Principal component analysis for multivariate familial data

Abstract

The use of a principal component analysis is considered for multivariate data on families with different numbers of siblings. The coefficients in principal components are given as the eigenvectors of the weighted sums of squares and products matrix from the sibling data. Asymptotic distributions of the eigenvalues and eigenvectors of the estimated covariance matrix are obtained for an elliptical population. Asymptotic distributions of statistics associated with reduction of dimensionality are also derived. The results can be used to construct approximate confidence intervals for the coefficients and variances of principal components.