Empirical likelihood confidence bands for mean functions of recurrent events with competing risks and a terminal event

Dauxois, Jean-Yves; Flesch, Alexis; Varron, Davit

ESAIM: Probability and Statistics, Tome 20 (2016), pp. 66-94.

In this paper, we consider recurrent events with competing risks in the presence of a terminal event and a censorship. We focus our attention on the mean functions which give the expected number of events of a specific type that have occurred up to a time $t$. Using heuristics from empirical likelihood theory, we propose a method to build simultaneous (in $t$) confidence regions for these functions. To establish the consistency of this estimation method (as well as its bootstrap calibration), we prove a weak convergence (as stochastic processes) of the associated empirical likelihood ratio processes. Our approach almost entirely relies on empirical process methods. In the proofs, we also establish some results in empirical processes theory that may present some independent interest. Then we carry out a simulation study of our confidence bands, we compare those obtained by empirical likelihood to the ones obtained by bootstrap. Finally, our procedure is applied on a real data set of nosocomial infections in an intensive care unit of a French hospital.

G. Adimari, Empirical likelihood type confidence intervals under random censorship. Ann. Inst. Statist. Math. 49 (1997) 447–466. | DOI | MR | Zbl

P.K. Andersen, Ø. Borgan, R.D. Gill and N. Keiding, Statistical models based on counting processes. Springer Ser. Stat. Springer-Verlag, New York (1993). | MR

Y. Bilias, M. Gu and Zh. Ying, Towards a general asymptotic theory for Cox model with staggered entry. Ann. Statist. 25 (1997) 662–682. | DOI | MR | Zbl

J. Cai and D.E. Schaubel, Analysis of recurrent event data. In Advances in survival analysis, Vol. 23 of Handbook of Statist. Elsevier, Amsterdam (2004) 603–623. | MR

J. Cai and D.E. Schaubel, Marginal means/rates models for multiple type recurrent event data. Lifetime Data Anal. 10 (2004) 121–138. | DOI | MR | Zbl

B.E. Chen and R.J. Cook, Tests for multivariate recurrent events in the presence of a terminal event. Biostatistics 5 (2004) 237–251. | Zbl

B.E. Chen, R.J. Cook, J.F. Lawless and M. Zhan, Statistical methods for multivariate interval-censored recurrent events. Stat. Med. 24 (2005) 671–691. | DOI | MR

J. Chen, L. Peng and Y. Zhao, Empirical likelihood based confidence intervals for copulas. J. Multivariate Anal. 100 (2009) 137–151 | DOI | MR | Zbl

S.X. Chen and I. Van Keilegom, A review on empirical likelihood methods for regression. TEST 18 (2009) 415–447. | DOI | MR | Zbl

S.X. Chen, L. Peng and Y. Qin, Effects of data dimension on empirical likelihood. Biometrika 96 (2009) 711–722. | DOI | MR | Zbl

R.J. Cook and J.F. Lawless, Marginal analysis of recurrent events and a terminating event. Stat. Med. 16 (1997) 911–924. | DOI

R.J. Cook and J.F. Lawless, The Statistical Analysis of Recurrent Events. Statistics for Biology and Health. Springer-Verlag (2007). | MR | Zbl

R.J. Cook, J.F. Lawless and C. Nadeau, Robust tests for treatment comparisons based on recurrent event responses. Biometrics 52 (1996) 557–571. | DOI | Zbl

R.J. Cook, J.F. Lawless, L. Lakhal-Chaieb and K. Lee, Robust estimation of mean functions and treatment effects for recurrent events under event-dependent censoring and termination: application to skeletal complications in cancer metastatic to bone. J. Amer. Statist. Assoc. 104 (2009) 60–75. | DOI | MR | Zbl

J-Y. Dauxois and S. Sencey, Non-parametric tests for recurrent events under competing risks. Scand. J. Stat. 36 (2009) 649–670. | DOI | MR | Zbl

L. Devroye and G. Lugosi, Combinatorial methods in density estimation. Springer, New York (2001). | MR | Zbl

P. Du, Nonparametric modeling of the gap time in recurrent event data. Lifetime Data Anal. 15 (2009) 256–277. | DOI | MR | Zbl

D. Ghosh, Accelerated rates regression models for recurrent failure time data. Lifetime Data Anal. 10 (2004) 247–261. | DOI | MR | Zbl

D. Ghosh and D.Y. Lin, Nonparametric analysis of recurrent events and death. Biometrics 56 (2000) 554–562. | DOI | MR | Zbl

D. Ghosh and D.Y. Lin, Marginal regression models for recurrent and terminal events. Statist. Sinica 12 (2002) 663–688. | MR | Zbl

D. Ghosh and D.Y. Lin, Semiparametric analysis of recurrent events data in the presence of dependent censoring. Biometrics 59 (2003) 877–885. | DOI | MR | Zbl

