Transitional Regression Models, with Application to Environmental Time Series

Abstract

Environmental epidemiologists often encounter time series data in the form of discrete or other nonnormal outcomes; for example, in modeling the relationship between air pollution and hospital admissions or mortality rates. We present a case study examining the association between pollen counts and meteorologic covariates. Although such time series data are inadequately described by standard methods for Gaussian time series, they are often autocorrelated, and warrant an analysis beyond those provided by ordinary generalized linear models (GLMs). Transitional regression models (TRMs), signifying nonlinear regression models expressed in terms of conditional means and variances given past observations, provide a unifying framework for two mainstream approaches to extending the GLM for autocorrelated data. The first approach 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 matrix. Although the two approaches coincide for the Gaussian GLM, which serves as a helpful introductory example, in general they yield fundamentally different models. We analyze the pollen study using TRM's of both types and present parameter estimates together with asymptotic and bootstrap standard errors. In several cases we find evidence of residual autocorrelation; however, when we relax the TRM to allow for a nonparametric smooth trend, the autocorrelation disappears. This kind of trade-off between autocorrelation and flexibility is to be expected, and has a natural interpretation in terms of the covariance function for a nonparametric smoother. We provide an algorithm for fitting these flexible TRM's that is relatively easy to program with the generalized additive model software in S-PLUS.