Exact likelihood ratio tests for penalised splines

Ciprian M Crainiceanu, David Ruppert, Gerda Claeskens, M. P. Wand

Research output: Contribution to journalArticle

Abstract

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 exact likelihood 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.

Original languageEnglish (US)
Pages (from-to)91-103
Number of pages13
JournalBiometrika
Volume92
Issue number1
DOIs
StatePublished - Mar 2005

Fingerprint

Penalized Splines
Likelihood Ratio Test
Splines
Linear Models
Smoothing Parameter
testing
Confidence Intervals
Random Coefficients
Linear regression
Polynomials
Decomposition
Local Power
Testing
Local Alternatives
Polynomial Regression
Additive Models
Spectral Decomposition
Variance Components
Random Error
Alternatives

Keywords

  • Linear mixed model
  • Penalised spline
  • Smoothing
  • Zero variance component

ASJC Scopus subject areas

  • Agricultural and Biological Sciences(all)
  • Agricultural and Biological Sciences (miscellaneous)
  • Statistics and Probability
  • Mathematics(all)
  • Applied Mathematics

Cite this

Exact likelihood ratio tests for penalised splines. / Crainiceanu, Ciprian M; Ruppert, David; Claeskens, Gerda; Wand, M. P.

In: Biometrika, Vol. 92, No. 1, 03.2005, p. 91-103.

Research output: Contribution to journalArticle

Crainiceanu, CM, Ruppert, D, Claeskens, G & Wand, MP 2005, 'Exact likelihood ratio tests for penalised splines', Biometrika, vol. 92, no. 1, pp. 91-103. https://doi.org/10.1093/biomet/92.1.91
Crainiceanu, Ciprian M ; Ruppert, David ; Claeskens, Gerda ; Wand, M. P. / Exact likelihood ratio tests for penalised splines. In: Biometrika. 2005 ; Vol. 92, No. 1. pp. 91-103.
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