The Annals of Statistics

Multivariate varying coefficient model for functional responses

Hongtu Zhu, Runze Li, and Linglong Kong

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Motivated by recent work studying massive imaging data in the neuroimaging literature, we propose multivariate varying coefficient models (MVCM) for modeling the relation between multiple functional responses and a set of covariates. We develop several statistical inference procedures for MVCM and systematically study their theoretical properties. We first establish the weak convergence of the local linear estimate of coefficient functions, as well as its asymptotic bias and variance, and then we derive asymptotic bias and mean integrated squared error of smoothed individual functions and their uniform convergence rate. We establish the uniform convergence rate of the estimated covariance function of the individual functions and its associated eigenvalue and eigenfunctions. We propose a global test for linear hypotheses of varying coefficient functions, and derive its asymptotic distribution under the null hypothesis. We also propose a simultaneous confidence band for each individual effect curve. We conduct Monte Carlo simulation to examine the finite-sample performance of the proposed procedures. We apply MVCM to investigate the development of white matter diffusivities along the genu tract of the corpus callosum in a clinical study of neurodevelopment.

Article information

Ann. Statist., Volume 40, Number 5 (2012), 2634-2666.

First available in Project Euclid: 4 February 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation 62G08: Nonparametric regression
Secondary: 62G20: Asymptotic properties

Functional response global test statistic multivariate varying coefficient model simultaneous confidence band weak convergence


Zhu, Hongtu; Li, Runze; Kong, Linglong. Multivariate varying coefficient model for functional responses. Ann. Statist. 40 (2012), no. 5, 2634--2666. doi:10.1214/12-AOS1045.

