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Single‐index varying coefficient model for functional responses

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Listed:
  • Xinchao Luo
  • Lixing Zhu
  • Hongtu Zhu

Abstract

Recently, massive functional data have been widely collected over space across a set of grid points in various imaging studies. It is interesting to correlate functional data with various clinical variables, such as age and gender, in order to address scientific questions of interest. The aim of this article is to develop a single‐index varying coefficient (SIVC) model for establishing a varying association between functional responses (e.g., image) and a set of covariates. It enjoys several unique features of both varying‐coefficient and single‐index models. An estimation procedure is developed to estimate varying coefficient functions, the index function, and the covariance function of individual functions. The optimal integration of information across different grid points is systematically delineated and the asymptotic properties (e.g., consistency and convergence rate) of all estimators are examined. Simulation studies are conducted to assess the finite‐sample performance of the proposed estimation procedure. Furthermore, our real data analysis of a white matter tract dataset obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI) study confirms the advantage and accuracy of SIVC model over the popular varying coefficient model.

Suggested Citation

  • Xinchao Luo & Lixing Zhu & Hongtu Zhu, 2016. "Single‐index varying coefficient model for functional responses," Biometrics, The International Biometric Society, vol. 72(4), pages 1275-1284, December.
    • Handle: RePEc:bla:biomet:v:72:y:2016:i:4:p:1275-1284
    DOI: 10.1111/biom.12526
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    References listed on IDEAS

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    1. Yimei Li & Hongtu Zhu & Dinggang Shen & Weili Lin & John H. Gilmore & Joseph G. Ibrahim, 2011. "Multiscale adaptive regression models for neuroimaging data," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 73(4), pages 559-578, September.
    2. Cook, R. Dennis & Forzani, Liliana, 2009. "Likelihood-Based Sufficient Dimension Reduction," Journal of the American Statistical Association, American Statistical Association, vol. 104(485), pages 197-208.
    3. Smith, Michael & Fahrmeir, Ludwig, 2007. "Spatial Bayesian Variable Selection With Application to Functional Magnetic Resonance Imaging," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 417-431, June.
    4. Reiss Philip T. & Huang Lei & Mennes Maarten, 2010. "Fast Function-on-Scalar Regression with Penalized Basis Expansions," The International Journal of Biostatistics, De Gruyter, vol. 6(1), pages 1-30, August.
    5. Yanyuan Ma & Liping Zhu, 2012. "A Semiparametric Approach to Dimension Reduction," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 107(497), pages 168-179, March.
    6. Yingcun Xia & Howell Tong & W. K. Li & Li‐Xing Zhu, 2002. "An adaptive estimation of dimension reduction space," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(3), pages 363-410, August.
    7. Michelle F. Miranda & Hongtu Zhu & Joseph G. Ibrahim, 2013. "Bayesian Spatial Transformation Models with Applications in Neuroimaging Data," Biometrics, The International Biometric Society, vol. 69(4), pages 1074-1083, December.
    8. C. Gössl & D. P. Auer & L. Fahrmeir, 2001. "Bayesian Spatiotemporal Inference in Functional Magnetic Resonance Imaging," Biometrics, The International Biometric Society, vol. 57(2), pages 554-562, June.
    9. Hongtu Zhu & Jianqing Fan & Linglong Kong, 2014. "Spatially Varying Coefficient Model for Neuroimaging Data With Jump Discontinuities," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(507), pages 1084-1098, September.
    10. Yanyuan Ma & Liping Zhu, 2014. "On estimation efficiency of the central mean subspace," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 76(5), pages 885-901, November.
    11. Liping Zhu & Tao Wang & Lixing Zhu & Louis Ferré, 2010. "Sufficient dimension reduction through discretization-expectation estimation," Biometrika, Biometrika Trust, vol. 97(2), pages 295-304.
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    Cited by:

    1. Xiong Cai & Liugen Xue & Xiaolong Pu & Xingyu Yan, 2021. "Efficient Estimation for Varying-Coefficient Mixed Effects Models with Functional Response Data," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 84(4), pages 467-495, May.
    2. Ruiyan Luo & Xin Qi, 2023. "Nonlinear function‐on‐scalar regression via functional universal approximation," Biometrics, The International Biometric Society, vol. 79(4), pages 3319-3331, December.
    3. Chen, Feifei & Jiang, Qing & Feng, Zhenghui & Zhu, Lixing, 2020. "Model checks for functional linear regression models based on projected empirical processes," Computational Statistics & Data Analysis, Elsevier, vol. 144(C).
    4. Rahul Ghosal & Arnab Maity, 2023. "Variable selection in nonlinear function‐on‐scalar regression," Biometrics, The International Biometric Society, vol. 79(1), pages 292-303, March.
    5. Xie, Haihan & Kong, Linglong, 2023. "Gaussian copula function-on-scalar regression in reproducing kernel Hilbert space," Journal of Multivariate Analysis, Elsevier, vol. 198(C).

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