We used recently developed statistical methods [12, 31] to estimate the variance explained by all autosomal markers. Specifically, we considered the following linear mixed model (1)y=Xβ+Wu+e,u~N(0,σu2I),e~N(0,σe2I),where y ∈ ℝn×1 is the response vector; X ∈ ℝn×c is the design matrix of fixed effects including the intercept and other covariates such as age, sex and principal components, β is the vector for regression coefficients of the covariates; W = [wim] ∈ ℝn×M is the standardized genotype matrix given by (2)wim=(gim−pm)2pm(1−pm),where gim ∈ {0, 1, 2} is the number of copies of the reference allele for the SNP of the individual and pm is the frequency of the reference allele; u is the random effect from, N(0,σu2I), and e is the residual error with variance σe2. Here n is the sample size, c is the number of fixed effects and M is the number of random effects. After integrating out u and e, we have (3)y~N(Xβ,WWTσu2+σe2I).The genetic relationship matrix (GRM) is defined as A=WWTM and the proportion of the phenotype variance explained by the genotyped markers in W, is given by h2=Mσu2Mσu2+σe2. For case-control studies, the estimated heritability (h2) is transformed on the liability scale [12].
