Conditional Poisson Regression with Random Effects for the Analysis of Multi-site Time Series Studies

The analysis of time series studies linking daily counts of a health indicator with environmental variables (e.g., mortality or hospital admissions with air pollution concentrations or temperature; or motor vehicle crashes with temperature) is usually conducted with Poisson regression models controlling for long-term and seasonal trends using temporal strata. When the study includes multiple zones, analysts usually apply a two-stage approach: first, each zone is analyzed separately, and the resulting zone-specific estimates are then combined using meta-analysis. This approach allows zone-specific control for trends. A one-stage approach uses spatio-temporal strata and could be seen as a particular case of the case-time series framework recently proposed. However, the number of strata can escalate very rapidly in a long time series with many zones. A computationally efficient alternative is to fit a conditional Poisson regression model, avoiding the estimation of the nuisance strata. To allow for zone-specific effects, we propose a conditional Poisson regression model with a random slope, although available frequentist software does not implement this model. Here, we implement our approach in the Bayesian paradigm, which also facilitates the inclusion of spatial patterns in the effect of interest. We also provide a possible extension to deal with overdispersed data. We first introduce the equations of the framework and then illustrate their application to data from a previously published study on the effects of temperature on the risk of motor vehicle crashes. We provide R code and a semi-synthetic dataset to reproduce all analyses presented.