variance in family relatedness [18], [27]. Therefore, all these methods explicitly or implicitly require a known pedigree. When the correlations between relatives are small, the first order approximation of the NCP in Rijsdijk et al [18] can be written in our notations as , which is the same as we derived (i.e. , see Equation (4) for ), even though our deviations were based on least squares regression analysis in unrelated samples whereas the derivations in Rijsdijk et al [18] were based on maximum likelihood approach in family data. This approximation is reasonably accurate when correlations between relatives are small for a pedigree-based design, which is not an issue for a population-based design where the genetic relationships between unrelated samples are very small as demonstrated in Yang et al [3]. We show by simulations (Text S1) that for a univariate analysis the LRT statistics calculated based on REML are highly consistent with the chi-squared test-statistics calculated by the Wald test using the sampling variance either observed from the simulations or predicted from our approximation theory (Figure S1).
