# Computing factor scores for factors when the PAF is based on tetrachoric correlations

I want to create factors from various binary items. Using the polycor package (Fox, 2006) and R-Essentials I created a tetrachoric correlation matrix in SPSS. The items are all exploratory, so I followed recommendations from Russell (2002) and performed a PAF with a Promax rotation on the data (instead of a PCA with Varimax, which already has been discussed here).

I now have a pattern matrix and a structure matrix. As the factors correlate with each other, some suggest using the structure matrix is best for computing factor scores (e.g. Gorsuch, 1983). But, as far as I could figure it out, there is no clear guideline yet (e.g. Grice, 2001). I know about the methods that exist to create scores in general (as discussed in the forum here) and I know some methods use the pattern, others the structure matrix.

Still, I am not sure which method is the best for creating factor scores from a PAF with Promax rotation that is based on tetrachoric correlations. Unfortunately, I could not find any recommendations for that in the internet... in the forum I found that for a PCA based on tetrachoric correlations classic methods won't do properly and one would have to use special estimations such as expected or maximum a posteriori (EAP, MAP). Is that the same for a PAF? What would be the correct way to compute factor scores? Is there a possibility to do that in SPSS?

References:

• Fox, J. (2006). Teacher’s corner: Structural Equation Modeling With the sem Package in R. Structural Equation Modeling: A Multidisciplinary Journal, 13(3), 465–486. https://doi.org/10.1207/s15328007sem1303_7

• Gorsuch, R. L. (1983). Factor Analysis (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum Associates, ISBN: 978-0898592023

• Grice, J. W. (2001). Computing and evaluating factor scores. Psychological methods, 6(4), 430.

• Russell, D. W. (2002). In Search of Underlying Dimensions: The Use (and Abuse) of Factor Analysis in Personality and Social Psychology Bulletin. Personality and Social Psychology Bulletin, 28(12), 1629–1646. https://doi.org/10.1177/014616702237645

• 1) One is not free to use pattern or structure matrix interchangeably in the formulas as they are. But the two matrices are clearly related via factor correlations, so if you know one you but know the other. 2) Unfortunately, standard methods to estimate f. scores are inappropriate in case of tetrachoric correlations because the input respondents' values do not correspond directly to those correlations. Special estimations (you mention some of them) are required. I don't know exactly the algorithms of them, sorry. – ttnphns Apr 28 '18 at 18:30
• Would it be possible to call R and load the mirt package through SPSS Syntax? – ruboin May 3 '18 at 11:27