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Exact likelihood ratio tests for penalised splines

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

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 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. Copyright 2005, Oxford University Press.

Suggested Citation

  • Ciprian Crainiceanu & David Ruppert & Gerda Claeskens & M. P. Wand, 2005. "Exact likelihood ratio tests for penalised splines," Biometrika, Biometrika Trust, vol. 92(1), pages 91-103, March.
    • Handle: RePEc:oup:biomet:v:92:y:2005:i:1:p:91-103
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    File URL: http://hdl.handle.net/10.1093/biomet/92.1.91
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    Cited by:

    1. Ana-Maria Staicu & Yingxing Li & Ciprian M. Crainiceanu & David Ruppert, 2014. "Likelihood Ratio Tests for Dependent Data with Applications to Longitudinal and Functional Data Analysis," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 41(4), pages 932-949, December.
    2. Chen, Haiqiang & Fang, Ying & Li, Yingxing, 2015. "Estimation And Inference For Varying-Coefficient Models With Nonstationary Regressors Using Penalized Splines," Econometric Theory, Cambridge University Press, vol. 31(4), pages 753-777, August.
    3. Rong Chen & Hua Liang & Jing Wang, 2011. "Determination of linear components in additive models," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 23(2), pages 367-383.
    4. Manuel Wiesenfarth & Tatyana Krivobokova & Stephan Klasen & Stefan Sperlich, 2012. "Direct Simultaneous Inference in Additive Models and Its Application to Model Undernutrition," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 107(500), pages 1286-1296, December.
    5. Sosso Feindouno & Michael Goujon, 2019. "Human Assets Index: Insights from a Retrospective Series Analysis," Social Indicators Research: An International and Interdisciplinary Journal for Quality-of-Life Measurement, Springer, vol. 141(3), pages 959-984, February.
    6. Zaili Fang & Inyoung Kim & Jeesun Jung, 2018. "Semiparametric Kernel-Based Regression for Evaluating Interaction Between Pathway Effect and Covariate," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 23(1), pages 129-152, March.
    7. Sue J. Welham & Brian R. Cullis & Michael G. Kenward & Robin Thompson, 2006. "The Analysis of Longitudinal Data Using Mixed Model L-Splines," Biometrics, The International Biometric Society, vol. 62(2), pages 392-401, June.
    8. repec:wyi:journl:002195 is not listed on IDEAS
    9. Sonja Greven & Ciprian Crainiceanu, 2013. "On likelihood ratio testing for penalized splines," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 97(4), pages 387-402, October.
    10. Annie Qu & Runze Li, 2006. "Quadratic Inference Functions for Varying-Coefficient Models with Longitudinal Data," Biometrics, The International Biometric Society, vol. 62(2), pages 379-391, June.
    11. Ugarte, M.D. & Goicoa, T. & Militino, A.F. & Durbán, M., 2009. "Spline smoothing in small area trend estimation and forecasting," Computational Statistics & Data Analysis, Elsevier, vol. 53(10), pages 3616-3629, August.
    12. J. D. Opsomer & G. Claeskens & M. G. Ranalli & G. Kauermann & F. J. Breidt, 2008. "Non‐parametric small area estimation using penalized spline regression," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 70(1), pages 265-286, February.
    13. Nathaniel E. Helwig, 2022. "Robust Permutation Tests for Penalized Splines," Stats, MDPI, vol. 5(3), pages 1-18, September.
    14. Ghosal, Rahul & Maity, Arnab, 2022. "A Score Based Test for Functional Linear Concurrent Regression," Econometrics and Statistics, Elsevier, vol. 21(C), pages 114-130.
    15. Giles Hooker, 2009. "Forcing Function Diagnostics for Nonlinear Dynamics," Biometrics, The International Biometric Society, vol. 65(3), pages 928-936, September.
    16. Azaïs, Jean-Marc & Ribes, Aurélien, 2016. "Multivariate spline analysis for multiplicative models: Estimation, testing and application to climate change," Journal of Multivariate Analysis, Elsevier, vol. 144(C), pages 38-53.
    17. Stefan Stremersch & Aurélie Lemmens, 2009. "Sales Growth of New Pharmaceuticals Across the Globe: The Role of Regulatory Regimes," Marketing Science, INFORMS, vol. 28(4), pages 690-708, 07-08.
    18. Yuanjia Wang & Huaihou Chen, 2012. "On Testing an Unspecified Function Through a Linear Mixed Effects Model with Multiple Variance Components," Biometrics, The International Biometric Society, vol. 68(4), pages 1113-1125, December.
    19. Zaixing Li & Lixing Zhu, 2010. "On Variance Components in Semiparametric Mixed Models for Longitudinal Data," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 37(3), pages 442-457, September.
    20. Stremersch, S. & Lemmens, A., 2008. "Sales Growth of New Pharmaceuticals Across the Globe: The Role of Regulatory Regimes," ERIM Report Series Research in Management ERS-2008-026-MKT, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
    21. Sweeney Elizabeth & Crainiceanu Ciprian & Gertheiss Jan, 2016. "Testing differentially expressed genes in dose-response studies and with ordinal phenotypes," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 15(3), pages 213-235, June.
    22. Oscar O. Melo & Carlos E. Melo & Jorge Mateu, 2016. "Beta spatial linear mixed model with variable dispersion using Monte Carlo maximum likelihood," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 70(1), pages 47-76, February.

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