Principal Component Analysis from the Multivariate Familial Correlation Matrix

This paper considers principal component analysis (PCA) in familial models, where the number of siblings can differ among families. S. Konishi and C. R. Rao (1992, Biometrika79, 631-641) used the unified estimator of S. Konishi and C. G. Khatri (1990, Ann. Inst. Statist. Math.42, 561-580) to develop a PCA derived from the covariance matrix. However, because of the lack of invariance to componentwise change of scale, an analysis based on the correlation matrix is often preferred. The asymptotic distribution of the estimated eigenvalues and eigenvectors of the correlation matrix are derived under elliptical sampling. A Monte Carlo simulation shows the usefulness of the asymptotic expressions for samples as small as N=25 families.