Many GWAS will study prognosis among the cases of disease, rather than adjusting for an index trait as a covariate. The susceptibility GWAS will typically be performed using logistic regression, rather than linear regression as developed here. Index event bias has a non-linear form in logistic models, but is approximately linear for the small effects typical of polygenic traits13; furthermore, small effects on linear and logistic scales are approximately proportional39. We therefore expect Eq. (3) to hold approximately when βGX and/or βGY are log odds ratios. Having already assumed no interaction between G and X in their effects on Y, we further expect Eq. (3) to hold when analysing only the cases of disease.