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  • [1] Aguirre, G. K., Zarahn, E. and D’esposito, M. (1998). The variability of human, BOLD hemodynamic responses. NeuroImage 8 360–369.
  • [2] Basser, P. J., Mattiello, J. and LeBihan, D. (1994). Estimation of the effective self- diffusion tensor from the NMR spin echo. Journal of Magnetic Resonance Ser. B 103 247–254.
  • [3] Basser, P. J., Mattiello, J. and LeBihan, D. (1994). MR diffusion tensor spectroscopy and imaging. Biophys. J. 66 259–267.
  • [4] Buzsáki, G. (2006). Rhythms of the Brain. Oxford Univ. Press, Oxford.
  • [5] Cardot, H. (2007). Conditional functional principal components analysis. Scand. J. Stat. 34 317–335.
  • [6] Cardot, H., Chaouch, M., Goga, C. and Labruère, C. (2010). Properties of design-based functional principal components analysis. J. Statist. Plann. Inference 140 75–91.
  • [7] Cardot, H. and Josserand, E. (2011). Horvitz–Thompson estimators for functional data: Asymptotic confidence bands and optimal allocation for stratified sampling. Biometrika 98 107–118.
  • [8] Chiou, J.-M., Müller, H.-G. and Wang, J.-L. (2004). Functional response models. Statist. Sinica 14 675–693.
  • [9] Degras, D. A. (2011). Simultaneous confidence bands for nonparametric regression with functional data. Statist. Sinica 21 1735–1765.
  • [10] Einmahl, U. and Mason, D. M. (2000). An empirical process approach to the uniform consistency of kernel-type function estimators. J. Theoret. Probab. 13 1–37.
  • [11] Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications. Monographs on Statistics and Applied Probability 66. Chapman & Hall, London.
  • [12] Fan, J., Yao, Q. and Cai, Z. (2003). Adaptive varying-coefficient linear models. J. R. Stat. Soc. Ser. B Stat. Methodol. 65 57–80.
  • [13] Fan, J. and Zhang, W. (1999). Statistical estimation in varying coefficient models. Ann. Statist. 27 1491–1518.
  • [14] Fan, J. and Zhang, W. (2000). Simultaneous confidence bands and hypothesis testing in varying-coefficient models. Scand. J. Stat. 27 715–731.
  • [15] Fan, J. and Zhang, W. (2008). Statistical methods with varying coefficient models. Stat. Interface 1 179–195.
  • [16] Faraway, J. J. (1997). Regression analysis for a functional response. Technometrics 39 254–261.
  • [17] Fass, L. (2008). Imaging and cancer: A review. Mol. Oncol. 2 115–152.
  • [18] Friston, K. J. (2007). Statistical Parametric Mapping: The Analysis of Functional Brain Images. Academic Press, London.
  • [19] Friston, K. J. (2009). Modalities, modes, and models in functional neuroimaging. Science 326 399–403.
  • [20] Goodlett, C. B., Fletcher, P. T., Gilmore, J. H. and Gerig, G. (2009). Group analysis of DTI fiber tract statistics with application to neurodevelopment. NeuroImage 45 S133–S142.
  • [21] Hall, P. and Hosseini-Nasab, M. (2006). On properties of functional principal components analysis. J. R. Stat. Soc. Ser. B Stat. Methodol. 68 109–126.
  • [22] Hall, P., Müller, H.-G. and Wang, J.-L. (2006). Properties of principal component methods for functional and longitudinal data analysis. Ann. Statist. 34 1493–1517.
  • [23] Hall, P., Müller, H.-G. and Yao, F. (2008). Modelling sparse generalized longitudinal observations with latent Gaussian processes. J. R. Stat. Soc. Ser. B Stat. Methodol. 70 703–723.
  • [24] Hastie, T. and Tibshirani, R. (1993). Varying-coefficient models. J. R. Stat. Soc. Ser. B Stat. Methodol. 55 757–796.
  • [25] Heywood, I., Cornelius, S. and Carver, S. (2006). An Introduction to Geographical Information Systems., 3rd ed. Prentice Hall, New York.
  • [26] Hoover, D. R., Rice, J. A., Wu, C. O. and Yang, L.-P. (1998). Nonparametric smoothing estimates of time-varying coefficient models with longitudinal data. Biometrika 85 809–822.
  • [27] Huang, J. Z., Wu, C. O. and Zhou, L. (2002). Varying-coefficient models and basis function approximations for the analysis of repeated measurements. Biometrika 89 111–128.
  • [28] Huang, J. Z., Wu, C. O. and Zhou, L. (2004). Polynomial spline estimation and inference for varying coefficient models with longitudinal data. Statist. Sinica 14 763–788.
  • [29] Huettel, S. A., Song, A. W. and McCarthy, G. (2004). Functional Magnetic Resonance Imaging. Sinauer, London.
  • [30] Kosorok, M. R. (2003). Bootstraps of sums of independent but not identically distributed stochastic processes. J. Multivariate Anal. 84 299–318.
  • [31] Kosorok, M. R. (2008). Introduction to Empirical Processes and Semiparametric Inference. Springer, New York.
  • [32] Li, Y. and Hsing, T. (2010). Uniform convergence rates for nonparametric regression and principal component analysis in functional/longitudinal data. Ann. Statist. 38 3321–3351.
  • [33] Lindquist, M., Loh, J. M., Atlas, L. and Wager, T. (2008). Modeling the hemodynamic response function in fMRI: Efficiency, bias and mis-modeling. NeuroImage 45 S187–S198.
  • [34] Lindquist, M. A. (2008). The statistical analysis of fMRI data. Statist. Sci. 23 439–464.
  • [35] Ma, S., Yang, L. and Carroll, R. J. (2012). A simultaneous confidence band for sparse longitudinal regression. Statist. Sinica 22 95–122.
  • [36] Mercer, J. (1909). Functions of positive and negative type, and their connection with the theory of integral equations. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 209 415–446.
  • [37] Niedermeyer, E. and da Silva, F. L. (2004). Electroencephalography: Basic Principles, Clinical Applications, and Related Fields. Williams & Wilkins, Baltimore.
  • [38] Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis, 2nd ed. Springer, New York.
  • [39] Rice, J. A. and Silverman, B. W. (1991). Estimating the mean and covariance structure nonparametrically when the data are curves. J. R. Stat. Soc. Ser. B Stat. Methodol. 53 233–243.
  • [40] Sun, J. and Loader, C. R. (1994). Simultaneous confidence bands for linear regression and smoothing. Ann. Statist. 22 1328–1345.
  • [41] Towle, V. L., Bolaños, J., Suarez, D., Tan, K., Grzeszczuk, R., Levin, D. N., Cakmur, R., Frank, S. A. and Spire, J. P. (1993). The spatial location of EEG electrodes: Locating the best-fitting sphere relative to cortical anatomy. Electroencephalogr. Clin. Neurophysiol. 86 1–6.
  • [42] van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes: With Applications to Statistics. Springer, New York.
  • [43] Wand, M. P. and Jones, M. C. (1995). Kernel Smoothing. Monographs on Statistics and Applied Probability 60. Chapman & Hall, London.
  • [44] Wang, L., Li, H. and Huang, J. Z. (2008). Variable selection in nonparametric varying-coefficient models for analysis of repeated measurements. J. Amer. Statist. Assoc. 103 1556–1569.
  • [45] Welsh, A. H. and Yee, T. W. (2006). Local regression for vector responses. J. Statist. Plann. Inference 136 3007–3031.
  • [46] Worsley, K. J., Taylor, J. E., Tomaiuolo, F. and Lerch, J. (2004). Unified univariate and multivariate random field theory. NeuroImage 23 189–195.
  • [47] Wu, C. O. and Chiang, C.-T. (2000). Kernel smoothing on varying coefficient models with longitudinal dependent variable. Statist. Sinica 10 433–456.
  • [48] Wu, C. O., Chiang, C.-T. and Hoover, D. R. (1998). Asymptotic confidence regions for kernel smoothing of a varying-coefficient model with longitudinal data. J. Amer. Statist. Assoc. 93 1388–1402.
  • [49] Wu, H. and Zhang, J.-T. (2006). Nonparametric Regression Methods for Longitudinal Data Analysis. Wiley, Hoboken, NJ.
  • [50] Yao, F. and Lee, T. C. M. (2006). Penalized spline models for functional principal component analysis. J. R. Stat. Soc. Ser. B Stat. Methodol. 68 3–25.
  • [51] Zhang, J.-T. and Chen, J. (2007). Statistical inferences for functional data. Ann. Statist. 35 1052–1079.
  • [52] Zhou, Z. and Wu, W. B. (2010). Simultaneous inference of linear models with time varying coefficients. J. R. Stat. Soc. Ser. B Stat. Methodol. 72 513–531.
  • [53] Zhu, H., Li, R. and Kong, L. (2012). Supplement to “Multivariate varying coefficient model for functional responses.” DOI:10.1214/12-AOS1045SUPP.
  • [54] Zhu, H., Zhang, H., Ibrahim, J. G. and Peterson, B. S. (2007). Statistical analysis of diffusion tensors in diffusion-weighted magnetic resonance imaging data. J. Amer. Statist. Assoc. 102 1085–1102.
  • [55] Zhu, H. T., Ibrahim, J. G., Tang, N., Rowe, D. B., Hao, X., Bansal, R. and Peterson, B. S. (2007). A statistical analysis of brain morphology using wild bootstrapping. IEEE Trans. Med. Imaging 26 954–966.
  • [56] Zhu, H. T., Styner, M., Tang, N. S., Liu, Z. X., Lin, W. L. and Gilmore, J. H. (2010). FRATS: Functional regression analysis of DTI tract statistics. IEEE Trans. Med. Imaging 29 1039–1049.

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