The polygenic score is now calculated aswhere and is the linear regression estimate as before. Clearly . The covariance is obtained by integrating over the distribution of . Allow again for selection of markers by their P-values as in equation 8 and denote the selection event by . Then using the symmetry of the distribution of the required covariance isThe probabilities in this expression are as follows. The selection probability is againThe probability density for nonzero isGiven some value of the probability that its estimator is also positive, and the marker is selected into the score, isSimilarly given the probability that its estimator is negative, and the marker is selected, isFinally the conditional mean of given is given by properties of the bivariate normal distribution as . The integral can be evaluated numerically, yielding values for power and accuracy from equations 2–5.
