Estimation with Interval Censored Data in Longitudinal Studies

In biostatistical applications interest often focuses on the estimation of the distribution of a time-until-event variable T. If one observes whether or not T exceeds an observed monitoring time at a random number of monitoring times, then the data structure is called interval censored data. We extend this data structure by allowing the presence of a possibly time-dependent covariate process that is observed until end of follow up. If one only assumes that the censoring mechanism satisfies coarsening at random, then, by the curve of dimensionality, typically no regular estimators will exist. To fight the curse of dimensionality we follow the approach of Robins and Rotnitzky (1992) by modeling parameters of the censoring mechanism. We model the right-censoring mechanism by modeling the hazard of the follow up time, conditional on T and the covariate process. For the monitoring mechanism we avoid modeling the joint distribution of the monitoring times by only modeling a univariate hazard of the pooled monitoring times, conditional on the follow up time, T, and the covariates process, which can be estimated by treating the pooled sample of monitoring times as i.i.d. In particular, it is assumed that the monitoring times and the right-censoring times only depend on T through the observed covariate process. We introduce inverse probability of censoring weighted (IPCW) estimator of the distribution of T and of smooth functionals thereof which are guaranteed to be consistent and asymptotically normal if we have available correctly specified semiparametric models for the two hazards of the censoring process. Furthermore, given such correctly specified models for these hazards of the censoring process, we propose a one-step estimator which will improve on the IPCW estimator if we correctly specify a lower-dimensional working model for the conditional distribution of T, given the covariate process, that remains consistent and asymptotically normal if this latter working model is misspecified. It is shown that the one-step estimator is efficient if each subject is at most monitored once and the working model contains the truth. In general, it is shown that the one-step estimator optimally uses the surrogate information if the working model contains the truth. It is not optimal in using the interval information provided by the current status indicators at the monitoring times, but simulations in Peterson, van der Laan (1997) show that the efficiency loss is small.

Smooth Hazard Function Estimation for Interval Censored Data with Time Varying Covariates

In this paper we propose methods for smooth hazard estimation of a time variable where that variable is interval censored. These methods allow one to model the transformed hazard in terms of either smooth (smoothing splines) or linear functions of time and other relevant time varying predictor variables. We illustrate the use of this method on a dataset of hemophiliacs where the outcome, time to seroconversion for HIV, is interval censored and left-truncated.

The Two-Interval Line-Segment Problem

In this paper, the NPMLE in the one-dimensional line segment problem is defined and studied, where line segments on the real line through two non-overlapping intervals are observed. The self-consistency equations for the NPMLE are defined and a quick algorithm for solving them is provided. Supnorm weak convergence to a Gaussian process and efficiency of the NPMLE is proved. The problem has a strong geological application in the study of the lifespan of species.

On the Accelerated Failure Time Model for Current Status and Interval Censored Data

This paper introduces a novel approach to making inference about the regression parameters in the accelerated failure time (AFT) model for current status and interval censored data. The estimator is constructed by inverting a Wald type test for testing a null proportional hazards model. A numerically efficient Markov chain Monte Carlo (MCMC) based resampling method is proposed to simultaneously obtain the point estimator and a consistent estimator of its variance-covariance matrix. We illustrate our approach with interval censored data sets from two clinical studies. Extensive numerical studies are conducted to evaluate the finite sample performance of the new estimators.

A Pseudolikelihood Approach for Simultaneous Analysis of Array Comparative Genomic Hybridizations (aCGH)

DNA sequence copy number has been shown to be associated with cancer development and progression. Array-based Comparative Genomic Hybridization (aCGH) is a recent development that seeks to identify the copy number ratio at large numbers of markers across the genome. Due to experimental and biological variations across chromosomes and across hybridizations, current methods are limited to analyses of single chromosomes. We propose a more powerful approach that borrows strength across chromosomes and across hybridizations. We assume a Gaussian mixture model, with a hidden Markov dependence structure, and with random effects to allow for intertumoral variation, as well as intratumoral clonal variation. For ease of computation, we base estimation on a pseudolikelihood function. The method produces quantitative assessments of the likelihood of genetic alterations at each clone, along with a graphical display for simple visual interpretation. We assess the characteristics of the method through simulation studies and through analysis of a brain tumor aCGH data set. We show that the pseudolikelihood approach is superior to existing methods both in detecting small regions of copy number alteration and in accurately classifying regions of change when intratumoral clonal variation is present.

Reduced Bayesian Hierarchical Models: Estimating Health Effects of Simultaneous Exposure to Multiple Pollutants

Quantifying the health effects associated with simultaneous exposure to many air pollutants is now a research priority of the US EPA. Bayesian hierarchical models (BHM) have been extensively used in multisite time series studies of air pollution and health to estimate health effects of a single pollutant adjusted for potential confounding of other pollutants and other time-varying factors. However, when the scientific goal is to estimate the impacts of many pollutants jointly, a straightforward application of BHM is challenged by the need to specify a random-effect distribution on a high-dimensional vector of nuisance parameters, which often do not have an easy interpretation. In this paper we introduce a new BHM formulation, which we call "reduced BHM", aimed at analyzing clustered data sets in the presence of a large number of random effects that are not of primary scientific interest. At the first stage of the reduced BHM, we calculate the integrated likelihood of the parameter of interest (e.g. excess number of deaths attributed to simultaneous exposure to high levels of many pollutants). At the second stage, we specify a flexible random-effect distribution directly on the parameter of interest. The reduced BHM overcomes many of the challenges in the specification and implementation of full BHM in the context of a large number of nuisance parameters. In simulation studies we show that the reduced BHM performs comparably to the full BHM in many scenarios, and even performs better in some cases. Methods are applied to estimate location-specific and overall relative risks of cardiovascular hospital admissions associated with simultaneous exposure to elevated levels of particulate matter and ozone in 51 US counties during the period 1999-2005.

Analyzing Bivariate Survival Data with Interval Sampling and Application to Cancer Epidemiology

In medical follow-up studies, ordered bivariate survival data are frequently encountered when bivariate failure events are used as the outcomes to identify the progression of a disease. In cancer studies interest could be focused on bivariate failure times, for example, time from birth to cancer onset and time from cancer onset to death. This paper considers a sampling scheme where the first failure event (cancer onset) is identified within a calendar time interval, the time of the initiating event (birth) can be retrospectively confirmed, and the occurrence of the second event (death) is observed sub ject to right censoring. To analyze this type of bivariate failure time data, it is important to recognize the presence of bias arising due to interval sampling. In this paper, nonparametric and semiparametric methods are developed to analyze the bivariate survival data with interval sampling under stationary and semi-stationary conditions. Numerical studies demonstrate the proposed estimating approaches perform well with practical sample sizes in different simulated models. We apply the proposed methods to SEER ovarian cancer registry data for illustration of the methods and theory.