I am trying to conduct a small experiment based on Likert style data. I have a total of 20 questions, 10 are referring to a latent construct of happiness, and the other 10 to a latent construct of depression.
I would like to identify (preferably using the PCA) the existence of those two latent constructs in my data. I am basing the technical part of the research on the following two references: 1 and 2. Initially I perform the KMO and Bartlett's tests and obtain satisfactory results. Then I run the PCA on the correlation matrix of the data. I've read about a rule of thumb when using the correlation based PCA, i.e., keeping those PCs that have eigenvalues larger than one. And in my case I have 3 such components.
My question is, how can I interpret the results of such a PCA. I know this might be a basic question but I am in need of some confirmation. Should I look an the rotations (I am using
prcomp in R) and see with which variables the component is mostly correlated? Because in the ideal world I would like to obtain two components, one which describes the construct of happiness (i.e. is correlated with those variables that are created from questions regarding happiness) and the other depression. What if the number of components is larger than 2? What is the interpretation than?