Exploratory factor analysis ($EFA$) is appropriate (psychometrically and otherwise) for examining the extent to which one may explain correlations among multiple items by inferring the common influence of (an) unmeasured (i.e., latent) factor(s). If this is not your specific intent, consider alternative analyses, e.g.:
- General linear modeling (e.g., multiple regression, canonical correlation, or (M)AN(C)OVA)
- Confirmatory factor analysis ($CFA$) or latent trait/class/profile analyses
- Structural equation ($SEM$) / partial least squares modeling
Dimensionality is the first issue $EFA$ can address. You can examine the eigenvalues of the covariance matrix (as by producing a scree plot via $EFA$) and conduct a parallel analysis to resolve the dimensionality of your measures. (See also some great advice and alternative suggestions from William Revelle.) You should do this carefully before extracting a limited number of factors and rotating them in $EFA$, or before fitting a model with a specific number of latent factors using $CFA$, $SEM$, or the like. If a parallel analysis indicates multidimensionality, but your general (first) factor vastly outweighs all others (i.e., has by far the largest eigenvalue / explains the majority of variance in your measures), consider bifactor analysis (Gibbons & Hedeker, 1992; Reise, Moore, & Haviland, 2010).
Many problems arise in $EFA$ and latent factor modeling of Likert scale ratings. Likert scales produce ordinal (i.e., categorical, polytomous, ordered) data, not continuous data. Factor analysis generally assumes any raw data input are continuous, and people often conduct factor analyses of matrices of Pearson product-moment correlations, which are only appropriate for continuous data. Here's a quote from Reise and colleagues (2010):
Ordinary confirmatory factor analytic techniques do not apply to dichotomous or polytomous data (Byrne, 2006). Instead, special estimation procedures are required (Wirth & Edwards, 2007). There basically are three options for working with polytomous item response data. The first is to compute a polychoric matrix and then apply standard factor analytic methods (see Knol & Berger, 1991). A second option is to use full-information item factor analysis (Gibbons & Hedeker, 1992). The third is to use limited information estimation procedures
designed specifically for ordered data such as weighted least squares with mean and variance adjustment (MPLUS; Muthén & Muthén, 2009).
I would recommend combining both the first and third approaches (i.e., use diagonally weighted least squares estimation on a polychoric correlation matrix), based on Wang and Cunningham's (2005) discussion of the problems with typical alternatives:
When confirmatory factor analysis was conducted with nonnormal ordinal data using maximum likelihood and based on Pearson product-moment correlations, the downward parameter estimates produced in this study were consistent with Olsson's (1979) findings. In other words, the magnitude of nonnormality in the observed ordinal variables is a major determinant of the accuracy of parameter estimates.
The results also support the findings of Babakus, et al. (1987). When maximum likelihood estimation is used with a polychoric correlation input matrix in confirmatory factor analyses, the solutions tend to result in unacceptable
and therefore significant chi-square values together with poor fit statistics.
The question remains as to whether researchers should use weighted least squares or diagonally weighted least squares estimators in estimating structural equation models with nonnormal categorical data. Neither weighted least squares nor diagonally weighted least squares estimation makes assumptions about the nature of the distribution of the variables and both methods produce asymptotically valid results. Nevertheless, because weighted least squares estimation is based on fourth-order moments, this approach frequently leads to practical problems and is very computationally demanding. This means that weighted least squares estimation may lack robustness when used to evaluate models of medium, i.e., with 10 indicators, to large size and small to moderate sample sizes.
It isn't clear to me whether the same concern with weighted least squares estimation applies to $DWLS$ estimation; regardless, the authors recommend that estimator. In case you don't have the means already:
- R (R Core Team, 2012) is free. You'll need an old version (e.g.,
2.15.2) for these packages:
psych package (Revelle, 2013) contains the
fa.parallel function can help identify the number of factors to extract.
lavaan package (Rosseel, 2012) offers $DWLS$ estimation for latent variable analysis.
semTools package contains the
mirt package (Chalmers, 2012) offers promising alternatives using item response theory.
I imagine Mplus (Muthén & Muthén, 1998-2011) would work too, but the free demo version won't accommodate more than six measurements, and the licensed version isn't cheap. It may be worth it if you can afford it though; people love Mplus, and the Muthéns' customer service via their forums is incredible!
As stated above, $DWLS$ estimation overcomes the problem of normality assumption violations (both univariate and multivariate), which is a very common problem, and almost ubiquitous in Likert scale rating data. However, it's not necessarily a pragmatically consequential problem; most methods aren't too sensitive to (heavily biased by) small violations (cf. Is normality testing 'essentially useless'?). chl's answer to this question raises more important, excellent points and suggestions regarding problems with extreme response style too; definitely an issue with Likert scale ratings and other subjective data.
- Babakus, E., Ferguson, J. C. E., & Jöreskog, K. G. (1987). The sensitivity of confirmatory maximum likelihood factor analysis to violations of measurement scale and distributional assumptions. Journal of Marketing Research, 24, 222–228.
- Byrne, B. M. (2006). Structural Equation Modeling with EQS. Mahwah, NJ:
- Chalmers, R. P. (2012). mirt: A multidimensional item response theory package for the R environment. Journal of Statistical Software, 48(6), 1–29. Retrieved from http://www.jstatsoft.org/v48/i06/.
- Gibbons, R. D., & Hedeker, D. R. (1992). Full-information item bi-factor analysis.
Psychometrika, 57, 423–436.
- Knol, D. L., & Berger, M. P. F. (1991). Empirical comparison between factor analysis and multidimensional item response models. Multivariate Behavioral Research, 26, 457–477.
- Muthén, L. K., & Muthén, B. O. (1998-2011). Mplus user's guide (6th ed.). Los Angeles, CA: Muthén & Muthén.
- Muthén, L. K., & Muthén, B. O. (2009). Mplus (Version 4.00). [Computer software]. Los Angeles, CA: Author. URL: http://www.statmodel.com.
- Olsson, U. (1979). Maximum likelihood estimates for the polychoric correlation coefficient. Psychometrika, 44, 443–460.
- R Core Team. (2012). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0, URL: http://www.R-project.org/.
- Reise, S. P., Moore, T. M., & Haviland, M. G. (2010). Bifactor models and rotations: Exploring the extent to which multidimensional data yield univocal scale scores. Journal of Personality Assessment, 92(6), 544–559. Retrieved from http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2981404/.
- Revelle, W. (2013). psych: Procedures for Personality and Psychological Research. Northwestern University, Evanston, Illinois, USA. Retrieved from http://CRAN.R-project.org/package=psych. Version = 1.3.2.
- Rosseel, Y. (2012). lavaan: An R Package for Structural Equation Modeling. Journal of Statistical Software, 48(2), 1–36. Retrieved from http://www.jstatsoft.org/v48/i02/.
- Wang, W. C., & Cunningham, E. G. (2005). Comparison of alternative estimation methods in confirmatory factor analyses of the General Health Questionnaire. Psychological Reports, 97, 3–10.
- Wirth, R. J., & Edwards, M. C. (2007). Item factor analysis: Current approaches
and future directions. Psychological Methods, 12, 58–79. Retrieved from http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3162326/.