Restricted Likelihood Ratio Tests for Functional Effects in the Functional Linear Model

References

• Aerts, M., Claeskens, G., and Wand, M. (2002), “Some Theory for Penalized Spline Generalized Additive Models,” Journal of Statistical Planning and Inference, 103, 455–470.
   Google Scholar
• Aneiros-Pérez, G., and Vieu, P. (2008), “Nonparametric Time Series Prediction: A Semi-Functional Partial Linear Modeling,” Journal of Multivariate Analysis, 99, 834–857.
   Web of Science ®Google Scholar
• Basser, P., Mattiello, J., and LeBihan, D. (1994), “MR Diffusion Tensor Spectroscopy and Imaging,” Biophysical Journal, 66, 259–267.
   PubMed Web of Science ®Google Scholar
• Basser, P., Pajevic, S., Pierpaoli, C., and Duda, J. (2000), “In Vivo Fiber Tractography Using DT-MRI Data,” Magnetic Resonance in Medicine, 44, 625–632.
   PubMed Web of Science ®Google Scholar
• Cardot, H., Ferraty, F., Mas, A., and Sarda, P. (2003), “Testing Hypotheses in the Functional Linear Model,” Scandinavian Journal of Statistics, 30, 241–255.
   Web of Science ®Google Scholar
• Cardot, H., Ferraty, F., and Sarda, P. (2003), “Spline Estimators for the Functional Linear Model,” Statistica Sinica, 13, 571–591.
   Web of Science ®Google Scholar
• Carter, C.K., and Kohn, R. (1994), “On Gibbs Sampling for State Space Models,” Biometrika, 81, 541–553.
   Web of Science ®Google Scholar
• Chen, D., Hall, P., and Müller, H.-G. (2011), “Single and Multiple Index Functional Regression Models With Nonparametric Link,” The Annals of Statistics, 39, 1720–1747.
   Web of Science ®Google Scholar
• Crainiceanu, C., Reiss, P., Goldsmith, J., Huang, L., Huo, L., and Scheipl, F. (2013), Refund: Regression With Functional Data, R package version 0.1-8.
   Google Scholar
• Crainiceanu, C., and Ruppert, D. (2004), “Likelihood Ratio Tests in Linear Mixed Models With One Variance Component,” Journal of the Royal Statistical Society, Series B, 66, 165–185.
   Google Scholar
• Crainiceanu, C., Ruppert, D., Claeskens, G., and Wand, M. (2005), “Exact Likelihood Ratio Tests for Penalised Splines,” Biometrika, 92, 91–103.
   Web of Science ®Google Scholar
• Crainiceanu, C.M., Staicu, A.M., and Di, C.-Z. (2009), “Generalized Multilevel Functional Regression,” Journal of the American Statistical Association, 104, 1550–1561.
   PubMed Web of Science ®Google Scholar
• Delsol, L., Ferraty, F., and Vieu, P. (2011), “Structural Test in Regression on Functional Variables,” Journal of Multivariate Analysis, 102, 422–447.
   Web of Science ®Google Scholar
• Di, C.-Z., Crainiceanu, C.M., Caffo, B.S., and Punjabi, N.M. (2009), “Multilevel Functional Principal Component Analysis,” Annals of Applied Statistics, 4, 458–488.
   Web of Science ®Google Scholar
• Eilers, P. H.C., Li, B., and Marx, B.D. (2009), “Multivariate Calibration With Single-Index Signal Regression,” Chemometrics and Intelligent Laboratory Systems, 96, 196–202.
   Web of Science ®Google Scholar
• Fahrmeir, L., and Lang, S. (2001a), “Bayesian Inference for Generalized Additive Mixed Models Based on Markov Random Field Priors,” Journal of the Royal Statistical Society, Series C, 50, 201–220.
   Web of Science ®Google Scholar
• ——— (2001b), “Bayesian Semiparametric Regression Analysis of Multicategorical Time–Space Data,” Annals of the Institute of Statistical Mathematics, 53, 11–30.
   Web of Science ®Google Scholar
• Ferraty, F., Goia, A., Salinelli, E., and Vieu, P. (2012), “Functional Projection Pursuit Regression,” Test, 22, 293–320.
   Web of Science ®Google Scholar
• Ferraty, F., González-Manteiga, W., Martínez-Calvo, A., and Vieu, P. (2011), “Presmoothing in Functional Linear Regression,” Statistica Sinica, 22, 69–94.
   Web of Science ®Google Scholar
• Ferraty, F., Hall, P., and Vieu, P. (2010), “Most-Predictive Design Points for Functional Data Predictors,” Biometrika, 97, 807–824.
   Web of Science ®Google Scholar
• Ferraty, F., and Vieu, P. (2006), Nonparametric Functional Data Analysis: Theory and Practice, New York: Springer.
   Google Scholar
• Gertheiss, J., Goldsmith, J., Crainiceanu, C., and Greven, S. (2013), “Longitudinal Scalar-on-Functions Regression With Application to Tractography Data,” Biostatistics, 14, 447–461.
