Exact Nonparametric Confidence Bands for the Survivor Function
-
David Matthews
Abstract
A method to produce exact simultaneous confidence bands for the empirical cumulative distribution function that was first described by Owen, and subsequently corrected by Jager and Wellner, is the starting point for deriving exact nonparametric confidence bands for the survivor function of any positive random variable. We invert a nonparametric likelihood test of uniformity, constructed from the Kaplan–Meier estimator of the survivor function, to obtain simultaneous lower and upper bands for the function of interest with specified global confidence level. The method involves calculating a null distribution and associated critical value for each observed sample configuration. However, Noe recursions and the Van Wijngaarden–Decker–Brent root-finding algorithm provide the necessary tools for efficient computation of these exact bounds. Various aspects of the effect of right censoring on these exact bands are investigated, using as illustrations two observational studies of survival experience among non-Hodgkin’s lymphoma patients and a much larger group of subjects with advanced lung cancer enrolled in trials within the North Central Cancer Treatment Group. Monte Carlo simulations confirm the merits of the proposed method of deriving simultaneous interval estimates of the survivor function across the entire range of the observed sample.
This research was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada. It was begun while the author was visiting the Department of Statistics, University of Auckland, and completed during a subsequent sojourn at the Medical Research Council Biostatistics Unit in Cambridge. The support of both institutions, in addition to that of NSERC and the University of Waterloo, is greatly appreciated.
References
1. OwenA. Nonparametric confidence bands for a distribution function. J Am Stat Assoc1995;90:516–21.10.1080/01621459.1995.10476543Search in Google Scholar
2. BerkR, JonesD. Goodness-of-fit test statistics that dominate the Kolmogorov statistics. Z Wahrscheinlichkeitstheorie Verwandte Geb1979;47:47–59.10.1007/BF00533250Search in Google Scholar
3. NoéM. The calculation of distributions of two-sided Kolmogorov–Smirnov-type statistics. Ann Math Stat1972;43:58–64.10.1214/aoms/1177692700Search in Google Scholar
4. JagerL, WellnerJ. A new goodness of fit test: the reversed Berk–Jones statistic. Technical Report 443, Department of Statistics, University of Washington, 2005.Search in Google Scholar
5. JagerL, WellnerJ. Goodness-of-fit tests via phi-divergences. Ann Stat2007;35:2018–53.10.1214/0009053607000000244Search in Google Scholar
6. KhmaladzeE, ShinjikashviliE. Calculation of noncrossing probabilities for Poisson processes and its corollaries. Adv Appl Probability2001;33:702–16.10.1017/S0001867800011083Search in Google Scholar
7. FreyJ. Optimal distribution-free confidence bands for a distribution function. J Stat Plan Inference2008;138:3086–98.10.1016/j.jspi.2007.12.001Search in Google Scholar
8. XuX, DingX, ZhaoS. A new confidence band for continuous cumulative distribution function. Austr N Z J Stat2009;51:305–18.10.1111/j.1467-842X.2009.00546.xSearch in Google Scholar
9. KaplanE, MeierP. Nonparametric estimation from incomplete observations. J Am Stat Assoc1958;53:457–81.10.1080/01621459.1958.10501452Search in Google Scholar
10. GillespieMJ, FisherL. Confidence bands for the Kaplan–Meier survival curve estimate. Ann Stat1979;7:920–4.10.1214/aos/1176344742Search in Google Scholar
11. HallW, WellnerJA. Confidence bands for a survival curve from censored data. Biometrika1980;67:133–43.10.1093/biomet/67.1.133Search in Google Scholar
12. NairV. Confidence bands for survival functions with censored data: a comparative study. Technometrics1984;14:265–75.10.1080/00401706.1984.10487964Search in Google Scholar
