Transitional regression models, with application to environmental time series

Environmental epidemiologists often encounter time series data in the form of discrete or other nonnormal outcomes: for example, in modeling the relat ionship between air pollution and hospital admissions or mortality rates. W e present a case study examining the association between pollen counts and meteorologic covariates. Although such time series data are inadequately de scribed by standard methods for Gaussian time series, they are often autoco rrelated, and warrant an analysis beyond those provided by ordinary general ized linear models (GLMs). Transitional regression models (TRMs), signifyin g nonlinear regression models expressed in terms of conditional means and v ariances given past observations, provide a unifying framework for two main stream approaches to extending the GLM for autocorrelated data. The first a pproach models current outcomes with a GLM that incorporates past outcomes as covariates, whereas the second models individual outcomes with marginal GLMs and then couples the error terms With an autoregressive covariance mat rix. Although the two approaches coincide for the Gaussian GLM, which serve s as a helpful introductory example, in general they yield fundamentally di fferent models. We analyze the pollen study using TRM's of both types and p resent parameter estimates together with asymptotic and bootstrap standard errors. In several cases we find evidence of residual autocorrelation; howe ver, when we relax the TRM to allow for a nonparametric smooth trend, the a utocorrelation disappears. This kind of trade-off between autocorrelation a nd flexibility is to be expected, and has a natural interpretation in terms of the covariance function for a nonparametric smoother. We provide an alg orithm for fitting these flexible TRM's that is relatively easy to program with the generalized additive model software in S-PLUS.