk. However, there remains a degree of heterogeneity induced by unobservable risk factors,
which may lead to distorted results. In order to model that unobserved heterogeneity in risk
among susceptible in addition to incorporation of a cured component, frailty cure models can be
the most appropriate choice. This article aims to estimate the cure fraction with frailty along with
the investigation of prognostic factors that influence the survival of cancer patients. A
retrospective data of 285 melanoma cancer patients is analyzed without frailty to Cox PH and
AFT models and with frailty to only AFT model. Model selection is ascertained using Akaike
Information Criterion. Estimates are obtained by using STATA software.
Keywords: cure frailty model, AFTM, Cox PH model, melanoma cancer, univariate frailty
models, frailty models
- Introduction
In clinical and epidemiological studies, the interest generally focuses on studying the effect of
concomitant information on the time to an event such as death or recurrence of a disease. But
now days with the development of advanced treatment technologies, more and more fatal
2
diseases are now curable. Therefore, in some clinical studies, a substantial proportion of patients
may never experience the event because the treatment has effectively cured the patients. We
refer to these subjects who never experience the event as cured (non-susceptible) and the
remaining subjects as uncured (susceptible) in the data. The main interest in such cases is to
estimate the proportion of cured patients, the failure time distribution of uncured patients, and the
possible effects of covariates, which can be achieved by using cure models that have historically
been utilized to analyze time to event data with a cured fraction. These models are becoming
very popular in clinical trials and medical research, especially in oncological studies, such as
breast cancer, leukemia and melanoma. In situations of existence of cured fraction in the
population, standard survival analysis techniques such as Cox proportional hazard model may
not be appropriate because they assume that all patients will eventually experience the event of
interest, given that the follow-up time is long enough. When the PH assumption of Cox model
does not satisfy, the AFT model is a useful alternative which regress the logarithm of survival
time over covariates. One advantage of the AFT model over the PH model is that, it measures the
effect of covariates directly on survival time instead of hazard. Thus, the interpretation of
covariate effects in the AFT model is much simpler than in the PH model. Although, both Cox
PH and AFT models measures the effect of explanatory variables on survival probabilities but
neither of them takes into account the unmeasured variability among patients beyond that of
measured covariates. Failing to account for this unmeasured variability may lead to distorted
results. Models which take into account the unobserved heterogeneity between individuals are
known as frailty models. Vaupel et al. (1979) introduced univariate frailty models (with a gamma
distribution) into survival analysis to account for unobserved heterogeneity or missing covariates
in the study population. Clayton (1978) promoted the model by its application to multivariate
3
situation on chronic disease incidence in families. These models are formulated on the idea that
different patients possesses different frailties and patients more ”frail” or ”prone” tend to have
the event earlier than those who are less frail. Several authors proposed various frailty models,
according to the distribution of frailties and estimation procedure. Hougaard (1986) considered a
parametric family of distributions for frailty and showed that it has Gamma and inverse Gaussian
distributions as special cases. Grover et al. (2014) applied frailty models to investigate the effect
of significant covariates on the survival times of advance liver disease patients. Govindarajulu et
al.(2011) has applied the methodology to choose between frailty and no-frailty models in
assessing genetic variability and found sex and birth year as significant covariates. However,
standard frailty models do not account for a cured proportion. This paper deals with the
extension of cure models in order to allow for heterogeneity among the cure fraction under risk
by using frailty cure models. An attractive feature of these models is the allowance for
heterogeneity in risk among those individuals experiencing the event of interest in addition to the
incorporation of a cured component. Price and Manatunga(2001) gave a good introduction to this
area by applying leukemia remission data to standard cure, standard frailty and frailty cure
models and found that gamma frailty mixture model provide a better fit to the data as compared
to standard cure model. Yu(2008) proposed a frailty mixture cure model for hospital readmission
data. A comparison of frailty cure model for recurrent event data with a flexible cured fraction
had been shown by Rondeau et. al.(2011). Peng and Zhang(2008) had presented a generalization
of mixture cure model by adding a frailty term to latency part. In this paper, our interest will
focus on estimating the cure fraction and frailty along with the identification of significant
predictors that contribute for the prevalence of melanoma cancer.
- Materials and Methods:
4
2.1 Cure frailty model
Cure models: In these models, population is divided into two sub-populations: cured or immunes
and uncured or susceptible. These are generally used to analyze survival data with long-term
survivors. Let p be the probability of an individual being either susceptible (0 < p< 1) with
baseline survival function S 0 (t) & (1-p) be the probability of an individual being cured . Then the
survival function at time t can be defined as:
S(t) = (1-p) + p S 0 (t) (1)
The only difference between cure models and standard frailty models is that the former models
assume that some individuals may not experience the event of interest after sufficient follow-up
whereas standard frailty model assumes that all individuals will eventually experience the event
of interest with varying risks greater than zero. Also, the proportion of cured individuals in cure
models is generally considered as individuals having lower frailty in frailty models.
