When there is no selection on or the regression of on has coefficient from equation 11. The converse regression of on has coefficient . The latter will also apply when there is ascertainment on , but the regression of on ascertained can also be written as , where denotes ascertainment, so thatThe desired quantity is the coefficient for the regression of ascertained on which is thusIn general the variance of genetic markers will differ from 1 under ascertainment but it will henceforth be assumed that its expectation over markers is approximately 1. A heuristic justification for this assumption is given in the Text S1. Based on this assumption, an effect on liability is transformed by the factor(13)to the observed case/control scale. Similarly to before, given the genetic variance-covariance matrix on the liability scale, the properties of the polygenic score can be calculated on the observed scale, substituting for and for throughout, and using as the sampling variance of .
