Suppose further that a proportion of the markers have no effect on (ie. ), and the remaining markers have effects drawn from . Then among the null markers the variance of , conditional on selection into the polygenic score, is obtained from properties of the truncated normal distribution as [40] Similarly, among the non-null markers the variance of , conditional on selection into the polygenic score, is(9)where , . The probability that a null marker is selected into the polygenic score is and the corresponding probability for a non-null marker is . Therefore the total variance of is(10)Note that when and , that is all markers are included in the score, then equation 10 reverts to , which is invariant to the proportion of null markers and does not assume a normal distribution for the non-null effects [23].
