Exact likelihood ratio tests for penalised splines

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

Research output: Contribution to journalArticlepeer-review

82 Scopus citations


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
Issue number1
StatePublished - Mar 2005


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

ASJC Scopus subject areas

  • Statistics and Probability
  • General Mathematics
  • Agricultural and Biological Sciences (miscellaneous)
  • General Agricultural and Biological Sciences
  • Statistics, Probability and Uncertainty
  • Applied Mathematics


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