Estimating Cure Rates from Survival Data: An Alternative to Two-Component Mixture Models

Journal of the American Statistical Association

Volumn 98, Issue 464, 2003, Pages 1063-1078

  Tsodikov, A D ^a   Ibrahim, J G ^b   Yakovlev, A Y ^c  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b IL   (United States)

^c UNIVERSITY OF ROCHESTER   (United States)

Author keywords

Bayesian methods; Biologically based models; Bounded cumulative hazard; Cure models; Hazard regression; Semiparametric inference; Survival data

Indexed keywords

EID: 1142289538     PISSN: 01621459     EISSN: None     Source Type: Journal    
DOI: 10.1198/01622145030000001007     Document Type: Review

References (95)

1. Andersen, P. K., Borgan, Ø., Gill, R. D., and Keiding, N. (1992), Statistical Models Based on Counting Processes, New York: Springer-Verlag.
2. Asselain, B., Fourquet, A., Hoang, T., Myasnikova, C., and Yakovlev, A. Yu. (1994), "Testing the Independence of Competing Risks: An Application to the Analysis of Breast Cancer Recurrence," Biometrical Journal, 36, 465-473.
3. Asselain, B., Fourquet, A., Hoang, T., Tsodikov, A., and Yakovlev, A. Yu. (1996), "A Parametric Regression Model of Tumor Recurrence: An Application to the Analysis of Clinical Data on Breast Cancer," Statistics & Probbability Letters, 29, 271-278.
4. Bartoszyński, R., Edler, L., Hanin, L., Kopp-Schneider, A., Pavlova, L., Tsodikov, A., Zorin, A., and Yakovlev, A. (2001), "Modeling Cancer Detection: Tumor Size as a Source of Information on Unobservable Stages of Carcinogenesis," Mathematical Biosciences, 171, 113-142.
5. Bentzen, S. M., Johansen, L. V., Overgaard, J., and Thames, H. D. (1991), "Clinical Radiobiology of Squamous Cell Carcinoma of the Oropharynx, " International Journal Radiation. Oncology-Biology-Physics, 20, 1197-1206.
6. Berkson, J., and Gage, R. P. (1952), "Survival Curves for Cancer Patients Following Treatment," Journal of the American Statistical Association, 47, 501-515.
7. Boag, J. M. (1949), "Maximum Likelihood Estimates of the Proportion of Patients Cured by Cancer Therapy," Journal of the Royal Statistical Society, Ser. B, 11, 15-44.
8. Broët, P., Rycke, Y. D., Tubert-Bitter, P., Lellouch, J., Asselain, B., and Moreau, T. (2001), "A Semiparametric Approach for the Two-Sample Comparison of Survival Times With Long-Term Survivors," Biometrics, 57, 844-852.
9. Cantor, A. B. (1997), "Cure Rate Estimation Using the Polynomial Gompertz Model," in The Communication of Statistics: Exchange and Dissemination, 1997 Spring Meeting of the International Biometric Society, Eastern North American Region, Abstracts, p. 144.
10. _ (2001), "Projecting the Standard Error of the Kaplan-Meier Estimator," Statistics in Medicine, 20, 2091-2097.
11. Cantor, A. B., and Shuster, J. J. (1992), "Parametric Versus Nonparametric Methods for Estimating Cure Rates Based on Censored Survival Data," Statistics in Medicine, 11, 931-937.
12. Chappell, R., Nondahl, D. M., and Fowler, J. F. (1995), "Modeling Dose and Local Control in Radiotherapy," Journal of the American Statistical Association, 90, 829-838.
13. Chen, M. H., Harrington, D. P., and Ibrahim, J. G. (2002a), "Bayesian Cure Rate Models for Malignant Melanoma: A Case Study of ECOG Trial E1690," Applied Statistics, 51, 135-150.
