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I have discussed this issue several times in this site, but I am asking it again for a final justification from the experts of our community. I wanted to extract four factors (I should call dimensions here I think) from a CATPCA along with the factor scores (note that, factor scores are not available if I use polychoric correlation matrix and factor analyze it) from a data set containing only categorical variables. I have 22 categorical variables overall.

Now, as far I know, CATPCA doesn't use rotation which can be handy to extract factors in standard factor analysis or PCA. I am having a little problem to get clear factors although it seems like a rotation may improve understanding of the factors. Or, in simple terms, I am a little desperate to extract neat factors so that I want to employ different rotations. (I should not be though!) :D

However, we can save the transformed variables in some out file and use them in a standard factor analysis as the variables are now quantified. Note that we need to mention the number of dimensions once while getting the optimally scaled (i.e. quantified) variables. I have reckoned that different number of dimensions mentioned produces different quantification. Using these quantified variables into further factor analysis will require to mention the number of factors again.

Is this a problem? Because as I want 4 factors to be extracted, while asking for the optimally scaled variables I am mentioning 4 dimensions and then planning to use the quantified variables in a standard factor analysis (to get the facility of rotation) where I need to mention the number of factors once again.

If this is a problem, then can we take the number of dimensions to be 22 for CATPCA to get optimally scaled variables (as there are 22 variables overall)? This is only to make sure we have no loss in percentage of variance explained. Then we can use the quantified variables from that for a standard factor analysis and get factor scores in SPSS with mentioning 4 factors this time. I totally don't know if it is a weird idea!

So, waiting for your kind direction.

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    $\begingroup$ Blain, your initial intention is correct. If you strongly believe there are 4 factors, specify 4 dimensions on CATPCA. This will quantify your categorical variables optimally to meet the expectation that there is 4 factors. Then (if you want FA model rather than PCA model and/or you want rotations) use standard FA with those quantified variables to extract the 4 factors and to rotate the loadings. $\endgroup$
    – ttnphns
    Aug 30, 2012 at 9:42
  • $\begingroup$ @ttnphns I am really grateful for your continuous support. But I have two more things to be clarified for further experiments. 1) What will be the factor scores in this case? Those obtained from standard FA on optimally scaled variables? (I am asking this just to make sure because CATPCA will also give me scores). 2) If I want the facility of rotation, but can't be sure of the number of factors at the beginning of the analysis, then should I use 22 dimensions in CATPCA to obtain optimally scaled variables and then use different number of factors to see what results in the best? $\endgroup$
    – Blain Waan
    Sep 1, 2012 at 14:47
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    $\begingroup$ Answers are on the surface, Blain. CATPCA is quatification-then-PCA. It gives you PC scores, the same that you obtain in standard PCA extraction with the quantified variables. To have FA scores, use FA extraction methods. If you think you have 3 factors, do the whole thing (CATPCA then FA) under 3 dimensions/factors. Want 5 factors? - do the whole thing with 5 dimensions/factors. Compare interpretations, to stay with the best of the solutions. $\endgroup$
    – ttnphns
    Sep 1, 2012 at 16:00

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I don't consider rotation as a way to better understand factors (PCs) based on loadings. Rather, rotation is a way to enforce variables to "mostly load" on one factor, which may have large repercussions on factor determination. However, if you a priori know what the factors are supposed to represent, and then are trying to confirm that the appropriate variables are loading correctly on factors, then whatever rotation schemes work are likely to be appropriate. Otherwise, it sounds like you are performing confirmatory factor analysis.

You never stated what the factor scores are going to be used for(?)

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