Tutorial in biostatistics: spline smoothing with linear mixed models.

The semi-parametric regression achieved via penalized spline smoothing can be expressed in a linear mixed models framework. This allows such models to be fitted using standard mixed models software routines with which many biostatisticians are familiar. Moreover, the analysis of complex correlated data structures that are a hallmark of biostatistics, and which are typically analysed using mixed models, can now incorporate directly smoothing of the relationship between an outcome and covariates. In this paper we provide an introduction to both linear mixed models and penalized spline smoothing, and describe the connection between the two. This is illustrated with three examples, the first using birth data from the U.K., the second relating mammographic density to age in a study of female twin-pairs and the third modelling the relationship between age and bronchial hyperresponsiveness in families. The models are fitted in R (a clone of S-plus) and using Markov chain Monte Carlo (MCMC) implemented in the package WinBUGS.