13. HollanderM, PeñaA. Families of confidence bands for the survival function under the general random censorship model and the Koziol-Green model. Can J Stat1989;17:59–74.10.2307/3314763Search in Google Scholar
14. HollanderM, McKeagueI, YangJ. Likelihood ratio-based confidence bands for survival functions. J Am Stat Assoc1997;92:215–26.10.1080/01621459.1997.10473619Search in Google Scholar
15. OwenA. Empirical likelihood. Boca Raton, FL: Chapman & Hall/CRC, 2001.Search in Google Scholar
16. PressW, FlanneryB, TeukolskyS, VetterlingW. Numerical recipes: the art of scientific computing. Cambridge, UK: Cambridge University Press, 1986.Search in Google Scholar
17. MatthewsD, FarewellV. Using and understanding medical statistics. Basel, Switzerland: S Karger AG, 4th completely revised and enlarged edition, 2007.Search in Google Scholar
18. LoprinziC, LaurieJ, WieandH, KrookJ, NovotnyP, KuglerJ, et al. Prospective evaluation of prognostic variables from patient-completed questionnaires. J Clin Oncol1994;12:601–7.10.1200/JCO.1994.12.3.601Search in Google Scholar PubMed
19. R Core Team. R: a language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing, 2011. Available at: http://www.R-project.org/. ISBN 3-900051-07-0.Search in Google Scholar
20. BorganØ, LiestølK. A note on confidence intervals and bands for the survival function based on transformations. Scand J Stat1990;17:35–41.Search in Google Scholar
21. FarewellV. Some comments on analysis techniques for censored water quality data. Environ Monit Assess1989;13:285–94.10.1007/BF00394234Search in Google Scholar PubMed
22. LawlessJ. Statistical models and methods for lifetime data, 2nd ed. Wiley Series in Probability and Statistics. Hoboken, NJ: John Wiley & Sons, 2003.10.1002/9781118033005Search in Google Scholar
©2013 by Walter de Gruyter Berlin / Boston
Articles in the same Issue
- Masthead
- Masthead
- Research Articles
- Sensitivity Analysis for Causal Inference under Unmeasured Confounding and Measurement Error Problems
- Assessing the Causal Effect of Policies: An Example Using Stochastic Interventions
- Novel Point Estimation from a Semiparametric Ratio Estimator (SPRE): Long-Term Health Outcomes from Short-Term Linear Data, with Application to Weight Loss in Obesity
- Exact Nonparametric Confidence Bands for the Survivor Function
- Semiparametric Regression Analysis of Clustered Interval-Censored Failure Time Data with Informative Cluster Size
- A Weighting Analogue to Pair Matching in Propensity Score Analysis
- Alternative Monotonicity Assumptions for Improving Bounds on Natural Direct Effects
- Estimation of Risk Ratios in Cohort Studies with a Common Outcome: A Simple and Efficient Two-stage Approach
- Distance-Based Mapping of Disease Risk
- The Balanced Survivor Average Causal Effect
- Commentary
- Principal Stratification: A Broader Vision
Articles in the same Issue
- Masthead
- Masthead
- Research Articles
- Sensitivity Analysis for Causal Inference under Unmeasured Confounding and Measurement Error Problems
- Assessing the Causal Effect of Policies: An Example Using Stochastic Interventions
- Novel Point Estimation from a Semiparametric Ratio Estimator (SPRE): Long-Term Health Outcomes from Short-Term Linear Data, with Application to Weight Loss in Obesity
- Exact Nonparametric Confidence Bands for the Survivor Function
- Semiparametric Regression Analysis of Clustered Interval-Censored Failure Time Data with Informative Cluster Size
- A Weighting Analogue to Pair Matching in Propensity Score Analysis
- Alternative Monotonicity Assumptions for Improving Bounds on Natural Direct Effects
- Estimation of Risk Ratios in Cohort Studies with a Common Outcome: A Simple and Efficient Two-stage Approach
- Distance-Based Mapping of Disease Risk
- The Balanced Survivor Average Causal Effect
- Commentary
- Principal Stratification: A Broader Vision
