While the following tests are useful when your variables' distributions make other approaches such as redundancy analysis (RDA) or canonical correspondence analysis (CCA) questionable, they may not have comparable power. It is often worth attempting to transform your data into a form that would allow the use of 'raw data' techniques rather than (dis)similarity-based techniques. You may also wish to consider distance-based RDA as a means to test the association of a (dis)similarity matrix of response data with a matrix of raw explanatory data.
As always, ensure that the (dis)similarity measure chosen is appropriate to your data set. If you are unsure, please use our (dis)similarity wizard for initial guidance. For more in-depth discussion, consult Legendre and Legendre (1998).
A Mantel test can be performed to find the correlation between two (dis)similarity matrices. Correlating a (dis)similarity matrix of response variables with one of explanatory variables computed for the same objects (samples, sites, etc.). Partial Mantel tests can be conducted if one wishes to 'partial out' or remove the effect of additional variables (such as geographic distance). These variables would be represented by additional(dis)similarity matrices
Following the conversion of (dis)similarity matrices into "object x variable" matrices by a method such as principal coordinates analysis (PCoA), Procrustes analysis allows the comparison of ordinations generated by methods such as non-metric dimensional scaling (NMDS). Unlike the Mantel test, visual inspection of the 'fit' of these ordinations (in this context, one generated from response data and the other explanatory data) is possible.
Legendre P, Legendre L. Numerical Ecology. 2nd ed. Amsterdam: Elsevier, 1998. ISBN 978-0444892508.