I have four datasets: morphological measurements for a set of species (M1), ecological measurements for the same set of species (E1), morphological measurements for a second set of species (M2), and ecological measurements for this second set of species (E2).

I am interested in finding the linear combinations of variables between M1 and E1, and between M2 and E2. That is, I'd like to know what combinations of morphological measurements are associated with what combination of ecological measurements--for each set of species separately. This seems like a good use of CCA (two separate CCAs).

But here's where things get tricky for me. I'd like to see whether the same linear combinations from one set of species do a good job of explaining the variation in the second set of matrices. And I'd like to see how they differ, if possible...e.g. variable 3 from M2 would be more heavily loaded on a given axis if we didn't constrain the second CCA by the linear combinations found from the first.

Is this making any sense? I'm not a statistician, so I admit my lack of experience up front. I could see simply running these as two separate CCAs, then comparing the results qualitatively. But that doesn't seem very rigorous. Should I be considering some other approach entirely?

Thanks for any input.

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    $\begingroup$ This should be possible with structural equation modeling, but I'm not sure how easy it would be. If you aren't very familiar w/ SEM, you may want to work w/ a statistical consultant. $\endgroup$ – gung - Reinstate Monica Mar 12 '14 at 15:30
  • $\begingroup$ This would only make sense if your two sets of species have the same size; this sounds a bit weird to me in the context of your research problem -- why would they (if not only by chance)? Apart from that, note that CCA does not necessarily "explain a lot of variance", so it's entirely possible that your canonical variates will explain very little variance even in the same dataset. In particular, if you have a lot of variables then CCA is very likely to overfit, resulting in very high correlations and very low variance. $\endgroup$ – amoeba Mar 12 '14 at 15:39
  • $\begingroup$ Thank you for the input. The two sets of species have the same sets of variables measured (i.e. if species are rows then both M1 & M2, and both E1 and E2 have the same numbers of columns). But they do not necessarily have the same number of rows. I'm aware that CCA does not always explain a lot of variance, but I've run it with one dataset (M1 vs E1) and it did a pretty good job. I want to see if the same ecomorphological relationships are evidenced in a separate study area. $\endgroup$ – forlooper Mar 12 '14 at 19:04
  • $\begingroup$ @amoeba Was this what you meant in terms of same size? $\endgroup$ – forlooper Mar 13 '14 at 17:17
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    $\begingroup$ @forlooper: You are right, what I said about same size did not really make sense. Regarding your main question: I am nor aware of any approach other than running two CCAs and then comparing them (e.g. explicitly checking how much variance projections from CCA2 explain in M1/E1, as you wrote yourself). If anything comes to mind, I will let you know. $\endgroup$ – amoeba Mar 14 '14 at 15:44

Aside from comparing the projection of one CCA on the other as mentioned by @amoeba in the comments. If the variables of both are the same, you could compare the loadings. The greater the difference in the loadings are, more information it carries.

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