The second analysis utilized the WF regression coefficient estimates (i.e., not only their signs) to estimate the amount of stratification bias. For each SNP j, let βj denote the GWAS estimate, and let βWF,j denote the WF estimate. Under the assumption that the causal effect of each SNP is the same within families as in the population, we can decompose the estimates as: β^j=βj+sj+Ujβ^WF,j=βj+Vj, where βj is the true underlying GWAS parameter for SNP j, sj is the bias due to stratification (defined to be orthogonal to βj and Uj), and Uj and Vj are the sampling variances of the estimates with E(Uj)=E(Vj)=0. Whenever sj ≠ 0, the GWAS estimate of β̂j is biased away from the population parameter βj. The proportion of variance in the GWAS coefficients accounted for by true genetic signals can be written as: Var(βj)Var(βj)+Var(sj).
