A Note on the Asymptotic Variance of Survival Function in the Semi-Markov Model

In Malani and Neilsen (1992) we have proposed alternative estimates of survival function (for time to disease) using a simple marker that describes time to some intermediate stage in a disease process. In this paper we derive the asymptotic variance of one such proposed estimator using two different methods and compare terms of order 1/n when there is no censoring. In the absence of censoring the asymptotic variance obtained using the Greenwood type approach converges to exact variance up to terms involving 1/n. But the asymptotic variance obtained using the theory of the counting process and results from Voelkel and Crowley (1984) on semi-Markov processes has a different term of order 1/n. It is not clear to us at this point why the variance formulae using the latter approach give different results.

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 HIERARCHICAL MODEL FOR CONSTRAINED DISTRIBUTED LAG FUNCTIONS: ESTIMATING THE TIME COURSE OF HOSPITALIZATION ASSOCIATED WITH AIR POLLUTION EXPOSURE

Numerous time series studies have provided strong evidence of an association between increased levels of ambient air pollution and increased levels of hospital admissions, typically at 0, 1, or 2 days after an air pollution episode. An important research aim is to extend existing statistical models so that a more detailed understanding of the time course of hospitalization after exposure to air pollution can be obtained. Information about this time course, combined with prior knowledge about biological mechanisms, could provide the basis for hypotheses concerning the mechanism by which air pollution causes disease. Previous studies have identified two important methodological questions: (1) How can we estimate the shape of the distributed lag between increased air pollution exposure and increased mortality or morbidity? and (2) How should we estimate the cumulative population health risk from short-term exposure to air pollution? Distributed lag models are appropriate tools for estimating air pollution health effects that may be spread over several days. However, estimation for distributed lag models in air pollution and health applications is hampered by the substantial noise in the data and the inherently weak signal that is the target of investigation. We introduce an hierarchical Bayesian distributed lag model that incorporates prior information about the time course of pollution effects and combines information across multiple locations. The model has a connection to penalized spline smoothing using a special type of penalty matrix. We apply the model to estimating the distributed lag between exposure to particulate matter air pollution and hospitalization for cardiovascular and respiratory disease using data from a large United States air pollution and hospitalization database of Medicare enrollees in 94 counties covering the years 1999-2002.