Is the purpose of latent variable models to model causality, where the causes are not observable i.e. latent?
Are latent variables modelling causes of the observable variables?
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Or, perhaps a better answer is, "it depends on what you mean by causality".
It is presumed, in the latent factor model, that the individual items measure the latent factor; that's really the point. So, on, e.g. the MMPI, the different questions are supposed to be measuring aspects of personality, and the purpose of factor analysis is to uncover those latent factors.
But does this mean that the personality factors "cause" the answers to the question?
The answer is philosophy, not statistics.
(StasK's edits:) To put it differently, there is nothing magic about latent variable models that immediately let you jump to the causality conclusions. It's just an extension of regression analysis to unobserved variables, that's all. If you trust regression with observational data to draw causal conclusions, you can likewise trust latent variable models. This may be discipline-specific, as some disciplines (biostat) only accept randomized experiments as the golden standard for establishing causality, while others (education research) may be happy with something less golden and less standard.