A Functional Varying-Coefficient Single-Index Model for Functional Response Data

Abstract

Motivated by the analysis of imaging data, we propose a novel functional varying-coefficient single-index model (FVCSIM) to carry out the regression analysis of functional response data on a set of covariates of interest. FVCSIM represents a new extension of varying-coefficient single-index models for scalar responses collected from cross-sectional and longitudinal studies. An efficient estimation procedure is developed to iteratively estimate varying coefficient functions, link functions, index parameter vectors, and the covariance function of individual functions. We systematically examine the asymptotic properties of all estimators including the weak convergence of the estimated varying coefficient functions, the asymptotic distribution of the estimated index parameter vectors, and the uniform convergence rate of the estimated covariance function and their spectrum. Simulation studies are carried out to assess the finite-sample performance of the proposed procedure. We apply FVCSIM to investigate the development of white matter diffusivities along the corpus callosum skeleton obtained from Alzheimer’s Disease Neuroimaging Initiative (ADNI) study. Supplementary material for this article is available online.

Keywords:

• Functional data analysis
• Image data analysis
• Single-index model
• Varying-coefficient model

Supplementary Material

The supplementary materials provide the technical conditions and theoretical proofs of the asymptotic results.

Acknowledgment

We are grateful for the many valuable suggestions from referees, associated editor, and editor. Data used in preparation of this article were obtained from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database (adni.loni.usc.edu). As such, the investigators within the ADNI contributed to the design and implementation of ADNI and/or provided data but did not participate in analysis or writing of this report. A complete listing of ADNI investigators can be found at: http://adni.loni.usc.edu/wp-content/uploads/how\_to\_apply/ADNI\_Acknowledgement\_List.pdf

Funding

Dr. Zhu’s work was partially supported by NIH grants MH086633 and 1UL1TR001111, NSF grants SES-1357666 and DMS-1407655, and a Senior investigator grant from the Cancer Prevention Research Institute of Texas. Dr. Li’s research work was partially supported by the National Medical Research Council NMRC/CBRG/0014/2012 and Academic Research Funding R-155-000-152-112. This material was based upon work partially supported by the NSF grant DMS-1127914 to the Statistical and Applied Mathematical Science Institute. The content is solely the responsibility of the authors and does not necessarily represent the official views of the NIH.

Notes

1 Data used in the preparation of this article were obtained from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database (adni.loni.usc.edu). The ADNI was launched in 2003 by the National Institute on Aging (NIA), the National Institute of Biomedical Imaging and Bioengineering (NIBIB), the Food and Drug Administration (FDA), private pharmaceutical companies, and nonprofit organizations, as a $60 million, 5-year public–private partnership. The primary goal of ADNI has been to test whether serial magnetic resonance imaging (MRI), positron emission tomography (PET), other biological markers, and clinical and neuropsychological assessment can be combined to measure the progression of mild cognitive impairment (MCI) and early Alzheimer’s disease (AD). Determination of sensitive and specific markers of very early AD progression is intended to aid researchers and clinicians to develop new treatments and monitor their effectiveness, as well as lessen the time and cost of clinical trials. The Principal Investigator of this initiative is Michael W. Weiner, MD, VA Medical Center and University of California, San Francisco. ADNI is the result of efforts of many coinvestigators from a broad range of academic institutions and private corporations, and subjects have been recruited from over 50 sites across the U.S. and Canada. The initial goal of ADNI was to recruit 800 subjects but ADNI has been followed by ADNI-GO and ADNI-2. To date these three protocols have recruited over 1500 adults, ages 55–90, to participate in the research, consisting of cognitively normal older individuals, people with early or late MCI, and people with early AD. The follow up duration of each group is specified in the protocols for ADNI-1, ADNI-2, and ADNI-GO. Subjects originally recruited for ADNI-1 and ADNI-GO had the option to be followed in ADNI-2. For up-to-date information, see www.adni-info.org.