Bounds for Self-consistent CDF Estimators for Univariate and Multivariate Censored Data

Abstract In this paper, lower bounds and upper bounds are given for the mass assigned to a set of maximal cliques in self-consistent estimates of CDF NPMLEs for multivariate (including univariate) interval censored data under the assumption that the censoring mechanism is ignorable for the purpose of likelihood inference. The bounds are applied to give upper bounds of the diameter and size of the polytope of CDF NPMLEs for multivariate censored data

Preschool children's vision screening in New Zealand: a retrospective evaluation of referral accuracy

Langeslag-Smith MA, Vandal AC, Briane V, et al. Preschool children's vision screening in New Zealand: a retrospective evaluation of referral accuracy. BMJ Open 2015;5:e009207. doi: 10.1136/bmjopen-2015-009207Objectives To assess the accuracy of preschool vision screening in a large, ethnically diverse, urban population in South Auckland, New Zealand. Design Retrospective longitudinal study. Methods B4 School Check vision screening records (n=5572) were compared with hospital eye department data for children referred from screening due to impaired acuity in one or both eyes who attended a referral appointment (n=556). False positive screens were identified by comparing screening data from the eyes that failed screening with hospital data. Estimation of false negative screening rates relied on data from eyes that passed screening. Data were analysed using logistic regression modelling accounting for the high correlation between results for the two eyes of each child. Primary outcome measure Positive predictive value of the preschool vision screening programme. Results Screening produced high numbers of false positive referrals, resulting in poor positive predictive value (PPV=31%, 95% CI 26% to 38%). High estimated negative predictive value (NPV=92%, 95% CI 88% to 95%) suggested most children with a vision disorder were identified at screening. Relaxing the referral criteria for acuity from worse than 6/9 to worse than 6/12 improved PPV without adversely affecting NPV. Conclusions The B4 School Check generated numerous false positive referrals and consequently had a low PPV. There is scope for reducing costs by altering the visual acuity criterion for referral.This work was supported by the Arthur D Bronlund Trust, CCRep and University of Auckland Faculty Research Development Fund Grants (3704420)

Order Theory and Nonparametric Maximum Likelihood for Interval Censored Data

Interval censored data arises when individuals can be subjected to periodic inspection at possibly 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 show that inference can be based on the set M of maximal antichains of the data, rather than on the real line, extending the reasoning of Turnbull (1976). We show how M can be used to obtain the nonparametric maximum likelihood estimators suggested by Turnbull (1976). We discuss some properties of self-consistent estimators of the cumulative distribution function 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 (Wellner & Zhan, 1997). Keywords: interval censored data,..

Weak Order Partitioning of Interval Orders With Applications to 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. Keywords: interval censoring, interval orders, weak orders,rank score,nonparametric inference 1 Introduction One of the most intuitively useful facets of a partial order (X; OE) is the set of its linear extensions, that is, the set of all linear or complete orders compatible with OE. This usefulness stems in ..