diseased and not diseased individuals, and if the individual with the highest predicted genetic risk has rank r1 = nd + nd' = n, AUC can be calculated directly from the mean rank of the diseased individuals (),(2)(see example in Figure S1). Equally, AUC can be calculated as AUC = 0.5(1 + D) where D is the Somers' rank correlation [17] between risk profile and disease status (1 = diseased, 0 = not diseased). Another equivalent definition of AUC is the probability that a randomly selected pair of diseased (d) and non-diseased (d') individuals are accurately classified [18]. The probability is the same as the probability that difference between the genetic liability of the d and d' individuals is greater than zero. This difference is approximately normally distributed with mean μd - μd' and variance . Using the liability threshold model and results of standard genetic selection theory [11] the means (μ) and variances (σ 2) of the genetic liability of d and d' individuals are where v = -iK/(1 – K). The genetic liabilities of the d and d' groups are each approximately normally distributed, the approximation being less accurate for high heritabilities.
