×

The wild bootstrap for multivariate Nelson-Aalen estimators. (English) Zbl 1429.62495

Summary: We rigorously extend the widely used wild bootstrap resampling technique to the multivariate Nelson-Aalen estimator under Aalen’s multiplicative intensity model. Aalen’s model covers general Markovian multistate models including competing risks subject to independent left-truncation and right-censoring. This leads to various statistical applications such as asymptotically valid confidence bands or tests for equivalence and proportional hazards. This is exemplified in a data analysis examining the impact of ventilation on the duration of intensive care unit stay. The finite sample properties of the new procedures are investigated in a simulation study.

MSC:

62P10 Applications of statistics to biology and medical sciences; meta analysis
62N05 Reliability and life testing
62G05 Nonparametric estimation
62G10 Nonparametric hypothesis testing
62N03 Testing in survival analysis and censored data

Software:

invGauss; R; sde
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Aalen OO (1978) Nonparametric inference for a family of counting processes. Ann Stat 6(4):701-726 · Zbl 0389.62025
[2] Aalen OO, Johansen S (1978) An empirical transition matrix for non-homogeneous Markov chains based on censored observations. Scand J Stat 5(3):141-150 · Zbl 0383.62058
[3] Aalen OO, Borgan Ø, Gjessing HK (2008) Survival and event history analysis: a process point of view. Springer, New York · Zbl 1204.62165
[4] Akritas MG (1986) Bootstrapping the Kaplan-Meier estimator. J Am Stat Assoc 81(396):1032-1038 · Zbl 0635.62032
[5] Allignol A, Schumacher M, Wanner C, Drechsler C, Beyersmann J (2011) Understanding competing risks: a simulation point of view. BMC Med Res Methodol 11:86
[6] Andersen PK, Væth M (1989) Simple parametric and nonparametric models for excess and relative mortality. Biometrics 45(2):523-535 · Zbl 0715.62225
[7] Andersen PK, Borgan Ø, Gill RD, Keiding N (1993) Statistical models based on counting processes. Springer, New York · Zbl 0769.62061
[8] Bagdonavičius V, Levuliené R, Nikulin MS (2010) Goodness-of-fit criteria for the Cox model from left truncated and right censored data. J Math Sci 167(4):436-443 · Zbl 1288.62032
[9] Bajorunaite R, Klein JP (2007) Two-sample tests of the equality of two cumulative incidence functions. Comput Stat Data Anal 51(9):4269-4281 · Zbl 1162.62422
[10] Beyersmann J, Gastmeier P, Grundmann H, Bärwolff S, Geffers C, Behnke M, Rüden H, Schumacher M (2006) Use of multistate models to assess prolongation of intensive care unit stay due to nosocomial infection. Infect Control Hosp Epidemiol 27(05):493-499
[11] Beyersmann J, Wolkewitz M, Allignol A, Grambauer N, Schumacher M (2011) Application of multistate models in hospital epidemiology: advances and challenges. Biom J 53(2):332-350 · Zbl 1209.62258
[12] Beyersmann J, Allignol A, Schumacher M (2012) Competing risks and multistate models with R. Springer, New York · Zbl 1304.62002
[13] Beyersmann J, Di Termini S, Pauly M (2013) Weak convergence of the wild bootstrap for the Aalen-Johansen estimator of the cumulative incidence function of a competing risk. Scand J Stat 40(3):387-402 · Zbl 1364.62103
[14] Bickel PJ, Freedman DA (1981) Some asymptotic theory for the bootstrap. Ann Stat 9(6):1196-1217 · Zbl 0449.62034
[15] Bie O, Borgan Ø, Liestøl K (1987) confidence intervals and confidence bands for the cumulative hazard rate function and their small sample properties. Scand J Stat 14(3):221-233 · Zbl 0627.62049
[16] Bluhmki T, Peter RS, Rapp K, König H-H, Becker C, Lindlbauer I, Rothenbacher D, Beyersmann J, Büchele G (2017) Understanding mortality of femoral fractures following low-impact trauma in persons with and without care need. J Am Med Dir Assoc 18(3):221-226
[17] Chen Wei, Wang Dehui, Li Yanfeng (2015) A class of tests of proportional hazards assumption for left-truncated and right-censored data. J Appl Stat 42(11):2307-2320 · Zbl 1514.62486
[18] Cox D (1972) Regression models and life tables (with discussion). J R Stat Soc 34:187-220 · Zbl 0243.62041