14. Chen, M. H., and Ibrahim, J. G. (2001), "Maximum Likelihood Methods for Cure Rate Models With Missing Covariates," Biometrics, 51, 43-52.
15. Chen, M. H., Ibrahim, J. G., and Lipsitz, S. R. (2002b), "Bayesian Methods for Missing Covariates in Cure Rate Models," Lifetime Data Analysis, 8, 117-146.
16. 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.
17. _ (2002c), "Bayesian Inference for Multivariate Survival Data With a Surviving Fraction," Journal of Multivariate Analysis, 80, 101-126.
18. Chen, M. H., Shao, Q. M., and Ibrahim, J. G. (2000), Monte Carlo Methods in Bayesian Computation, New York: Springer-Verlag.
19. Cheng, S., Wei, L., and Ying Z. (1995), "Analysis of Transformation Models With Censored Data," Biometrika, 82, 835-845.
20. Clayton, D. G. (1978), "A Model for Association in Bivariate Life Tables and Its Application in Epidemiological Studies in Familial Tendency in Chronic Disease Incidence," Biometrika, 65, 141-151.
21. Cox, D. R. (1972), "Regression Models and Life Tables" (with discussion), Journal of the Royal Statistical Society, Ser. B, 34, 187-220.
22. Dempster, A., Laird, N., and Rubin, D. (1977), "Maximum Likelihood From Incomplete Data via the EM Algorithm," Journal of the Royal Statistical Society, Ser. B, 39, 1-38.
23. Etezadi-Amoli, L., and Ciampi, A. (1987), "Extended Hazard Regression for Censored Survival Data With Covariates: A Spline Approximation for the Baseline Hazard Function," Biometrics, 43, 181-192.
24. Farewell, V. T. (1982), "The Use of Mixture Models for the Analysis of Survival Data With Long-Term Survivors," Biometrics, 38, 1041-1046.
25. Feller, W. (1971), An Introduction to Probability Theory and Its Applications, Vol. 2, New York: Wiley.
26. Fleming, T., and Lin, D. (2000), "Survival Analysis in Clinical Trials: Past Developments and Future Directions," Biometrics, 56, 971-983.
27. Gamel, J. W., McLean, I. W., and Rosenberg, S. H. (1990), "Proportion Cured and Mean Log Survival Time as Functions of Tumour Size," Statistics in Medicine, 9, 999-1006.
28. Gamel, J. W., Meyer, J. S., Feuer, E., and Miller, B. A. (1996), "The Impact of Stage and Histology on the Long-Term Clinical Course of 163,808 Patients With Breast Carcinoma," Cancer, 77, 1459-1469.
29. Gamel, J. W., and Vogel, R. L. (1993), "A Model of Long-Term Survival Following Adjuvant Therapy for Stage 2 Breast Cancer," British Journal of Cancer, 68, 1167-1170.
30. Gamel, J. W., Vogel, R. L., and McLean, I. W. (1993), "Assessing the Impact of Adjuvant Therapy on Cure Rate for Stage 2 Breast Carcinoma," British Journal of Cancer, 68, 115-118.
31. Gamel, J. W., Vogel, R. L., Valagussa, P., and Bonadonna, G. (1994), "Parametric Survival Analysis of Adjuvant Therapy for Stage II Breast Cancer," Cancer, 74, 2483-2490.
32. Goldman, A. I. (1984), "Survivorship Analysis When Cure Is a Possibility: A Monte-Carlo Study," Statistics in Medicine, 3, 153-163.
33. Goldman, A. I., and Hillman, D. W. (1992), "Exemplary Data: Sample Size and Power in the Design of Event-Time Clinical Trials," Controlled Clinical Trials, 13, 256-271.
34. Gordon, N. H. (1990), "Application of the Theory of Finite Mixtures for the Estimation of 'Cure' Rates of Treated Cancer Patients," Statistics in Medicine, 9, 397-407.
35. Greenhouse, J. B., and Wolfe, R. A. (1984), "A Competing Risk Derivation of a Mixture Model for the Analysis of Survival Data," Communications in Statistics - Theory and Methods, 25, 3133-3154.
