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Suppose I have 3 manifest variables, x, y and z. I would like to use path analysis to determine the direction of influence. Is there a test that lets me distinguish between the following models?

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and

enter image description here

In the first, z is caused directly by x, and indirectly by x through y. In the second, x is caused by y and z.

I fear that it is not, given that my SEM model is saturated in both cases, but I would like to be able to distinguish these situations.

Would it make a difference if there were a measurement component to the model, as below?

enter image description here

which should be distinguished from, say, this:

enter image description here

Placidia
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1 Answers1

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I think I know the answer to my question. I tried a few models like the ones shown above, using the sample data sets from lavaan in R. Typically, models that simply replace a two-headed arrow (correlation) with a one-way arrow (regression) have the same number of degrees of freedom and cannot be distinguished from each other. A model like $X \rightarrow Y \rightarrow Z$ has one more degree of freedom from model $X \rightarrow Y \rightarrow Z, X \rightarrow Z$, and these can be distinguished.

Intuitively, if it were possibly to distinguish the models I outlined in my question, correlation would show causality, and this XKCD comic would no longer be funny.

Placidia
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    More generally, this is the problem of 'equivalent models' (e.g., http://www2.gsu.edu/~mkteer/equival.html). You cannot distinguish equivalent models based on their fit (since it is the same). This is unfortunately an often ignored issue (i.e., just because a particular model fits does not automatically imply that we have established how the variables are related to each other). – Wolfgang Nov 25 '15 at 16:43
  • That's great. That's just the kind of reference I was hoping to find. Thanks. – Placidia Nov 25 '15 at 21:23
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    The answer by @Placidia isn't always true. (a) X -> Y -> Z implies a correlation between X and Z, (b) X -> Y Z does not, nor does (c) X -> Y – Jeremy Miles Nov 26 '15 at 00:00