[19] Curley MAQ, Wypij D, Watson RS, Grant MJC, Asaro LA, Cheifetz IM, Dodson BL, Franck LS, Gedeit RG, Angus DC, Matthay MA (2015) Protocolized sedation vs usual care in pediatric patients mechanically ventilated for acute respiratory failure: a randomized clinical trial. J Am Med Assoc 313(4):379-389 and for the RESTORE Study Investigators and the PALISI Network
[20] Dabrowska DM, Ho W-T (2000) Confidence bands for comparison of transition probabilities in a Markov chain model. Lifetime Data Anal 6(1):5-21 · Zbl 0948.62064
[21] de Wit M, Gennings C, Jenvey WI, Epstein SK (2008) Randomized trial comparing daily interruption of sedation and nursing-implemented sedation algorithm in medical intensive care unit patients. Crit Care 12(3):R70
[22] Dobler D, Pauly M (2014) Bootstrapping Aalen-Johansen processes for competing risks: handicaps, solutions, and limitations. Electron J Stat 8(2):2779-2803 · Zbl 1309.62079
[23] Dobler D, Beyersmann J, Pauly M (2017) Non-strange weird resampling for complex survival data. Biometrika 104(3):699-711 · Zbl 07072236
[24] Dudek A, Goćwin M, Leśkow J (2008) Simultaneous confidence bands for the integrated hazard function. Comput Stat 23(1):41-62
[25] Eefting M, de Wreede LC, Halkes CJM, Peter A, Kersting S, Marijt EWA, Veelken H, Putter H, Schetelig J, Falkenburg JHF (2016) Multi-state analysis illustrates treatment success after stem cell transplantation for acute myeloid leukemia followed by donor lymphocyte infusion. Haematologica 101(4):506-514
[26] Efron B (1979) Bootstrap methods: another look at the jackknife. Ann Stat 7(1):1-26 · Zbl 0406.62024
[27] Efron B (1981) Censored data and the bootstrap. J Am Stat Assoc 76(374):312-319 · Zbl 0461.62039
[28] Gill R, Schumacher M (1987) A simple test of the proportional hazards assumption. Biometrika 74(2):289-300 · Zbl 0621.62047
[29] Gill RD, Johansen S (1990) A survey of product-integration with a view toward application in survival analysis. Ann Stat 18(4):1501-1555 · Zbl 0718.60087
[30] Gillam MH, Ryan P, Salter A, Graves SE (2012) Multi-state models and arthroplasty histories after unilateral total hip arthroplasties. Acta Orthop 83(3):220-226
[31] Glidden DV (2002) Robust inference for event probabilities with non-Markov event data. Biometrics 58(2):361-368 · Zbl 1209.62047
[32] Good PI (2005) Permutation, parametric, and bootstrap tests of hypotheses, 3rd edn. Springer, New York · Zbl 1076.62043
[33] Grambsch PM, Therneau TM (1994) Proportional hazards tests and diagnostics based on weighted residuals. Biometrika 81(3):515-526 · Zbl 0810.62096
[34] Hall P, Wilson SR (1991) Two guidelines for bootstrap hypothesis testing. Biometrics 47(2):757-762
[35] Hall WJ, Wellner JA (1980) Confidence bands for a survival curve from censored data. Biometrika 67(1):133-143 · Zbl 0423.62078
[36] Hess KR (1995) Graphical methods for assessing violations of the proportional hazards assumption in Cox regression. Stat Med 14(15):1707-1723
[37] Hieke S, Bertz H, Dettenkofer M, Schumacher M, Beyersmann J (2013) Initially fewer bloodstream infections for allogeneic vs. autologous stem-cell transplants in neutropenic patients. Epidemiol Infect 141(01):158-164
[38] Horvath L, Yandell BS (1987) Convergence rates for the bootstrapped product-limit process. Ann Stat 15(3):1155-1173 · Zbl 0637.62014
[39] Hyun S, Sun Y, Sundaram R (2009) Assessing cumulative incidence functions under the semiparametric additive risk model. Stat Med 28(22):2748-2768
[40] Iacus SM (2014) sde: simulation and inference for stochastic differential equations. https://CRAN.R-project.org/package=sde. R package version 2.0.13. Accessed 8 Nov 2017
[41] Janssen A, Pauls T (2003) How do bootstrap and permutation tests work? Ann Stat 31(3):768-806 · Zbl 1028.62027
[42] Jepsen P, Vilstrup H, Andersen PK (2015) The clinical course of cirrhosis: the importance of multistate models and competing risks analysis. Hepatology 62(1):292-302
[43] Koziol JA, Byar DP (1975) Percentage points of the asymptotic distributions of one and two sample K-S statistics for truncated or censored data. Technometrics 17(4):507-510 · Zbl 0326.62032
[44] Kraus D (2007) Data-driven smooth tests of the proportional hazards assumption. Lifetime Data Anal 13(1):1-16 · Zbl 1121.62086
[45] Lin DY (1991) Goodness-of-fit analysis for the Cox regression model based on a class of parameter estimators. J Am Stat Assoc 86(415):725-728 · Zbl 0733.62048
