[en] Cure survival models are used when we desire to acknowledge explicitly that an unknown proportion of the population studied will never experience the event of interest. An extension of the promotion time cure model enabling the inclusion of time-varying covariates as regressors when modelling (simultaneously) the probability and the timing of the monitored event is presented. Our proposal enables us to handle non-monotone population hazard functions without a specific parametric assumption on the baseline hazard. This extension is motivated by and illustrated on data from the German Socio-Economic Panel by studying the transition to second and third births in West Germany.
Disciplines :
Mathematics
Author, co-author :
Lambert, Philippe ; Université de Liège - ULiège > Faculté des sciences sociales > Méthodes quantitatives en sciences sociales
Bremhorst, Vincent; Université catholique de Louvain, Louvain-la-Neuve, Belgium
Language :
English
Title :
Inclusion of time-varying covariates in cure survival models with an application in fertility studies
Publication date :
2020
Journal title :
Journal of the Royal Statistical Society. Series A, Statistics in Society
Aalen, O. (1992) Modelling heterogeneity in survival analysis by the compound Poisson distribution. Ann. Appl. Probab., 2, 951–972.
Andersen, P., Borgan, Ø., Gill, R. and Keiding, N. (1993) Statistical Models based on Counting Processes. New York: Springer.
Andersen, P. and Gill, R. (1982) Cox's regression model for counting processes: a large sample study. Ann. Statist., 10, 1100–1120.
Atchadé, Y. and Rosenthal, J. (2005) On adaptive Markov chain Monte Carlo algorithms. Bernoulli, 11, 815–828.
Bartus, T., Murinkó, L., Szalma, I. and Szél, B. (2013) The effect of education on second births in Hungary: a test of the time-squeeze, self-selection, and partner-effect hypotheses. Demog. Res., 28, 1–32.
Berkson, J. and Gage, R. (1952) Survival curve for cancer patients following treatment. J. Am. Statist. Ass., 47, 501–515.
Boag, J. (1949) Maximum likelihood estimates of the proportion of patients cured by cancer therapy (with discussion). J. R. Statist. Soc. B, 11, 15–53.
Bremhorst, V., Kreyenfeld, M. and Lambert, P. (2016) Fertility progression in Germany: an analysis using flexible non parametric cure survival models. Demog. Res., 35, 505–534.
Bremhorst, V., Kreyenfeld, M. and Lambert, P. (2019) Nonparametric double additive cure survival models: an application to the estimation of the nonlinear effect of age at first parenthood on fertility progression. Statist. Modllng, 19, 248–275.
Bremhorst, V. and Lambert, P. (2016) Flexible estimation in cure survival models using Bayesian P-splines. Computnl Statist. Data Anal., 93, 270–284.
Brown, E. and Ibrahim, J. (2003) Bayesian approaches to joint cure-rate and longitudinal models with applications to cancer vaccine trials. Biometrics, 59, 686–693.
Chen, M., Ibrahim, J. and Sinha, D. (1999) A new Bayesian model for survival data with a surviving fraction. J. Am. Statist. Ass., 94, 909–919.
Chen, M., Ibrahim, J. and Sinha, D. (2004) A new joint model for longitudinal and survival data with a cure fraction. J. Multiv. Anal., 91, 18–34.
Chi, Y. and Ibrahim, J. (2006) Joint models for multivariate longitudinal and multivariate survival data. Biometrics, 7, 432–445.
Clayton, D. (1978) A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika, 65, 141–151.
Duchateau, L. and Janssen, P. (2008) The Frailty Model. New York: Springer.
Eilers, P. H. C. and Marx, B. D. (1996) Flexible smoothing with B-splines and penalties (with discussion). Statist. Sci., 11, 89–121.
Eilers, P. H. C. and Marx, B. D. (2010) Splines, knots, and penalties. Computnl Statist., 2, 637–653.
Fleming, T. and Harrington, D. (1991) Counting Processes and Survival Analysis. New York: Wiley.
Geweke, J. (1992) Evaluating the accuracy of sampling-based approaches to calculating posterior moments. In Bayesian Statistics 4 (eds J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith). Oxford: Clarendon.
Govindarajulu, U. S., Lin, H., Lunetta, K. L. and D’Agostino, R. B. (2011) Frailty models: applications to biomedical and genetic studies. Statist. Med., 30, 2754–2764.
Gressani, O. and Lambert, P. (2018) Fast Bayesian inference in semi-parametric P-spline cure survival models using Laplace approximations. Computnl Statist. Data Anal., 124, 151–167.
Haario, H., Saksman, E. and Tamminen, J. (2001) An adaptive Metropolis algorithm. Bernoulli, 7, 223–242.
Hougaard, P. (1986) Survival models for heterogeneous populations derived from stable distributions. Biometrika, 73, 387–396.
Hougaard, P. (1995) Frailty models for survival data. Liftim. Data Anal., 1, 255–273.