36. Gregori, G., Hanin, L., Luebeck, G., Moolgavkar, S., and Yakovlev, A. (2002), "Testing Goodness of Fit With Stochastic Models of Carcinogenesis," Mathematical Biosciences, 175, 13-29.
37. Hanin, L. G. (2001), "Iterated Birth and Death Process as a Model of Radiation Cell Survival," Mathematical Biosciences, 169, 89-107.
38. Hanin, L. G., Zaider, M., and Yakovlev, A. Y. (2001), "Distribution of the Number of Clonogens Surviving Fractionated Radiotherapy: A Long-Standing Problem Revisited," International Journal of Radiation Biology, 77, 205-213.
39. Haybittle, J. L. (1959), "The Estimation of the Proportion of Patients Cured After Treatment for Cancer of the Breast," British Journal of Radiology, 32, 725-733.
40. _ (1965), "A Two-Parameter Model for the Survival Curve of Treated Cancer Patients," Journal of the American Statistical Association, 60, 16-26.
41. Hjort, N. (1990), "Goodness of Fit Tests in Models for Life History Based on Cumulative Hazard Rates," The Annals of Statistics 18, 1221-1258.
42. Hoang, T., Tsodikov, A., Yakovlev, A. Yu., and Asselain, B. (1995), "Modeling Breast Cancer Recurrence," in Mathematical Population Dynamics: Analysis of Heterogeneity, Vol. 2, eds. O. Arino, D. Axelrod, and M. Kimmel, Winnipeg: Wuerz Publications, pp. 283-296.
43. Hougaard, P. (1986), "A Class of Multivariate Failure Time Distributions," Biometrika, 73, 671-678.
44. Ibrahim, J. G., and Chen, M. H. (2000), "Power Prior Distributions for Regression Models," Statistical Science, 15, 46-60.
45. Ibrahim, J. G., Chen, M. H., and Sinha, D. (2001a), Bayesian Survival Analysis, New York: Springer-Verlag.
46. _ (2001b), "Bayesian Semi-Parametric Models for Survival Data With a Cure Fraction," Biometrics, 57, 383-388.
47. Kalbfleisch, J. D., and Prentice, R. L. (1980), The Statistical Analysis of Failure Time Data, New York: Wiley.
48. Klebanov, L. B., Rachev, S. T., and Yakovlev, A. Yu. (1993a), "A Stochastic Model of Radiation Carcinogenesis: Latent Time Distributions and Their Properties," Mathematical Biosciences, 113, 51-75.
49. _ (1993b), "On the Parametric Estimation of Survival Functions," Statistics & Decisions, 3, 83-102.
50. Klein, J. (1992), "Semiparametric Estimation of Random Effects Using the Cox Model Based on the EM Algorithm," Biometrics, 48, 795-806.
51. Kruglikov, I. L., Pilipenko, N. I., Tsodikov, A. D., and Yakovlev, A. Y. (1997), "Assessing Risk With Doubly Censored Data: An Application to the Analysis of Radiation-Induced Thyropathy," Statistics & Probability Letters, 32, 223-230.
52. Kuk, A. Y. C., and Chen, C.-H. (1992), "A Mixture Model Combining Logistic Regression With Proportional Hazards Regression," Biometrika, 79, 531-541.
53. Lange, K., Hunter, D. R., and Yang, I. (2000), "Optimization Transfer Using Surrogate Objective Functions" (with discussion), Journal of Computational and Graphical Statistics, 9, 1-59.
54. Laska, E. M., and Meisner, M. J. (1992), "Nonparametric Estimation and Testing in a Cure Model," Biometrics, 48, 1223-1234.
55. Maller, R. A., and Zhou, S. (1992), "Estimating the Proportion of Immunes in a Censored Sample," Biometrika, 79, 731-739.
56. _ (1994), "Testing for Sufficient Follow-up and Outliers in Survival Data," Journal of the American Statistical Association, 89, 1499-1506.
