Exploratory factor and principal component analyses: some new aspects

Exploratory Factor Analysis (EFA) and Principal Component Analysis (PCA) are popular techniques for simplifying presentation of, and investigating structure of, an (n×p) data matrix. However, these fundamentally different techniques are frequently confused, and the differences between them are obscured, because they give similar results in some practical cases. We therefore investigate conditions under which factor scores and principal component scores are expected to be close to each other, by considering EFA as a matrix decomposition so that it can be directly compared with the data matrix decomposition underlying PCA. We propose a QR decomposition as an alternative to the usual singular value decomposition for PCA, provide iterative algorithms for the computation, and derive a matrix condition that has to be satisfied for the two techniques to give similar results. Throughout, we consider separately the cases n > p and p ≥ n. All derived algorithms and matrix conditions are illustrated on two data sets, one for each of these two cases.