PCA vignette Principal components analysis with snpStats

Usually, principal components analysis is carried out by calculating the eigenvalues and eigenvectors of the correlation matrix. With N cases and P variables, if we write X for the N × P matrix which has been standardised so that columns have zero mean and unit standard deviation, we find the eigenvalues and eigenvectors of the P × P matrix X.X (which is N or (N − 1) times the correlation matrix depending on which denominator was used when calculating standard deviations). The first eigenvector gives the loadings of each variable in the first principal component, the second eigenvector gives the loadings in the second component, and so on. Writing the first C component loadings as columns of the P ×C matrix B, the N×C matrix of subjects’ principal component scores, S, is obtained by applying the factor loadings to the original data matrix, i.e. S = X.B. The sum of squares and products matrix, S.S = D, is diagonal with elements equal to the first C eigenvalues of the X.X matrix, so that the variances of the principal components can obtained by dividing the eigenvalues by N or (N − 1). This standard method is rarely feasible for genome-wide data since P is very large indeed and calculating the eigenvectors of X.X becomes impossibly onerous. However, the calculations can also be carried out by calculating the eigenvalues and eigenvectors of the N × N matrix X.X. The (non-zero) eigenvalues of this matrix are the same as those of X.X, and its eigenvectors are proportional to the principal component scores defined above; writing the first C eigenvectors of X.X as the columns of the N × C matrix, U , then U = S.D−1/2. Since for many purposes we are not too concerned about the scaling of the principal components, it will often be acceptable to use the eigenvectors, U , in place of the more conventionally scaled principal components. However some attention should be paid to the corresponding eigenvalues since, as noted above, these are proportional to the