Under the assumption that a quantitative trait is affected by multiple genetic variants, we can express phenotypes in a sample of unrelated individuals by a multi-SNP model as (1)y=Xb+e where y = {yi} is an n × 1 vector of phenotypes, with n being the sample size, X = {xij} is an n × N genotype matrix, with xij = –2pj, 1 – 2pj or 2 – 2pj for the jth SNP of the ith individual, with pj being the allele frequency of a SNP j and N being the number of SNPs fitted in the model, and b = {bj}, an N × 1 vector of joint SNP effects. For simplicity, we subtract the mean of the phenotype from yi, such that we do not need to fit the intercept in the model. We therefore can estimate the joint effects of multiple SNPs by the least-squares approach as (2)b^=(X′X)−1X′yand var(b^)=σJ2(X′X)−1 where σJ2 is the residual variance in the joint analysis.