D.J. Hand, The statistical analysis of recurrent events by Richard R. Cook, Jerald F. Lawless. Int. Stat. Rev. 76 (2008) 142–143. | DOI

N.L. Hjort, I.W. Mckeague and I. Van Keilegom, Extending the scope of empirical likelihood. Ann. Statist. 37 (2009) 1079–1111. | DOI | MR | Zbl

B. Jing, J. Yuan and W. Zhou, Jackknife empirical likelihood. J. Amer. Statist. Assoc. 104 (2009) 1224–1232. | DOI | MR | Zbl

A. Keziou and S. Leoni-Aubin, On empirical likelihood for semiparametric two-sample density ratio models. J. Statist. Plann. Inference 138 (2008) 915–928. | DOI | MR | Zbl

J.F. Lawless, The analysis of recurrent events for multiple subjects. J. Roy. Statist. Soc. Ser. C 44 (1995) 487–498.

J.F. Lawless and C. Nadeau, Some simple robust methods for the analysis of recurrent events. Technometrics 37 (1995) 158–168. | DOI | MR | Zbl

G. Li, R. Li and M. Zhou, Empirical likelihood in survival analysis. In Contemporary multivariate analysis and design of experiments. Vol. 2 of Ser. Biostat. World Sci. Publ., Hackensack, NJ (2005) 337–349. | MR | Zbl

D.Y. Lin and Z. Ying, Nonparametric tests for the gap time distributions of serial events based on censored data. Biometrics 57 (2001) 369–375. | DOI | MR | Zbl

D.Y. Lin, L.J. Wei and Zhiliang Ying. Accelerated failure time models for counting processes. Biometrika 85 (1998) 605–618. | DOI | MR | Zbl

D.Y. Lin, W. Sun and Z. Ying, Nonparametric estimation of the gap time distributions for serial events with censored data. Biometrika 86 (1999) 59–70. | DOI | MR | Zbl

D.Y. Lin, L.J. Wei, I. Yang and Z. Ying, Semiparametric regression for the mean and rate functions of recurrent events. J. R. Stat. Soc. Ser. B Stat. Methodol. 62 (2000) 711–730. | DOI | MR | Zbl

M. Miloslavsky, S. Keleş, M.J. Van Der Laan and S. Butler. Recurrent events analysis in the presence of time-dependent covariates and dependent censoring. J. R. Stat. Soc. Ser. B 66 (2004) 239–257. | DOI | MR | Zbl

A.B. Owen, Empirical likelihood ratio confidence intervals for a single functional. Biometrika 75 (1988) 237–249. | DOI | MR | Zbl

A.B. Owen, Empirical likelihood for linear models. Ann. Statist. 19 (1991) 1725–1747. | MR | Zbl

A.B. Owen, Empirical likelihood ratio confidence regions. Ann. Statist. 18 (1990) 90–120. | MR | Zbl

A.B. Owen, Empirical Likelihood. Monographs on Statistics and Applied Probability. Taylor & Francis (2001). | Zbl

A. Pakes and D. Pollard, Simulation and the asymptotics of optimization estimators. Econometrica 57 (1989) 1027–1057. | DOI | MR | Zbl

D. Pollard, Empirical processes: theory and applications. Vol. 2 of NSF-CBMS Regional Conference Series in Probability and Statistics. Institute of Mathematical Statistics, Hayward, CA (1990). | MR | Zbl

J. Ren, Weighted empirical likelihood ratio confidence intervals for the mean with censored data. Ann. Inst. Statist. Math. 53 (2001) 498–516. | DOI | MR | Zbl

J. Ren, Smoothed weighted empirical likelihood ratio confidence intervals for quantiles. Bernoulli 14 (2008) 725–748. | MR | Zbl

J. Ren, Weighted empirical likelihood in some two-sample semiparametric models with various types of censored data. Ann. Statist. 36 (2008) 147–166. | MR | Zbl

D.R. Thomas and G.L. Grunkemeier, Confidence interval estimation of survival probabilities for censored data. J. Amer. Statist. Assoc. 70 (1975) 865–871. | DOI | MR | Zbl

A.W. Van der Vaart and J.A. Wellner, Weak convergence and empirical processes. Springer (1996). | MR | Zbl

D. Varron, Empirical likelihood confidence bands for functional parameters in plug-in estimation. Preprint (2014). | MR

Q. Wang and B. Jing, Empirical likelihood for a class of functionals of survival distribution with censored data. Ann. Inst. Statist. Math. 53 (2001) 517–527. | DOI | MR | Zbl

W. Wang and M.T. Wells, Nonparametric estimation of successive duration times under dependent censoring. Biometrika 85 (1998) 561–572. | DOI | MR | Zbl