57. _ (1995), "Testing for the Presence of Immune or Cured Individuals in Censored Survival Data," Biometrics, 51, 1197-1205.
58. _ (1996), Survival Analysis With Long-Term Survivors, Chichester, U.K.: Wiley.
59. Miller, R. G. (1981), Survival Analysis, New York: Wiley.
60. Murphy, S. (2000), "On Profile Likelihood," Journal of the American Statistical Association, 95, 449-485.
61. Myasnikova, E. M., Asselain, B., and Yakovlev, A. Yu. (2000), "Spline-Based Estimation of Cure Rates: An Application to the Analysis of Breast Cancer Data," Mathematical and Computer Modelling, 32, 217-228.
62. Nielsen, G., Gill, R., Andersen, P., and Sørensen, T. (1992), "A Counting Process Approach to Maximum Likelihood Estimation in Frailty Models," Scandinavian Journal of Statistics, 19, 25-43.
63. Oakes, D. (1989), "Bivariate Survival Models Induced by Frailties," Journal of the American Statistical Association, 84, 487-493.
64. Peng, Y., and Dear, K. B. G. (2000), "A Nonparametric Mixture Model for Cure Rate Estimation," Biometrics, 56, 237-243.
65. Pepe, M. S., and Fleming, T. R. (1989), "Weighted Kaplan-Meier Statistics: A Class of Distance Tests for Censored Survival Data," Biometrics, 45, 497-507.
66. Press, W., Flannery, B., Teukolsky, S., and Vetterling, W. (1994), Numerical Recipes in Pascal: The Art of Scientific Computing, New York: Cambridge University Press.
67. Rachev, S. T., Wu, C., and Yakovlev, A. Y. (1995), "A Bivariate Limiting Distribution of Tumor Latency Time," Mathematical Biosciences, 127, 127-147.
68. Sposto, R., Sather, H. N., and Baker, S. A. (1992), "A Comparison of Tests of the Difference in the Proportion of Patients Who Are Cured," Biometrics, 48, 87-99.
69. Sy, J. P., and Taylor, J. M. G. (2000), "Estimation in a Cox Proportional Hazards Cure Model," Biometrics, 56, 227-236.
70. _ (2001), "Standard Errors for the Cox Proportional Hazards Cure Model," Mathematical and Computer Modelling, 33, 1237-1252.
71. Taylor, J. M. G. (1995), "Semi-Parametric Estimation in Failure Time Mixture Models," Biometrics, 51, 899-907.
72. Trelford, J., Tsodikov, A. D., and Yakovlev, A. Y. (2001), "Modeling Post-Treatment Development of Cervical Carcinoma: Exophytic or Endophytic - Does It Matter?" Mathematical and Computer Modelling, 33, 1439-1443.
73. Tsodikov, A. (1998a), "A Proportional Hazards Model Taking Account of Long-Term Survivors," Biometrics, 54, 1508-1516.
74. _ (1998b), "Asymptotic Efficiency of a Proportional Hazards Model With Cure," Statistics & Probability Letters, 39, 1998, 237-244.
75. _ (2001), "Estimation of Survival Based on Proportional Hazards When Cure Is a Possibility," Mathematical and Computer Modelling, 33, 1227-1236.
76. _ (2002), "Semiparametric Models of Long- and Short-Term Survival: An Application to the Analysis of Breast Cancer Survival in Utah by Age and Stage," Statistics in Medicine, 21, 895-920.
77. _ (2003), "Semiparametric Models: A Generalized Self-Consistency Approach," submitted to Journal of the Royal Statistical Society, Ser. B, 65, 759-774.
78. Tsodikov, A. D., Asselain, B., Fourquet, A., Hoang, T., and Yakovlev, A. Yu. (1995), "Discrete Strategies of Cancer Post-Treatment Surveillance: Estimation and Optimization Problems," Biometrics, 51, 437-447.
79. Tsodikov, A., Dicello, J., Zaider, M., Zorin, A., and Yakovlev, A. (2001), "Analysis of a Hormesis Effect in the Leukemia Caused Mortality Among Atomic Bomb Survivors," Human and Ecological Risk Assessment, 7, 829-847.
