Generalized linear mixed models in time series studies of air pollution

Generalized additive models (GAMs) with natural cubic splines (NS) as smoothing functions have become standard analytical tools in time series studies of health effects of air pollution. A GAM with NS as a smoother is reduced to a generalized linear model and is denoted by GLM+NS in literature. The amount of smoothing is controlled by the parameter degrees of freedom (df) in the fitted NS. While a large amount of smoothing could result in a less biased parameter estimate, over-smoothing may attenuate important signals in the data. In practice, this issue is often addressed by sensitivity analyses with different df values. Smoothing can also be achieved by assuming the parameters of the splines as random effects with an appropriate distribution. In time series studies of health effects of air pollution the outcome variable, daily mortality (or morbidity), is commonly assumed to follow a Poisson distribution. With assuming parameters of the natural cubic spline smoother random, a generalized linear mixed model is resulted and, denoted by GLMM+NS. We investigate the validity of this mixed modeling approach through a simulation study. Our simulation results show that for small true pollution effects, fitting a GLMM+NS model results in absolute biases similar to those obtained from the fitting of a GLM+NS, but provides larger empirical standard deviations than a GLM+NS. It can be noted that as the variability in the observed time series data can be different over the study period, the assumption of a variable amount of smoothing over a study period is more realistic and the larger standard deviation may reflect reality. We provide an application of GLMM+NS utilizing data from the Allegheny County Air Pollution Study with interest in estimating the relative risk of cardiopulmonary hospital admissions for a 20 μg/m³ increase in PM₁₀.