Smoothing spline based tests for non-linearity in a partially linear model |
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| Institution: | 1. College of Mathematics and Statistics, Institute of Statistical Sciences, Shenzhen University, Shenzhen, China;2. Shenzhen-Hong Kong Joint Research Center for Applied Statistical Sciences, Shenzhen University, Shenzhen, China;3. Department of Biostatistics and Computational Biology, University of Rochester, NY, USA;1. Department of Gastroenterology and Hepatology, Erasmus MC, Rotterdam, the Netherlands;3. Department of Medicine & Therapeutics, Chinese University of Hong Kong, Hong Kong SAR, China;4. Hepatology Unit, University of Pisa, Pisa, Italy;6. Centre de Recherche Biomédicale Bichat-Beaujon, Université Paris-Diderot, Clichy, France;5. Department of Gastroenterology, Hepatology and Endocrinology, Hannover Medical School, Hannover, Germany;7. Liver Research Unit, Chang Gung Memorial Hospital, Chang Gung University College of Medicine, Taipei, Taiwan;12. UHN Liver Clinic, Toronto Western and General Hospital, University Health Network Toronto, Toronto, Canada;8. Department of Public Health, Erasmus MC, Rotterdam, the Netherlands;1. Division of Viral Hepatitis, National Center for HIV/AIDS, Viral Hepatitis, STD and TB Prevention, Centers for Disease Control and Prevention, Atlanta, GA, USA;2. Arctic Investigations Program, Division of Preparedness and Emerging Infections, National Center for Emerging and Zoonotic Infectious Diseases, Centers for Disease Control and Prevention, Anchorage, AK, USA;3. Liver Disease and Hepatitis Program, Alaska Native Tribal Health Consortium, Anchorage, AK, USA;1. Department of Ophthalmology, University of California, San Francisco, San Francisco, California, USA;2. Glaucoma Research Unit, Faculty of Medicine, Chulalongkorn University and King Chulalongkorn Memorial Hospital, Thai Red Cross Society, Bangkok, Thailand;3. Fundación Oftalmológica Los Andes, Universidad de los Andes, Santiago, Chile;4. Ngoc Nguyen Eye Clinic, San Jose, California, USA;5. Glaucoma Center of San Francisco, San Francisco, California, USA |
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| Abstract: | This paper deals with testing for non-linearity in a regression model with one possibly non-linear component being estimated non-parametrically using smoothing splines. We propose two new variance–covariance based tests for detecting non-linearity applying a likelihood ratio hypothesis testing approach. The first test is for the inclusion of a possibly non-linear component and the second one is for linearity of a possibly non-linear component. The tests are based on a stochastic model in state space form given by Wahba (J. Roy. Statist. Soc. Ser. B 40 (1978) 364), Wecker and Ansley (J. Amer. Statist. Assoc. 78 (1983) 81) and de Jong and Mazzi (Modeling and smoothing unequally spaced sequence data, University of York and University of British Columbia, Unpublished paper) for which smoothing splines provide an optimal estimate. Pitrun (A smoothing spline approach to non-linear interface for time series, Department of Econometrics and Business Statistics, Monash University, Unpublished Ph.D. thesis) derived the variance–covariance structure of this model, which allows the use of a marginal likelihood approach. This leads naturally to marginal-likelihood based likelihood ratio tests for non-linearity. Small sample properties of the new tests have been investigated via Monte Carlo studies. |
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