Current Status Data Research Articles

Score Statistics for Current Status Data: Comparisons with Likelihood Ratio and Wald Statistics

In this paper we introduce three natural “score statistics” for testing the hypothesis that F (t0) takes on a fixed value in the context of nonparametric inference with current status data. These three new test statistics have natural intepretations in terms of certain (weighted) L2 distances, and are also connected to natural “one-sided” scores. We compare these new test statistics with an analogue of the classical Wald statistic and the likelihood ratio statistic introduced in Banerjee and Wellner (2001) for the same testing problem. Under classical “regular” statistical problems the likelihood ratio, score, and Wald statistics all have the same chi-squared limiting distribution under the null hypothesis. In sharp contrast, in this non-regular problem all three statistics have different limiting distributions under the null hypothesis. Thus we begin by establishing the limit distribution theory of the statistics under the null hypothesis, and discuss calculation of the relevant critical points for the test statistics. Once the null distribution theory is known, the immediate question becomes that of power. We establish the limiting behavior of the three types of statistics under local alternatives. We have also compared the power of these five different statistics via a limited Monte-Carlo study. Our conclusions are: (a) the Wald statistic is less powerful than the the likelihood ratio and score statistics; and (b) one of the score statistics may have more power than the likelihood ratio statistic for some alternatives.

Locally efficient estimation of regression parameters using current status data

In biostatistics applications interest often focuses on the estimation of the distribution of a time-variable T. If one only observes whether or not T exceeds an observed monitoring time C, then the data structure is called current status data, also known as interval censored data, case I. We consider this data structure extended to allow the presence of both time-independent covariates and time-dependent covariate processes that are observed until the monitoring time. We assume that the monitoring process satisfies coarsening at random. Our goal is to estimate the regression parameter β of the regression model T = Z ⊤ β + ε . The curse of dimensionality implies no globally efficient nonparametric estimator with good practical performance at moderate sample sizes exists. We present an estimator of the parameter β that attains the semiparametric efficiency bound if we correctly specify (a) a model for the monitoring mechanism and (b) a lower-dimensional model for the conditional distribution of T given the covariates. In addition, our estimator is robust to model misspecification. If only (a) is correctly specified, the estimator remains consistent and asymptotically normal. We conclude with a simulation experiment and a data analysis.

Statistical analysis of current status data with informative observation times

Current status data arise when each study subject is observed only once and the survival time of interest is known only to be either less or greater than the observation time. Such data often occur in, for example, cross-sectional studies, demographical investigations and tumorigenicity experiments and several semi-parametric and non-parametric methods for their analysis have been proposed. However, most of these methods deal only with the situation where observation time is independent of the underlying survival time completely or given covariates. This paper discusses regression analysis of current status data when the observation time may be related to the underlying survival time and inference procedures are presented for estimation of regression parameters under the additive hazards regression model. The procedures can be easily implemented and are applied to two motivating examples.

On the testability of the CAR assumption

In recent years a popular nonparametric model for coarsened data is an assumption on the coarsening mechanism called coarsening at random (CAR). It has been conjectured in several papers that this assumption cannot be tested by the data, that is, the assumption does not restrict the possible distributions of the data. In this paper we will show that this conjecture is not always true; an example will be current status data. We will also give conditions when the conjecture is true, and in doing so, we will introduce a generalized version of the CAR assumption. As an illustration, we retrieve the well-known result that the CAR assumption cannot be tested in the case of right-censored data.

Estimating a Survival Distribution with Current Status Data and High-dimensional Covariates

We consider the inverse problem of estimating a survival distribution when the survival times are only observed to be in one of the intervals of a random bisection of the time axis. We are particularly interested in the case that high-dimensional and/or time-dependent covariates are available, and/or the survival events and censoring times are only conditionally independent given the covariate process. The method of estimation consists of regularizing the survival distribution by taking the primitive function or smoothing, estimating the regularized parameter by using estimating equations, and finally recovering an estimator for the parameter of interest.

