Statistics and Computing

, Volume 25, Issue 5, pp 997–1008 | Cite as

Restricted likelihood ratio tests for linearity in scalar-on-function regression



We propose a procedure for testing the linearity of a scalar-on-function regression relationship. To do so, we use the functional generalized additive model (FGAM), a recently developed extension of the functional linear model. For a functional covariate \(X(t)\), the FGAM models the mean response as the integral with respect to \(t\) of \(F\{X(t),t\}\) where \(F(\cdot ,\cdot )\) is an unknown bivariate function. The FGAM can be viewed as the natural functional extension of generalized additive models. We show how the functional linear model can be represented as a simple mixed model nested within the FGAM. Using this representation, we then consider restricted likelihood ratio tests for zero variance components in mixed models to test the null hypothesis that the functional linear model holds. The methods are general and can also be applied to testing for interactions in a multivariate additive model or for testing for no effect in the functional linear model. The performance of the proposed tests is assessed on simulated data and in an application to measuring diesel truck emissions, where strong evidence of nonlinearities in the relationship between the functional predictor and the response are found.


Functional data analysis  Functional regression Generalized additive model  P-spline Restricted likelihood ratio test  P-spline analysis of variance 


  1. Ait-Saïdi, A., Ferraty, F., Kassa, R., Vieu, P.: Cross-valiyeard estimations in the single-functional index model. Statistics 42(6), 475–494 (2008)MathSciNetCrossRefMATHGoogle Scholar
  2. Asencio, M., Hooker, G., Gao, H.O.: Functional convolution models. Statist. Model. 4(4), 1–21, (2014)
  3. Bates, D., Maechler, M., Bolker, B.: lme4: Linear mixed-effects models using S4 classes. R package version 1.0–4. (2013)
  4. Cardot, H., Ferraty, F., Mas, A., Sarda, P.: Testing hypotheses in the functional linear model. Scand. J. Stat. 30(1), 241–255 (2003)MathSciNetCrossRefMATHGoogle Scholar
  5. Chen, D., Hall, P., Müller, H.G.: Single and multiple index functional regression models with nonparametric link. Ann. Stat. 39(3), 1720–1747 (2011)CrossRefMATHGoogle Scholar
  6. Clark, N.N., Gautam, M., Wayne, W.S., Lyons, G.W., Thompson, G.J.: Heavy-duty vehicle chassis dynamometer testing for emissions inventory, air quality modeling, source apportionment, and air toxics emissions inventory. Technical Report CRC Rep. No. E55/59, Coordinating Research Council, Inc. (CRC), (2007)
  7. Clark, N.N., Vora, K.A., Wang, L., Gautam, M., Wayne, W.S., Thompson, G.J.: Expressing cycles and their emissions on the basis of properties and results from other cycles. Environ. Sci. Technol. 44(15), 5986–5992 (2010)CrossRefGoogle Scholar
  8. Core Team, R.: A language and environment for statistical computing. R Foundation for Statistical Computing, (2012)
  9. Crainiceanu, C.M., Ruppert, D.: Likelihood ratio tests in linear mixed models with one variance component. J. R. Stat. Soc. 66(1), 165–185 (2004)MathSciNetCrossRefMATHGoogle Scholar
  10. Crainiceanu, C.M., Reiss, P.T., Goldsmith, J., Huang, L., Huo, L., Scheipl, F.: Refund: regression with functional data. R package version 0.1-7, (2013)
  11. 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)
  12. Febrero-Bande, M., Galeano, P., González-Manteiga, W.: Generalized additive models for functional data. TEST 22(2), 278–292 (2013)MathSciNetCrossRefMATHGoogle Scholar
  13. Ferraty, F., Vieu, P.: Nonparametric Functional Data Analysis: Theory and Practice. Springer Verlag, Berlin (2006a)Google Scholar
  14. Ferraty, F., Vieu, P.: npfda NonParametric functional data analysis. (2006b)
  15. Gabrys, R., Horváth, L., Kokoszka, P.: Tests for error correlation in the functional linear model. J. Am. Stat. Assoc. 105(491), 1113–1125 (2010)CrossRefGoogle Scholar
  16. García-Portugués, E., González-Manteiga, W., Febrero-Bande, M.: A goodness-of-fit test for the functional linear model with scalar response. J. Comput. Graph. Stat. (2013, page to appear)Google Scholar
  17. Goldsmith, J., Bobb, J., Crainiceanu, C.M., Caffo, B., Reich, D.: Penalized functional regression. J. Comput. Graph. Stat. 20(4), 830–851 (2011)MathSciNetCrossRefGoogle Scholar
  18. Greven, S., Crainiceanu, C.M., Küchenhoff, H., Peters, A.: Restricted likelihood ratio testing for zero variance components in linear mixed models. J. Comput. Graph. Stat. 17(4), 870–891 (2008)CrossRefGoogle Scholar
  19. Guillas, S., Lai, M.J.: Bivariate splines for spatial functional regression models. J. Nonparametr. Stat. 22(4), 477–497 (2010)MathSciNetCrossRefMATHGoogle Scholar
