In my understanding, exploratory factor analysis (EFA) is used for discovering latent structure among variables. It consists of two interrelated parts:
- determine the number of latent factors $k$,
- determine the loadings of factors onto variables.
However, it has been suggested (in the comments in another thread) -- unless I misinterpreted it -- that EFA is primarily about part 2, while part 1 can be taken as given. That does not sound quite right to me. If we know what the variables in the dataset measure and we know $k$ ahead of seeing the actual data, then with some subject-matter expertise* we probably can guess what the factors could be and consequently guess something about the loadings, too, e.g. which loadings will be close to zero. If so, we could do confirmatory factor analysis (CFA) right away instead of doing EFA.
Could you offer any example where we realistically know the number of latent factors $k$ ahead of seeing the data but have no idea about factor loadings?
*Yes, this relies on subject-matter expertise and on the number of factors $k$ not being too high. If we have a dataset where we do not know what the variables are measuring, or it comes from an area in which we have zero subject matter expertise, then knowing $k$ will not help with factor loadings.