Clarifying and extending permutation tests on brain map correspondence through mixed-effects modeling

Permutation-based methods such as the spin test, Brain Surrogate Maps with Autocorrelated Spatial Heterogeneity (BrainSMASH), and the simple permutation-based intermodal correspondence (SPICE) test are widely used to assess correspondence between brain maps while accounting for their spatial autocorrelation. However, these methods define and evaluate correspondence in fundamentally different ways, making their results difficult to compare or interpret jointly. We address these limitations by introducing a two-factor mixed-effects model that decomposes brain map variability into components arising from inter-subject variability and spatial variability across brain locations. This formulation provides a principled way to characterize distinct sources of variability in brain maps and to formally link each correspondence test to the specific component it targets. Within this framework, we further provide the analytical expressions of the null distributions of the spin test, BrainSMASH, and SPICE in terms of model parameters. This unified framework clarifies the fundamental distinctions among the permutation tests, reveals their implicit assumptions, and provides a principled way to compare and interpret their results. Beyond clarifying existing methods, the modeling framework naturally motivates a bootstrap-based method that enables simultaneous inference of multiple forms of correspondence arising from different components of brain map variability. Through extensive simulations and empirical analyses of both structural (cortical thickness vs. sulcal depth) and functional (language vs. motor contrast) brain maps, we demonstrate that the bootstrap-based method achieves well-calibrated type I error, substantially higher statistical power, and provides a robust and comprehensive characterization of different forms of brain map correspondence.