Exact likelihood ratio tests for penalised splines
1 Department of Biostatistics, Johns Hopkins University, 615 N. Wolfe Street, Baltimore, Maryland 221205, U.S.A. ccrainic{at}jhsph.edu, 2 School of Operations Research and Industrial Engineering, Cornell University, Ithaca, New York 14853, U.S.A. davidr{at}orie.cornell.edu, 3 Departement OR & Business Statistics, Katholieke Universiteit Leuven, Naamsestraat 69, 3000 Leuven, Belgium claeskens{at}econ.kuleuven.ac.be, 4 Department of Statistics, School of Mathematics, University of New South Wales, Sydney 2052, Australia wand{at}maths.unsw.edu.au
Penalised-spline-based additive models allow a simple mixed model representation where the variance components control departures from linear models. The smoothing parameter is the ratio of the random-coefficient and error variances and tests for linear regression reduce to tests for zero random-coefficient variances. We propose exactlikelihood and restricted likelihood ratio tests for testing polynomial regression versus a general alternative modelled by penalised splines. Their spectral decompositions are used as the basis of fast simulation algorithms. We derive the asymptotic local power properties of the tests under weak conditions. In particular we characterise the local alternatives that are detected with asymptotic probability one. Confidence intervals for the smoothing parameter are obtained by inverting the tests for a fixed smoothing parameter versus a general alternative. We discuss F and R tests and show that ignoring the variability in the smoothing parameter estimator can have a dramatic effect on their null distributions. The powers of several known tests are investigated and a small set of tests with good power properties is identified. The restricted likelihood ratio test is among the best in terms of power.
Key Words: Linear mixed model; Penalised spline; Smoothing; Zero variance component
Received December 2002. Revised May 2004.
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