Nelson-Aalen and Kaplan-Meier Estimators in Competing Risks

Didier Alain Njamen-Njomen, Joseph Ngatchou-Wandji

Department of Mathematics and Computer Sciences, Faculty of Sciences, University of Maroua, Maroua, Cameroon;Department of Mathematics, Faculty of Sciences, University of Yaounde 1, Yaounde, Cameroon ;.
University of Lorraine, Lorraine, France.

DOI: 10.4236/am.2014.54073                

Abstract

In this paper, stochastic processes developed by Aalen [1] [2] are adapted to the Nelson-Aalen and Kaplan-Meier [3] estimators in a context of competing risks. We focus only on the probability distributions of complete downtime individuals whose causes are known and which bring us to consider a partition of individuals into sub-groups for each cause. We then study the asymptotic properties of nonparametric estimators obtained.

Keywords

Censored Data; Counting Process; Competitive Risk; Non-Parametric Estimators; The Cumulative Incidence Function; Risk Function Specific Cause; Conditional Distribution Function

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Aalen, O.O. (1978) Nonparametric Estimation of Partial Transition Probabilities in Multiple Decrement Models. The Annals of Statistics, 6, 534-545. http://dx.doi.org/10.1214/aos/1176344198
[2]	Aalen, O.O. (1978) Nonparametric Inference for a Family of Counting Processes. The Annals of Statistics, 6, 701-726. http://dx.doi.org/10.1214/aos/1176344247
[3]	Kaplan, E.L. and Meier, P. (1958) Nonparametric Estimation from Incomplete Observations. Journal of the American Statistical Association, 53, 457-481. http://dx.doi.org/10.1080/01621459.1958.10501452
[4]	Andersen, P.K., Borgan, ?., Gill, R.D. and Keiding, N. (1993) Statistical Models Based on Counting Processes. Springer Series in Statistics, Spring-Verlag, New York,. http://dx.doi.org/10.1007/978-1-4612-4348-9
[5]	Tsiatis, A. (1975) A Nonidentifiability Aspect of the Problem of Competing Risks. Proceeding of the National Academy of Sciences of the United States of America, 72, 20-22. http://dx.doi.org/10.1073/pnas.72.1.20
[6]	Heckman, J. and Honoré, B. (1989) The Identifiability of the Competing Risks Models. Biometrika, 76, 325-330. http://dx.doi.org/10.1093/biomet/76.2.325
[7]	Fleming, T. and Harrington, D. (1990) Counting Processes and Survival Analysis. John Wiley & Sons, Inc, Hoboken.
[8]	Prentice, R.L., Kalbfleisch, J.D., Peterson, A.V., Flournoy, N., Farewell, V.T. and Breslow, N.E. (1978) The Analysis of Failure Times in the Presence of Competing Risks. Biometrics, 34, 541-554. http://dx.doi.org/10.2307/2530374
[9]	Breslow, N. and Crowley, J. (1974) A Large Sample Study of the Life Table and Product-Limit Estimates under Random Censorship. The Annals of Statistics, 2, 437-453. http://dx.doi.org/10.1214/aos/1176342705
[10]	F?ldes, A. and Rejt?, L. (1981) Strong Uniform Consistency for Nonparametric Survival Curve Estimators from Randomly Censored Data. The Annals of Statistics, 9, 122-129. http://dx.doi.org/10.1214/aos/1176345337
[11]	Major, P. and Rejt?, L. (1998) Strong Embedding of the Estimator of the Distribution Function under Random Censorship. The Annals of Statistics, 16, 1113-1132. http://dx.doi.org/10.1214/aos/1176350949
[12]	F?ldes, A. and Rejt?, L. (1981) A LIL-Type Result for the Product-Limit Estimator. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 56, 75-86.
[13]	Gill, R. (1983) Large Sample Behavior of the Product-Limit Estimator on the Whole Line. The Annals of Statistics, 11, 49-58. http://dx.doi.org/10.1214/aos/1176346055
[14]	Cs?rg?, S. and Horváth, L. (1981) On the Koziol-Green Model for Random Censorship. Biometrika, 68, 391-401.
