The selection of smoothing parameters by generalized cross-validation (GCV) becomes complicated when dealing with correlated data. In this paper, we develop an automatic algorithm for selection of smoothing parameters in non-parametric longitudinal models by combining the BRUTO algorithm of Hastie (1989) and the modifications to GCV due to Altman (1990) to handle the correlation. The algorithm is detailed and illustrated via analysis of a panic-attack data set.
1998 John Wiley & Sons, Ltd.
KEY WORDS generalized estimating equations; quasi-likelihood; correlated data; local-scoring, smoothing; cross validation; BRUTO
1. Introduction
Most of the work on non-parametric models for longitudinal data so far deals with Gaussian data in the time series context. Altman and Diggle and Hutchinson studied smoothing parameter selection in models with correlated Gaussian errors for kernel smoothers and smoothing splines, respectively. Hurvich and Zeger proposed a frequency domain selection criterion for regression problems with autocorrelated errors. The above models predominantly dealt with a single predictor.
Recently, Berhane and Tibshirani developed a class of generalized additive models for longitudinal data in the style of the generalized-estimating-equations (GEE) approach of Liang and Zeger. Hastie proposed an automatic smoothing parameter selection procedure, known as BRUTO, that uses a modified version of the GCV criterion by exploiting the advantages of the backfitting algorithm. This procedure is not directly applicable to longitudinal data, since the use of GCV is known to lead to undersmoothing or oversmoothing, depending on the nature of the underlying correlation. In this paper, we extend the BRUTO algorithm into the longitudinal setting. The remainder of the paper is organized as follows. In Section 2, we summarize the main ideas of the generalized additive modelling technique for longitudinal data. In Section 3, we introduce a new automatic procedure for smoothing parameter selection in the style of the BRUTO algorithm of Hastie. The new procedure is illustrated in Section 4 via analysis of a panic-attack data set.