80. Tsodikov, A. D., Hasenclever, D., and Loeffler, M. (1998a), "Regression With Bounded Outcome Score: Evaluation of Power by Bootstrap and Simulation in a Chronic Myelogenous Leukaemia Clinical Trial," Statistics in Medicine, 17, 1909-1922.
81. Tsodikov, A., Loeffler, M., and Yakovlev, A. Yu. (1997), "Assessing the Risk of Secondary Leukemia in Patients Treated for Hodgkin's Disease. A Report From the International Database on Hodgkin's Disease," Journal of Biological Systems, 5, 433-444.
82. _ (1998b), "A Cure Model With Time-Changing Risk Factor: An Application to the Analysis of Secondary Leukemia. A Report From the International Database on Hodgkin's Disease," Statistics in Medicine, 17, 27-40.
83. Tsodikov, A. D., and Müller, W. (1998), "Modeling Carcinogenesis Under a Time-Changing Exposure," Mathematical Biosciences, 152, 179-191.
84. Tucker, S. L. (1999), "Modeling the Probability of Tumor Cure After Fractionated Radiotherapy," in Mathematical Models in Medical and Health Sciences, eds. M. A. Horn, O. Simonett, and G. Webb, Nashville: Vanderbilt University Press, pp. 1-15.
85. Tucker, S. L., and Taylor, J. M. G. (1996), "Improved Models of Tumour Cure," International Journal of Radiation Biology, 70, 539-553.
86. Tucker, S. L., Thames, H. D., and Taylor, J. M. G. (1990), "How Well Is the Probability of Tumor Cure After Fractionated Irradiation Described by Poisson Statistics?" Radiation Research, 124, 273-282.
87. Wassel, J. T., and Moeschberger, M. L. (1993), "A Bivariate Survival Model With Modified Gamma Frailty for Assessing the Impact of Interventions," Statistics in Medicine, 12, 241-248.
88. Yakovlev, A. Yu. (1994), "Letter to the Editor," Statistics in Medicine, 13, 983-986.
89. _ (1996), "Threshold Models of Tumor Recurrence," Mathematical and Computer Modelling, 23, 153-164.
90. Yakovlev, A. Y., Asselain, B., Bardou, V. J., Fourquet, A., Hoang, T., Rochefordiere, A., and Tsodikov, A. D. (1993), "A Simple Stochastic Model of Tumor Recurrence and Its Application to Data on Premenopausal Breast Cancer," in Biometrie et Analyse de Donnees Spatio-Temporelles, No. 12, eds. B. Asselain, M. Boniface, C. Duby, C. Lopez, J. P. Masson, and J. Tranchefort, Rennes, France: Société Française de Biométrie, pp. 66-82.
91. Yakovlev, A. Yu., and Tsodikov, A. D. (1996), Stochastic Models of Tumor Latency and Their Biostatistical Applications, Singapore: World Scientific.
92. Yakovlev, A. Y., Tsodikov, A. D., Boucher, K., and Kerber, R. (1999), "The Shape of the Hazard Function in Breast Carcinoma: Curability of the Disease Revisited," Cancer, 85, 1789-1798.
93. Yamaguchi, K. (1992), "Accelerated Failure-Time Regression Model With a Regression Model of Surviving Fraction: An Analysis of Permanent Employment in Japan," Journal of the American Statistical Association, 87, 284-292.
94. Zaider, M., Zelefsky, M. J., Hanin, L. G., Tsodikov, A. D., Yakovlev, A. Y., and Leibel, S. A. (2001), "A Survival Model for Fractionated Radiotherapy With an Application to Prostate Cancer," Physics in Medicine and Biology, 46, 2745-2758.
95. Zorin, A. V., Tsodikov, A. D., Zharinov, G. M., and Yakovlev, A. Y. (2001), "The Shape of the Hazard Function in Cancer of the Cervix Uteri," Journal of Biological Systems, 9, 221-233.