If we knew the genotypes at the causal variants, we could fit model [3] and estimate the genetic variance σg2. Instead we will use a modified version (A*) of the relationship matrix based on the SNPs A. Although we will use REML to estimate σg2, the requirements of A* to obtain an unbiased estimate of σg2 are more easily understood for the method illustrated in Fig. 3. In this method Δyjk2=(yj−yk)2 for each pair of subjects is regressed on Gjk. The slope of this regression is −2σg2. If we replace Gjk by an estimate Ajk∗ such that E(Gjk∣Ajk∗)=Ajk∗ then E(Δyjk2)=E(a+bGjk)=a+bAjk∗, and the regression of Δyjk2 on Ajk∗ is still b=−2σg2, so the estimate of σg2 remains the same −b/2. To obtain an unbiased estimate of Gjk with the required property, we use linear regression of Gjk on Ajk. We cannot calculate G, so instead we use one set of SNPs to mimic causal variants using the following steps:
