The Garden of Forking Paths: Reinterpreting Haseman-Elston Regression for a Genotype-by-Environment Model

Haseman-Elston regression (HE-reg) has been known as a classic tool for detecting an additive genetic variance component. However, in this study we find that HE-reg can capture GxE under certain conditions, so we derive and reinterpret the analytical solution of HE-reg. In the presence of GxE, it leads to a natural discrepancy between linkage and association results, the latter of which is not able to capture GxE if the environment is unknown. Considering linkage and association as symmetric designs, we investigate how the symmetry can and cannot hold in the absence and presence of GxE, and consequently we propose a pair of statistical tests, Symmetry Test I and Symmetry Test II, both of which can be tested using summary statistics. Test statistics, and their statistical power issues are also investigated for Symmetry Tests I and II. Increasing the number of sib pairs is important to improve statistical power for detecting GxE.