Abstract

The mixture cure model is the most popular model used to analyse the major event with a potential cure fraction. But in the real world there may exist a potential risk from other non-curable competing events. In this paper, we study the accelerated failure time model with mixture cure model via kernel-based nonparametric maximum likelihood estimation allowing non-curable competing risk. An EM algorithm is developed to calculate the estimates for both the regression parameters and the unknown error densities, in which a kernel-smoothed conditional profile likelihood is maximised in the M-step, and the resulting estimates are consistent. Its performance is demonstrated through comprehensive simulation studies. Finally, the proposed method is applied to the colorectal clinical trial data.

References

1. 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. doi: 10.1111/j.2517-6161.1949.tb00020.x [Crossref], [Web of Science ®], [Google Scholar]
2. Crowder, M. (2001). Classical competing risk. London: Chapman and Hall/CRC. [Crossref], [Google Scholar]
3. David, H., & Moeschberge, M. (1978). The theory of competing risks. London: Griffn. [Google Scholar]
4. Fine, J., & Gray, R. (1999). A proportional hazards model for the subdistribution of a competing risk. Journal of the American Statistical Association, 94(446), 496–509. doi: 10.1080/01621459.1999.10474144 [Taylor & Francis Online], [Web of Science ®], [Google Scholar]
5. Fusaro, R., Bacchetti, P., & Jewell, N. (1996). A competing risks analysis of presenting aids diagnoses trends. Biometrics, 52, 211–225. doi: 10.2307/2533157 [Crossref], [Web of Science ®], [Google Scholar]
6. Gamel, J., Weller, E., Wesley, M., & Feuer, E. (2000). Parametric cure models of relative and cause-specific survival for grouped survival times. Computer Methods and Programs in Biomedicine, 61(2), 99–110. doi: 10.1016/S0169-2607(99)00022-X [Crossref], [Web of Science ®], [Google Scholar]
7. Gaynor, J., Feuer, E., Tan, C., Wu, D., Little, C., Straus, D., …Brennan, M. (1993). On the use of cause-specific failure and conditional failure probabilities: Examples from clinical oncology data. Journal of the American Statistical Association, 88(422), 400–409. doi: 10.1080/01621459.1993.10476289 [Taylor & Francis Online], [Web of Science ®], [Google Scholar]
8. González, J. R., Fernandez, E., Moreno, V., Ribes, J., Peris, M., Navarro, M., …Borrás, J. M. (2005). Sex differences in hospital readmission among colorectal cancer patients. Journal of Epidemiology & Community Health, 59(6), 506–511. doi: 10.1136/jech.2004.028902 [Crossref], [Web of Science ®], [Google Scholar]
9. Kalbfleisch, J.D., & Prentice, R.L. (2002). The statistical analysis of failure time data. New York: John Wiley & Sons. [Crossref], [Google Scholar]
10. Klein, J. (2006). Modelling competing risks in cancer studies. Statistics in Medicine, 25(6), 1015–1034. doi: 10.1002/sim.2246 [Crossref], [Web of Science ®], [Google Scholar]
11. Kleinbaum, D., & Klein, M. (2006). Survival analysis: A self-learning text. New York: Springer Science & Business Media. [Google Scholar]
12. Kuk, A. (1992). A semiparametric mixture model for the analysis of competing risks data. Australin Jounal of Statistics, 34(2), 169–180. [Crossref], [Google Scholar]
13. Lambert, P., Thompson, J., Weston, C., & Dickman, P. (2006). Estimating and modeling the cure fraction in population-based cancer survival analysis. Biostatistics (Oxford, England), 8(3), 576–594. doi: 10.1093/biostatistics/kxl030 [Crossref], [Web of Science ®], [Google Scholar]
14. Larson, M., & Dinse, G. (1985). A mixture model for the regression analysis of competing risks data. Applied Statistics, 34(3), 201–211. doi: 10.2307/2347464 [Crossref], [Web of Science ®], [Google Scholar]
