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== incomplete ==
The main idea...
A combination of Mantel correlation and multiple regression, multiple regression on distance matrices (MRM; Manly, 1986; Smouse et al., 1986; Legendre et al., 1994) allows a regression-type analysis of two or more (dis)similarity matrices, using permutations to determine the significance of the coefficients of determination. One matrix must contain (dis)similarities calculated from response data, such as OTU abundances, and the other matrices must contain (dis)similarities calculated from explanatory data (e.g. environmental parameters or space). MRM has been used as a method to disentangle the influence of space and environmental factors in ecological data (Lichstein, 2006).
Results and interpretation
The results of MRM are largely comparable to the Mantel test and MLR. Users are directed to those endpoints for further information.
Key assumptions
An appropriate (dis)similarity coefficient must be chosen for each data-type analysed.
The regression analysis need not be linear. Non-parametric and non-linear methods may be used, however, the assumptions of these methods must also be met.
Warnings
All hypotheses must be in terms of the (dis)similarities between objects (see Legendre 2005, Tuomisto & Ruokolainen, 2006, Legendre et al. 2008).
Confidence interval estimation can be severely affected by extreme (dis)similarities (i.e. approaching 0 or 1) between non-identical objects.
Be sure to consider whether a raw data approach is more appropriate to your questions (see Legendre 2005, Legendre et al. 2008)
An appropriate permutational scheme must be chosen. The (dis)similarity values themselves are interdependent and thus not truly exchangeable. One approach is to permute the objects in the raw response data and, after each permutation, calculate a new (dis)similarity matrix for re-analysis. Permutation of residuals may also be considered.
As raised by Manly (1986), the impact of transformations on (dis)similarity matrices can have large effects on regression coefficients and whether the transformation has been 'successful' is difficult to determine.
Spatial relationships should be modelled with care: Euclidean, polynomial, or other models are required in different circumstances. For spatial analysis, consider also principal coordinates of neighbour matrices (PCNM).
Implementations
References
Legendre P (2005) Analyzing beta diversity: partitioning the spatial variation of community composition data. Ecol Monogr. 75: 435–50.
Legendre P, Borcard D, Peres-Neto P (2008) Analyzing or explaining beta diversity? Comment. Ecology. 89: 3238–3244.
Legendre P, Lapointe F, Casgrain P (1994) Modeling brain evolution from behavior: A permutational regression approach. Evolution. 48: 1487-1499.
Lichstein JW (2006) Multiple regression on distance matrices: a multivariate spatial analysis tool. Plant Ecol. 188:117 –131.
Manly BF (1986) Randomization and regression methods for testing for associations with geographical, environmental and biological distances between populations. Res Popul Ecol. 28: 201–218.
Smouse PE, Long JC, Sokal RR (1986) Multiple regression and correlation extensions of the Mantel test of matrix correspondence. Syst Zool. 35: 627 –632.
Tuomisto H, Ruokolainen K (2006) Analyzing or explaining beta diversity? Understanding the targets of different methods of analysis. Ecology. 87: 2697–2708.