# Factor analysis - CATPCA combined with conventional PCA

I have some concerns regarding factor analysis and especially about combining the factor analysis for an ordinal scale (categorical data) - CATPCA with conventional PCA. Basically, I need to enter my set of variables in CATPCA, take the transformed variables from CATPCA and introduce them in conventional PCA (because, for example, I need to rotate the factor to obtain a reasonable solution).

My concerns are:

1. After introducing the transformed variables in conventional PCA, is it okay just to follow the steps for conventional PCA and not taking in consideration that the initial variables have been transformed (standardized, from what I have observed) in CATPCA? I mean, can I go through evaluating MSA, communalities, factor loadings and so on, as in a regular case?

2. After obtaining the factors, if I want to assess the reliability, is it okay to take into consideration the initial variables (not the transformed ones in CATPCA which I also use in conventional PCA), assuming that I would like to generate the summated scales (for each factor) based on the initial variables and further use the summated scale for different constructs for correlation and regression?

3. Is it okay to calculate summated scales if I am undertaking an exploratory study and some of the factors that result are somewhat different from what I have found in the literature? (I have read that summated scales are not of much use if the solution obtained from the PCA is not supported by literature, for example if I have one or two factors that combine some variables in a different form than from I have found in the literature, thus I also name them differently)

4. Does it have importance in conventional PCA which of the criteria: communality and loading problems is taken into consideration for deleting variables in the conventional PCA? (should I first delete the variables that have a low communality or the variables that have cross-loadings?)

5. If I have a sample of over 350 respondents, is it acceptable to consider loadings over 0.50 as being practically significant or should I consider loadings over 0.30 as being significant (which would lead, for example, in the case of a variable that loads with 0.60 on a factor and respectively 0.30 on another factor to be considered as having a cross-loading)?

6. If I have a certain number of constructs and I would like to use the same methodology for each construct, is it okay to use CATPCA + conventional PCA even for the constructs for which I do not need to rotate the solution of the ones that are unidimensional? (I want to use a combination of CATPCA+PCA for a number of reasons, including to rotate the solution, but some of the constructs I am analyzing are unidimensional or, even I they are multidimensional, the solution may be okay without rotation)

7. Is it accepted to replace missing values with the mode in CATPCA while replacing the missing values with the mean (for the initial variables, not the transformed ones) for summated scales, correlation and regression? (This would also imply that I would use the initial variables and not the transformed ones for calculating summated scales, for correlation and for regression)

8. For summated scales: can I use all the variables that load on a factor or is it necessary to use only the variables with high loading (over 0.70, for example)?

9. Can I calculate a latent variable (construct) after the formula: Sum of (summated scale for factor i * variance explained by factor i), i taking values from 1 to n (n being the number of factors)?

Thank you!

• CATPCA transforms categorical variables into interval under the hypothesis that there is m components. CATPCA is equivalent to taking those transformed variables into conventional PCA and doing it with the extraction of m components. But in your case it sounds that you need factor analysis, not PCA. So, input the transformed vars and do FA as usual, but with the extraction of strictly m factors. It is a good approach: CATPCA-based transformation of vars + FA then = nonlinear FA. – ttnphns Jun 30 '14 at 3:30
• Loading is the correlation between factor and variable. If small correlations dominated between variables, loadings are expected to be not high. Then even loading of 0.35 or so may be considered "significant". There is no clear-cut rule. If a variable is loaded 0.30 by one factor and 0.60 another, I will certainly ignore 0.30: 0.60 is much higher. – ttnphns Jun 30 '14 at 3:37
• Doing all subsequent linear analyses, including Reliability, is logical on the transformed variables only. But I'm not in your shoes to recommend categorically, and you describe your situation too laconically. Your points 3, 4, 6 - sorry, I didn't understand them well enough. Maybe you rewrite them in more detail? – ttnphns Jun 30 '14 at 3:44
• Thank you so much for your answer! Yes, by PCA I meant factor analysis (I know they're actually not the same). I am using SPSS for my analyses. I wrote more details to questions 3, 4 and 6. – Lucy Jun 30 '14 at 8:59
• Lucy, after your editing the question has become even more bloated and encyclopedic (for me). Each of your point might be discussed as a separate question... Please, define your priorities of interest. – ttnphns Jun 30 '14 at 9:40