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.
CoxDR.Regression models and life-tables.J Royal Stat Soc B1972;
34: 187–220.
2.
SchemperM.Cox analysis of survival data with non-proportional hazard functions.Statistician1992;
41: 455–465.
3.
SchoenfeldD.Partial residuals for the proportional hazards regression model.Biometrika1982;
69: 239–241.
4.
PettittABin DaudI.Investigating time dependence in Cox’s proportional hazards model.Appl Stat1990;
39: 313–329.
5.
GrambschPMTherneauTM.Proportional hazards tests and diagnostics based on weighted residuals.Biometrika1994;
81: 515–526.
6.
KleinJPMoeschbergerML.Survival analysis: Statistical methods for censored and truncated data.
New York, NY:
Springer Verlag, 2003.
7.
HessKR.Graphical methods for assessing violations of the proportional hazards assumption in Cox regression.Stat Med1995;
14: 1707–1723.
8.
PrenticeRL.Linear rank tests with right censored data.Biometrika1978;
65: 167–179.
9.
KalbfleischJDPrenticeRL.The statistical analysis of failure time data.
Hoboken, NJ:
John Wiley & Sons, 2011.
10.
Etezadi-AmoliJCiampiA.Extended hazard regression for censored survival data with covariates: a spline approximation for the baseline hazard function.Biometrics1987;
43: 181–192.
11.
Louzada-NetoF.Extended hazard regression model for reliability and survival analysis.Lifetime Data Analysis1997;
3: 367–381.
12.
Louzada-NetoF.Polyhazard models for lifetime data.Biometrics1999;
55: 1281–1285.
13.
MackenzieG.Regression models for survival data: the generalized time-dependent logistic family.Statistician1996;
45: 21–34.
14.
Louzada-NetoFCremascoCPMacKenzieG.Sampling-based inference for the generalized time-dependent logistic hazard model.J Stat Theory Appl2010;
9: 169–184.
15.
MilaniEATomazellaVLDiasTC, et al.
The generalized time-dependent logistic frailty model: an application to a population-based prospective study of incident cases of lung cancer diagnosed in Northern Ireland.Brazil J Probabil Stat2015;
29: 132–144.
16.
BoagJW.Maximum likelihood estimates of the proportion of patients cured by cancer therapy.J Royal Stat Soc B1949;
11: 15–53.
17.
BerksonJGageRP.Survival curve for cancer patients following treatment.J Am Stat Assoc1952;
47: 501–515.
18.
HougaardP.Modelling heterogeneity in survival data.J Appl Probabil1991;
28: 695–701.
19.
VaupelJWMantonKGStallardE.The impact of heterogeneity in individual frailty on the dynamics of mortality.Demography1979;
16: 439–454.
20.
ClaytonD.A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence.Biometrika1978;
65: 141–151.
21.
AndersenP.Statistical models based on counting processes.
New York, NY:
Springer Verlag, 1993.
22.
HougaardP.Frailty models for survival data.Lifetime Data Analysis1995;
1: 255–273.
23.
SinhaDDeyD.Semiparametric Bayesian analysis of survival data.J Am Stat Assoc1997;
92: 1195–1212.
24.
OakesD.A model for association in bivariate survival data.J Royal Stat Soc B1982;
44: 414–422.
25.
AalenOO.Heterogeneity in survival analysis.Stat Med1988;
7: 1121–1137.
26.
HougaardPMyglegaardPBorch-JohnsenK.Heterogeneity models of disease susceptibility, with application to diabetic nephropathy.Biometrics1994;
50: 1178–1188.
27.
PriceDLManatungaAK.Modelling survival data with a cured fraction using frailty models.Stat Med2001;
20: 1515–1527.
28.
PengYTaylorJMGYuB.A marginal regression model for multivariate failure time data with a surviving fraction.Lifetime Data Analysis2007;
13: 351–369.
29.
YuBPengY.Mixture cure models for multivariate survival data.Computat Stat Data Analys2008;
52: 1524–1532.
30.