Jullion, A. and Lambert, P. (2007) Robust specification of the roughness penalty prior distribution in spatially adaptive Bayesian P-splines models. Computnl Statist. Data Anal., 51, 2542–2558.
Kim, S., Xi, Y. and Chen, M. (2009) A new latent cure rate marker model for survival data. Ann. Appl. Statist., 3, 1124–1146.
Kim, S., Zeng, D., Li, Y. and Spiegelman, D. (2013) Joint modelling of longitudinal and cure survival data. J. Statist. Theory Pract., 7, 324–344.
Kreyenfeld, M. (2002) Time squeeze, partner effect or self-selection?: An investigation into the positive effect of women's education on second birth risks in West Germany. Demog. Res., 7, no. 2, 15–48.
Kuk, A. and Chen, M. (1992) A mixture model combining logistic regression with proportional hazards regression. Biometrika, 79, 531–541.
Lambert, P. (2007) Archimedean copula estimation using Bayesian splines smoothing techniques. Computnl Statist. Data Anal., 51, 6307–6320.
Lambert, P. and Bremhorst, V. (2018) Estimation and identification issues in the promotion time cure model when the same covariates enter the cure probability and time-to-event model components. Biometr. J., 61, 275–279.
Lang, S. and Brezger, A. (2004) Bayesian P-splines. J. Computnl Graph. Statist., 13, 183–212.
Li, L. and Lee, J.-H. (2017) A latent promotion time cure rate model using dependent tail-free mixtures. J. R. Statist. Soc. A, 180, 891–905.
Li, C. and Taylor, J. (2002) A semi-parametric accelerated failure time cure model. Statist. Med., 21, 3235–3247.
Liang, K.-Y., Self, S. G., Bandeen-Roche, K. J. and Zeger, S. L. (1995) Some recent developments for regression analysis of multivariate failure time data. Liftim. Data Anal., 1, 403–415.
Liu, H. and Shen, Y. (2009) A semi-parametric accelerated failure time cure model. J. Am. Statist. Ass., 104, 1168–1178.
Lopes, C. and Bolfarine, H. (2012) Random effects in promotion time cure rate models. Computnl Statist. Data Anal., 56, 75–87.
López-Cheda, A., Cao, R., Jácome, M. and Van Keilegom, I. (2017) Nonparametric incidence estimation and bootstrap bandwidth selection in mixture cure models. Computnl Statist. Data Anal., 105, 144–165.
Lu, W. (2010) Efficient estimation for an accelerated failure time model with a cure fraction. Statist. Sin., 20, 661–674.
Ní Bhrolcháin, M. (1986) Women's paid work and the timing of births: longitudinal evidence. Eur. J. Popln, 2, 43–70.
Peng, Y. and Dear, K. (2000) A nonparametric mixture model for cure rate estimation. Biometrics, 56, 237–243.
Roberts, G. and Rosenthal, J. (2001) Optimal scaling for various Metropolis Hastings algorithms. Eur. J. Popln, 16, 357–367.
Sy, J. and Taylor, J. (2000) Estimation in a Cox proportional hazards cure model. Biometrics, 56, 227–236.
Taylor, J. (1995) Semi-parametric estimation in failure time mixture models. Biometrics, 51, 899–907.
Tsodikov, A. (1998) A proportional hazard model taking account of long-term survivors. Biometrics, 54, 1508–1516.
Tsodikov, A. (2002) Semi-parametric model of long- and short-term survival: an application to the analysis of breast cancer survival in Utah by age and stage. Statist. Med., 21, 895–920.
Wagner, G., Frick, J. and Schupp, J. (2007) The German Socio-Economic Panel Study (SOEP): scope, evolution and enhancements. Schmoll. Jahrb., 127, 139–169.
Wang, L., Du, P. and Liang, H. (2012) Two-component mixture cure rate model with spline estimated nonparametric components. Biometrics, 68, 726–735.
Yakovlev, A. and Tsodikov, A. (1996) Stochastic Models for Tumor of Latency and Their Biostatistical Applications. Singapore: World Scientific Publishing.
Yin, G. and Ibrahim, J. (2005) Cure rate models: a unified approach. Can. J. Statist., 33, 559–570.
Zeng, D., Yin, G. and Ibrahim, J. (2006) Semiparametric transformation models for survival data with a cure fraction. J. Am. Statist. Ass., 101, 670–684.
Zhang, J. and Peng, Y. (2007) A new estimation method for the semiparametric accelerated failure time mixture cure model. Statist. Med., 26, 3157–3171.
Zhang, J., Peng, Y. and Li, H. (2013) A new semiparametric estimation method for accelerated hazards mixture cure model. Computnl Statist. Data Anal., 59, 95–102.
Zhou, J., Zhang, J., McLain, A. C. and Cai, B. (2016) A multiple imputation approach for semiparametric cure model with interval censored data. Computnl Statist. Data Anal., 99, 105–114.