The geometry of G × E : how scaling and endogenous treatment effects shape interaction direction

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Abstract
Gene-environment interaction ( G × E ) studies hold promise for identifying genetic loci mediating the effects of environmental risk on disease. However, interpretation of G × E effects is often confounded by two fundamental issues: the dependence of interaction estimates on outcome scale and the presence of endogenous treatment effects, in which genetic liability influences environmental exposure. These factors can induce spurious G × E signals—even when genetic and environmental contributions are purely additive on an unobserved scale. In this work, we demonstrate that any monotone convex transformation of an outcome induces sign-consistent G × E effects: the sign of the interaction term aligns with the sign of the corresponding main genetic effect. We further show that endogenous treatment effects, modeled as threshold-based interventions, generate G × E effects with a similar directional signature. Exploiting this property, we propose a simple diagnostic: sign consistency across G × E estimates can identify artifacts driven by outcome scaling or exposure endogeneity. We validate our framework in the UK Biobank using transcriptome-wide interaction studies (TxEWAS) across multiple trait–environment pairs, observing widespread sign consistency in some settings—suggesting confounding by scaling or treatment bias. Our results provide both a theoretical foundation and a practical tool for interpreting G × E findings, enabling researchers to distinguish biologically meaningful interactions from those induced by statistical artifacts.

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