Exploratory factor and principal component analyses: some new aspects
Exploratory Factor Analysis (EFA) and Principal Component Analysis (PCA) are popular techniques for simplifying the presentation of, and investigating the structure of, an (nxp) 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 they 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. Correspondingly, we propose an extended version of PCA, called the EFA-like PCA, which mimics the EFA matrix decomposition in the sense that they contain the same unknowns. We provide iterative algorithms for estimating the EFA-like PCA parameters, and derive conditions that have to be satisfied for the two techniques to give similar results. Throughout, we consider separately the cases n > p and pa parts per thousand yenn. All derived algorithms and matrix conditions are illustrated on two data sets, one for each of these two cases.
data matrix decomposition svd and qr factorization projected gradients optimality conditions procrustes problems