Statistics and Computing

, Volume 23, Issue 2, pp 209–220

Exploratory factor and principal component analyses: some new aspects

Authors

    • Department of Mathematics and StatisticsThe Open University
  • Steffen Unkel
    • Department of Mathematics and StatisticsThe Open University
  • Wojtek Krzanowski
    • School of Engineering, Mathematics and Physical SciencesUniversity of Exeter
Article

DOI: 10.1007/s11222-011-9303-7

Cite this article as:
Trendafilov, N.T., Unkel, S. & Krzanowski, W. Stat Comput (2013) 23: 209. doi:10.1007/s11222-011-9303-7

Abstract

Exploratory Factor Analysis (EFA) and Principal Component Analysis (PCA) are popular techniques for simplifying the presentation of, and investigating the 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 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 pn. All derived algorithms and matrix conditions are illustrated on two data sets, one for each of these two cases.

Keywords

Data matrix decompositionSVD and QR factorizationProjected gradientsOptimality conditionsProcrustes problems

Copyright information

© Springer Science+Business Media, LLC 2011