Abstract
Penalty splines such as smoothing spline and Pβspline, as well as unpenalized regression splines, have become increasingly popular methods in contemporary nonβparametric and semiparametric regressions, particularly for data arising from longitudinal, multilevel, and spatiotemporal settings. In the recent decade, the development of the Markov chain Monte Carlo (MCMC) methods has facilitated applications of flexible spline fittings via Bayesian hierarchical formulation. In this paper, we study three spline smoothing methods in the context of spatiotemporal modeling of rates and Bayesian disease mapping. We explore their potentials for fully Bayesian (FB) rate and risk trends ensemble estimation and spatiotemporal relative risks inference. In particular, our study presents and compares Bayesian hierarchical formulations of regression Bβspline, smoothing spline, and Pβspline, explores the connections and distinctions among them, and sheds light on their varying capabilities as βdataβdrivenβ smoothers for risk trends exploration and sequential disease mapping. The methods are motivated and illustrated through a Bayesian analysis of adverse medical events (commonly known as iatrogenic injures) to hospitalized children and youth in British Columbia, Canada. Copyright Β© 2007 John Wiley & Sons, Ltd.