The defective generalized Gompertz distribution and its use in the analysis of lifetime data in presence of cure fraction, censored data and covariates


Abstract


Survival analysis methods are widely used in studies where the variable of interest is related to the time until the occurrence of an event. The usual methods assume that all individuals under study are subject to this event, but there are practical situations where this assumption is unrealistic. In some cases it is possible that a percentage of individuals are immune to the event of interest or, especially in cancer clinical trials, they were cured from their disease after a given treatment. In the literature, this percentage is usually referred as "cure fraction". In the present paper, we have proposed a model based on a modification of the generalized Gompertz distribution introduced by El-Gohary et al. (2013) to account for the presence of a cure fraction. We also considered the presence of censored data and covariates. Maximum likelihood and Bayesian methods for estimation of the model parameters are presented. A simulation study is provided to evaluate the performance of the maximum likelihood method in estimating parameters. In the Bayesian analysis, posterior distributions of the parameters are estimated using the Markov chain Monte Carlo (MCMC) method. An example involving a real data set is presented.

Keywords: Survival analysis; Gompertz distribution; censored data; maximum likelihood estimation; Bayesian inference; defective distributions

References


Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In Proceedings of the 2nd International Symposium on Information Theory, pages 267-281.

Balka, J., Desmond, A. F. and McNicholas, P. D. (2011). Bayesian and likelihood inference for cure rates based on defective inverse Gaussian regression models. Journal of Applied Statistics, 38(1):127-144.

Boag, J. W. (1949). Maximum likelihood estimates of the proportion of patients cured by cancer therapy. Journal of the Royal Statistical Society. Series B (Methodological), 11(1):15-53.

Cancho, V. G. and Bolfarine, H. (2001). Modeling the presence of immunes by using the exponentiated-Weibull model. Journal of Applied Statistics, 28(6):659-671.

Cantor, A. B., Shuster, J. J. (1992) Parametric versus nonparametric methods for estimating cure rates based on censored survival data. Statistics in Medicine, 11(7):931-937.

El-Gohary, A., Alshamrani, A. and Al-Otaibi, A. N. (2013). The generalized Gompertz distribution. Applied Mathematical Modelling, 37(1):13-24.

Farewell, V. T. (1982). The use of mixture models for the analysis of survival data with long-term survivors. Biometrics, 38(4):1041-1046.

Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A. and Rubin, D. B. (2013). Bayesian Data Analysis, 3 edn. Chapman and Hall/CRC.

Ghitany, M. E., Maller, R. A. and Zhou, S. (1994). Exponential mixture models with long-term survivors and covariates. Journal of Multivariate Analysis, 49(2):218-241.

Gieser, P. W., Chang, M. N., Rao, P. V, Shuster, J. J. and Pullen, J. (2014). Modelling cure rates using the Gompertz model with covariate information. Statistics in Medicine, 17(8):831-839.

Henningsen, A. and Toomet, O. (2011). maxLik: A package for maximum likelihood estimation in R. Computational Statistics, 26(3):443-458.

Kalbeisch, J. D., Prentice, R. L. (1980) The statistical analysis of failure time data, John Wiley & Sons

Lambert, P. C., Thompson, J. R., Weston, C. L. and Dickman, P. W. (2007). Estimating and modeling the cure fraction in population-based cancer survival analysis. Biostatistics, 8(3):576-594.

Lunn, D. J., Thomas, A., Best, N. and Spiegelhalter, D. (2000). WinBUGS-a Bayesian modelling framework: concepts, structure, and extensibility. Statistics and Computing, 10(4):325-337.

Morbiducci, M., Nardi, A. and Rossi, C. (2003). Classification of "cured" individuals in survival analysis: the mixture approach to the diagnostic prognostic problem. Computational Statistics & Data Analysis, 41(3):515-529.

Oehlert, G. W. (1992). A note on the delta method. The American Statistician, 46(1):27-29.

Rocha, R., Nadarajah, S., Tomazella, V., Louzada, F. and Eudes, A. (2015). New defective models based on the Kumaraswamy family of distributions with application to cancer data sets. Statistical Methods in Medical Research, 1-23.

Rocha, R., Nadarajah, S., Tomazella, V. and Louzada, F. (2017). A new class of defective models based on the Marshall-Olkin family of distributions for cure rate modeling. Computational Statistics & Data Analysis, 107: 48-63.

Rocha, R. F., Tomazella, V. L. D. and Louzada, F. (2014). Bayesian and classic inference for the Defective Gompertz Cure Rate Model. Revista Brasileira de Biometria, 32(1):104-114.

Santos, M. R., Achcar, J. A. and Martinez, E. Z. (2017). Bayesian and maximum likelihood inference for the defective Gompertz cure rate model with covariates: an application to the cervical carcinoma study. Ciencia e Natura, in press.

Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and van der Linde, A. (2014). The deviance information criterion: 12 years on (with discussion). Journal of the Royal Statistical Society: Series B (Statistical Methodology), 76(3):485-493.


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