   PubMed Web of Science ®Google Scholar
• Gertheiss, J., Maity, A., and Staicu, A.-M. (2013), “Variable Selection in Generalized Functional Linear Models,” Stat, 2, 86–101.
   PubMedGoogle Scholar
• Goldsmith, J., Bobb, J., Crainiceanu, C., Caffo, B., and Reich, D. (2011), “Penalized Functional Regression,” Journal of Computational and Graphical Statistics, 20, 831–851.
   Web of Science ®Google Scholar
• Goldsmith, J., Crainiceanu, C.M., Caffo, B., and Reich, D. (2012), “Longitudinal Penalized Functional Regression for Cognitive Outcomes on Neuronal Tract Measurements,” Journal of the Royal Statistical Society, Series C, 61, 453–469.
   Web of Science ®Google Scholar
• Goldsmith, J., Wand, M.P., and Crainiceanu, C.M. (2011), “Functional Regression via Variational Bayes,” Electronic Journal of Statistics, 5, 572–602.
   PubMed Web of Science ®Google Scholar
• Gong, G., and Samaniego, F.J. (1981), “Pseudo Maximum Likelihood Estimation: Theory and Applications,” The Annals of Statistics, 861–869.
   Web of Science ®Google Scholar
• Goutis, C., and Fearn, T. (1996), “Partial Least Squares Regression on Smooth Factors,” Journal of the American Statistical Association, 91, 627–632.
   Web of Science ®Google Scholar
• Greven, S., Crainiceanu, C., Küchenhoff, H., and Peters, A. (2008), “Restricted Likelihood Ratio Testing for Zero Variance Components in Linear Mixed Models,” Journal of Computational and Graphical Statistics, 17, 870–891.
   Web of Science ®Google Scholar
• Gronwall, D. M.A. (1977), “Paced Auditory Serial-Addition Task: A Measure of Recovery From Concussion,” Perceptual and Motor Skills, 44, 367–373.
   PubMed Web of Science ®Google Scholar
• Hastie, T., and Tibshirani, R. (2000), “Bayesian Backfitting” (with discussion), Statistical Science, 15, 196–223.
   Web of Science ®Google Scholar
• Horváth, L., and Kokoszka, P. (2012), Inference for Functional Data With Applications (vol. 200), New York: Springer.
   Google Scholar
• James, G.M., Hastie, T.J., and Sugar, C.A. (2000), “Principal Component Models for Sparse Functional Data,” Biometrika, 87, 587–602.
   Web of Science ®Google Scholar
• James, G.M., and Silverman, B. (2005), “Functional Adaptive Model Estimation,” Journal of the American Statistical Association, 100, 565–576.
   Web of Science ®Google Scholar
• James, G.M., Wang, J., and Zhu, J. (2009), “Functional Linear Regression That’s Interpretable,” The Annals of Statistics, 37, 2083–2108.
   Web of Science ®Google Scholar
• Kong, D., Staicu, A., and Maity, A. (2013), “Classical Testing in Functional Linear Models,” North Carolina State University Department of Statistics Technical Reports, 2647, 1–23.
   Google Scholar
• Lang, S., and Brezger, A. (2004), “Bayesian P-Splines,” Journal of Computational and Graphical Statistics, 13, 183–212.
   Web of Science ®Google Scholar
• LeBihan, D., Mangin, J., Poupon, C., and Clark, C. (2001), “Diffusion Tensor Imaging: Concepts and Applications,” Journal of Magnetic Resonance Imaging, 13, 534–546.
   PubMed Web of Science ®Google Scholar
• Lindquist, M., and McKeague, I. (2009), “Logistic Regression With Brownian-Like Predictors,” Journal of the American Statistical Association, 104, 1575–1585.
   Web of Science ®Google Scholar
• Marx, B., and Eilers, P. (1998), “Direct Generalized Additive Modeling With Penalized Likelihood,” Computational Statistics & Data Analysis, 28, 193–209.
   Web of Science ®Google Scholar
• ——— (1999), “Generalized Linear Regression on Sampled Signals and Curves: A P-Spline Approach,” Technometrics, 41, 1–13.
   Web of Science ®Google Scholar
• McLean, M.W., Hooker, G., Staicu, A.-M., Scheipl, F., and Ruppert, D. (2012), “Functional Generalized Additive Models,” Journal of Computational and Graphical Statistics, DOI: 10.1080/10618600.2012.729985. Available at http://amstat.tandfonline.com/.
   Web of Science ®Google Scholar
• Molenberghs, G., and Verbeke, G. (2007), “Likelihood Ratio, Score, and Wald Tests in a Constrained Parameter Space,” The American Statistician, 61, 22–27.
   Web of Science ®Google Scholar
• Mori, S., and Barker, P. (1999), “Diffusion Magnetic Resonance Imaging: Its Principle and Applications,” The Anatomical Record, 257, 102–109.
   PubMedGoogle Scholar
• Müller, H.-G., and Stadtmüller, U. (2005), “Generalized Functional Linear Models,” The Annals of Statistics, 33, 774–805.
