More precisely, consider a pair of traits expressed as a linear combination of genetic effects and an error term that includes environmental and unmodelled genetic effects:(1)where is a matrix of coefficients, is a -vector of coded genetic markers, and is a pair of random errors that are independent of . Now suppose that the genetic effects on are estimated from a sample of size and used to construct a polygenic score to be tested for association to in an independent sample of size . Define the polygenic score to beSome important statistical properties of can be expressed in terms of and , expressions for which are derived in the Methods. The coefficient of determination for the polygenic score on the second trait is(2)which is the squared correlation between the score and the trait. The prediction mean square error is(3)The asymptotic non-centrality parameter of the test for association of with is(4)on 1df, and the power of the two-tailed test of association at significance level is(5)Binary traits are assumed to arise from a liability threshold model [24] leading to calculation of the
