Categorical Principal Component Analysis - using Count, Continuous, Ordinal variables together

Question

I have some variables and I want to reduce their number for further analysis. I initially thought of combining them using factor analysis. But since the variables are of all kinds (rating, count, ordinal, continuous dollar amount) I am thinking about using CATPCA for it. However, I have some questions about the technique -

1. Is CATPCA the right approach or Latent Class Factor Analysis would be the better one?
2. By using the different kinds of variables together - how will the object scores be interpreted? For example - The rating question is on a 10 point scale but the dollar amount ranges from 0 to 1000s, so what will the object score obtained after CATPCA represent? And is it comparable to the Factor Score obtained from traditional PCA?
3. For ordinal variables - Do all of them need to be in the same form, e.g. 1 meaning lowest and 10 meaning highest? Or can they be used in whatever form they are?
4. Is it good to categorize the count and continuous variables based on their distribution and then treat then as ordinal/categorical variables?

I would highly appreciate any help you can provide!

Answer 1

There have been questions on this site addressing related issues as your questions. To find them, make the local search about "CATPCA" and "Optimal scaling".

1. I don't know what is Latent Class Factor Analysis. According the traditional classification of such methods, factor analysis produces continuous latents and latent class analysis produces categorical latents.
2. The object scores are usual PCA scores. CATPCA is usual PCA (of the correlation matrix) but the PCA is performed after the embedded optimal scaling algorithm quantifies (transforms, generally nonlinearly) categorical input variables into scale (interval) variables; the "optimality" target function is to maximize variance accounted by m first components. Be aware that CATPCA does not allow for truly continuous variables input. It accepts only descrete positive values. So, your Dollars variable should be discretized first; in SPSS, there is a good option to do it right in CATPCA procedure. Discretization amounts to binning data into categories 1, 2, 3, 4, etc; therefore, your Dollars variable becomes initially comparable with your likert-scale variable. If the categories are not equally spaced because you deliberately drop some of the categories, as in 1, 2, 4, 5, this will produce the same quantification as 1, 2, 3, 4 under "ordinal" transformation, but different quantification under "numeric" (=linear) transformation.
3. In SPSS, CATPCA can take any descrete positive values, but 1, 2, 3,... coding for ordinal variables is recommended. So, make them all start with "1", but different variables may potentially be of different number of scale points.
4. See point 2: continuous variables must always be categorized into 1, 2, 3... scale, no fractional values allowed; you may prefer any method of binning you like. CATPCA does not work with truly continuous variables.