Suppose that there are n patients (i=1, 2,.., n) suffering from melanoma cancer with survival and
hazard functions as S(t) and h(t) respectively. Let w the unobservable frailty effect.
Then the individual hazard & survival function for the j th individual conditional on frailty effect
w is given by:
jjjwthwth*)()/(0
(2)
and
jtStSjjj)()/(
(3)
5
The concept of frailty provides us a suitable way to access unobserved heterogeneity among
patients. Several distributions are there to access the frailty, but the most common frailty
distribution is the Gamma distribution (Wienke, 2011). The Gamma distribution is a very
convenient distribution from an analytical and computational point of view, as it is easy to derive
its closed form expressions of survival and hazard functions. Also, it is a flexible distribution that
takes a variety of shapes as k varies: when k = 1, it is identical to the well-known exponential
distribution; when k is large, it takes a bell-shaped form reminiscent of a normal distribution.
Suppose that w follows gamma distribution with mean 1 and variance θ(Price et al.2001, Grover
et al.2013) , then the density function of w will be,
1
11
1
)(
w
ew
wg
, w>0,θ>0 (4)
Then the population survival function is obtained by integrating out the unobservable w as:
dwwgtStS
w
)()()(
0
=
dwwwtSw
0
1
11
1
)/exp()(
=
dwtSww)}1
)((lnexp{
1
1
0
11
1
6
1
)](ln1[
tS (5)
After replacing the survival function of susceptibles in equation (1) by equation (5) we have:
1
0)](1[)1()(
thpptS
The cure fraction p is associated to covariates by the logistic link
)’exp(1
)’exp(
)(
Zb
Zb
Zp
,where Z is a set of covariates.
We have considered two AFT models viz. Weibull and log-logistic for modeling univariate
frailty. As frailty is based on principle that more frail patients will experience the event earlier
than the less frail patients, so the larger the values of θ, greater are the heterogeneity between
groups and individuals with higher value of w has a high risk of developing the disease. If a
frailty term is not significant in the model, it implies that there is no significant frailty effect and
a frailty model is not considered to be distinctly different from a non frailty model.
2.2 Maximum likelihood estimation
Let t i be the true failure time of individual i(i=1,2,…,n) , c i be the censoring time and d i be the
corresponding censoring indicator. Define y i = min(t i ,c i ) with
)(,0
)(,1
censoredctif
uncensoredctif
d
ii
ii
i
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So, the contribution to the likelihood for a sample of n observations is
n
i
d
i
d
inn
ii
tStfyyyL
1
1
21))(())((),…,,(
where the baseline survivor function is assumed to follow Weibull distribution with shape
parameter γ and scale parameter α, so that,
iid
i
d
n
i
inntppttyyyL
11
1
1
21))1()1(())exp((),…,,(
Model selection is ascertained using Akaike’s information criterion (AIC)(Akaike, 1974). Lower
AIC value gives the best model fit.
- Results
We apply cure frailty models to the melanoma data from the Eastern Cooperative Oncology
Group (ECOG) phase III clinical trial e1684 (Kirkwood et al.1996, Chen et al.1999, Chen et
al.2002, Corbierre et al.2007). In this study, a total of 287 patients with high-risk melanoma were
accrued between 1984 and 1990, out of which 87(31%) are censored and 167(69%) experience
the event. Three covariates treatment (0=control,1=treatment), gender (0=male,1=female) and
age (continuous variable which is centered to the mean) are included in the analyses.
In addition to Cox proportional hazard model, Weibull AFTM and loglogistic AFTM are
considered with and without frailty for examining the effects of various predictors on the
survival of patients suffering from melanoma cancer. The frailty models are considered for
measuring the unobserved heterogeneity among patients. Modeling a frailty effect is not only a
function of unobserved heterogeneity but also of observed covariates. In all the considered
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models, 5% is the level of significance. The discussed models have been implemented in
statistical software STATA 11.0. The proportionality assumption has been checked by using the
graphical and Scoen-feld methods. Figure 1, presents the proportionality assumption graphs for
the covariates: treatment and sex both of which satisfy Cox PH assumption. Table 1 presents the
results of schoen-feld method, according to which the global significance (all covariates
combined) is 0.39, implying that the PH assumption should be accepted. We also tested each
coefficient separately: rho varied from 0 to 0.1, and the p-values varied from 0.1 to 0.9.
