Importance resampling is an approach that uses exponential tilting to reduce the resampling necessary for the construction of nonparametric bootstrap confidence intervals. The properties of bootstrap importance confidence intervals are well established when the data is a smooth function of means and when there is no censoring. However, in the framework of survival or time-to-event data, the asymptotic properties of importance resampling have not been rigorously studied, mainly because of the unduly complicated theory incurred when data is censored. This paper uses extensive simulation to show that, for parameter estimates arising from fitting Cox proportional hazards models, importance bootstrap confidence intervals can be constructed if the importance resampling probabilities of the records for the n individuals in the study are determined by the empirical influence function for the parameter of interest. Our results show that, compared to uniform resampling, importance resampling improves the relative mean-squared-error (MSE) efficiency by a factor of nine (for n = 200). The efficiency increases significantly with sample size, is mildly associated with the amount of censoring, but decreases slightly as the number of bootstrap resamples increases. The extra CPU time requirement for calculating importance resamples is negligible when compared to the large improvement in MSE efficiency. The method is illustrated through an application to data on chronic lymphocytic leukemia, which highlights that the bootstrap confidence interval is the preferred alternative to large sample inferences when the distribution of a specific covariate deviates from normality. Our results imply that, because of its computational efficiency, importance resampling is recommended whenever bootstrap methodology is implemented in a survival framework. Its use is particularly important when complex covariates are involved or the survival problem to be solved is part of a larger problem; for instance, when determining confidence bounds for models linking survival time with clusters identified in gene expression microarray data.
[1]
Debashis Kushary,et al.
Bootstrap Methods and Their Application
,
2000,
Technometrics.
[2]
E. Mammen.
The Bootstrap and Edgeworth Expansion
,
1997
.
[3]
William H. Press,et al.
Numerical recipes in C
,
2002
.
[4]
TWO-SAMPLE NONPARAMETRIC TILTING METHOD
,
1997
.
[5]
G. Jasso.
Review of "International Encyclopedia of Statistical Sciences, edited by Samuel Kotz, Norman L. Johnson, and Campbell B. Read, New York, Wiley, 1982-1988"
,
1989
.
[6]
Robert Tibshirani,et al.
Bootstrap Methods for Standard Errors, Confidence Intervals, and Other Measures of Statistical Accuracy
,
1986
.
[7]
C. Mcgilchrist,et al.
Regression with frailty in survival analysis.
,
1991,
Biometrics.
[8]
Cedric A. B. Smith.
Discussion of Paper by D.V. Lindley
,
1982
.
[9]
Bradley Efron,et al.
Censored Data and the Bootstrap
,
1981
.
[10]
Peter Hall,et al.
On importance resampling for the bootstrap
,
1991
.
[11]
D. Collet.
Modelling Survival Data in Medical Research
,
2004
.
[12]
A. Tsiatis.
A Large Sample Study of Cox's Regression Model
,
1981
.
[13]
Nancy Reid,et al.
Influence functions for proportional hazards regression
,
1985
.
[14]
David R. Cox,et al.
Regression models and life tables (with discussion
,
1972
.
[15]
Deborah Burr,et al.
A Comparison of Certain Bootstrap Confidence Intervals in the Cox Model
,
1994
.
[16]
M. Johns.
Importance Sampling for Bootstrap Confidence Intervals
,
1988
.
[17]
F. A. Seiler,et al.
Numerical Recipes in C: The Art of Scientific Computing
,
1989
.