Skip to main content
SpringerLink
Log in

Factor Analysis

  • Chapter
  • First Online:
Multivariate Analysis

Abstract

The explorative factor analysis is a procedure of multivariate analysis which aims at identifying structures in large sets of variables. Large sets of variables are often characterized by the fact that as the number of variables increases, it may be assumed that more and more variables are correlated. The exploratory factor analysis aims to structure the relationships in a large set of variables to the extent that it identifies groups of variables that are highly correlated with each other and separates them from less correlated groups. The groups of highly correlated variables are called factors. Apart from the structuring function, factor analysis is also used for data reduction. At the end of the chapter, there is also a brief outlook on confirmatory factor analysis, in which predefined factor structures are examined.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Correlations form the basis of factor analysis. For readers who are not sufficiently familiar with the term, the concept of correlations is explained in detail in Sect. 1.2.2.

  2. 2.

    On the website www.multivariate-methods.info, we provide supplementary material (e.g., Excel files) to deepen the reader’s understanding of the methodology.

  3. 3.

    Standardized variables have an average of 0 and a variance of 1. For the standardization of variables see also Sect. 1.2.1.

  4. 4.

    For a brief summary of the basics of statistical testing see Sect. 1.2.2.

  5. 5.

    The determinant of the correlation matrix in the application example is det = 0.001. Furthermore, ln(0.00096) = −6.9485.

  6. 6.

    If a variable xj is transformed into a standardized variable zj, the mean value of zj = 0 and the variance of zj = 1. This results in a considerable simplification in the representation of the following relationships. See the explanations on standardization in Sect. 1.2.1.

  7. 7.

    Please note that example 3 does not correspond to the application example in Sect. 7.2.1.

  8. 8.

    Standardized variables with a unit variance of 1 are assumed. For the decomposition of the variance of an output variable, see also the explanations in Sect. 7.2.2.4.2 and especially Fig. 7.13.

  9. 9.

    Remember that we use standardized variables and therefore the variance of each variable is 1 and the total variance in the data set is 5.

  10. 10.

    Mathematically the eigenvalues are calculated first (this is a standard problem of mathematics) and then the components or factors are derived.

  11. 11.

    On the problem of interpreting factors or main components, also compare Sect. 7.2.3 and the case study in Sect. 7.3.3.4.

  12. 12.

    In contrast, the case study shows major differences between PCA and PAF (cf. Sect. 7.3.3.4).

  13. 13.

    In contrast to PAF, the aim of the ML, GLS and ULS methods is to determine the factor loadings in such a way that the difference between the empirical correlation matrix (R) and the model-theoretical correlation matrix (\({\hat{\mathbf{R}}}\)) is minimal. In alpha factorization, Cronbach's alpha is maximized, and image factorization is based on the image of a variable. The listed procedures are all implemented in SPSS, with PCA included as a further extraction procedure (see Fig. 7.21).

  14. 14.

    Note that PCA, in contrast, requires a collective term for correlating variables. Cf. the presentation on the problem of the interpretation of factors in Sect. 7.2.3 and the notes on the case study in Sect. 7.3.3.4.

  15. 15.

    Missing values are a frequent and unfortunately unavoidable problem when conducting surveys (e.g. because people cannot or do not want to answer some of the questions, or as a result of mistakes by the interviewer). The handling of missing values in empirical studies is discussed in Sect. 1.5.2.

  16. 16.

    In the following the same data set is used as in the case study of discriminant analysis (Chap. 4), logistic regression (Chap. 5) and cluster analysis (Chap. 8). Thus, it is easier to demonstrate the similarities and differences between the methods.

  17. 17.

    By reducing the number of variables to 9, the number of valid cases in the case study changes to 117.

  18. 18.

    Cluster analysis is the central methodological instrument for identifying similarly perceived objects. The cluster analysis presented in this book (cf. Chap. 8) is also based on the data set used in this case study (Table 7.24) and confirms the result of a two-cluster solution that is emerging here.

  19. 19.

    See the general comments in Sect. 7.2.2.4.

References

  • Child, D. (2006). The essentials of factor analysis (3rd ed.). London: Bloomsbury Academic.

