Principal components and factor analysis: A comparative study

 

A comparison between Principal Component Analysis (PCA) and Factor Analysis (FA) is performed both theoretically and empirically for a random matrix X:(nxp), where n is the number of observations and both coordinates may be very large. The comparison surveys the asymptotic properties of the factor scores, of the singular values and of all other elements involved, as well as the characteristics of the methods utilized for detecting the true dimension of X. In particular, the norms of the FA scores, whichever their number, and the norms of their covariance matrix are shown to be always smaller and to decay faster as n goes to infinity.  This causes the FA scores, when utilized as regressors and/or instruments, to produce more efficient slope estimators in instrumental variable estimation. Moreover, as compared to PCA, the FA scores and factors exhibit a higher degree of consistency because the difference between the estimated and their true counterparts is smaller, and so is also the corresponding variance. Finally, FA usually selects a much less encumbering number of scores than PCA, greatly facilitating the search and identification of the common components of X.