Accurate bootstrap confidence limits for the cumulative hazard and survivor functions under random censoring

Journal of the American Statistical Association

Volumn 92, Issue 440, 1997, Pages 1356-1374

  Strawderman, Robert L ^a   Wells, Martin T ^b  

^a UNIVERSITY OF MICHIGAN   (United States)

^b Department of Obstetrics Gynecology   (United States)

Author keywords

Asymptotic U statistic; Bias corrected adjustment; Edgeworth expansion; Kaplan Meier estimator; Martingale; Nelson Aalen estimator; Percentile t interval; Second order correctness

Indexed keywords

EID: 0031323521     PISSN: 01621459     EISSN: 1537274X     Source Type: Journal    
DOI: 10.1080/01621459.1997.10473657     Document Type: Article

References (52)

1. Abramovitch, L., and Singh, K.(1985), "Edgeworth-Corrected Pivotal Statistics and the Bootstrap,"The Annals of Statistics, 13, 116-132.
2. Akritas, M.(1986), "Bootstrapping the Kaplan-Meier Estimator,"Journal of the American Statistical Association, 81, 1032-1038.
3. Andersen, P. K., Borgan, O., Gill, R., and Keiding, N.(1993), Statistical Models Based on Counting Processes, New York: Springer-Verlag.
4. Babu, G. J., and Singh, K.(1984), "On One-Term Edgeworth Correction by Efron's Bootstrap,"Sankyha, Ser. A, 46, 219-232.
5. Beran, R(1987), "Prepivoting to Reduce Level Error of Confidence Sets,"Biometrika, 74, 457-468.
6. Beran, R (1988), "Prepivoting Test Statistics: A Bootstrap View of Asymptotic Refinements,"Journal of the American Statistical Association, 83,687-697.
7. Bhattacharya, R N., and Ghosh, J. K.(1978), "On the Validity of Formal Edgeworth Expansion,"The Annals of Statistics, 6, 434-451.
8. Bickel, P. J.(1988), Discussion of "Theoretical Comparison of Bootstrap Confidence Intervals" by P. Hall, The Annals of Statistics, 16,959-961.
9. Bickel, P. J., and Freedman, D. A.(1981), "Some Asymptotics for the Bootstrap,"The Annals of Statistics, 9, 1196-1217.
10. Bickel, P. J., Gotze, E, and van Zwet, W. R(1986), "The Edgeworth Expansion for U Statistics of Degree Two,"The Annals of Statistics, 14,1463-1484.
11. Bie, Ø., Borgan, Ø., and Liestlø, K.(1987), "Confidence Intervals and Confidence Bands for the Cumulative Hazard Rate Function and Their Small Sample Properties,"Scandinavian Journal of Statistics, 14, 221-233.
12. Breslow, N. E., and Crowley, J.(1974), "A Large Sample Study of the Life Table and Product-Limit Estimates Under Random Censorship,"The Annals of Statistics, 2, 437-443.
13. Brookmeyer, R., and Crowley, J.(1982), "A Confidence Interval for the Median Survival Time,"Biometrics, 38, 29-41.
14. Chan, S. K.(1994), "Asymptotic Expansions for Martingales and Improvement of the P-Value Estimate in the Two-Sample Problem in Survival Analysis," unpublished Ph.D. dissertation, University of Chicago.
15. Chang, A. Y, Kim, K., Glick, J., Anderson, T., Karp, D., and Johnson, D.(1993), "Phase II Study of Taxol, Merbarone, and Piroxantrone in Stage IV Non-Small-Cell Lung Cancer: The Eastern Cooperative Oncology Group Results,"Journal of the National Cancer Institute, 85,388-393.
16. Chen, K., and Lo, S. H.(1996), "On Bootstrap Accuracy With Censored Data,"The Annals of Statistics, 24, 569-595.
17. Dasgupta, R. (1984), "On Large Deviation Probabilities of U-Statistics in Non l.I.D. Case," Sankhya, Ser. A, 46, 110-116.
18. DiCiccio, T., and Efron, B.(1992), "More Accurate Confidence Intervals in Exponential Families,"Biometrika, 79, 231-245.
19. DiCiccio, T., and Efron, B. (1996), "Bootstrap Confidence Intervals" (with discussion), Statistical Science, II, 189-228.
20. DiCiccio, T. J., and Romano, J. P.(1988), "The Automatic Percentile Method: Accurate Confidence Limits in Parametric Models,"Canadian Journal of Statistics, 17, 155-169.
21. DiCiccio, T. J., and Romano, J. P. (1995), "On Bootstrap Procedures for Second-Order Accurate Confidence Limits in Parametric Models,"Statistica Sinica, 5, 141-160.
22. DiCiccio, T., and Tibshirani, R.(1987), "Bootstrap Confidence Intervals and Bootstrap Approximations,"Journal of the American Statistical Association,82, 163-170.
