Biostatistics (2004), 5, 3, pp. 341-360
Biostatistics Vol. 5 No. 3 © Oxford University Press 2004; all rights reserved.
Hierarchical bivariate time series models: a combined analysis of the effects of particulate matter on morbidity and mortality
Department of Biostatistics, Bloomberg School of Public Health, 615 N. Wolfe Street, The Johns Hopkins University, Baltimore, MD 21205-3179, USA
fdominic{at}jhsph.edu
Department of Environmental Health, Harvard School of Public Health, 615 N. Wolfe Street, The Johns Hopkins University, Baltimore, MD 21205-3179, USA
Department of Biostatistics, Bloomberg School of Public Health, 615 N. Wolfe Street, The Johns Hopkins University, Baltimore, MD 21205-3179, USA
Department of Environmental Health, Harvard School of Public Health, 615 N. Wolfe Street, The Johns Hopkins University, Baltimore, MD 21205-3179, USA
Department of Epidemiology, Bloomberg School of Public Health, 615 N. Wolfe Street, The Johns Hopkins University, Baltimore, MD 21205-3179, USA
To whom correspondence should be addressed.
In this paper we develop a hierarchical bivariate time series model to characterize the relationship between particulate matter less than 10 microns in aerodynamic diameter (
10) and both mortality and hospital admissions for cardiovascular diseases. The model is applied to time series data on mortality and morbidity for 10 metropolitan areas in the United States from 1986 to 1993. We postulate that these time series should be related through a shared relationship with 10.
At the first stage of the hierarchy, we fit two seemingly unrelated Poisson regression models to produce city-specific estimates of the log relative rates of mortality and morbidity associated with exposure to 10 within each location. The sample covariance matrix of the estimated log relative rates is obtained using a novel generalized estimating equation approach that takes into account the correlation between the mortality and morbidity time series. At the second stage, we combine information across locations to estimate overall log relative rates of mortality and morbidity and variation of the rates across cities.
Using the combined information across the 10 locations we find that a 10 µg/m3 increase in average 10 at the current day and previous day is associated with a 0.26% increase in mortality (95% posterior interval –0.37, 0.65), and a 0.71% increase in hospital admissions (95% posterior interval 0.35, 0.99). The log relative rates of mortality and morbidity have a similar degree of heterogeneity across cities: the posterior means of the between-city standard deviations of the mortality and morbidity air pollution effects are 0.42 (95% interval 0.05, 1.18), and 0.31 (95% interval 0.10, 0.89), respectively. The city-specific log relative rates of mortality and morbidity are estimated to have very low correlation, but the uncertainty in the correlation is very substantial (posterior mean = 0.20, 95% interval –0.89, 0.98).
With the parameter estimates from the model, we can predict the hospitalization log relative rate for a new city for which hospitalization data are unavailable, using that city's estimated mortality relative rate. We illustrate this prediction using New York as an example.
Keywords: Air pollution; Generalized additive models; Generalized estimating equations; Hierarchical models; Log relative rate; Particulate matter