As a first step we estimate the genetic background of unrelated individuals (unrelated cases, unrelated controls, and trio probands) using a dimension reduction technique. Let xij be the minor allele count for the ith subject and the jth SNP in a matrix X. Center and scale the columns of X by subtracting the mean and dividing by the standard deviation. Assuming a sample size of N, traditional PCA decomposes X Xt using eigenvalue decomposition to obtain the eigenvectors, (u1, ...,uN), and eigenvalues, λ1≥, ...≥λN . Rescaled eigenvectors map the ith subject into an s-dimensional space according to (1)(λ11∕2u1(i),…,λs1∕2us(i)).
