Hello, In structural equation modeling can we favor a model over the rival one because of better multiple correlation? I have two rival models. First is slightly have better fit values than the second, but the second has higher multiple correlation values. In this case can I favor the second? Thanks in advance.
Hi Ceren,
you are mixing two important and interesting questions:
1) What to strive for: prediction or (causal) explanation or in other words: what is more important: Fit or predictive value?
2) With the goal of correct causal specification, how to judge differences in fit?
Wrt. 1: The answer depends on the very problem you want to solve: It is really a pure prediction task (e.g. when wanting to have some predictive information about an applicant in order to make a decision about whether you employ him/her; or whether to take an umbrella when the clouds have a dark color)? In these situations, you have information in X that reduces the range of possibilities for Y, thus, reducing uncertainty. While job performance has a large range (from disasters to excellence), having a highly intelligent person reduces this range (i.e. it is likely that s/he will perform better).
No need for causal correctness here. Even if you have a completely spurious correlation, the correlation lets you predict Y based on X. For instance, if you find out that "shoe size" of a person predicts its athletic success as a runner, choosing a large runner for sports team may be a good decision.
This is different if your goal is causal. Here you want to test or explore what the effect in Y would be if you intervene on X. If X indeed has a non-substantial effect, specifying a model that actually reveals its small effect still implies that you have the correct model for that task. That is, causal tasks strive for correctness of some causal truth while prediction strives for maximal information gain from a pragmatic point of view. In a causal model you do not want to have confounding variables that upward bias your "effect" - but from a predictive point of view, this does not matter.
Wrt. 2: If you have the causal goal, differences in fit of two model that both show a substantial misfit can be misleading, as the model with the "better" fit may still by substantially misspecified or worse, even more misspecified that the worse looking model.
Compare this with the following situation: You want to buy a car: The salesman presents two cars as his special offers: One has only three wheels, and the other has even only 2 wheels. Would you buy the 3-wheel-car as it is "better" (with regard to the fit to the tasks of a car--driving)?
The practice to test series of models and choose the one with the "better fit" concurs with this analogy. The better model can still be wrong.
Best,
Holger
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