[15]	Ying, Z. (1989) A Note on the Asymptotic Properties of the Product-Limit Estimator on the Whole Line. Statistics & Probability Letters, 7, 311-314. http://dx.doi.org/10.1016/0167-7152(89)90113-2
[16]	Chen, K. and Lo, S.-H. (1997) On the Rate of Uniform Convergence of the Product-Limit Estimator: Strong and Weak Laws. The Annals of Statistics, 25, 1050-1087. http://dx.doi.org/10.1214/aos/1069362738
[17]	Latouche, A. (2004) Modèles de Régression en Présence de Compétition. Thèse de Doctorat, Université de Paris, Paris.
[18]	Belot, A. (2009) Modélisation Flexible des Données de Survie en Présence de Risques Concurrents et Apports de la Méthode du Taux en Excès. Thèse de Doctorat, Université de la Méditerranée, Marseille.
[19]	Fine, J.P. and Gray, R.J. (1999) A Proportional Hazards Model for the Subdistribution of a Competing Risk. Journal of the American Statistical Association, 99, 496-509. http://dx.doi.org/10.1080/01621459.1999.10474144
[20]	Aalen, O.O. and Johansen, S. (1978) An Empirical Transition Matrix for Non-Homogeneous Markov Chains Based on Censored Observations. Scandinavian Journal of Statistics, 5, 141-150.
[21]	Giné, E. and Guillou, E. (1999) Laws of the Iterated Logarithm for Censored Data. The Annals of Probability, 27, 2042-2067. http://dx.doi.org/10.1214/aop/1022874828
[22]	Cox, D. and Oakes, D. (1984) Analysis of Survival Data. Chapman and Hall, London.
[23]	Kalbfleisch, J. and Prentice, R. (1980) The Statistical Analysis of Failure Time Data. John Wiley, New York.
[24]	Nelson, W. (1969) Hazard Plotting for Incomplete Observations. Journal of Quality Technology, 1, 27-52.
[25]	Nelson, W. (1972) A Short Life Test for Comparing a Sample with Previous Accelerated Test Results. Technometrics, 14, 175-185. http://dx.doi.org/10.1080/00401706.1972.10488894
[26]	Cs?rg?, S. (1996) Universal Gaussian Approximations under Random Censorship. The Annals of Statistics, 24, 27442778. http://dx.doi.org/10.1214/aos/1032181178
[27]	Satten, G.A. and Datta, S. (1999) Kaplan-Meier Representation of Competing Risk Estimates. Statistics & Probability Letters, 42, 299-304. http://dx.doi.org/10.1016/S0167-7152(98)00220-X
[28]	Datta, S. and Satten, G.A. (2000) Estimating Future Stage Entry and Occupation Probabilities in a Multistage Model Based on Randomly Right-Censored Data. Statistics & Probability Letters, 50, 89-95. http://dx.doi.org/10.1016/S0167-7152(00)00086-9
[29]	Gill, R. and Johansen, S. (1990) A Survey of Product-Integration with a View toward Application in Survival Analysis. The Annals of Statistics, 18, 1501-1555. http://dx.doi.org/10.1214/aos/1176347865
[30]	Breuils, C. (2003) Analyse de Durées de Vie: Analyse Séquentielle du Modèle des Risques Proportionnels et Tests d’Homogénéité. Thèse de Doctorat, Université de Technologie de Compiègne, Compiègne.
[31]	Deheuvels, P. and Einmahl, J. (1996) On the Strong Limiting Behavior of Local Functionals of Empirical Processes Based upon Censored Data. The Annals of Statistics, 24, 504-525. http://dx.doi.org/10.1214/aop/1042644729
[32]	Deheuvels, P. and Einmahl, J. (2000) Functional Limit Laws for the Increments of Kaplan-Meier Product-Limit Processes and Applications. The Annals of Statistics, 28, 1301-1335. http://dx.doi.org/10.1214/aop/1019160336
[33]	Stute, W. (1994) Strong and Weak Representations of Cumulative Hazard Function and Kaplan-Meier Estimators on Increasing Sets. Journal of Statistical Planning and Inference, 42, 315-329. http://dx.doi.org/10.1016/0378-3758(94)00032-8
[34]	Giné, E. and Guillou, A. (2001) On Consistency of Kernel Density Estimators for Randomly Censored Data. Annales de l’Institut Henri Poincare (B) Probability and Statistics, 37, 503-522.