I have carried out an exploratory factor analysis (PCA). Using factor scores resulting of such EFA, I have carried out a new factor analysis. The problems is that in the factor analysis of the factor scores, the order in which the variables ( the factor scores) are introduced affects to the results. A different order generates a different factor solution.
From what you say the second analysis you did is not the confirmatory analysis, because there are placed according to your exploratory analysis. The order of the indicators may affect because the first item may be a golden item of the factor.
Hello Victor,
That's certainly an interesting situation; in EFA, the factor solution should be invariant with respect to the order in which variables are listed.
But, to understand better, it would help to know the following:
1. The software you used for the principal components extraction.
2. What rotation method you used for the final solution (and how many components you decided to retain).
3. The method for computing component scores.
4. What type of analysis (and software) you were applying to the component scores (another PCA extraction using EFA?).
5. The different variable entry sequence and the different "solutions" that follow.
Some general observations:
1. PCA isn't the same as common factor analysis; it assumes no unique, specific, or error variance (which is pretty unrealistic for a lot of applications). Compared to common factor analysis, PCA loading estimates tend to be biased towards stronger values unless you have a lot of variables in your set.
2. Generally, factoring of factors (e.g., evaluating a second-order factor solution) is simpler by using SEM software than by the process you appear to have followed.
Good luck with your work!
Thank you for the answers. Here I provide more specific information about my question. In the first PCA I applied a varimax rotation and, as consequence, the factor scores obtained are orthogonal. The correlations between all the factor scores are equal to zero (r=0.00). PCA is based on the correlation matrix, but in this case all the values in the correlation matrix are zero (out of the diagonal). So, in this case, in the second order analysis (also using PCA varimax rotated), what uses PCA to obtain the factors instead the correlation matrix?
In the first PCA 5 factors (components) are retained. In the second-order PCA 2 factors (components) are retained. Every different combination in the order in which the 5 factors are listed generates a different factor solution. As many factor solutions as possible combinations of the initial 5 factors.
If the 5 factor scores in the initial PCA are obtained with an oblique rotation (direct oblimin), in that case the correlation matrix of the factor scores includes values different to zero and the order in which the variables (factor scores) are listed in the second PCA do not affects to the results (the factor solution is invariant regardless the order in which the variables are listed).
Thus, the problem seems to be related to what uses PCA to calculate the factors when all the values in the correlation matrix are equal to zero.
The software used for both analysis is SPSS.
EFA (PCA) varimax rotated is used in the two analysis.
The method used to obtain factor scores (component scores) is Regression.
Hello Victor,
Since you're essentially factoring unrelated variables (orthogonal factors), what you're seeing is analogous to random noise (due, I suspect, only to rounding errors in the construction of the component/factor scores). This is not unlike running parallel analysis on vectors of random numbers (which itself will tend to yield one or more "factors" having eigenvalues greater than 1). For this reason, I think it unlikely that any of the "second order" solutions will be particularly meaningful or replicable in future samples.
At this point, I concur with Mathew Kerry; you're looking at an artifact (though not because of having run PCA).
If you really want to consider a higher-order factor structure, you must use a form of oblique rotation so that the resultant factors can have some correlation.
Good luck with your work!
Your first PCA is not at factor analysis. It creates a sort of index which can be useful for some predictions, like a regression analysis. It is questionable if it is meaninfuld to develop a second order model based on this.
Try instead to begin with a real factor analysis using oblique rotation, as David mentioned. However, using factor scores from the first factor analysis as a basis for the secondary factor analysis is not the best way, since these are just approximations for the latent factors. You could instead make a single model encompassing both first- and second order factors.
You can also order the SPSS-program to confine the number of factors to be drawn to two (or three, or whatever), perform an oblique rotation, and see how meaningful the results are.
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