   Web of Science ®Google Scholar
• Ngo, L., and Wand, M. (2004), “Smoothing With Mixed Model Software,” Journal of Statistical Software, 9, 1–54.
   Google Scholar
• Ozturk, A., Smith, S., Gordon-Lipkin, E., Harrison, D., Shiee, N., Pham, D., Caffo, B., Calabresi, P., and Reich, D. (2010), “MRI of the Corpus Callosum in Multiple Sclerosis: Association With Disability,” Multiple Sclerosis, 16, 166–177.
   PubMed Web of Science ®Google Scholar
• Preda, C., and Saporta, G. (2005), “Pls Regression on a Stochastic Process,” Computational Statistics & Data Analysis, 48, 149–158.
   Web of Science ®Google Scholar
• Preda, C., Saporta, G., and Lévéder, C. (2007), “Pls Classification of Functional Data,” Computational Statistics, 22, 223–235.
   Web of Science ®Google Scholar
• Ramsay, J. J.O., Hooker, G., and Graves, S. (2009), Functional Data Analysis With R and MATLAB, New York: Springer.
   Google Scholar
• Ramsay, J.O., and Silverman, B.W. (2005), Functional Data Analysis, New York: Springer.
   Google Scholar
• Reiss, P., and Ogden, R. (2007), “Functional Principal Component Regression and Functional Partial Least Squares,” Journal of the American Statistical Association, 102, 984–996.
   Web of Science ®Google Scholar
• ——— (2010), “Functional Generalized Linear Models With Images as Predictors,” Biometrics, 66, 61–69.
   PubMed Web of Science ®Google Scholar
• Ruppert, D. (2002), “Selecting the Number of Knots for Penalized Splines,” Journal of Computational and Graphical Statistics, 11, 735–757.
   Web of Science ®Google Scholar
• Ruppert, D., Wand, M., and Carroll, R. (2003), Semiparametric Regression (vol. 12), New York: Cambridge University Press.
   Google Scholar
• Scheipl, F., Greven, S., and Kuechenhoff, H. (2008), “Size and Power of Tests for a Zero Random Effect Variance or Polynomial Regression in Additive and Linear Mixed Models,” Computational Statistics & Data Analysis, 52, 3283–3299.
   Web of Science ®Google Scholar
• Self, S., and Liang, K. (1987), “Asymptotic Properties of Maximum Likelihood Estimators and Likelihood Ratio Tests Under Nonstandard Conditions,” Journal of the American Statistical Association, 82, 605–610.
   Web of Science ®Google Scholar
• Staniswalis, J., and Lee, J. (1998), “Nonparametric Regression Analysis of Longitudinal Data,” Journal of the American Statistical Association, 444, 1403–1418.
   Google Scholar
• Stram, D., and Lee, J. (1994), “Variance Components Testing in the Longitudinal Mixed Effects Model,” Biometrics, 1171–1177.
   PubMed Web of Science ®Google Scholar
• Tzeng, J., and Zhang, D. (2007), “Haplotype-Based Association Analysis via Variance-Components Score Test,” American Journal of Human Genetics, 81, 927.
   PubMed Web of Science ®Google Scholar
• Verbeke, G., and Molenberghs, G. (2003), “The Use of Score Tests for Inference on Variance Components,” Biometrics, 59, 254–262.
   PubMed Web of Science ®Google Scholar
• Wand, M. (2003), “Smoothing and Mixed Models,” Computational Statistics, 18, 223–250.
   Web of Science ®Google Scholar
• Wood, S. (2012), “On p-Values for Smooth Components of an Extended Generalized Additive Model,” Biometrika. Available at http://biomet.oxfordjournals.org/citmgr?gca=biomet;ass048v1.
   Web of Science ®Google Scholar
• Xiao, L., Ruppert, D., Zipunnikov, V., and Crainiceanu, C. (2013), “Fast Covariance Estimation for High-Dimensional Functional Data,” ArXiv e-prints. Available at http://adsabs.harvard.edu/abs/2013arXiv1306.5718X.
   Google Scholar
• Yao, F., Müller, H., and Wang, J. (2005), “Functional Data Analysis for Sparse Longitudinal Data,” Journal of the American Statistical Association, 100, 577–590.
   Web of Science ®Google Scholar
• Yao, F., Müller, H.G., Clifford, A., Dueker, S., Follett, J., Lin, Y., Buchholz, B., and Vogel, J. (2003), “Shrinkage Estimation for Functional Principal Component Scores With Application to the Population,” Biometrics, 59, 676–685.
   PubMed Web of Science ®Google Scholar
• Zhang, D., and Lin, X. (2003), “Hypothesis Testing in Semiparametric Additive Mixed Models,” Biostatistics, 4, 57–74.
   PubMed Web of Science ®Google Scholar
• Zhang, D., and Lin, X. (2008), “Variance Component Testing in Generalized Linear Mixed Models for Longitudinal/Clustered Data and Other Related Topics,” in Random Effect and Latent Variable Model Selection, New York: Springer, 19–36.
   Google Scholar