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Hi All,

I've been working with a model that I thought made perfect sense, but then someone made a suggestion to change the model into one that I think is both meaningless and impossible. I've attached diagrams of both the original (sensible0.pdf) and the suggested modification (sensible1.pdf).

The problem I see with something like sensible1 is that nothing defines the latent variable L. Put another way, there is no measurement model for L. Am I missing something, or is sensible1 inestimable?

I'd like to be as confident as I can, so your feedback is greatly appreciated.

Sensible0 only makes sense if there is a typo, specifically if the latent variable L is supposed to have variance. Without it, there is no way to estimate any of the paths involving L. Assuming you meant to include that variance, then that's a pretty standard latent regression.

Sensible1 implies the following:

-the set of x variables are uncorrelated, as they are all exogenous.

-the latent variable L exists to somehow simplify the regression of Y on the set of x variables

-at least one constraint is made among the regressions involving L, much like constraining a factor loading. This makes the model possible.

If you meant for L to have a variance in sensible1 as well, then the above conditions still apply. You'll need a second constraint to identify the model, applied to some combination of the latent variance and the set of regressions involving L. I don't see a whole lot of utility in sensible1, as it can be replaced by the set of x variables directly predicting y.

While I won't presume that whoever suggested your model made this error, some researchers mistakenly specify models like sensible1 when they really want sensible0 because it makes more sense conceptually for their construct. The manifest variables aggregate, and thus cause, the latent variable, which then predicts other things.

Hope this helps.

Thanks for the response, Ryne.

The original model, sensible0, was supposed to have variance on the latent variable L. My apologies for the potentially confusing typo.

Your comments on sensible1 were very helpful. The first two comments definitely occurred to me. I think the third comment is the key to having sensible1 make ... sense. My best guess for a good constraint among the regressions involving L would be either constraining two of the parameters to be the same, or to fix one of the parameters to a constant. Fixing a parameter to a constant would scale the latent variable.

I believe the person who suggested sensible1 did so as a consequence of observing somewhat modest correlations among the x variables. Hence, it seemed less reasonable to define a latent factor as the minimal covariances of the x variables.

The benefit of having one path from L to y would be to see the effect of all the x variables on y. Of course, if the x variables are minimally correlated in the data (and zero correlated in the model), one might question the appropriateness of examining the aggregate effect in the L-to-y path, but it could be justified in some situations.

Thanks for your help! No need to reply further, unless I've obviously misunderstood something.

One more comment might be useful.

The fact that the x variables in sensible1 are uncorrelated in the model provides an "opportunity for misfit". If your theory does not involve testing whether the x variables are or are not correlated, then it might be useful to allow them to correlate freely.

If the x variables all intercorrelate freely, L does not need to have a variance. It could be a dummy variable whose only purpose in life was to allow you to read off the aggregated effect of the x variables on Y.