SMOOTHING SPLINE MODELS FOR THE ANALYSIS OF NESTED AND CROSSED SAMPLES OF CURVES

We introduce a class of models for an additive decomposition of groups of curves stratified by crossed and nested factors, generalizing smoo thing splines to such samples by associating them with a corresponding mixed-effects model. The models are also useful for imputation of mis sing data and exploratory analysis of variance. We prove that the best linear unbiased predictors (BLUPs) from the extended mixed-effects mo del correspond to solutions of a generalized penalized regression wher e smoothing parameters are directly related to variance components, an d we show that these solutions are natural cubic splines. The model pa rameters are estimated using a highly efficient implementation of the EM algorithm for restricted maximum likelihood (REML) estimation based on a preliminary eigenvector decomposition. Variability of computed e stimates can be assessed with asymptotic techniques or with a novel hi erarchical bootstrap resampling scheme for nested mixed-effects models . Our methods are applied to menstrual cycle data from studies of repr oductive function that measure daily urinary progesterone; the sample of progesterone curves is stratified by cycles nested within subjects nested within conceptive and nonconceptive groups.