Accurate Bootstrap Confidence Limits for the Cumulative Hazard and Survivor Functions under Random Censoring

Abstract In clinical trials and other settings, confidence intervals for survival probabilities at a fixed point in time are often required. Typically, such intervals are constructed on the cumulative hazard scale; the quantiles required for computing these intervals are generally obtained using a first-order normal theory approximation. The appropriateness of such intervals for small sample sizes or under moderate to heavy censoring is suspect. We demonstrate that significant improvement is possible for randomly censored data using the nonparametric bootstrap. In particular, using Edgeworth expansion, it is shown that the percentile-t and BC a bootstrap methodologies based on the studentized Nelson-Aalen estimator yield second-order-correct confidence limits for the cumulative hazard function at a fixed time point. An explicit formula for the correction term of the Edgeworth expansion is found. Easy-to-compute estimators for the components of the correction term are also found and lead to simple analytic...

[1]  Y. Maesono.  Edgeworth expansions of a studentized U-statistic and a jackknife estimator of variance , 1997 .

[2]  B. Efron,et al.  Bootstrap confidence intervals , 1996 .

[3]  Kani Chen,et al.  On bootstrap accuracy with censored data , 1996 .

[4]  Tze Leung Lai,et al.  BOOTSTRAP METHODS FOR TRUNCATED AND CENSORED DATA , 1996 .

[5]  P. Mykland.  Martingale expansions and second order inference , 1995 .

[6]  Per A. Mykland,et al.  Bartlett Type Identities for Martingales , 1994 .

[7]  Richard D. Gill,et al.  Lectures on survival analysis , 1994 .

[8]  Niels Keiding,et al.  Statistical Models Based on Counting Processes , 1993 .

[9]  P. Mykland.  Asymptotic Expansions for Martingales , 1993 .

[10]  T Anderson,et al.  Phase II study of taxol, merbarone, and piroxantrone in stage IV non-small-cell lung cancer: The Eastern Cooperative Oncology Group Results. , 1993, Journal of the National Cancer Institute.

[11]  Minggao Gu,et al.  On the Edgeworth expansion and bootstrap approximation for the cox regression model under random censorship , 1992 .

[12]  B. Efron,et al.  More accurate confidence intervals in exponential families , 1992 .

[13]  Per A. Mykland,et al.  Asymptotic Expansions and Bootstrapping Distributions for Dependent Variables: A Martingale Approach , 1992 .

[14]  P. Hall.  The Bootstrap and Edgeworth Expansion , 1992 .

[15]  S. Konishi.  Normalizing Transformatins and Bootstrap Confidence Intervals , 1991 .

[16]  D. Harrington,et al.  Counting Processes and Survival Analysis , 1991 .

[17]  Thomas J. DiCiccio,et al.  The automatic percentile method: Accurate confidence limits in parametric models , 1989 .

[18]  P. Hall.  Theoretical Comparison of Bootstrap Confidence Intervals , 1988 .

[19]  Discussion: Theoretical Comparison of Bootstrap Confidence Intervals , 1988 .

[20]  R. Beran.  Prepivoting Test Statistics: A Bootstrap View of Asymptotic Refinements , 1988 .

[21]  R. Beran.  Prepivoting to reduce level error of confidence sets , 1987 .

[22]  B. Yandell,et al.  Convergence Rates for the Bootstrapped Product-Limit Process , 1987 .

[23]  Robert Tibshirani,et al.  Bootstrap confidence intervals and bootstrap approximations , 1987 .

[24]  B. Efron.  Better Bootstrap Confidence Intervals , 1987 .

[25]  F. Götze,et al.  The Edgeworth Expansion for $U$-Statistics of Degree Two , 1986 .

[26]  K. Singh,et al.  The product-limit estimator and the bootstrap: Some asymptotic representations , 1986 .

[27]  Ørnulf Borgan,et al.  Confidence intervals and confidence bands for the cumulative hazard rate function and their small sample properties , 1986 .

[28]  M. Akritas.  Bootstrapping the Kaplan—Meier Estimator , 1986 .

[29]  N. Hjort.  Bootstrapping Cox's Regression Model. , 1985 .

[30]  K. Singh,et al.  EDGEWORTH CORRECTED PIVOTAL STATISTICS AND THE BOOTSTRAP , 1985 .

[31]  D P Byar,et al.  A comparison of reflected versus test-based confidence intervals for the median survival time, based on censored data. , 1984, Biometrics.

[32]  David P. Harrington,et al.  Nonparametric estimation of the survival distribution in censored data , 1984 .

[33]  C. Link,et al.  Confidence intervals for the survival function using Cox's proportional-hazard model with covariates. , 1984, Biometrics.

[34]  Ron Brookmeyer,et al.  A Confidence Interval for the Median Survival Time , 1982 .

[35]  D. Freedman,et al.  Some Asymptotic Theory for the Bootstrap , 1981 .

[36]  K. Singh,et al.  On the Asymptotic Accuracy of Efron's Bootstrap , 1981 .

[37]  Bradley Efron,et al.  Censored Data and the Bootstrap , 1981 .

[38]  B. Efron.  Bootstrap Methods: Another Look at the Jackknife , 1979 .

[39]  N. Reid.  Influence functions for censored data , 1979 .

[40]  V. Susarla,et al.  Large Sample Theory for a Bayesian Nonparametric Survival Curve Estimator Based on Censored Samples , 1978 .

[41]  J. Ghosh,et al.  ON THE VALIDITY OF THE FORMAL EDGEWORTH EXPANSION , 1978 .

[42]  N. Breslow,et al.  A Large Sample Study of the Life Table and Product Limit Estimates Under Random Censorship , 1974 .