15. Li, C., & Taylor, J. (2002). A semi-parametric accelerated failure time cure model. Statistics in Medicine, 21(21), 3235–3247. doi: 10.1002/sim.1260 [Crossref], [Web of Science ®], [Google Scholar]
16. Lu, W. (2010). Efficient estimation for an accelerated failure time model with a cure fraction. Statistica Sinica, 20, 661–674. [Web of Science ®], [Google Scholar]
17. Lu, W., & Peng, L. (2008). Semiparametric analysis of mixture regression models with competing risks data. Lifetime Data Analysis, 14(3), 231–252. doi: 10.1007/s10985-007-9077-6 [Crossref], [Web of Science ®], [Google Scholar]
18. Ng, S., & McLachlan, G. (2003). An em-based semi-parametric mixture model approach to the regression analysis of competing-risks data. Statistics in Medicine, 22(7), 1097–1111. doi: 10.1002/sim.1371 [Crossref], [Web of Science ®], [Google Scholar]
19. Ohneberg, K., Schumacher, M., & Beyersmann, J. (2017). Modelling two cause-specific hazards of competing risks in one cumulative proportional odds model?. Statistics in Medicine, 36(27), 4353–4363. doi: 10.1002/sim.7437 [Crossref], [Web of Science ®], [Google Scholar]
20. Peng, Y. (2003). Fitting semiparametric cure models. Computational Statistics & Data Analysis, 41(3), 481–490. doi: 10.1016/S0167-9473(02)00184-6 [Crossref], [Web of Science ®], [Google Scholar]
21. Peng, Y., & Dear, K. B. (2000). A nonparametric mixture model for cure rate estimation. Biometrics, 56(1), 237–243. doi: 10.1111/j.0006-341X.2000.00237.x [Crossref], [Web of Science ®], [Google Scholar]
22. Peng, Y., Dear, K. B., & Denham, J. (1998). A generalized f mixture model for cure rate estimation. Statistics in Medicine, 17(8), 813–830. doi: 10.1002/(SICI)1097-0258(19980430)17:8<813::AID-SIM775>3.0.CO;2-# [Crossref], [Web of Science ®], [Google Scholar]
23. Pintilie, M. (2007). Analysing and interpreting competing risk data. Statistics in Medicine, 26(6), 1360–1367. doi: 10.1002/sim.2655 [Crossref], [Web of Science ®], [Google Scholar]
24. Sy, J., & Taylor, J. (2000). Estimation in a cox proportional hazards cure model. Biometrics, 56(1), 227–236. doi: 10.1111/j.0006-341X.2000.00227.x [Crossref], [Web of Science ®], [Google Scholar]
25. Tai, B., Machin, D., White, I., & Gebski, V. (2001). Competing risks analysis of patients with osteosarcoma: A comparison of four different approaches. Statistics in Medicine, 20(5), 661–684. doi: 10.1002/sim.711 [Crossref], [Web of Science ®], [Google Scholar]
26. Van Der VaartJon, A., & Wellner, J. (1996). Weak convergence and empirical processes. New York, NY: Springer. [Crossref], [Google Scholar]
27. Xu, L., & Zhang, J. (2009). An alternative estimation method for the semiparametric accelerated failure time mixture cure model. Communications in Statistics-Simulation and Computation, 38(9), 1980–1990. doi: 10.1080/03610910903180657 [Taylor & Francis Online], [Web of Science ®], [Google Scholar]
28. Yu, B., & Tiwari, R. (2007). Application of em algorithm to mixture cure model for grouped relative survival data. Journal of Data Science, 5, 41–51. [Google Scholar]
29. Yu, B., Tiwari, R., Cronin, K., & Feuer, E. (2004). Cure fraction estimation from the mixture cure models for grouped survival data. Statistics in Medicine, 23, 1733–1747. doi: 10.1002/sim.1774 [Crossref], [Web of Science ®], [Google Scholar]
30. Zeng, D., & Lin, D. (2007). Efficient estimation for the accelerated failure time model. Journal of the American Statistical Association, 102(480), 1387–1396. doi: 10.1198/016214507000001085 [Taylor & Francis Online], [Web of Science ®], [Google Scholar]
31. Zhang, J., & Peng, Y. (2007). A new estimation method for the semiparametric accelerated failure time mixture cure model. Statistics in Medicine, 26(16), 3157–3171. doi: 10.1002/sim.2748 [Crossref], [Web of Science ®], [Google Scholar]