Some scientific papers report results of parallel analysis of principal axis factor analysis in a way inconsistent with my understanding of the methodology. What am I missing? Am I wrong or are they.
- Data: The performance of 200 individual humans has been observed on 10 tasks. For each individual and each task, one has a performance score. The question now is to determine how many factors are the cause for the performance on the 10 tasks.
- Method: parallel analysis to determine the number of factors to retain in a principal axis factor analysis.
- Example for reported result: “parallel analysis suggests that only factors with eigenvalue of 2.21 or more should be retained”
That is nonsense, isn’t it?
From the original paper by Horn (1965) and tutorials like Hayton et al. (2004) I understand that parallel analysis is an adaptation of the Kaiser criterion (eigenvalue > 1) based on random data. However, the adaptation is not to replace the cut-off 1 by another fixed number but an individual cut-off value for each factor (and dependent on the size of the data set, i.e. 200 times 10 scores). Looking at the examples by Horn (1965) and Hayton et al. (2004) and the output of R functions fa.parallel in the psych package and parallel in the nFactors package, I see that parallel analysis produces a downward sloping curve in the Scree plot to compare to the eigenvalues of the real data. More like “Retain the first factor if its eigenvalue is > 2.21; additionally retain the second if its eigenvalue is > 1.65; …”.
Is there any sensible setting, any school of thought, or any methodology that would render “parallel analysis suggests that only factors with eigenvalue of 2.21 or more should be retained” correct?
Hayton, J.C., Allen, D.G., Scarpello, V. (2004). Factor retention decisions in exploratory factor analysis: a tutorial on parallel analysis. Organizational Research Methods, 7(2):191-205.
Horn, J.L. (1965). A rationale and test for the number of factors in factor analysis. Psychometrika, 30(2):179-185.