The analysis of designed experiments and longitudinal data by using smoothing splines

In designed experiments and in particular longitudinal studies, the aim may be to assess the effect of a quantitative variable such as time on treatme nt effects. Modelling treatment effects can be complex in the presence of o ther sources of variation. Three examples are presented to illustrate an ap proach to analysis in such cases. The first example is a longitudinal exper iment on the growth of cows under a factorial treatment structure where ser ial correlation and variance heterogeneity complicate the analysis. The sec ond example involves the calibration of optical density and the concentrati on of a protein DNase in the presence of sampling variation and variance he terogeneity. The final example is a multienvironment agricultural field exp eriment in which a yield-seeding rate relationship is required for several varieties of lupins. Spatial variation within environments, heterogeneity b etween environments and variation between varieties all need to be incorpor ated in the analysis. In this paper, the cubic smoothing spline is used in conjunction with fixed and random effects, random coefficients and variance modelling to provide simultaneous modelling of trends and covariance struc ture. The key result that allows coherent and flexible empirical model buil ding in complex situations is the linear mixed model representation of the cubic smoothing spline. An extension is proposed in which trend is partitio ned into smooth and nonsmooth components. Estimation and inference, the ana lysis of the three examples and a discussion of extensions and unresolved i ssues are also presented.