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.

Performance of the Halex in Logitudinal Studies of Older Adults

Goal: The Halex is an indicator of health status that combines self-rated health and activity limitations, which has been used by NCHS to predict future years of healthy life. The scores for each health state were developed based on strong assumptions, notably that a person in excellent health with ADL disabilities is as healthy as a person in poor health with no disabilities. Our goal was to examine the performance of the Halex as a longitudinal measure of health for older adults, and to improve the scoring if necessary. Methods: We used data from the Cardiovascular Health Study (CHS) to compare the relationship of baseline health to health 2 years later. Subject ages ranged from 65 to 103 (mean age 75). A total of 40,827 transitions were available for analysis. We examined whether Halex scores at time 0 were related monotonically to scores two years later, and iterated the original scores to improve the fit over time. Findings: The original Halex scores were not consistent over time. Persons in excellent health with ADL limitations were much healthier 2 years later than people in poor health with no limitations, even though they had been assumed to have identical health. People with ADL limitations had higher scores than predicted. The assumptions made in creating the Halex were not upheld in the data. Conclusions: The new iterated scores are specific to older adults, are appropriate for longitudinal data, and are relatively assumption-free. We recommend the use of these new scores for longitudinal studies of older adults that use the Halex health states.

Underestimation of Standard Errors in Multi-Site Time Series Studies

Multi-site time series studies of air pollution and mortality and morbidity have figured prominently in the literature as comprehensive approaches for estimating acute effects of air pollution on health. Hierarchical models are generally used to combine site-specific information and estimate pooled air pollution effects taking into account both within-site statistical uncertainty, and across-site heterogeneity. Within a site, characteristics of time series data of air pollution and health (small pollution effects, missing data, highly correlated predictors, non linear confounding etc.) make modelling all sources of uncertainty challenging. One potential consequence is underestimation of the statistical variance of the site-specific effects to be combined. In this paper we investigate the impact of variance underestimation on the pooled relative rate estimate. We focus on two-stage normal-normal hierarchical models and on under- estimation of the statistical variance at the first stage. By mathematical considerations and simulation studies, we found that variance underestimation does not affect the pooled estimate substantially. However, some sensitivity of the pooled estimate to variance underestimation is observed when the number of sites is small and underestimation is severe. These simulation results are applicable to any two-stage normal-normal hierarchical model for combining information of site-specific results, and they can be easily extended to more general hierarchical formulations. We also examined the impact of variance underestimation on the national average relative rate estimate from the National Morbidity Mortality Air Pollution Study and we found that variance underestimation as much as 40% has little effect on the national average.

A BAYESIAN SHRINKAGE MODEL FOR INCOMPLETE LONGITUDINAL BINARY DATA WITH APPLICATION TO THE BREAST CANCER PREVENTION TRIAL

We consider inference in randomized studies, in which repeatedly measured outcomes may be informatively missing due to drop out. In this setting, it is well known that full data estimands are not identified unless unverified assumptions are imposed. We assume a non-future dependence model for the drop-out mechanism and posit an exponential tilt model that links non-identifiable and identifiable distributions. This model is indexed by non-identified parameters, which are assumed to have an informative prior distribution, elicited from subject-matter experts. Under this model, full data estimands are shown to be expressed as functionals of the distribution of the observed data. To avoid the curse of dimensionality, we model the distribution of the observed data using a Bayesian shrinkage model. In a simulation study, we compare our approach to a fully parametric and a fully saturated model for the distribution of the observed data. Our methodology is motivated and applied to data from the Breast Cancer Prevention Trial.