Nonparametric regression with current status data

We apply nonparametric regression to current status data, which often arises in survival analysis and reliability analysis. While no parametric assumption on the distributions has been imposed, most authors have employed parametric models like linear models to measure the covariate effects on failure times in regression analysis with current status data. We construct a nonparametric estimator of the regression function by modifying the maximum rank correlation (MRC) estimator. Our estimator can deal with the cases where the other estimators do not work. We present the asymptotic bias and the asymptotic distribution of the estimator by adapting a result on equicontinuity of degenerate U-processes to the setup of this paper.

Testing Independence for Bivariate Current Status Data

This article develops a nonparametric procedure for testing marginal independence based on bivariate current status data. Asymptotic properties of the proposed tests are derived, and their finite-sample performance is studied via simulations. The method is applied to analyze data from a community-based study of cardiovascular epidemiology in Taiwan.

A model to estimate risk of infection with human herpesvirus 8 associated with transfusion from cross-sectional data.

In cross-sectional studies of infectious diseases, the data typically consist of: age at the time of study, status (presence or absence) of infection, and a chronology of events possibly associated with the disease. Motivated by a study of how human herpesvirus 8 (HHV-8) is transmitted among children with sickle cell anemia in Uganda, we have developed a flexible parametric approach for combining current-status data with a history of blood transfusions. We model heterogeneity in transfusion-associated risk by a child-specific random effect. We present an extension of the model to accommodate the fact that there is no gold standard for HHV-8 infection and infection status was assessed by a serological assay. The parameters are estimated via maximum likelihood. Finally, we present results from applying various parameterizations of the model to the Ugandan study.

Consistency of Concave Regression with an Application to Current-Status Data

We consider the problem of nonparametric estimation of a concave regression function F. We show that the supremum distance between the least square s estimatorand F on a compact interval is typically of order(log(n)/n)2/5. This entails rates of convergence for the estimator’s derivative. Moreover, we discuss the impact of additional constraints on F such as monotonicity and pointwise bounds. Then we apply these results to the analysis of current status data, where the distribution function of the event times is assumed to be concave.

Maximum likelihood estimation of ordered multinomial parameters

The pool adjacent violator algorithm Ayer et al. (1955, The Annals of Mathematical Statistics, 26, 641-647) has long been known to give the maximum likelihood estimator of a series of ordered binomial parameters, based on an independent observation from each distribution (see Barlow et al., 1972, Statistical Inference under Order Restrictions, Wiley, New York). This result has immediate application to estimation of a survival distribution based on current survival status at a set of monitoring times. This paper considers an extended problem of maximum likelihood estimation of a series of 'ordered' multinomial parameters p(i)= (p(1i),p(2i),.,p(mi)) for 1 <or=i <ro=k, where ordered means that p(j1) <or=p(j2) <or=<or=p(jk) for each j with 1 <or=j <or=m-1. The data consist of k independent observations X(1),., X(k) where X(i) has a multinomial distribution with probability parameter p(i) and known index n(i)\geq 1. By making use of variants of the pool adjacent violator algorithm, we obtain a simple algorithm to compute the maximum likelihood estimator of p(1),., p(k), and demonstrate its convergence. The results are applied to nonparametric maximum likelihood estimation of the sub-distribution functions associated with a survival time random variable with competing risks when only current status data are available (Jewell et al. 2003, Biometrika, 90, 183-197).

Goodness-of-fit methods for additive-risk models in tumorigenicity experiments.

In tumorigenicity experiments, a complication is that the time to event is generally not observed, so that the time to tumor is subject to interval censoring. One of the goals in these studies is to properly model the effect of dose on risk. Thus, it is important to have goodness of fit procedures available for assessing the model fit. While several estimation procedures have been developed for current-status data, relatively little work has been done on model-checking techniques. In this article, we propose numerical and graphical methods for the analysis of current-status data using the additive-risk model, primarily focusing on the situation where the monitoring times are dependent. The finite-sample properties of the proposed methodology are examined through numerical studies. The methods are then illustrated with data from a tumorigenicity experiment.

