Nonparametric inference for competing risks current status data with continuous, discrete or grouped observation times

Current status data with two competing risks and time-dependent missing failure types

In competing risks data, in practice, there may be lack of information or uncertainty about the true failure type, termed as 'missing failure type', for some subjects. We consider a general pattern of missing failure type in which we observe, if not the true failure type, a set of possible failure types containing the true one. In this work, we focus on both parametric and non-parametric estimation based on current status data with two competing risks and the above-mentioned missing failure type. Here, the missing probabilities are assumed to be time-dependent, that is, dependent on both failure and monitoring time points, in addition to being dependent on the true failure type. This makes the missing mechanism non-ignorable. We carry out maximum likelihood estimation and obtain the asymptotic properties of the estimators. Simulation studies are conducted to investigate the finite sample properties of the estimators. Finally, the methods are illustrated through a data set on hearing loss.

Misclassified group-tested current status data

Group testing, introduced by Dorfman (1943), has been used to reduce costs when estimating the prevalence of a binary characteristic based on a screening test of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$k$\end{document} groups that include \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$n$\end{document} independent individuals in total. If the unknown prevalence is low and the screening test suffers from misclassification, it is also possible to obtain more precise prevalence estimates than those obtained from testing all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$n$\end{document} samples separately (Tu et al., 1994). In some applications, the individual binary response corresponds to whether an underlying time-to-event variable \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$T$\end{document} is less than an observed screening time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$C$\end{document}, a data structure known as current status data. Given sufficient variation in the observed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$C$\end{document} values, it is possible to estimate the distribution function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$F$\end{document} of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$T$\end{document} nonparametrically, at least at some points in its support, using the pool-adjacent-violators algorithm (Ayer et al., 1955). Here, we consider nonparametric estimation of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$F$\end{document} based on group-tested current status data for groups of size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$k$\end{document} where the group tests positive if and only if any individual’s unobserved \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$T$\end{document} is less than the corresponding observed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$C$\end{document}. We investigate the performance of the group-based estimator as compared to the individual test nonparametric maximum likelihood estimator, and show that the former can be more precise in the presence of misclassification for low values of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$F(t)$\end{document}. Potential applications include testing for the presence of various diseases in pooled samples where interest focuses on the age-at-incidence distribution rather than overall prevalence. We apply this estimator to the age-at-incidence curve for hepatitis C infection in a sample of U.S. women who gave birth to a child in 2014, where group assignment is done at random and based on maternal age. We discuss connections to other work in the literature, as well as potential extensions.

A pool-adjacent-violators type algorithm for non-parametric estimation of current status data with dependent censoring

A likelihood based approach to obtaining non-parametric estimates of the failure time distribution is developed for the copula based model of Wang et al. (Lifetime Data Anal 18:434-445, 2012) for current status data under dependent observation. Maximization of the likelihood involves a generalized pool-adjacent violators algorithm. The estimator coincides with the standard non-parametric maximum likelihood estimate under an independence model. Confidence intervals for the estimator are constructed based on a smoothed bootstrap. It is also shown that the non-parametric failure distribution is only identifiable if the copula linking the observation and failure time distributions is fully-specified. The method is illustrated on a previously analyzed tumorigenicity dataset.

Estimation of a discrete monotone distribution

We study and compare three estimators of a discrete monotone distribution: (a) the (raw) empirical estimator; (b) the "method of rearrangements" estimator; and (c) the maximum likelihood estimator. We show that the maximum likelihood estimator strictly dominates both the rearrangement and empirical estimators in cases when the distribution has intervals of constancy. For example, when the distribution is uniform on {0, … , y}, the asymptotic risk of the method of rearrangements estimator (in squared ℓ(2) norm) is y/(y + 1), while the asymptotic risk of the MLE is of order (log y)/(y + 1). For strictly decreasing distributions, the estimators are asymptotically equivalent.