zbMATH — the first resource for mathematics

The Kaplan-Meier integral in the presence of covariates: a review. (English) Zbl 1383.62215
Ferger, Dietmar (ed.) et al., From statistics to mathematical finance. Festschrift in honour of Winfried Stute. Cham: Springer (ISBN 978-3-319-50985-3/hbk; 978-3-319-50986-0/ebook). 25-41 (2017).
Summary: In a series of papers, Winfried Stute introduced and studied the Kaplan-Meier integral as an estimator of parameters of the joint distribution of survival times and covariates based on right censored survival times. We present a review of this work and show that his estimator has an inverse probability of censoring weighting (IPCW) representation. We further investigate large sample bias and efficiency. As a central application in a biostatistical context, Kaplan-Meier integrals are used to estimate transition probabilities in a non-Markov illness-death model. We extend already existing approaches by introducing a novel estimator that also works in the presence of additional left truncation. This application illustrates that Winfried Stute’s work can successfully be used to develop inferential statistical methods in complex survival models.
For the entire collection see [Zbl 1383.62010].

MSC:
 62N02 Estimation in survival analysis and censored data 62N05 Reliability and life testing 62P10 Applications of statistics to biology and medical sciences; meta analysis 62-02 Research exposition (monographs, survey articles) pertaining to statistics 62-03 History of statistics
Stute, Winfried
dynpred
Full Text:
References:
 [1] Akritas, M. G. (1994). Nearest neighbor estimation of a bivariate distribution under random censoring. $$Ann. Statist.$$ , 22:1299-1327. · Zbl 0819.62028 [2] Akritas, M. G. (2000). The central limit theorem under censoring. $$Bernoulli$$ , pages 1109-1120. · Zbl 0979.60015 [3] Allignol, A., Beyersmann, J., Gerds, T., and Latouche, A. (2014). A competing risks approach for nonparametric estimation of transition probabilities in a non-Markov illness-death model. $$Lifetime Data Analysis$$ , 20:495-513. · Zbl 1359.62471 [4] Andersen, P. and Perme, M. (2008). Inference for outcome probabilities in multi-state models. $$Lifetime Data Analysis$$ , 14(4):405-431. · Zbl 1302.62226 [5] Andersen, P. K., Borgan, O., Gill, R. D., and Keiding, N. (1993). $$Statistical Models Based on Counting Processes$$ . Springer Series in Statistics. Springer, New York. · Zbl 0769.62061 [6] Anderson, J., Cain, K., and Gelber, R. (2008). Analysis of survival by tumor response and other comparisons of time-to-event by outcome variables. $$Journal of Clinical Oncology$$ , 26(24):3913-3915. [7] Begun, J. M., Hall, W. J., Huang, W.-M., and Wellner, J. A. (1983). Information and asymptotic efficiency in parametric-nonparametric models. $$The Annals of Statistics$$ , 11:432-452. · Zbl 0526.62045 [8] Bender, R., Augustin, T., and Blettner, M. (2005). Generating survival times to simulate Cox proportional hazards models. $$Statistics in medicine$$ , 24:1713-1723. [9] Bickel, P. J., Klaassen, C. A., Ritov, Y., and Wellner, J. A. (1993). $$Efficient and Adaptive Estimation for Semiparametric Models$$ . Johns Hopkins. · Zbl 0786.62001 [10] De Uña Álvarez, J. and Rodriguez-Campos, M. (2004). Strong consistency of presmoothed Kaplan-Meier integrals when covariables are present. $$Statistics$$ , 38:483-496. · Zbl 1055.62052 [11] de Uña-Álvarez, J. and Meira-Machado, L. (2015). Nonparametric estimation of transition probabilities in the non-markov illness-death model: A comparative study. $$Biometrics$$ , 71(2):364-375. · Zbl 1390.62246 [12] Gerds, T. and Schumacher, M. (2006). Consistent estimation of the expected Brier score in general survival models with right-censored event times. $$Biometrical Journal$$ , 48(6):1029-1040. [13] Gerds, T. A. (2002). $$Nonparametric efficient estimation of prediction error for incomplete data models$$ . PhD thesis, Albert-Ludwig Universität Freiburg. · Zbl 1042.62530 [14] Gilbert, P. B., McKeague, I. W., and Sun, Y. (2008). The 2-sample problem for failure rates depending on a continuous mark: an application to vaccine efficacy. $$Biostatistics$$ , 9(2):263-276. · Zbl 1143.62078 [15] Gill, R. D. (1980). Censoring and stochastic integrals. Mathematical Centre Tracts 124, Mathematisch Centrum, Amsterdam. · Zbl 0456.62003 [16] Gill, R. D., Van der Laan, M. J., and Robins, J. M. (1995). Coarsening at random: Characterizations, conjectures and counter-examples. In Lin, D. Y. and Fleming, T. R., editors, $$Proceedings of the First Seattle Symposium in Biostatistics$$ , pages 255-294. Springer Lecture Notes in Statistics. · Zbl 0918.62003 [17] Graf, E., Schmoor, C., Sauerbrei, W. F., and Schumacher, M. (1999). Assessment and comparison of prognostic classification schemes for survival data. $$Statist. Med.