A Wild Bootstrap Approach for the Aalen–Johansen Estimator

Abstract

We suggest a wild bootstrap resampling technique for nonparametric inference on transition probabilities in a general time-inhomogeneous Markov multistate model. We first approximate the limiting distribution of the Nelson-Aalen estimator by repeatedly generating standard normal wild bootstrap variates, while the data is kept fixed. Next, a transformation using a functional delta method argument is applied. The approach is conceptually easier than direct resampling for the transition probabilities. It is used to investigate a non-standard time-to-event outcome, currently being alive without immunosuppressive treatment, with data from a recent study of prophylactic treatment in allogeneic transplanted leukemia patients. Due to non-monotonic outcome probabilities in time, neither standard survival nor competing risks techniques apply, which highlights the need for the present methodology. Finite sample performance of time-simultaneous confidence bands for the outcome probabilities is assessed in an extensive simulation study motivated by the clinical trial data. Example code is provided in the web-based Supplementary Materials.

Nonparametric tests for transition probabilities in nonhomogeneous Markov processes

“…The issue of nonparametric comparison of transition probabilities in general nonhomogeneous Markov processes has received little attention in the literature. To the best of our knowledge, the only fully nonparametric approach for comparing the transitions probabilitis for a particular transtion in general non-homogeneous Markov processes is a graphical procedure proposed by Bluhmki et al (2018). This proposal is based on the construction of a simultaneous confidence band for the difference between the transition probabilities of two groups.…”

“…This can be done by replacing the influence function of the standard Aalen-Johansen estimator with the influence function of any other well-behaved and asymptotically linear estimator of the transition probabilities in our proposed testing procedures. Such adaptations are not trivial within the framework of the graphical testing procedure proposed by Bluhmki et al (2018).…”

Dynamic inference for non‐Markov transition probabilities under random right censoring

“…Even though the bootstrap procedure is useful but not strictly necessary for inference on the expected length of stay, it plays an essential role in time-simultaneous inference on the transition probabilities t → P IJ (s, t). This is due to the unknown stochastic behaviour of the limit process U which, again in the context of creating confidence bands, is also the reason for the inevitableness of resampling procedures for Aalen-Johansen estimators even if the Markov assumption is fulfilled; see Bluhmki et al (2018) for theoretical justifications and their practical performance. For the derivation of reasonable time-simultaneous 1 − α ∈ (0, 1) confidence bands for P IJ (s, ·) on a time interval [t 1 , t 2 ] ⊂ [s, τ ] based on P IJ (s, ·) in combination with the bootstrap, we focus on the process…”

“…For the non-Markov estimator, in addition to the Hall-Wellner and equal precision bands using the transformation φ 2 (p), a naive confidence band based on a constant weight function, w(t) ≡ 1, and an identity transformation, φ(p) = p, is also constructed. In addition, for the Aalen-Johansen estimates, EP bands based on φ 2 (p), are constructed via the wild bootstrap using the R code which has been made available in the supplement to Bluhmki et al (2018). In all cases, for each bootstrap B = 1000 samples are generated.…”

Wild bootstrap logrank tests with broader power functions for testing superiority

“…It is the aim of the present paper to address these points accordingly. In particular, employing multiplier wild bootstrap resampling (Lin, 1997;Beyersmann et al, 2013;Dobler and Pauly, 2014;Bluhmki et al, 2018b) instead of permutation enables us to directly cope with the complex limit distribution without making the above studentization detour. We rigorously analyze the asymptotic properties of the resulting procedure, specifically preserving the asymptotic optimality of the Brendel et al (2014) method (Section 3.2 below).…”