Generalized Normative Modeling: A One-Step Hierarchical Kernel Framework for Multi-Site Brain Charts with Self-Correcting Z-Scores

Normative modeling expresses individual brain phenotypes as z-scores relative to a population norm, but in multi-site studies batch effects contaminate these z-scores and undermine their biomarker value. Existing approaches either harmonize data before fitting a normative model (ComBat+Normative), letting residual site effects leak into z-scores, or use parametric one-step methods (GAMLSS, HBR) that cannot flexibly model multivariate covariate interactions. We propose Generalized Normative Modeling (GNM), a onestep hierarchical framework that jointly estimates the global trajectory and site-specific effects via NUFFT-accelerated kernel regression with GCV bandwidth selection. Because the z-score is the ratio of batch-corrected residual to batch-corrected scale, residual site variance cancels algebraically -- a property we term self-correction. On ABIDE I cortical thickness (387 HC, 11 sites, 68 ROIs) and HarMNqEEG log-power spectra (1,564 subjects, 14 sites, 18 channels x 235 frequency bins), GNM produced the most site-invariant z-scores and best age-signal preservation among four methods. This work provides an open-source MATLAB toolbox with a declarative formula interface (https://github.com/LMNonlinear/Generalized-Normative-Modeling), enabling reliable individual-level inference in pooled multi-site cohorts and advancing the use of normative deviations as clinical biomarkers in precision psychiatry and neurology.