PCA with correlated variables

Question

I'm analyzing data from around 10 survey questions focused on regulatory issues. I've noticed these questions are highly correlated (of course since they are all about regulation), and I'm concerned about the implications of simply summing the responses to create an index. My worry is that this approach might exaggerate differences between responses, especially since they're ordinal. For instance, the perceived difference between firms rated 4 and 5 on a regulation scale could be artificially inflated once you simply add them across all questions.

I have two main questions:

1. Is my concern about the potential for distortion by summing responses justified?
2. Assuming my concern is valid, would Principal Component Analysis (PCA) be an appropriate method to address this issue? I've come across advice suggesting the removal of highly correlated variables, but I'm inclined to think that in this context, retaining them is necessary.

Answer 1

For the purposes you are considering, this would actually be desirable. When creating a composite, it is generally helpful for the items to be highly correlated with each other to ensure that they reliably measure the same latent construct. That said, it would be useful to quantify that reliability in some way, as typical regression methods (which I assume you are using) assume perfect measurement of the predictors. One could use something like Cronbach's alpha or McDonald's omega to measure the reliability of the items together. If the reliability is high, can simply use it straightforwardly. If the reliability is low, You would either have to include that uncertainty into estimation (where some kind of structural equation model may be more ideal) or find a way to modify the indicators (perhaps one isn't as important for your purposes and can be omitted).

For more on using Likert scale tests, this is a useful read. They note in this article that the creator of Likert scales used both sum and average scores, but this can depend on the context.

A PCA isn't entirely wrong, but in using PCA (I assume before fitting to a regression) one loses interpretation of the original variables and loses other information in the process.