Multivariate Spline Estimation and Inference for Image-On-Scalar Regression

TL;DR

This paper introduces a flexible multivariate spline method for image-on-scalar regression, enabling accurate estimation and inference of coefficient functions on irregular image domains, with proven statistical properties and practical algorithms.

Contribution

It develops a novel spline-based approach for image-on-scalar regression that handles irregular domains and provides valid inference, with proven consistency and asymptotic normality.

Findings

Method achieves root-n consistency and asymptotic normality.

Provides efficient covariance estimation and confidence intervals.

Demonstrates good finite-sample performance in simulations and real data.

Abstract

Motivated by recent data analyses in biomedical imaging studies, we consider a class of image-on-scalar regression models for imaging responses and scalar predictors. We propose using flexible multivariate splines over triangulations to handle the irregular domain of the objects of interest on the images, as well as other characteristics of images. The proposed estimators of the coefficient functions are proved to be root-n consistent and asymptotically normal under some regularity conditions. We also provide a consistent and computationally efficient estimator of the covariance function. Asymptotic pointwise confidence intervals and data-driven simultaneous confidence corridors for the coefficient functions are constructed. Our method can simultaneously estimate and make inferences on the coefficient functions while incorporating spatial heterogeneity and spatial correlation. A highly efficient and scalable estimation algorithm is developed. Monte Carlo simulation studies are conducted to examine the finite-sample performance of the proposed method, which is then applied to the spatially normalized positron emission tomography data of the Alzheimer's Disease Neuroimaging Initiative.