23. Efron, B.(1979), "Bootstrap Methods: Another Look at the Jackknife," The Annals of Statistics, 7, 1-26.
24. Efron, B. (1981), "Censored Data and the Bootstrap,"Journal of the American Statistical Association, 76, 312-319.
25. Efron, B. (1987), "Better Bootstrap Confidence Intervals,"Journal of the American Statistical Association, 82, 171-200.
26. Fleming, T. R., and Harrington, D. P.(1984), "Nonparametric Estimation of the Survival Distribution in Censored Data,"Communications in Statistics, Part A-Theory and Methods, 13, 2469-2486.
27. Fleming, T. R, and Harrington, D. P.(1991), Counting Processes and Survival Analysis, New York: Wiley.
28. Ghosh, M., and Dasgupta, R.(1982), "Berry-Esseen Theorems for UStatistics in the Non-lID Case,"in Nonparametric Statistical Inference, Amsterdam: North-Holland.
29. Gill, R.(1994), "Lectures on Survival Analysis," in Lectures on Probability Theory: Ecole d'Ete de Probabilites de Saint-Flour XXII-I992, ed. P. Bernard, Berlin: Springer-Verlag.
30. Gross, S., and Lai, T. L.(1996), "Bootstrap Methods for Truncated and Censored Data,"Statistica Sinica, 6, 509-530.
31. Gu, M.(1992), "On the Edgeworth Expansion and Bootstrap Approximation for the Cox Regression Model Under Random Censorship,"Canadian Journal of Statistics, 20, 399-414.
32. Hall, P.(1988), "Theoretical Comparison of Bootstrap Confidence Intervals,"The Annals of Statistics, 16, 927-953.
33. Hall, P. (1992), The Bootstrap and Edgeworth Expansion, New York: Springer-Verlag.
34. Hjort, N.(1985), "Bootstrapping Cox's Regression Model," Statistics Technical Report 241, Stanford University.
35. Horvath, L., and Yandell, B. S.(1987), "Convergence Rates for the Bootstrapped Product-Limit Process,"The Annals of Statistics, 15, 1155-1174.
36. Konishi, S.(1991), "Normalizing Transformations and Bootstrap Confidence Intervals,"The Annals of Statistics, 19, 2209-2225.
37. Lai, T. L., and Wang, J. Q.(1993), "Edgeworth Expansions for Symmetric Statistics With Applications to Bootstrap Methods,"Statistica Sinica, 3,517-542.
38. Link, C.(1984), "Confidence Intervals for the Survival Function Using Cox's Proportional Hazards Model With Covariates,"Biometrics, 40,601-610.
39. Lo, S-W., and Singh, K.(1985), "The Product-Limit Estimator and the Bootstrap: Some Asymptotic Representations,"Theory of Probability and Related Fields, 71, 455-465.
40. Loh, W-Y (1991), "Bootstrap Calibration for Confidence Interval Construction and Selection,"Statistica Sinica, 1,477-491.
41. Maesono, Y(1994), "Edgeworth Expansions of Studentized U Statistic and a Jackknife Estimator of Variance," Technical Report SRR-038-94, Kyushu University, Japan.
42. Maesono, Y (1995), "On the Edgeworth Expansions for Functionals of U Statistics," Technical Report SRR-015-95, Kyushu University, Japan.
43. Mykland, P.(1992), "Asymptotic Expansions and Bootstrapping Distribu tions for Dependent Variables: A Martingale Approach,"The Annals of Statistics, 20, 62-854.
44. Mykland, P. (1993), "Asymptotic Expansions for Martingales,"Annals of Probability,21,800-818.
45. Mykland, P. (1994), "Bartlett Identities for Martingales,"The Annals of Statistics,22, 21-38.
46. Mykland, P. (1995), "Martingale Expansions and Second-Order Inference," The Annals of Statistics, 23, 707-739.
47. Reid, N.(1981), "Influence Functions for Censored Data,"The Annals of Statistics, 9, 78-92.
48. Shorack, G., and Wellner, J. A.(1986), Empirical Processes With Applica- tions to Statistics, New York: Wiley.
49. Singh, K.(1981), "On the Asymptotic Accuracy of Efron's Bootstrap,"The Annals of Statistics, 9, 1187-1195.
50. Slud, E. v., Byar, D. P., and Green, S. B.(1984), "A Comparison of Reflected Versus Test-Based Confidence Intervals for the Median Survival Time Based on Censored Data,"Biometrics, 40, 587.
51. Susarla, v., and Van Ryzin, J.(1978), "Large Sample Theory for a Bayesian Nonparametric Survival Curve Estimator Based on Censored Samples,"The Annals of Statistics, 6, 755-768.
52. U.S.Renal Data System(1995), USRDS 1995 Annual Data Report, Bethesda, MD: The National Institutes of Health, National Institute of Diabetes and Digestive and Kidney Diseases.