CalsavaraVFTomazellaVLDFogoJC.The effect of frailty term in the standard mixture model. Chilean J Stat2013;
4: 95–109.
31.
CalsavaraVFRodriguesASTomazellaVLD, et al.
Frailty models power variance function with cure fraction and latent risk factors negative binomial.Commun Stat-Theory Meth2017;
46: 9763–9776.
32.
BalkaJDesmondAFMcNicholasPD.Review and implementation of cure models based on first hitting times for Wiener processes.Lifetime Data Analysis2009;
15: 147–176.
33.
BalkaJDesmondAFMcNicholasPD.Bayesian and likelihood inference for cure rates based on defective inverse Gaussian regression models.J Appl Stat2011;
38: 127–144.
34.
RochaRNadarajahSTomazellaV, et al.
Two new defective distributions based on the Marshall–Olkin extension.Lifetime Data Analysis2016;
22: 216–240.
35.
RochaRNadarajahSTomazellaV, et al.
A new class of defective models based on the Marshall–Olkin family of distributions for cure rate modeling.Computat Stat Data Analys2017;
107: 48–63.
36.
RochaRNadarajahSTomazellaV, et al.
New defective models based on the Kumaraswamy family of distributions with application to cancer data sets.Stat Meth Med Res2017;
26: 1737–1755.
37.
ScudilioJCalsavaraVFRochaR, et al.
Defective models induced by gamma frailty term for survival data with cured fraction.J Appl Stat2019;
46: 484–507.
38.
CalsavaraVFRodriguesASRochaR, et al.
Defective regression models for cure rate modeling with interval-censored data.Biometric J2019; 61: 841--859.
39.
CalsavaraVFRodriguesASRochaR, et al.
Zero-adjusted defective regression models for modeling lifetime data.J Appl Stat2019;
46: 2434–2459.
40.
Coordenação de Prevenção e Vigilância. Instituto Nacional de Cancer José Alencar Gomes da Silva. Estimativa 2018: Incidência de Cancer no Brasil. Coordenação de Prevenção e Vigilância – Rio de Janeiro, http://www1.inca.gov.br/estimativa/2018/ (2017).
41.
ErvikMLamFFerlayJ, et al. Cancer today Lyon, France: International agency for research on cancer. Cancer Today, http://gco iarc fr/today (2016, accessed 01 February 2019).
42.
TweedieMCK. An index which distinguishes between some important exponential families. In: Statistics: applications and new directions. Proc. Indian statistical institute golden jubilee international conference, pp.579–604.
43.
HougaardP.Survival models for heterogeneous populations derived from stable distributions.Biometrika1986;
73: 387–396.
R Core Team.R: A language and environment for statistical computing.
Vienna, Austria:
R Foundation for Statistical Computing, 2018.
46.
MallerRZhouX.Survival analysis with long-term survivors.
Hoboken, NJ:
John Wiley & Sons, 1996.
47.
KaplanELMeierP.Nonparametric estimation from incomplete observations.J Am Stat Assoc1958;
53: 457–481.
48.
AndradeCTdMagedanzAMPCBEscobosaDM, et al.
The importance of a database in the management of healthcare services.Einstein (São Paulo)2012;
10: 360–365.
49.
Gershenwald JE, Scolyer RA, Hess KR, et al. Melanoma staging: evidence-based changes in the American joint committee on cancer eighth edition cancer staging manual. CA: A Cancer J Clin 2017; 67: 472--492.
50.
Eggermont AM and Dummer R. The 2017 complete overhaul of adjuvant therapies for high-risk melanoma and its consequences for staging and management of melanoma patients. Eur J Cancer 2017; 86: 101--105.
51.
AsciertoPAFlahertyKGoffS.Emerging strategies in systemic therapy for the treatment of melanoma.Am Soc Clin Oncol Educ Book2018;
38: 751–758.
52.
PuzaCJBresslerESTerandoAM, et al.
The emerging role of surgery for patients with advanced melanoma treated with immunotherapy.J Surg Res2019;
236: 209–215.
Supplementary Material
Please find the following supplemental material available below.
For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.
For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.
