## Brazilian Journal of Probability and Statistics

### Long-term survival models with latent activation under a flexible family of distributions

#### Abstract

In this paper, we propose a new cure rate survival model under a flexible family of distributions. Our approach enables different underlying activation mechanisms that lead to the event of interest. The number of competing causes of the event of interest follows a power series distribution. This model includes the standard mixture cure model and the promotion time cure model. The model is parametrized in terms of the cured fraction, which is then linked to covariates. We carried out a simulation study to assess some properties of our proposal. An illustrative example with a real data set is provided to illustrate the models.

#### Article information

Source
Braz. J. Probab. Stat., Volume 27, Number 4 (2013), 585-600.

Dates
First available in Project Euclid: 9 September 2013

https://projecteuclid.org/euclid.bjps/1378729989

Digital Object Identifier
doi:10.1214/12-BJPS186

Mathematical Reviews number (MathSciNet)
MR3105045

Zentralblatt MATH identifier
1298.62165

#### Citation

Cancho, Vicente G.; de Castro, Mário; Dey, Dipak K. Long-term survival models with latent activation under a flexible family of distributions. Braz. J. Probab. Stat. 27 (2013), no. 4, 585--600. doi:10.1214/12-BJPS186. https://projecteuclid.org/euclid.bjps/1378729989

#### References

• Berkson, J. and Gage, R. P. (1952). Survival curve for cancer patients following treatment. Journal of the American Statistical Association 47, 501–515.
• Boag, J. W. (1949). Maximum likelihood estimates of the proportion of patients cured by cancer therapy. Journal of the Royal Statistical Society, Ser. B 11, 15–53.
• Cancho, V. G., de Castro, M. and Dey, D. K. (2012). Supplement to “Long-term survival models with latent activation under a flexible family of distributions.” DOI:10.1214/12-BJPS186SUPP.
• Cancho, V. G., Louzada-Neto, F. and Barriga, G. D. C. (2011). The Poisson-exponential lifetime distribution. Computational Statistics & Data Analysis 55, 677–686.
• Chahkandi, M. and Ganjali, M. (2009). On some lifetime distributions with decreasing failure rate. Computational Statistics & Data Analysis 53, 4433–4440.
• Chen, M.-H., Ibrahim, J. G. and Sinha, D. (1999). A new Bayesian model for survival data with a surviving fraction. Journal of the American Statistical Association 94, 909–919.
• Cooner, F., Banerjee, S. and McBean, A. M. (2006). Modelling geographically referenced survival data with a cure fraction. Statistical Methods in Medical Research 15, 307–324.
• Cooner, F., Banerjee, S., Carlin, B. P. and Sinha, D. (2007). Flexible cure rate modeling under latent activation schemes. Journal of the American Statistical Association 102, 560–572.
• Corless, R. M., Gonnet, G. H., Hare, D. E. G., Jeffrey, D. J. and Knuth, D. E. (1996). On the Lambert $W$ function. Advances in Computational Mathematics 5, 329–359.
• de Castro, M., Cancho, V. G. and Rodrigues, J. (2009). A Bayesian long-term survival model parametrized in the cured fraction. Biometrical Journal 51, 443–455.
• Dunn, P. K. and Smyth, G. K. (1996). Randomized quantile residuals. Journal of Computational and Graphical Statistics 5, 236–244.
• Flores D., J., Borges, P., Cancho, V. G. and Louzada, F. (2013). The complementary exponential power series distribution. Brazilian Journal of Probability and Statistics 27, 565–584.
• Ibrahim, J. G., Chen, M.-H. and Sinha, D. (2001). Bayesian Survival Analysis. New York: Springer.
• Johnson, N. L., Kemp, A. W. and Kotz, S. (2005). Univariate Discrete Distributions, 3rd ed. Hoboken: Wiley.
• Kim, S., Chen, M.-H. and Dey, D. K. (2011). A new threshold regression model for survival data with a cure fraction. Lifetime Data Analysis 17, 101–122.
• Klein, J. P. and Moeschberger, M. L. (2003). Survival Analysis, 2nd ed. New York: Springer.
• Kuk, A. Y. C. and Chen, C. H. (1992). A mixture model combining logistic regression with proportional hazards regression. Biometrika 79, 531–541.
• Li, C. S., Taylor, J. M. and Sy, J. P. (2001). Identifiability of cure models. Statistics and Probability Letters 54, 389–395.
• Louzada-Neto, F., Roman, M. and Cancho, V. G. (2011). The complementary exponential geometric distribution: Model, properties and a comparison with its counterpart. Computational Statistics & Data Analysis 55, 2516–2524.
• Maller, R. A. and Zhou, X. (1996). Survival Analysis with Long-Term Survivors. New York: Wiley.
• Marshall, A. W. and Olkin, I. (1997). A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika 84, 641–652.
• Morais, A. L. and Barreto-Souza, W. (2011). A compound class of Weibull and power series distributions. Computational Statistics & Data Analysis 55, 1410–1425.
• R Development Core Team (2011). R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing.
• Rigby, R. A. and Stasinopoulos, D. M. (2005). Generalized additive models for location, scale and shape (with discussion). Applied Statistics 54, 507–554.
• Rodrigues, J., de Castro, M., Cancho, V. G. and Balakrishnan, N. (2009a). COM–Poisson cure rate survival models and an application to a cutaneous melanoma data. Journal of Statistical Planning and Inference 139, 3605–3611.
• Rodrigues, J., Cancho, V. G., de Castro, M. and Louzada-Neto, F. (2009b). On the unification of the long-term survival models. Statistics & Probability Letters 79, 753–759.
• Rodrigues, J., Balakrishnan, N., Cordeiro, G. M. and de Castro, M. (2011). A unified view on lifetime distributions arising from selection mechanisms. Computational Statistics & Data Analysis 55, 3311–3319.
• Scheike, T. (2009). timereg package. R package version 1.1-0. With contributions from T. Martinussen and J. Silver. R package version 1.1-6.
• Sy, J. P. and Taylor, J. M. G. (2000). Estimation in a Cox proportional hazards cure model. Biometrics 56, 227–236.
• Tournoud, M. and Ecochard, R. (2007). Application of the promotion time cure model with time-changing exposure to the study of HIV/AIDS and other infectious diseases. Statistics in Medicine 26, 1008–1021.
• Tsodikov, A. D., Ibrahim, J. G. and Yakovlev, A. Y. (2003). Estimating cure rates from survival data: An alternative to two-component mixture models. Journal of the American Statistical Association 98, 1063–1078.
• Yakovlev, A. Y. and Tsodikov, A. D. (1996). Stochastic Models of Tumor Latency and Their Biostatistical Applications. Singapore: World Scientific.