Confidence intervals for the cumulative incidence function via constrained NPMLE. (English) Zbl 1437.62361

Summary: The cumulative incidence function (CIF) displays key information in the competing risks setting, which is common in medical research. In this article, we introduce two new methods to compute non-parametric confidence intervals for the CIF. First, we introduce non-parametric profile-likelihood confidence intervals. The method builds on constrained non-parametric maximum likelihood estimation (NPMLE), for which we derive closed-form formulas. This method can be seen as an extension of that of D. R. Thomas and G. L. Grunkemeier [J. Am. Stat. Assoc. 70, 865–871 (1975; Zbl 0331.62028)] to the competing risks setting, when the CIF is of interest instead of the survival function. Second, we build on constrained NPMLE to introduce constrained bootstrap confidence intervals. This extends an interesting approach introduced by S. Barber and C. Jennison [Biometrics 55, No. 2, 430–436 (1999; Zbl 1059.62603)] to the competing risks setting. A simulation study illustrates how these methods can perform as compared to benchmarks implemented in popular software. The results suggest that more accurate confidence intervals than usual Wald-type ones can be obtained in the case of small to moderate sample sizes and few observed events. An application to melanoma data is provided for illustration purpose.


62N01 Censored data models
62G15 Nonparametric tolerance and confidence regions
62G07 Density estimation
62P10 Applications of statistics to biology and medical sciences; meta analysis
Full Text: DOI


[1] Aalen, O.; Borgan, Ø.; Gjessing, Hk; Gjessing, S., Survival and event history analysis: a process point of view (2008), Berlin: Springer, Berlin · Zbl 1204.62165
[2] Aalen, Oo; Johansen, S., An empirical transition matrix for non-homogeneous markov chains based on censored observations, Scand J Stat, 5, 141-150 (1978) · Zbl 0383.62058
[3] Allignol, A.; Schumacher, M.; Beyersmann, J., Empirical transition matrix of multi-state models: the etm package, J Stat Softw, 38, 4, 1-15 (2011)
[4] Andersen, Pk; Borgan, Ø.; Gill, Rd; Keiding, N., Statistical models based on counting processes (1993), New York: Springer, New York
[5] Andersen, Pk; Geskus, Rb; De Witte, T.; Putter, H., Competing risks in epidemiology: possibilities and pitfalls, Int J Epidemiol, 41, 3, 861-870 (2012)
[6] Andersen, Pk; Skovgaard, Lt, Regression with linear predictors (2010), Berlin: Springer, Berlin
[7] Barber, S.; Jennison, C., Symmetric tests and confidence intervals for survival probabilities and quantiles of censored survival data, Biometrics, 55, 2, 430-436 (1999) · Zbl 1059.62603
[8] Beyersmann, J.; Allignol, A.; Schumacher, M., Competing risks and multistate models with R (2011), Berlin: Springer, Berlin
[9] Borgan, Ø.; Liestøl, K., A note on confidence intervals and bands for the survival function based on transformations, Scand J Stat, 17, 35-41 (1990)
[10] Braun, Tm; Yuan, Z., Comparing the small sample performance of several variance estimators under competing risks, Stat Med, 26, 5, 1170-1180 (2007)
[11] Canty A, Ripley BD (2017) Boot: Bootstrap R (S-Plus) Functions. R package version 1.3-20
[12] Choudhury, Jb, Non-parametric confidence interval estimation for competing risks analysis: application to contraceptive data, Stat Med, 21, 8, 1129-1144 (2002)
[13] Davison, Ac; Hinkley, Dv, Bootstrap methods and their application (1997), Cambridge: Cambridge University Press, Cambridge
[14] Drzewiecki, K.; Andersen, Pk, Survival with malignant melanoma: a regression analysis of prognostic factors, Cancer, 49, 2414-2419 (1982)
[15] Efron, B., Better bootstrap confidence intervals, J Am Stat Assoc, 82, 397, 171-185 (1987) · Zbl 0622.62039
[16] Fay, Mp; Brittain, Eh; Proschan, Ma, Pointwise confidence intervals for a survival distribution with small samples or heavy censoring, Biostatistics, 14, 4, 723-736 (2013)
[17] Gerds TA (2017) Prodlim: Product-limit estimation for censored event history analysis. R package version 1.6.1
[18] Geskus, Rb, Data analysis with competing risks and intermediate states (2015), Boca Raton: CRC Press, Boca Raton
[19] Hollander, M.; Mckeague, Iw; Yang, J., Likelihood ratio-based confidence bands for survival functions, J Am Stat Assoc, 92, 437, 215-226 (1997) · Zbl 1090.62560
[20] Jennison, C.; Page, C.; Lepage, R., Bootstrap tests and confidence intervals for a hazard ratio when the number of observed failures is small, with applications to group sequential survival studies, Computing science and statistics, 89-97 (1992), New York: Springer-Verlag, New York
[21] Johansen, S., The product limit estimator as maximum likelihood estimator, Scand J Stat, 5, 195-199 (1978) · Zbl 0385.62059
[22] Latouche, A.; Allignol, A.; Beyersmann, J.; Labopin, M.; Fine, Jp, A competing risks analysis should report results on all cause-specific hazards and cumulative incidence functions, J Clin Epidemiol, 66, 6, 648-653 (2013)
[23] Li, G., On nonparametric likelihood ratio estimation of survival probabilities for censored data, Stat Probab Lett, 25, 2, 95-104 (1995) · Zbl 0851.62026
[24] Lin, D., Non-parametric inference for cumulative incidence functions in competing risks studies, Stat Med, 16, 8, 901-910 (1997)
[25] Logan, Br; Zhang, M-J, The use of group sequential designs with common competing risks tests, Stat Med, 32, 6, 899-913 (2013)
[26] Martinussen, T.; Scheike, Th, Dynamic regression models for survival data (2006), New York: Springer, New York · Zbl 1096.62119
[27] Owen, Ab, Empirical likelihood (2001), New York: Chapman and Hall/CRC, New York
[28] Pfeiffer, Rm; Gail, Mh, Absolute risk: methods and applications in clinical management and public health (2017), Boca Raton: CRC Press, Boca Raton
[29] Schumacher, M.; Ohneberg, K.; Beyersmann, J., Competing risk bias was common in a prominent medical journal, J Clin Epidemiol, 80, 135-136 (2016)
[30] Therneau TM (2015) A package for survival analysis in S. version 2.41-3
[31] Therneau, Tm; Grambsch, Pm, Modeling survival data: extending the Cox model (2000), Berlin: Springer, Berlin · Zbl 0958.62094
[32] Thomas, Dr; Grunkemeier, Gl, Confidence interval estimation of survival probabilities for censored data, J Am Stat Assoc, 70, 352, 865-871 (1975) · Zbl 0331.62028
[33] Zhou, M., Empirical likelihood method in survival analysis (2016), Boca Raton: CRC Press, Boca Raton · Zbl 1341.62031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.