Conditional-prevalence correction for liability-scale analyses of covariate-adjusted binary GWAS

Adjustment for heritable covariates in genome-wide association studies of disease can induce collider bias, creating spurious genetic associations. Following bias correction, interpretation on the liability scale, which represents continuous disease susceptibility, requires an appropriate conversion from log odds. We derive this conversion under a Gaussian liability model for small genetic effects. The direct effect, excluding pathways through the adjusted covariates, and the collider-bias component share a common conversion factor. We characterize how this factor depends on variation in disease risk across covariate values and extend the derivation to case-control sampling. In simulations, the proposed conversion reduced the effect-size inflation associated with scaling based solely on disease prevalence. These results link existing collider-bias corrections to estimation of direct liability effects and clarify the interpretation of covariate-adjusted associations in population and case-control studies.