Current status and right-censored data structures when observing a marker at the censoring time

We study nonparametric estimation with two types of data structures. In the first data structure n i.i.d. copies of $(C,N(C))$ are observed, where N is a finite state counting process jumping at time-variables of interest and C a random monitoring time. In the second data structure n i.i.d. copies of $(C\wedge T,I(T\leq C),N (C\wedge T))$ are observed, where N is a counting process with a final jump at time T (e.g., death). This data structure includes observing right-censored data on T and a marker variable at the censoring time. In these data structures, easy to compute estimators, namely (weighted)-pool-adjacent-violator estimators for the marginal distributions of the unobservable time variables, and the Kaplan-Meier estimator for the time T till the final observable event, are available. These estimators ignore seemingly important information in the data. In this paper we prove that, at many continuous data generating distributions the ad hoc estimators yield asymptotically efficient estimators of $\sqrt{n}$-estimable parameters.

Nonparametric estimation from current status data with competing risks

SUMMARY A great deal of recent attention has focused on the estimation of survival distributions based on current status data, an extreme form of interval censored data. This particular data structure arises in a wide variety of applications where cross-sectional observation either naturally occurs or is preferred to more traditional forms of follow-up. Here we consider current status data in the context of competing risks. We briefly consider simple parametric models as a backdrop to nonparametric procedures. We make some brief comparisons and remarks regarding the nonparametric maximum likelihood estimator. The ideas are illustrated on the data of Krailo & Pike (1983) which considers estimation of the age distribution at both natural and operative menopause. We also consider the case where there is exact observation of failure times due to one of the competing risks when failure occurs prior to the monitoring time. Basic techniques of survival analysis focus on estimation of the survival distribution based on various forms of censored data. An extreme form of censoring arises where the only information on studied individuals is their survival status at a single monitoring time. This particular data structure is known as current status data. Nonparametric estimation of the survival function and semiparametric techniques for related regression models, based on current status data, have been much studied of late. Here, we consider estimation of sub-survival functions based on current status observations in the presence of competing risks. For simplicity, we describe only two competing risks, but the ideas are easily extended when there are more than two possible causes of failure. Let )j denote the cause-specific hazard function for cause j = 1, 2 (Kalbfleisch & Prentice, 2002, p. 252). If T and J are the random variables that measure time to failure and cause of failure, respectively, the two sub-distribution functions of interest are

Exploring the endangered species criteria: rethinking the IUCN Red List Criteria

For preserving global biodiversity, we should evaluate the level of endangerment of a species based on information such as number and distribution of individuals, fluctuations and decline in abundance and distribution, and risk of extinction. We should then design conservation measures accordingly. The Species Survival Commission of IUCN has devoted itself to the study of IUCN Red List Criteria. The IUCN 40th Council Meeting adopted the so called modified Mace Lande Species Criteria as the standard criteria for the IUCN Red List in November 1994. The Red List is intended to focus national and local conservation actions on the species that most need support. Although the IUCN Red List is not an international law, it has deep impact on conservation decision making in governmental and nongovernmental organizations, as well as on conservation law and policy making in many countries. The IUCN Red List Criteria also have had a profound influence on the theory of conservation biology. However, when we tried to apply the IUCN Red List Criteria to assess the status of endangered aquatic wildlife in China, we found several problems: (1) How do we distinguish those species that are naturally rare and those species whose population and habitat are declining due to human activity? (2) Can we apply the same criteria to different taxonomic groups, particularly those with different life histories? (3) How do we evaluate changes in population density that occur in the core habitat versus those that occur in peripheral habitats of the same species? (4) How should we treat the extirpation of local populations in a metapopulation? (5) How to deal with those species that can be successfully bred in captivity? (6) If we do not have accurate historical and current population and habitat status data, how do we evaluate the level of endangerment of those species? We suggest solutions to those problems in this paper. Firstly, we set up simplified endangered species criteria categories: Least concern, Concern, Threatened, and Extinct, of which, Concern, Threatened, and are further divided into two levels: ordinary and highly. We further suggest distinguishing ecologically endangered species versus evolutionarily endangered species, and evaluating species with different life histories, for examples, the K strategists and r strategists, with different criteria. Third, we suggest the introduction of Economically Endangered as a criterion of Threatened for r strategy species. Fourth, more conservation attention should be allocated to endemic species. Finally, we introduce the metapopulation concept into evaluation of endangered aquatic wildlife.

