2 Common Effects G $\times$ E Interaction Model

A common effects G $\times$ E model can also be fitted to covariance matrices computed conditionally on environmental exposure by simply fixing the environment-specific effects of the general model to zero, and comparing the two using a $\chi^{2}$ difference test. The information from pairs discordant for environmental exposure allows for this comparison. A critical sub-model of the common effects G $\times$ E model is one which tests the hypothesis that exposure group heterogeneity is solely due to heteroscedasticity, or group differences in random environmental variance, rather than group differences in genetic variance. To fit this model, the genetic parameters are simply equated across groups, while allowing the random environmental effects to take on different values. If this model does not fit worse than the full common effects model, then there is evidence for heteroscedasticity. A second sub-model of the common effects G $\times$ E interaction model is one which constrains the environmental parameters to be equal across exposure groups, while allowing the genetic variance components to differ. If this model is not significantly worse than the full common effects model, then there is evidence to suggest that the environmental interaction only involves a differential expression of genetic, but not environmental, influences.