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Review
. 2019 Jan;25(1):97-127.
doi: 10.1007/s10985-018-9423-x. Epub 2018 Mar 6.

The wild bootstrap for multivariate Nelson-Aalen estimators

Tobias Bluhmki  1 , Dennis Dobler  2 , Jan Beyersmann  1 , Markus Pauly  1
Affiliations
Review

The wild bootstrap for multivariate Nelson-Aalen estimators

Tobias Bluhmki et al. Lifetime Data Anal. 2019 Jan.

Abstract

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.

Keywords: Conditional central limit theorem; Counting process; Equivalence test; Kolmogorov–Smirnov test; Proportional hazards; Survival analysis; Weak convergence.

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Figures

Fig. 1
Fig. 1
Illness-death model with recovery and transition hazards α01,α10,α02, and α12 at time t
Fig. 2
Fig. 2
95% confidence bands based on standard normal multipliers for the cumulative hazard of end-of-stay from the data example in Sect. 6. The solid black lines are the Nelson–Aalen estimators separately for ‘no ventilation’ (state 0, right plot) and ‘ventilation’ (state 1, left plot)
Fig. 3
Fig. 3
95% equal precision confidence bands based on standard normal multipliers and 95% log-transformed pointwise confidence intervals for the cumulative hazard of end-of-stay from the data example in Sect. 6. The solid black lines are the Nelson–Aalen estimators separately for ‘no ventilation’ (state 0, right plot) and ‘ventilation’ (state 1, left plot)
Fig. 4
Fig. 4
95% confidence bands from relation (4.5) based on standard normal multipliers and 95% linear pointwise confidence intervals for difference of the two cumulative hazards of end-of-stay from the data example in Sect. 6. The solid black lines is the difference ‘ventilation vs. no ventilation’ of the Nelson–Aalen estimators within the two ventilation groups
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References

    1. Aalen OO. Nonparametric inference for a family of counting processes. Ann Stat. 1978;6(4):701–726.
    1. Aalen OO, Johansen S. An empirical transition matrix for non-homogeneous Markov chains based on censored observations. Scand J Stat. 1978;5(3):141–150.
    1. Aalen OO, Borgan Ø, Gjessing HK. Survival and event history analysis: a process point of view. New York: Springer; 2008.
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    1. Allignol A, Schumacher M, Wanner C, Drechsler C, Beyersmann J. Understanding competing risks: a simulation point of view. BMC Med Res Methodol. 2011;11:86. - PMC - PubMed

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