Polygenic-by-Environment Adjustment for Binary GWAS with Out-of-Fold Block-PRS and Low-Rank Bilinear Models

Binary-trait genome-wide association studies (GWAS) typically adjust for covariates and additive polygenic background, but environmental variables can also modulate how aggregate genetic liability is expressed. Unmodelled polygenic gene-environment interaction (G$\times$E) can reduce association power and complicate calibration, while existing methods are limited: SNP$\times$exposure scans are expensive and often underpowered, and scalable adjustment methods generally omit exposure-dependent polygenic effects. We propose a cross-fitted, leave-one-chromosome-out pipeline that combines out-of-fold block-level polygenic scores with a low-rank bilinear neural adjustment. The model decomposes the phenotype logit into environmental main effects, additive polygenic effects, and exposure-modulated polygenic interaction terms, which are then included as covariates in chromosome-wise logistic score tests. In simulations with polygenic interaction, the proposed method maintains calibration while achieving up to 4.5\% higher genome-wide power and up to 6.9\% higher mean causal $\chi^2$ than additive block-PRS adjustment when interaction variance is present. The learned gate also recovers the simulated environmental modulation function, suggesting a practical route to interaction-aware nuisance adjustment for binary GWAS.