# AIC with Mantel's tests

Mantel's tests are commonly used to compare genetic distances (say, between a number of individuals) with true or hypothesized landscape distances between those same individuals. For example, “does simple linear distance better correlate with observed genetic distance than a distance based on avoiding some landscape feature (e.g., water bodies, high elevations)?”. More information about the details of the test can be found here: Spatial Analysis in Ecology.

My question, however, has to do with using AIC and model selection methods to choose between competing models for the landscape distances. For example, if you know the genetic distances, you may hypothesize 3 different models for creating the landscape distances:

1. Linear distance
2. Distance based on avoiding high elevation
3. Distance based on avoiding high elevation and water bodies

Currently, choosing between these models based on the outcome of the Mantel’s tests has been a bit ad hoc—basically comparing the magnitude of Mantel’s $r$ and confidence intervals for each model. I’d like to use an AIC/model selection framework for this, but there appear to be a few problems:

1. I’ve been advised that because of non-independence in the data structure (that is, there aren’t $n(n-1)/2$ independent observations), AIC would not be appropriate. A related issue is how would $n$ be calculated for AIC in this case?
2. My original thought was that model 3 would have more parameters than models 2 or 1, and therefore $K$ in the AIC calculation would reflect that. But, it seems clear to me now that because the inputs are simply two matrices, the number of parameters in all models would be equal (regardless of the number of variables used to create the underlying landscape distance values), and therefore AIC probably wouldn’t tell you anything more than the raw $r$ values from the Mantel’s test.

So, my questions are:

• Can anyone confirm #1 (above)? Is there any way around this issue? I’ve been advised that some form of AIC might be developed, but that it would need to reflect the non-independence of the observations.

• Is my observation in #2 (above) correct? That is, will $K$ in the AIC formula always be the same for a series of Mantel’s tests?

• What might be a way around this issue? One thing that is being used is multiple matrix regression with some sort of selection (e.g., stepwise), to identify the best combination of variables to include in the landscape distance calculation, before running the final Mantel’s tests. But, I’m guessing that this approach has many of the same problems with stepwise regression in general, and why AIC methods have been embraced for model selection.