Order Theory and Nonparametric Analysis for Interval Censored Data

Abstract

Interval censored data arises when individuals can be subjected to periodic inspection at random moments, and their status (e.g. failed or functioning) is ascertained at each inspection. We exploit the order theoretic properties of interval orders to develop a new language describing interval censored data. We propose a method by which the set of linear extensions of an interval order may be partitioned into sets of linear extensions of weak orders, using so-called marked configurations of the interval order. The technique relies heavily on the natural linear ordering of maximal antichains in interval orders. We also propose a method whereby sets from this partition can be generated with known probability so as to permit efficient cluster or staged sampling. These techniques, among other uses, may be applied to generate sampling estimates of average rank score statistics for interval censored data similar in construction to that proposed by Prentice (1978) for right-censored data. In order to address the above problem we must determine all sets which form minimal covers of maximal antichains for interval orders. Finding minimal covers generalizes the minimum clique cover problem. We produce an algorithm enumerating all minimal covers using the minimal elements of the interval order and also characterize maximal removable sets, which are the complements of minimal covers We use this characterization to provide bounds on the maximum number of minimal covers for an interval order with a given number of maximal antichains. Finally, we determine nonparametric maximum likelihood estimators (NPMLE) of the cumulative distribution function (CDF) on the set of maximal antichains M of the data rather than the real line, extending the reasoning of Peto (1973) and Turnbull (1976). We discuss some properties of self-consistent estimators of the CDF in light of the structure of M. We show the identity between self-consistency augmented by Kuhn-Tucker conditions and Fenchel duality, which characterize the NPMLE on M. We port to M recently developed isotonic regression techniques to estimate the NPMLE. We correct some misapprehensions which have gained currency in recent literature on interval censored data. Keywords: interval censored data; survival analysis; interval order; nonparametric maximum likelihood; maximal antichains; self-consistency; isotonic regression; linear extensions; order partition.