[46] Lin DY (1997) Non-parametric inference for cumulative incidence functions in competing risks studies. Stat Med 16(8):901-910
[47] Lin DY, Wei LJ, Ying Z (1993) Checking the Cox model with cumulative sums of martingale-based residuals. Biometrika 80(3):557-572 · Zbl 0788.62094
[48] Lin DY, Fleming TR, Wei LJ (1994) Confidence bands for survival curves under the proportional hazards model. Biometrika 81(1):73-81 · Zbl 0800.62708
[49] Lin DY, Wei LJ, Yang I, Ying Z (2000) Semiparametric regression for the mean and rate functions of recurrent events. J R Stat Soc Ser B (Stat Methodol) 62(4):711-730 · Zbl 1074.62510
[50] Liu L, Logan B, Klein JP (2008) Inference for current leukemia free survival. Lifetime Data Anal 14(4):432-446 · Zbl 1302.62234
[51] Liu RY (1988) Bootstrap procedures under some non-iid models. Ann Stat 16(4):1696-1708 · Zbl 0655.62031
[52] Lo SH, Singh K (1986) The product-limit estimator and the bootstrap: some asymptotic representations. Probab Theory Relat Fields 71(3):455-465 · Zbl 0561.62032
[53] Mayet A, Legleye S, Falissard B, Chau N (2012) Cannabis use stages as predictors of subsequent initiation with other illicit drugs among french adolescents: use of a multi-state model. Addict Behav 37(2):160-166
[54] Pauly M, Brunner E, Konietschke F (2015) Asymptotic permutation tests in general factorial designs. J R Stat Soc Ser B (Stat Methodol) 77(2):461-473 · Zbl 1414.62339
[55] R Core Team (2016) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. https://www.R-project.org/. Accessed 8 Nov 2017
[56] Sauaia A, Moore EE, Johnson JL, Ciesla DJ, Biffl WL (2009) Validation of postinjury multiple organ failure scores. Shock 31(5):438-447
[57] Scheike TH, Martinussen T (2004) On estimation and tests of time-varying effects in the proportional hazards model. Scand J Stat 31(1):51-62 · Zbl 1053.62111
[58] Scheike TH, Zhang M-J (2003) Extensions and applications of the Cox-Aalen survival model. Biometrics 59(4):1036-1045 · Zbl 1274.62678
[59] Schmoor C, Schumacher M, Finke J, Beyersmann J (2013) Competing risks and multistate models. Clin Cancer Res 19(1):12-21
[60] Schoenfeld DA, Bernard GR, Network ARDS (2002) Statistical evaluation of ventilator-free days as an efficacy measure in clinical trials of treatments for acute respiratory distress syndrome. Crit Care Med 30(8):1772-1777
[61] Schumacher M (1984) Two-sample tests of Cramér-von Mises- and Kolmogorov-Smirnov-type for randomly censored data. Int Stat Rev 52(3):263-281 · Zbl 0569.62038
[62] Silvia Munoz-Price L, Frencken Jos F, Sergey Tarima, Marc Bonten (2016) Handling time-dependent variables: antibiotics and antibiotic resistance. Clin Infect Dis 62(12):1558-1563. https://doi.org/10.1093/cid/ciw191 ISSN 1537-6591 · doi:10.1093/cid/ciw191
[63] Shu Y, Klein JP, Zhang M-J (2007) Asymptotic theory for the Cox semi-Markov illness-death model. Lifetime Data Anal 13(1):91-117 · Zbl 1331.62439
[64] Stewart RM, Park PK, Hunt JP, McIntyre RC Jr, McCarthy J, Zarzabal LA, Michalek JE; for the NIH/NHLBI ARDS Clinical Trials Network (2009) Less is more: improved outcomes in surgical patients with conservative fluid administration and central venous catheter monitoring. J Am Coll Surg 208(5):725-735
[65] Trof RJ, Beishuizen A, Cornet AD, de Wit RJ, Girbes ARJ, Groeneveld ABJ (2012) Volume-limited versus pressure-limited hemodynamic management in septic and nonseptic \[\text{ shock }^*\] shock∗. Crit Care Med 40(4):1177-1185
[66] von Cube M, Schumacher M, Wolkewitz M (2017) Basic parametric analysis for a multi-state model in hospital epidemiology. BMC Med Res Methodol 17(1):111
[67] Wellek S (2010) Testing statistical hypotheses of equivalence and noninferiority, 2nd edn. CRC Press, Boca Raton · Zbl 1219.62002
[68] Whitt W (1980) Some useful functions for functional limit theorems. Math Oper Res 5(1):67-85 · Zbl 0428.60010
[69] Wolkewitz M, Vonberg RP, Grundmann H, Beyersmann J, Gastmeier P, Bärwolff S, Geffers C, Behnke M, Rüden H, Schumacher M (2008) Risk factors for the development of nosocomial pneumonia and mortality on intensive care units: application of competing risks models. Crit Care 12(2):R44
[70] Wu CFJ (1986) Jackknife, bootstrap and other resampling methods in regression analysis. Ann Stat 14(4):1261-1295 · Zbl 0618.62072
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.