Table1: Schoen-Feld Test
Variables rho chi2 Df Prob>chi2
treatment 0.0001 0 1 0.9988
age
0.1228
9 2.7 1 0.1003
sex
0.0320
8 0.2 1 0.6529
global test 3.01 3 0.3907
-4-202
ln(analysis time)
treatment = 0treatment = 1
-4-202
ln(analysis time)
sex = 0sex = 1
Figure1: Proportionality assumption test graphs for treatment and sex
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Prognostic
factors
Model1(Without frailty) Model2(Univariate Frailty)(gamma)
Cox
AFTM AFTM
Weibull Log-logistic Weibull Log-logistic
Coef
f HR
p-
value
Coef
f TR
p-
value
Coef
f TR
p-
value
Coef
f TR p-value
Coef
f TR
p-
value
Treatment -0.16 0.70 0.01* 0.64 1.90 0.01* 0.70 2.02 0.01* 0.53 1.71 0.01* 0.56 1.76 0.00*
Age 0.00 1.01 0.35 -0.01 0.99 0.20 -0.01 0.99 -0.43 -0.01 0.99 0.48 0.00 1.00 0.97
Sex -0.01 0.98 0.91 0.00 1.00 1.00 0.01 1.01 0.97 -0.06 0.94 0.77 0.02 1.02 0.93
Theta 7.74 1.73
LL 2024.83 1035.38 998.55 946.35 949.90
AIC 2030.83 1041.38 1004.55 952.35 955.90
Cure
fraction(1-p) 0.54 0.34 0.33 0.38 0.36
Table2. Comparison of coefficient and corresponding hazard/time ratios of various survival models
with and without frailty
*Significant at 5%
Table2 presents the estimated hazard ratio for Cox PH model without frailty and corresponding
time ratio for Weibull and log logistic AFT models with and without frailty. From the results of
table2, the treatment effect is found to be significant in both Cox PH and AFT models. Patients
with IFN treatment have increasing survival time by a factor of 70% as compared to those
without receiving the treatment. The time ratio for treatment under Weibull models decreases by
20% when frailties are included in the model as compared to models with no frailties. Also,
according to AIC criterion, the log-logistic AFTM is found to be the best among all models
under no frailty effect.
From the univariate unshared frailty comparison of Weibull AFT and loglogistic AFT models,
we noted that treatment is also significant across these models. The value of theta is found to be
highest in Weibull AFTM in model 2, concluding frailty to be significant. According to AIC
value, Weibull AFTM is found to be the best among all other competing models under unshared
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frailty effect. The cure rate is 0.38 in Weibull AFTM and 0.36 in log-logistic AFTM with frailty
which is lesser as compared to models without frailty, confirming that Weibull AFTM with
gamma frailty is the best amongst all the distributions in estimating cure rate and frailty among
patients.
- Discussion
In this paper we have used frailty models as an alternative approach for modeling survival data
with a cure fraction. Ordinary survival models are based on the assumption that studied
population is homogeneous in nature. But in medical studies, individuals differ greatly due to the
influence of various explanatory variables which leads to one of the most important sources of
variability in biological and medical applications. In a heterogeneous population, the individual
hazard may rise over time while the population hazard can fail. It virtually ensures that
population hazard decline over time regardless of the shape of the hazards face by individuals
separately (Cleves et. al.2008). With the passage of time, the covariates become less of a factor
and frailty becomes more of a factor in determining the chance of being disease free.
We have specifically shown the importance and utility of frailty models in the data of melanoma
cancer patients with a cured fraction. The significance of predictors has been tested with and
without incorporating frailty to Cox PH and AFTM models and with frailty to only AFT models.
Two distributions Weibull and loglogistic had been considered in AFTM under both the cases:
with and without frailty. The IFN treatment is found to be significant across all the considered
models viz. no frailty and unshared frailty, which match with the findings of Kirkwood et
al.2001. Moreover, there is a decrease of at least 20% in hazard ratios of treatment when frailties
are included in model as compared to the model with no frailties. Similarly, the time ratio of men
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versus women decreases by 6% when frailties are included in the model. Based on AIC criterion
Weibull AFTM with frailty is found to be the best among all the models. Moreover, the cure
fraction is also found to be the largest in Weibull AFTM as compared to loglogistic AFTM. The
cure rate is found to be 0.38 in Weibull AFTM with frailty as compared to 0.34 in Weibull
AFTM without frailty implying an increase of 4% in the cure rate with frailty. These findings
reveal that gamma frailty mixture model provide a better fit to the data as compared to standard
cure model which match with the findings of Price et.al.(2001). The results of frailty models
have confirmed much of what is known about the natural course and factors affecting melanoma
cancer. However, by using these models we have learnt much about how different prognostic
factors explain unobserved heterogeneity which differs at individual level.
In summary, our analysis suggests that frailty models can be useful in analyzing survival models
with a cure fraction. We have shown that Weibull AFT model with gamma frailty provides a
better fit to the melanoma data as compared to standard cure model. This study supports the fact
that all individuals are at risk to develop cancer but with heterogeneity in the risk. This model
can also be used to compare and evaluate short and long-term effects of different medical
treatments on the recurrence pattern of cancer patients.