Case-control current status data

SUMMARY In this paper, we show that the distribution function of survival times is identified, up to a one-parameter family of distribution functions, based on information from case control current status data. With supplementary information on the population frequency of cases relative to controls, a simple weighted version of the nonparametric maximum likelihood estimator for prospective current status data provides a natural estimator for case-control samples. Following the parametric results of Scott & Wild (1997), we show that this estimator is, in fact, the nonparametric maximum likelihood estimator.

A multistate model for the genetic analysis of the ageing process.

In this paper a multivariate frailty model is suggested that can be used in the genetic analysis of the ageing process as a whole, simplified to consisting of the states 'healthy', 'disabled' and 'deceased'. The model allows us to evaluate simultaneously the relative magnitude of genetic and environmental influences on frailty variables corresponding to the period of good health and to the life span. The frailty variables can be interpreted as susceptibility to illness or death. The model can be applied to data on groups of related individuals (twins, siblings, a litter). One of the major advantages of this model is that it allows one to include groups of individuals where some or all members of the group are already deceased at the time of observation. The current health status of the living individuals and the exact life span of individuals who are already deceased is the only information necessary for the application of the model. Questions concerning the identifiability of the model based on current health status data and estimation strategies are discussed in the context of specifying the model for twins. Finally, the results of a sample analysis of twin data on prostate cancer are presented.

Efficient estimation in additive hazards regression with current status data

SUMMARY Current status data arise when the exact timing of an event is unobserved, and it is only known at a given point in time whether or not the event has occurred. Recently Lin et al. (1998) studied the additive semiparametric hazards model for current status data. They showed that the analysis of current status data under the additive hazards model reduces to ordinary Cox regression under the assumption that a proportional hazards model may be used to describe the monitoring intensity. This analysis does not make efficient use of data, and in some cases it may not be appropriate to assume a proportional hazards model for the monitoring times. We study the semiparametric hazards model for current status data but make use of the semiparametric efficient score function. The suggested approach has the advantages that it is efficient in that it reaches the semiparametric information bound, and it does not involve any modelling of the monitoring times.

Sharing the Risk of Home-ownership: A Portfolio Approach

This paper examines the relationships among the desirability of home-ownership, price risks and the return of housing investment. It argues that price risks may undermine the desirability of ownership, using both a theoretical and an empirical approach. The theoretical model studies aspatially the equilibrium house prices in a two-good, pure exchange economy, where investors (consumers) differ in their level of risk aversion. It shows that larger price risks and higher level of risk aversion decrease the attractiveness of owning a house. The empirical component uses current status data and a proportional hazard model to show the effect of risks and returns on households' hazard rates to be owners. The implication points to the inadequacy of the conventional housing finance arrangements, which provide little help for households to diversify or hedge the risk of home-ownership.

Bayesian Models for Multivariate Current Status Data with Informative Censoring

Multivariate current status data, consist of indicators of whether each of several events occur by the time of a single examination. Our interest focuses on inferences about the joint distribution of the event times. Conventional methods for analysis of multiple event-time data cannot be used because all of the event times are censored and censoring may be informative. Within a given subject, we account for correlated event times through a subject-specific latent variable, conditional upon which the various events are assumed to occur independently. We also assume that each event contributes independently to the hazard of censoring. Nonparametric step functions are used to characterize the baseline distributions of the different event times and of the examination times. Covariate and subject-specific effects are incorporated through generalized linear models. A Markov chain Monte Carlo algorithm is described for estimation of the posterior distributions of the unknowns. The methods are illustrated through application to multiple tumor site data from an animal carcinogenicity study.

Likelihood Ratio Tests for Monotone Functions

We study the problem of testing for equality at a fixed point in the setting of nonparametric estimation of a monotone function.The likelihood ratio test for this hypothesis is derived in the particular case of interval censoring (or current status data)and its limiting distribution is obtained. The limiting distribution is that of the integral of the difference of the squared slope processes corresponding to a canonical version of the problem involving Brownian motion $+t^2$ and greatest convex minorants thereof. Inversion of the family of tests yields pointwise confidence intervals for the unknown distribution function.We also study the behavior of the statistic under local and fixed alternatives.