For unrelated individuals, where the phenotypic correlation between individuals is small, mixed linear model analysis using the REML approach is asymptotically equivalent to simple regression analysis of pairwise phenotypic similarity/difference on pairwise genetic similarity, as measured by identity-by-descent (IBD) or identity-by-state (IBS) at genome-wide markers [17]–[20]. Under such circumstance, a regression of the cross-product of the phenotypes is equivalent to using both the squared difference and squared sum of the pairwise phenotypes, and using the cross-product is equivalent to using maximum likelihood [19]. The model for the regression-based analysis can be written as(2)where with and being the phenotypes of individuals i and j (), is the ij-th element of the GRM A, and is the residual of this regression. There are observations (contrasts) in the regression. The regression coefficient b is equivalent to becauseIn such a simple regression, the sampling variance of the estimate of the regression coefficient is(3)If the samples are unrelated and the phenotypes have been standardized with mean of 0 and variance of 1, then and . Since is small, there is hardly any variance in that
