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Modern Psychometrics with R pp 17–61Cite as

Factor Analysis

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Abstract

This chapter introduces exploratory and confirmatory factor analysis. It starts with a section on correlation coefficients since factor analytic techniques are based on covariance/correlation matrices. Special emphasis is on tetrachoric/polychoric correlations for ordinal input data. This is followed by elaborations on exploratory factor analysis including practical aspects such as determining the number of factors and rotation techniques to facilitate factor interpretation. A recent development is Bayesian exploratory factor analysis which, in addition to the loadings, also estimates the number of factors and allows them to be correlated. This approach is explored in a separate section. The second part of this chapter consists of a detailed treatment of confirmatory factor analysis which lays the groundwork for structural equation models presented in the next chapter. In confirmatory factor analysis, the number of factors and the assignment of indicators to factors are determined by substantive considerations. Several extensions in terms of multigroup, longitudinal, and multilevel settings are presented. The chapter concludes with a Bayesian approach to confirmatory factor analysis.

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Notes

  1. 1.

    Eigenvalues will be introduced in Sect. 6.1.1.

  2. 2.

    This call gives a warning that the matrix is not positive definite.

  3. 3.

    In EFA, residuals are defined by \((\mathbf R - \hat {\mathbf P})\), where R is the sample correlation matrix and \(\hat {\mathbf P}\) the estimated model correlation matrix.

  4. 4.

    An overview of rotation techniques and corresponding comparisons can be found in Browne (2001).

  5. 5.

    Thanks to Rémi Piatek and Sylvia Frühwirth-Schnatter for their support with this application.

  6. 6.

    In their original paper, Conti et al. (2014) use the more restrictive assumption of at least three manifest variables per active factor, to rule out potential identification problems due to extreme cases with zero correlation between some factors. With correlated factors, however, the weaker assumption of two manifest variables per factor is sufficient for identification.

  7. 7.

    We switch the notation for the input data (Y instead of X) in order to be consistent with the standard SEM model formulation presented in the next chapter.

  8. 8.

    Note that compared to Eq. (2.4) we slightly change the notation (i.e., Ψ instead of Φ, and Θ instead of Ψ) in order to be consistent with the names of the output objects in the lavaan package (Rosseel, 2012), which is used throughout this chapter.

  9. 9.

    At the time this book was written, lavaan allows for two-level structures only. Also, thanks to Yves Rosseel for sharing the code.

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Authors and Affiliations

  1. Department of Psychology, Harvard University, Cambridge, MA, USA

    Patrick Mair

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Mair, P. (2018). Factor Analysis. In: Modern Psychometrics with R. Use R!. Springer, Cham. https://doi.org/10.1007/978-3-319-93177-7_2

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