Computing Science and Statistics pp 89-97 | Cite as

# Bootstrap Tests and Confidence Intervals for a Hazard Ratio When the Number of Observed Failures is Small, With Applications to Group Sequential Survival Studies

## Abstract

We present a small sample approximation to the distribution of the efficient score statistic for testing the null hypothesis *λ*=*λ* _{0}, where *λ* is the hazard ratio in a proportional hazards model. Here, “small sample” refers to a small number of failures in the observed data. Our basic approximation is to the conditional distribution of observations’ group memberships, given the observed number of failures and number of censored observations between successive failures; the implied distribution of the score statistic is found by simulation. Our method can be incorporated into sequential procedures for monitoring survival data and it successfully overcomes inaccuracies in the usual normal approximations that arise when only a few subjects have failed at early analyses.

A simulation study to assess the accuracy of our approximation required bootstrap simulation nested within experimental replications. Here, the computational demands were alleviated substantially by conducting the bootstrap tests themselves sequentially; using an innovative form of stochastic curtailment reduced the required computation by a factor of up to 25, 50 or even more.

## Keywords

Normal Approximation Sequential Test Bootstrap Test Logrank Statistic Tail Point## Preview

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## References

- Barnard, G. A. (1963) Discussion of paper by M. S. Bartlett.
*J. R. Statist. Soc*,**B 25**, 294.MathSciNetGoogle Scholar - Brown, B. M. (1971) Martingale central limit theorems.
*Ann. Math. Statist*.,**42**, 59–66.MathSciNetMATHCrossRefGoogle Scholar - Cox, D. R. (1972) Regression models and life tables (with discussion).
*J. R. Statist. Soc*. B,**34**, 187–220.MATHGoogle Scholar - DeMets, D. L. and Gail, M. H. (1985) Use of logrank tests and group sequential methods at fixed calendar times.
*Biometrics*,**41**, 1039–1044.CrossRefGoogle Scholar - Efron, B. (1979) Bootstrap methods: another look at the jackknife.
*Ann. Statist*.,**7**, 1–26.MathSciNetMATHCrossRefGoogle Scholar - Efron, B. (1981) Censored data and the bootstrap.
*J. Amer. Statist. Assoc*,**76**, 312–319.MathSciNetMATHCrossRefGoogle Scholar - Gail, M. H., DeMets, D. L. and Slud, E. V. (1982) Simulation studies on increments of the two-sample logrank score test for survival data, with application to group sequential boundaries. In
*Survival Analysis*, Monograph Series 2, (J. Crowley and R. Johnson eds), 287–301. Hayward, California: IMS Lecture Notes.CrossRefGoogle Scholar - Harrington, D. P., Fleming, T. R. and Green, S. J. (1982) Procedures for serial testing in censored survival data. In
*Survival Analysis*, Monograph Series 2, (J. Crowley and R. Johnson eds), 269–286. Hayward, California: IMS Lecture Notes.CrossRefGoogle Scholar - Jennison, C. and Tumbull, B. W. (1984) Repeated confidence intervals for group sequential clinical trials.
*Controlled Clinical Trials*,**5**, 33–45.CrossRefGoogle Scholar - Jennison, C. and Turnbull, B. W. (1989) Interim analyses: the repeated confidence interval approach (with discussion). J. Roy. Statist Soc., B,
**51**, 305–361.MathSciNetMATHGoogle Scholar - Jennison, C. and Turnbull, B. W. (1991). Exact calculations for sequential t,
*x*^{2}and F tests. To appear in*Biometrika*.Google Scholar - Lan, K. K. G., Simon, R. and Halperin, M. (1982) Stochastically curtailed tests in long-term clinical trials.
*Sequential Analysis*,**1**, 207–219.MathSciNetMATHCrossRefGoogle Scholar - Marriott, F. H. C. (1979) Barnard’s Monte Carlo test-how many simulations?
*Appl. Statist*.,**28**, 75–77.CrossRefGoogle Scholar - Pocock, S. J. (1977) Group sequential methods in the design and analysis of clinical trials.
*Biometrika*,**64**, 191–199.CrossRefGoogle Scholar - Reid, N. (1981) Estimating the median survival time.
*Biometrika*,**68**, 601–608.MathSciNetMATHCrossRefGoogle Scholar - Schultz, J. R., Nichol, F. R., Elfring, G. L. and Weed, S. D. (1973) Multiple-stage procedures for drug screening.
*Biometrics*,**29**, 293–300.CrossRefGoogle Scholar - Slud, E. V. and Wei, L. J. (1982) Two-sample repeated significance tests based on the modified Wilcoxon statistic.
*J. Amer. Statist. Assoc*,**77**, 862–868.MathSciNetMATHCrossRefGoogle Scholar - Tsiatis, A. A. (1981) The asymptotic joint distribution of the efficient scores test for the proportional hazards model calculated over time.
*Biometrika*,**68**, 311–315.MathSciNetMATHCrossRefGoogle Scholar - Tsiatis, A. A. (1982) Group sequential methods for survival analysis with staggered entry. In
*Survival Analysis*, Monograph Series 2, (J. Crowley and R. Johnson eds), 257–268. Hayward, California: IMS Lecture Notes.CrossRefGoogle Scholar