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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?

Masher
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    Lots of related threads here e.g. http://stats.stackexchange.com/questions/11713/whether-to-use-efa-or-pca-to-assess-dimensionality-of-a-set-of-likert-items You will get, I trust, more detailed replies but in the interim one big red flag. You are maklng extraordinarily strong assumptions that you can just take Likert style data (exactly what you mean by that needs more detail) and then treat them as if they were just measured. If you haven't encountered it yet, know that there's an entire spectrum of reactions ranging from (at least) "completely invalid" to "defensible with caution". – Nick Cox Apr 07 '16 at 15:20
  • Thanks for your comment. I am using the 1-7 Likert scale in my research, if that is what you meant by more detail. – Masher Apr 07 '16 at 16:06
  • OK, but watch out for legions of researchers, some gatekeepers in your path, who will warn that's not interval scale measurement. – Nick Cox Apr 07 '16 at 16:12
  • I will watch out, thanks. Do you have any idea about the interpretation of the PCs when their number is different than the underlying assumptions? – Masher Apr 07 '16 at 16:33
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    I don't think there is a useful general answer to that beyond sometimes the data make you change your mind. That's good. – Nick Cox Apr 07 '16 at 16:35

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I think EFA (or CFA) would be a better fit for this analysis. The reason I'm stating this is because you already have a proposed model (two latent constructs, each associated with ten items) and factor analysis would allow you to assess how well your data fits the model. Additionally, it will give you meaningful information about how much the variance in each item is accounted for in the model.

ksfowler
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  • Thanks for your answer. But shouldn't the PCA confirm that two PCs do indeed represent the latent constructs? – Masher Apr 07 '16 at 15:21