Joint Spherical-Harmonics Regression for PheWAS: Global Maps, Residual Localization, and Spherical-Cap Enrichment

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Abstract
We present an application for analyzing variant-by-phenotype association summaries from PheWAS [10] using spherical harmonics (SH) [5, 4]. The method embeds phenotypes onto the unit sphere, fits a joint SH map across all variants via weighted ridge regression [3, 8, 9] with a Laplacian penalty, and then assesses per-variant residual localization beyond this global structure. Rotation-invariant descriptors (degree power spectrum, l 95 , entropy, and a localization index) summarize spatial complexity. We detect localized genetic effects with a sign-aware spherical-cap enrichment test that scans cap radii around SH extrema and identifies phenotypes driving hotspots via inverse-variance fixed-effect meta-analysis [7]. Model selection uses BIC [2], and significance for high-degree structure uses a nested F -test within the linear-model framework [8, 9] with BH-FDR across variants [1]. The implementation provides 2D maps and an optional 3D globe, supports standard long and matrix data formats, and exports all artifacts in a single ZIP for reproducibility.