    Google Scholar 

  • Cureton, E. E., & D’Agostino, R. B. (1993). Factor analysis: An applied approach. Hillsdate: Lawrence Erlbaum Associates.

    Google Scholar 

  • Dziuban, C. D. & Shirkey, E. C. (1974). When is a correlation matrix appropriate for factor analysis? Some decision rules. Psychological bulletin, 81(6), 358.

    Google Scholar 

  • Guttman, L. (1953). Image theory for the structure of quantitative variates. Psychometrika, 18(4), 277–296.

    Article  Google Scholar 

  • Harrington, D. (2009). Confirmatory factor analysis. Oxford: Oxford University Press.

    Google Scholar 

  • Kaiser, H. F. & Rice, J. (1974). Little jiffy, mark IV. Educational and psychological measurement, Vol. 34, Number One, 111–117.

    Google Scholar 

  • Loehlin, J. (2004). Latent variable models: An introduction to factor, path, and structural equation analysis (4th ed.). New Jersey: Psychology Press.

    Book  Google Scholar 

Further reading

  • Bartholomew, D. J., Knott, M., & Moustaki, I. (2011). Latent variable models and factor analysis: A unified approach. (Vol. 904). John Wiley & Sons.

    Google Scholar 

  • Costello, A. B., & Osborne, J. W. (2005). Best practices in exploratory factor analysis: Four recommendations for getting the most from your analysis. Practical Assessment, Research and Evaluation, 10(7), 1–9.

    Google Scholar 

  • Harman, H. (1976). Modern factor analysis (3rd ed.). Chicago: The University of Chicago Press.

    Google Scholar 

  • Kaiser, H. F. (1970). A second generation little jiffy. Psychometrika, 35(4), 401–415.

    Article  Google Scholar 

  • Kim, J. O., & Mueller, J. (1978). Introduction to factor analysis: What it is and how to do it. Beverly Hills: SAGE Publications.

    Book  Google Scholar 

  • Stewart, D. (1981). The application and misapplication of factor analysis. Journal of Marketing Research, 18(1), 51–62.

    Article  Google Scholar 

  • Tabachnick, B. G., & Fidell, L. S. (2007). Using multivariate statistics (5th ed.). Boston, MA: Allyn & Bacon.

    Google Scholar 

  • Thompson, B. (2004). Exploratory and confirmatory factor analysis – Understanding concepts and applications. Washington DC: American Psychological Association.

    Book  Google Scholar 

  • Yong, A. G., & Pearce, S. (2013). A beginner’s guide to factor analysis: Focusing on exploratory factor analysis. Tutorials in Quantitative Methods for Psychology, 9(2), 79–94.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Business-to-Business Marketing, Marketing Center Münster, University of Münster, Münster, Germany

    Klaus Backhaus

  2. Otto-von-Guericke-University Magdeburg, Magdeburg, Germany

    Bernd Erichson

  3. Chair for Value-Based-Marketing, Marketing Center Münster, University of Münster, Münster, Germany

    Sonja Gensler

  4. Chair of Marketing and Innovation, University of Trier, Trier, Germany

    Rolf Weiber

  5. Munich, Germany

    Thomas Weiber

Authors
  1. Klaus Backhaus
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bernd Erichson
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sonja Gensler
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Rolf Weiber
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Thomas Weiber
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Klaus Backhaus .

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Der/die Herausgeber bzw. der/die Autor(en), exklusiv lizenziert durch Springer Fachmedien Wiesbaden GmbH, ein Teil von Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Backhaus, K., Erichson, B., Gensler, S., Weiber, R., Weiber, T. (2021). Factor Analysis. In: Multivariate Analysis. Springer Gabler, Wiesbaden. https://doi.org/10.1007/978-3-658-32589-3_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-658-32589-3_7

  • Published:

  • Publisher Name: Springer Gabler, Wiesbaden

  • Print ISBN: 978-3-658-32588-6

  • Online ISBN: 978-3-658-32589-3

  • eBook Packages: Business and Economics (German Language)

Publish with us

Policies and ethics

Search

Navigation