Cascade screening following a polygenic risk score test: what is the disease risk of a sibling conditional on a high score of a proband?

Polygenic risk scores (PRSs) for predicting disease risk have become increasingly accurate, leading to rising popularity of PRS tests. Consider an individual whose PRS test has placed him/her at the top q-quantile of genetic risk. Recently, Reid et al. (Circ Genom Precis Med. 2021;14:e003262) have investigated whether such a finding should motivate cascade screening in the probands siblings. Specifically, using data from the UK biobank, Reid et al. computed the empirical probability of a sibling of the proband to also have a PRS at the top q-quantile. In this short note, I use the liability threshold model to compute this probability analytically (for either a sibling of the proband or for a more distant relative), showing excellent agreement with the empirical results of Reid et al., including that this probability is disease-independent. Further, I compute the probability of the relative of the proband to be affected, as a function of the quantile threshold q, the proportion of variance explained by the score, and the disease prevalence.