$$ , 18:2529-2545. [18] Grüger, J., Kay, R., and Schumacher, M. (1991). The validity of inference based on incomplete observations in disease state models. $$Biometrics$$ , 47:595-605. [19] Hudson, H. M., Lô, S. N., John Simes, R., Tonkin, A. M., and Heritier, S. (2014). Semiparametric methods for multistate survival models in randomised trials. $$Statistics in medicine$$ , 33(10):1621-1645. [20] Keiding, N. (1992). Independent delayed entry. In Klein, J. and Goel, P., editors. $$Survival analysis: state of the art, Kluwer, Dordrecht$$ , pages 309-326. · Zbl 0761.62156 [21] Malani, H. M. (1995). A modification of the redistribution to the right algorithm using disease markers. $$Biometrika$$ , 82:515-526. · Zbl 0845.62079 [22] Meira-Machado, L., de Uña Álvarez, J., and Suárez, C. (2006). Nonparametric estimation of transition probabilities in a non-markov illness-death model. $$Lifetime Data Analysis$$ , 12(3):325-344. · Zbl 1356.62127 [23] Neuhaus, G. (2000). A method of constructing rank tests in survival analysis. $$Journal of Statistical Planning and inference$$ , 91(2):481-497. · Zbl 0965.62082 [24] Orbe, J., Ferreira, E., and Núñez-Antón, V. (2002). Comparing proportional hazards and accelerated failure time models for survival analysis. $$Statistics in medicine$$ , 21(22):3493-3510. [25] Orbe, J., Ferreira, E., and Nunez-Anton, V. (2003). Censored partial regression. $$Biostatistics$$ , 4:109-121. · Zbl 1139.62307 [26] Reeds, J. A. (1976). $$On the definition of von Mises Functionals$$ . PhD thesis, Havard University, Cambridge, Massachusetts. [27] Robins, J. M. and Rotnitzky, A. (1992). Recovery of information and adjustment for dependent censoring using surrogate markers. In Jewell, N. P., Dietz, K., and Farewell, V. T., editors, $$AIDS Epidemiology, Methodological Issues$$ , pages 297-331. Birkh”auser, Boston. [28] Rotnitzky, A. and Robins, J. M. (1995). Semiparametric regression estimation in the presence of dependent censoring. $$Biometrika$$ , 82:805-820. · Zbl 0861.62030 [29] Satten, G. and Datta, S. (2001). The kaplan-meier estimator as an inverse-probability-of-censoring weighted average. $$The American Statistician$$ , 55(3):207-210. · Zbl 1182.62191 [30] Stute, W. (1993). Consistent estimation under random censorship when covariables are present. $$J. Multivariate Anal.$$ , 45:89-103. · Zbl 0767.62036 [31] Stute, W. (1996). Distributional convergence under random censorship when covariables are present. $$Scand. J. Statist.$$ , 23:461-71. · Zbl 0903.62045 [32] Stute, W. (1999). Nonlinear censored regression. $$Statistica Sinica$$ , 9:1089-1102. · Zbl 0940.62061 [33] Tsai, W. and Crowley, J. (1998). A note on nonparametric estimators of the bivariate survival function under univariate censoring. $$Biometrika$$ , 85(3):573-580. · Zbl 0954.62037 [34] Tsiatis, A. (1975). A nonidentifiability aspect of the problem of competing risks. $$Proc. Natl. Acad. Sci.$$ , 72:20-22. · Zbl 0299.62066 [35] Van der Laan, M. and Robins, J. (2003). $$Unified Methods for Censored Longitudinal Data and Causality$$ . Springer. · Zbl 1013.62034 [36] van der Laan, M. J., Hubbard, A. E., and Robins, J. (2002). Locally efficient estimation of a multivariate survival function in longitudinal studies. $$Journal of the American Statistical Association$$ , 97:494-507. · Zbl 1073.62568 [37] Van der Vaart, A. W. (1991). On differentiable functionals. $$Ann. Statist.$$ , 19:178-204. · Zbl 0732.62035 [38] Van der Vaart, A. W. (1998). $$Asymptotic Statistics$$ . Cambridge University Press. · Zbl 0910.62001 [39] van Houwelingen, J. and Putter, H. (2012). $$Dynamic Prediction in Clinical Survival Analysis$$ . CRC Press. · Zbl 1272.62004
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.