Nonparametric generalized fiducial inference for survival functions under censoring

Abstract

Fiducial Inference, introduced by Fisher in the 1930s, has a long history, which at times aroused passionate disagreements. However, its application has been largely confined to relatively simple parametric problems. In this paper, we present what might be the first time fiducial inference, as generalized by Hannig et al. (2016), is systematically applied to estimation of a nonparametric survival function under right censoring. We find that the resulting fiducial distribution gives rise to surprisingly good statistical procedures applicable to both one sample and two sample problems. In particular, we use the fiducial distribution of a survival function to construct pointwise and curvewise confidence intervals for the survival function, and propose tests based on the curvewise confidence interval. We establish a functional Bernstein-von Mises theorem, and perform thorough simulation studies in scenarios with different levels of censoring. The proposed fiducial based confidence intervals maintain coverage in situations where asymptotic methods often have substantial coverage problems. Furthermore, the average length of the proposed confidence intervals is often shorter than the length of competing methods that maintain coverage. Finally, the proposed fiducial test is more powerful than various types of log-rank tests and sup log-rank tests in some scenarios. We illustrate the proposed fiducial test comparing chemotherapy against chemotherapy combined with radiotherapy using data from the treatment of locally unresectable gastric cancer.

Discussion of ‘Nonparametric generalized fiducial inference for survival functions under censoring’

“…All of this, seen in retrospect, is excellently presented and exemplified by Fraser (1968) for classical linear models. We believe that Cui and Hannig (2019) have taken the first important step for similar results in the nonparametric case. Their main technical result proves that the nonparametric fiducial is asymptotically a confidence distribution.…”

OptBand: optimization-based confidence bands for functions to characterize time-to-event distributions

“…We compare our proposed OptBand against FD-I (Cui and Hannig 2019a ), EL (Hollander et al. 1997 ), EP (Nair 1984 ), and HW (Hall and Wellner 1984 ) bands.…”

“…( 2011 ) derived bands which target the highest confidence density region (HCDR), but this approach requires the standard specifications and tuning procedures that accompany a Markov chain Monte Carlo process and can be computationally burdensome. Finally, Cui and Hannig ( 2019a ) introduced a nonparametric fiducial approach to confidence bands, which has been shown to be robust and efficient in small samples; several auxiliary works (Cui and Hannig 2019b ; Martin 2019 ) explored the implications of this work and how fiducial inference fits in the context of modern statistics.…”

Approximately valid probabilistic inference on a class of statistical functionals

“…On the other hand, "nonparametric" can refer to methods whose inferential target is the underlying distribution, or some other infinite-dimensional object (e.g., Wasserman 2006). This can be done in a probabilistic way using Bayesian nonparametrics (Ghosal and van der Vaart 2017;Ghosh and Ramamoorthi 2003;Hjort et al 2010) or generalized fiducial (Cui and Hannig 2019). The downside is that these methods are indirect in the sense that their inferential target is typically much more complicated than the quantity of interest so, e.g., if the quantity of interest is low-dimensional, then it may be computationally/statistically inefficient to first infer an infinite-dimensional unknown and then carry out an extreme marginalization step from high to low dimensions.…”