Tables 3, 4 and Supplementary Tables 5–8 show similar patterns when the incidence and prognosis traits are binary and prognosis is analysed in cases only (Methods). Supplementary Tables 9–12 also show similar patterns when the prognosis is quantitative and analysed either in cases only or in the full sample with adjustment for case/control status (Methods). These results confirm that our approach is applicable when incidence is analysed by logistic regression, and that it maintains the correct type-1 error rate when there is no genetic correlation between incidence and prognosis, and otherwise has a lower type-1 error rate than the unadjusted analysis, except under strong negative genetic correlation or no non-genetic correlation. While the relative strength of genetic and non-genetic confounding is unknown in practice, we might expect them to act in the same direction, and the genetic confounding not to dominate the non-genetic. These are the scenarios in which our approach does best; furthermore, the type-1 errors are more consistent under different scenarios under our approach than the unadjusted analysis. Turning to power, there is again a modest reduction in
