Abstract
A scalar-response functional model describes the association between a scalar response and a set of functional covariates. An important problem in the functional data literature is to test nullity or linearity of the effect of the functional covariate in the context of scalar-on-function regression. This article provides an overview of the existing methods for testing both the null hypotheses that there is no relationship and that there is a linear relationship between the functional covariate and scalar response, and a comprehensive numerical comparison of their performance. The methods are compared for a variety of realistic scenarios: when the functional covariate is observed at dense or sparse grids and measurements include noise or not. Finally, the methods are illustrated on the Tecator data set.
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References
Borggaard, C., Thodberg, H.H.: Optimal minimal neural interpretation of spectra. Anal. Chem. 64(5), 545–551 (1992)
Cai, T.T., Hall, P., et al.: Prediction in functional linear regression. Ann. Stat. 34(5), 2159–2179 (2006)
Cardot, H., Ferraty, F., Sarda, P.: Functional linear model. Stat. Probab. Lett. 45(1), 11–22 (1999)
Cardot, H., Ferraty, F., Mas, A., Sarda, P.: Testing hypotheses in the functional linear model. Scand. J. Stat. 30(1), 241–255 (2003)
Cardot, H., Goia, A., Sarda, P.: Testing for no effect in functional linear regression models, some computational approaches. Commun. Stat. Simul. Comput. 33(1), 179–199 (2004)
Crainiceanu, C.M., Ruppert, D.: Likelihood ratio tests in linear mixed models with one variance component. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 66(1), 165–185 (2004). https://doi.org/10.1111/j.1467-9868.2004.00438.x
Crambes, C., Kneip, A., Sarda, P.: Smoothing splines estimators for functional linear regression. Ann. Stat. 37, 35–72 (2009)
Delsol, L., Ferraty, F., Vieu, P.: Structural test in regression on functional variables. J. Multivar. Anal. 102(3), 422–447 (2011). https://doi.org/10.1016/j.jmva.2010.003. http://linkinghub.elsevier.com/retrieve/pii/S0047259X10002095. Accessed 6 Oct 2015
Febrero-Bande, M., González-Manteiga, W.: Generalized additive models for functional data. Test 22(2), 278–292 (2013)
Febrero-Bande, M., Oviedo de la Fuente, M.: Statistical computing in functional data analysis: the R package fda.usc. J. Stat. Softw. 51(4), 1–28 (2012). http://www.jstatsoft.org/v51/i04/. Accessed 6 Oct 2015
Ferraty, F., Vieu, P.: Nonparametric Functional Data Analysis: Theory and Practice. Springer, Berlin (2006)
García-Portugués, E., González-Manteiga, W., Febrero-Bande, M.: A goodness-of-fit test for the functional linear model with scalar response. J. Comput. Gr. Stat. 23(3), 761–778 (2014). https://doi.org/10.1080/10618600.2013.812519
Goldsmith, J., Bobb, J., Crainiceanu, C.M., Caffo, B., Reich, D.: Penalized functional regression. J. Comput. Gr. Stat. 20, 830–851 (2011)
González-Manteiga, W., González-Rodríguez, G., Martínez-Calvo, A., García-Portugués, E.: Bootstrap independence test for functional linear models (2012). arXiv preprint arXiv:1210.1072
Greven, S., Crainiceanu, C.M., Küchenhoff, H., Peters, A.: Restricted likelihood ratio testing for zero variance components in linear mixed models. J. Comput. Gr. Stat. 17(4), 870–891 (2008). https://doi.org/10.1198/106186008X386599
Hall, P., Horowitz, J.L., et al.: Methodology and convergence rates for functional linear regression. Ann. Stat. 35(1), 70–91 (2007)
Horváth, L., Reeder, R.: A test of significance in functional quadratic regression. Bernoulli 19(5A), 2120–2151 (2013). https://doi.org/10.3150/12-BEJ446. http://projecteuclid.org/euclid.bj/1383661216. Accessed 6 Oct 2015
Kim, J., Maity, A., Staicu, A.M.: General functional concurrent model. Technical report #2660, North Carolina State University (2016). http://www.stat.ncsu.edu/information/library/mimeo.php
Kong, D., Staicu, A.M., Maity, A.: Classical testing in functional linear models. J. Nonparametr. Stat. 28(4), 813–838 (2016)
McLean, M.W., Hooker, G., Staicu, A.M., Scheipl, F., Ruppert, D.: Functional generalized additive models. J. Comput. Gr. Stat. 23(1), 249–269 (2014)
McLean, M.W., Hooker, G., Ruppert, D.: Restricted likelihood ratio tests for linearity in scalar-on-function regression. Stat. Comput. 25(5), 997–1008 (2015). https://doi.org/10.1007/s11222-014-9473-1
Morris, J.S.: Functional regression. Ann. Rev. Stat. Appl. 2, 321–359 (2015)
Müller, H.G., Stadtmüller, U.: Generalized functional linear models. Ann. Stat. 33, 774–805 (2005)
Pinheiro, J., Bates, D., DebRoy, S., Sarkar, D., R Core Team.: nlme: Linear and nonlinear mixed effects models. R package version 3.1-128 (2016). https://CRAN.R-project.org/package=nlme. Accessed 3 Jan 2017
Ramsay, J.O., Dalzell, C.J.: Some tools for functional data analysis. J. R. Stat. Soc. Ser. B (Methodol.) 53, 539–572 (1991)
Ramsay, J.O., Silverman, B.: Functional Data Analysis. Springer, New York (1997)
Ramsay, J.O., Silverman, B.: Functional Data Analysis. Springer, New York (2005)
Reiss, P.T., Huang, L., Mennes, M.: Fast function-on-scalar regression with penalized basis expansions. Int. J. Biostat. 6(1) (2010)
Reiss, P.T., Goldsmith, J., Shang, H.L., Ogden, R.T.: Methods for scalar-on-function regression. Int. Stat. Rev. 85(2), 228–249 (2017)
Scheipl, F., Bolker, B., R Core Team.: RLRsim: Exact (restricted) likelihood ratio tests for mixed and additive models. R package version 3.1-3 (2016). https://cran.r-project.org/package=RLRsim. Accessed 3 Jan 2017
Su, Y.R., Di, C.Z., Hsu, L.: Hypothesis testing in functional linear models. Biometrics 73, 551–561 (2017)
Swihart, B.J., Goldsmith, J., Crainiceanu, C.M.: Restricted likelihood ratio tests for functional effects in the functional linear model. Technometrics 56(4), 483–493 (2014)
Wood, S.N., Scheipl, F., Faraway, J.J.: Straightforward intermediate rank tensor product smoothing in mixed models. Stat. Comput. 1–20 (2013)
Yao, F., Müller, H.G.: Functional quadratic regression. Biometrika 97(1), 49–64 (2010)
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Tekbudak, M.Y., Alfaro-Córdoba, M., Maity, A. et al. A comparison of testing methods in scalar-on-function regression. AStA Adv Stat Anal 103, 411–436 (2019). https://doi.org/10.1007/s10182-018-00337-x
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DOI: https://doi.org/10.1007/s10182-018-00337-x