Statistics and Computing

, Volume 23, Issue 2, pp 209–220 | Cite as

Exploratory factor and principal component analyses: some new aspects

  • Nickolay T. Trendafilov
  • Steffen Unkel
  • Wojtek Krzanowski
Article

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 decomposition SVD and QR factorization Projected gradients Optimality conditions Procrustes problems 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Berry, M.W., Pulatova, S.A., Stewart, G.W.: Algorithm 844: computing sparse reduced-rank approximations to sparse matrices. ACM Trans. Math. Softw. 31, 252–269 (2005) MathSciNetMATHCrossRefGoogle Scholar
  2. De Leeuw, J.: Least squares optimal scaling of partially observed linear systems. In: van Montfort, K., Oud, J., Satorra, A. (eds.) Recent Developments on Structural Equation Models: Theory and Applications, pp. 121–134. Kluwer Academic, Dordrecht (2004) Google Scholar
  3. Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. The John Hopkins University Press, Baltimore (1996) MATHGoogle Scholar
  4. Harman, H.H.: Modern Factor Analysis, 3rd edn. University of Chicago Press, Chicago (1976) Google Scholar
  5. Hotelling, H.: Analysis of a complex of statistical variables into principal components. J. Educ. Psychol. 24(6), 417–441 (1933). Continued on 24(7), 498–520 CrossRefGoogle Scholar
  6. Jolliffe, I.T.: Principal Component Analysis, 2nd edn. Springer, New York (2002) MATHGoogle Scholar
  7. Magnus, I.R., Neudecker, H.: Matrix Differential Calculus with Applications in Statistics and Econometrics, 3rd edn. Wiley, Chichester (2007) Google Scholar
  8. Mulaik, S.A.: Looking back on the indeterminacy controversies in factor analysis. In: Maydeu-Olivares, A., McArdle, J.J. (eds.) Contemporary Psychometrics: A Festschift for Roderick P. McDonald, pp. 173–206. Lawrence Erlbaum, Mahwah (2005) Google Scholar
  9. Mulaik, S.A.: Foundations of Factor Analysis, 2nd edn. Chapman and Hall/CRC, Boca Raton (2010) MATHGoogle Scholar
  10. Pearson, K.: On lines and planes of closest fit to systems of points in space. Philos. Mag. 2, 559–572 (1901) CrossRefGoogle Scholar
  11. Rao, C.R.: Principal component and factor analyses. In: Maddala, G.S., Rao, C.R. (eds.) Handbook of Statistics, 14, pp. 489–505. Elsevier, Amsterdam (1996) Google Scholar
  12. Schneeweiss, H., Mathes, H.: Factor analysis and principal components. J. Multivar. Anal. 55, 105–124 (1995) MathSciNetMATHCrossRefGoogle Scholar
  13. Spearman, C.: “General intelligence,” objectively determined and measured. Am. J. Psychol. 15, 201–292 (1904) CrossRefGoogle Scholar
  14. Stewart, G.W.: Matrix Algorithms, Vol I: Basic Decompositions. Philadelphia, SIAM (1998) MATHCrossRefGoogle Scholar
  15. Thurstone, L.L.: Multiple Factor Analysis. University of Chicago Press, Chicago (1947) MATHGoogle Scholar
  16. Trendafilov, N.T.: The dynamical system approach to multivariate data analysis, a review. J. Comput. Graph. Stat. 15, 628–650 (2006) MathSciNetCrossRefGoogle Scholar
  17. Trendafilov, N.T., Unkel, S.: Exploratory factor analysis of data matrices with more variables than observations. J. Comput. Graph. Stat. (2011). doi:10.1198/jcgs.2011.0921
  18. Unkel, S., Trendafilov, N.T.: Simultaneous parameter estimation in exploratory factor analysis: an expository review. Int. Stat. Rev. 78, 363–382 (2010) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Nickolay T. Trendafilov
    • 1
  • Steffen Unkel
    • 1
  • Wojtek Krzanowski
    • 2
  1. 1.Department of Mathematics and StatisticsThe Open UniversityMilton KeynesUK
  2. 2.School of Engineering, Mathematics and Physical SciencesUniversity of ExeterExeterUK

Personalised recommendations