• Kernel;
  • Multiscale adaptive regression;
  • Neuroimaging data;
  • Propagation–separation;
  • Smoothing;
  • Sphere;
  • Test statistics

Summary.  Neuroimaging studies aim to analyse imaging data with complex spatial patterns in a large number of locations (called voxels) on a two-dimensional surface or in a three-dimensional volume. Conventional analyses of imaging data include two sequential steps: spatially smoothing imaging data and then independently fitting a statistical model at each voxel. However, conventional analyses suffer from the same amount of smoothing throughout the whole image, the arbitrary choice of extent of smoothing and low statistical power in detecting spatial patterns. We propose a multiscale adaptive regression model to integrate the propagation– separation approach with statistical modelling at each voxel for spatial and adaptive analysis of neuroimaging data from multiple subjects. The multiscale adaptive regression model has three features: being spatial, being hierarchical and being adaptive. We use a multiscale adaptive estimation and testing procedure to utilize imaging observations from the neighbouring voxels of the current voxel to calculate parameter estimates and test statistics adaptively. Theoretically, we establish consistency and asymptotic normality of the adaptive parameter estimates and the asymptotic distribution of the adaptive test statistics. Our simulation studies and real data analysis confirm that the multiscale adaptive regression model significantly outperforms conventional analyses of imaging data.