### 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 language | English (US) |
---|---|

Pages (from-to) | 91-103 |

Number of pages | 13 |

Journal | Biometrika |

Volume | 92 |

Issue number | 1 |

DOIs | |

State | Published - Mar 2005 |

### Fingerprint

### 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

*Biometrika*,

*92*(1), 91-103. https://doi.org/10.1093/biomet/92.1.91

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

Research output: Contribution to journal › Article

*Biometrika*, vol. 92, no. 1, pp. 91-103. https://doi.org/10.1093/biomet/92.1.91

}

TY - JOUR

T1 - Exact likelihood ratio tests for penalised splines

AU - Crainiceanu, Ciprian M

AU - Ruppert, David

AU - Claeskens, Gerda

AU - Wand, M. P.

PY - 2005/3

Y1 - 2005/3

N2 - 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.

AB - 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.

KW - Linear mixed model

KW - Penalised spline

KW - Smoothing

KW - Zero variance component

UR - http://www.scopus.com/inward/record.url?scp=15844423998&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=15844423998&partnerID=8YFLogxK

U2 - 10.1093/biomet/92.1.91

DO - 10.1093/biomet/92.1.91

M3 - Article

AN - SCOPUS:15844423998

VL - 92

SP - 91

EP - 103

JO - Biometrika

JF - Biometrika

SN - 0006-3444

IS - 1

ER -