  20. Horváth, L., Reeder, R.W.: A test of significance in functional quadratic regression. Bernoulli. (2014, to appear)Google Scholar
  21. James, G.M., Silverman, B.W.: Functional adaptive model estimation. J. Am. Stat. Assoc. 100(470), 565–577 (2005)MathSciNetCrossRefMATHGoogle Scholar
  22. Kong, D., Staicu, A.M., Maity, A.: Classical testing in functional linear models. Technical report, North Carolina State University, staicu/papers/ClassicalTest\_FLM\_KSM.pdf. (2013, submitted)
  23. Li, Y., Wang, N., Carroll, R.J.: Generalized functional linear models with semiparametric single-index interactions. J. Am. Stat. Assoc. 105(490), 621–633 (2010)MathSciNetCrossRefGoogle Scholar
  24. Marx, B.D., Eilers, P.H.C.: Multidimensional penalized signal regression. Technometrics 47(1), 13–22 (2005)MathSciNetCrossRefGoogle Scholar
  25. McLean, M.W., Scheipl, F., Hooker, G., Greven, S., Ruppert, D.: Bayesian functional generalized additive models with sparsely observed covariates. (2013, submitted)
  26. McLean, M.W., Hooker, G., Staicu, A.M., Scheipl, F., Ruppert, D.: Functional generalized additive models. J. Comput. Graph. Stat. 23(1), 249–269 (2014). doi:10.1080/10618600.2012.729985 MathSciNetCrossRefGoogle Scholar
  27. Müller, H.G., Wu, Y., Yao, Y.: Continuously additive models for nonlinear functional regression. Biometrika. 100(3), doi:10.1093/biomet/ast004. (2013)
  28. Müller, H.G., Wu, Y., Yao, Y.: Continuously additive models for nonlinear functional regression. Biometrika. 100(3), doi:10.1093/biomet/ast004. (2013)
  29. Patterson, H.D., Thompson, R.: Recovery of inter-block information when block sizes are unequal. Biometrika 58(3), 545–554 (1971)MathSciNetCrossRefMATHGoogle Scholar
  30. Pinheiro, J.C., Bates, D.M.: Linear mixed-effects models: basic concepts and examples. Springer, New York (2000)Google Scholar
  31. Pinheiro, J.C., Bates, D.M., DebRoy, S., Deepayan Sarkar., Core Team, R.: nlme: Linear and Nonlinear Mixed Effects Models, R package version 3.1-111 (2013)Google Scholar
  32. Ramsay, J.O., Silverman, B.W.: Functional Data Analysis, 2nd edn. Springer, New York (2005)Google Scholar
  33. Reiss, P.T., Ogden, R.T.: Smoothing parameter selection for a class of semiparametric linear models. J. R. Stat. Soc. 71(2), 505–523 (2009) Google Scholar
  34. Ruppert, D., Wand, M.P., Carroll, R.J.: Semiparametric regression. Cambridge University Press, New York (2003)CrossRefMATHGoogle Scholar
  35. SAS Institute Inc.: SAS/STAT9.2 User’s Guide (2008)Google Scholar
  36. Scheipl, F., Greven, S., Küchenhoff, H.: Size and power of tests for a zero random effect variance or polynomial regression in additive and linear mixed models. Comput. Stat. Data Anal. 52(7), 3283–3299 (2008)CrossRefMATHGoogle Scholar
  37. Swihart Bruce J., Goldsmith Jeff., Crainiceanu Ciprian M.: Restricted likelihood ratio tests for functional effects in the functional linear model. Technometrics, (2014, to appear)Google Scholar
  38. Wang, X., Ruppert, D.: Optimal prediction in an additive functional model. (2013, submitted)
  39. Wang, Y.: Smoothing Splines: Methods and Applications. CRC Press, Boca Raton (2011)CrossRefGoogle Scholar
  40. Wang, Y., Chen, H.: On testing an unspecified function through a linear mixed effects model with multiple variance components. Biometrics 68(4), 1113–1125 (2012). doi:10.1111/j.1541-0420.2012.01790.x MathSciNetCrossRefMATHGoogle Scholar
  41. Wood, S.N.: Generalized Additive Models: An Introduction with R. CRC Press, Boca Raton (2006a)Google Scholar
  42. Wood, S.N.: Low-rank scale-invariant tensor product smooths for generalized additive mixed models. Biometrics 62(4), 1025–1036 (2006b)MathSciNetCrossRefMATHGoogle Scholar
  43. Wood, S.N.: Fast stable restricted maximum likelihood and marginal likelihood estimation of semiparametric generalized linear models. J. R. Stat. Soc. 73(1), 3–36 (2011)MathSciNetCrossRefGoogle Scholar
  44. Wood, S.N.: On p values for smooth components of an extended generalized additive model. Biometrika 100(1), 221–228 (2013)MathSciNetCrossRefMATHGoogle Scholar
  45. Wood, S.N., Scheipl, F., Faraway, J.J.: Straightforward intermediate rank tensor product smoothing in mixed models. Stat. Comput. 23(3), 341–360 (2013)MathSciNetCrossRefGoogle Scholar
  46. Zhang, J.-T., Chen, J.: Statistical inferences for functional data. Ann. Stat. 35(3), 1052–1079 (2007)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Mathew W. McLean
    • 1
  • Giles Hooker
    • 2
  • David Ruppert
    • 3
  1. 1.Institute for Applied Mathematics and Computational ScienceTexas A&M UniversityCollege StationUSA
  2. 2.Department of Biological Statistics and Computational BiologyCornell UniversityIthacaUSA
  3. 3.School of Operations Research and Information EngineeringCornell UniversityIthacaUSA

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