Heritable environments: bias due to conditioning on a collider in models with polygenic scores

The application of polygenic scores has transformed our ability to investigate whether and how genetic and environmental factors jointly contribute to the variation of complex traits. Modelling the complex interplay between genes and environment, however, raises serious methodological challenges. Here we illustrate the largely unrecognised impact of gene-environment dependencies on the identification of the effects of genes and their variation across environments. We show that controlling for heritable covariates in regression models that include polygenic scores as independent variables introduces endogenous selection bias when one or more of these covariates depends on unmeasured factors that also affect the outcome. This results in the problem of conditioning on a collider, which in turn leads to spurious associations and effect sizes. Using graphical and simulation methods we demonstrate that the degree of bias depends on the strength of the gene-covariate correlation and of hidden heterogeneity linking covariates with outcomes, regardless of whether the main analytic focus is mediation, confounding, or gene x covariate (commonly gene x environment) interactions. We offer potential solutions, highlighting the importance of causal inference. We also urge further caution when fitting and interpreting models with polygenic scores and non-exogenous environments or phenotypes and demonstrate how spurious associations are likely to arise, advancing our understanding of such results.