Generalized Scalar-on-Image Regression Models via Total Variation

ABSTRACT

The use of imaging markers to predict clinical outcomes can have a great impact in public health. The aim of this article is to develop a class of generalized scalar-on-image regression models via total variation (GSIRM-TV), in the sense of generalized linear models, for scalar response and imaging predictor with the presence of scalar covariates. A key novelty of GSIRM-TV is that it is assumed that the slope function (or image) of GSIRM-TV belongs to the space of bounded total variation to explicitly account for the piecewise smooth nature of most imaging data. We develop an efficient penalized total variation optimization to estimate the unknown slope function and other parameters. We also establish nonasymptotic error bounds on the excess risk. These bounds are explicitly specified in terms of sample size, image size, and image smoothness. Our simulations demonstrate a superior performance of GSIRM-TV against many existing approaches. We apply GSIRM-TV to the analysis of hippocampus data obtained from the Alzheimers Disease Neuroimaging Initiative (ADNI) dataset. Supplementary materials for this article are available online.

KEYWORDS:

• Excess risk
• Functional regression
• Generalized scalar-on-image regression
• Prediction
• Total variation

Supplementary Materials

The online supplementary materials contain the appendices for the article.

Acknowledgments

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. The authors would like to thank Dr. Yalin Wang for sharing the processed data. We would also like to thank the Editor, Associate Editor, and referees for constructive comments.

Funding

The research of Xiao Wang is supported by NSF grants DMS1042967 and CMMI1030246. The research of Hongtu Zhu is partially supported by NIH grants MH086633 and 1UL1TR001111, NSF grants SES-1357666 and DMS-1407655, and a grant from Cancer Prevention Research Institute of Texas. This material was based upon work partially supported by the NSF grant DMS-1127914 to the Statistical and Applied Mathematical Science Institute.