# Comparing independent MRDM models (multiple regression on distance matrices)

I am conducting Multiple Regressions on Distance Matrices (MRM or MRDM) to estimate the influence of 3 independent variables (IVs) on intraspecific $$\beta$$-diversity (my response variable DV) between several geographical locations. All variables are in the form of a matrix of pairwise distances between locations.

My 3 IVs matrices are pairwise geographical distance, pairwise connectivity and a species-specific pairwise genetic index. I am using the method developed by Legendre et al. (2014) and implemented in the function MRM() from the R package ecodist (Goslee & Urban 2007) to know if some of my IVs are significant predictors of intraspecific $$\beta$$-diversity.

Here is the situation:

• I need to replicate such an analysis for several animal species. Sampling locations are identical between species but, for some species, a few locations are missing. Thus, DV and IVs distance matrices do not necessarily have the same size from one MRM() analyse to another, and each analysis does not necessarily base in the same number of observations.
• Two of my IVs matrices (geographical distance and connectivity) are identical from one MRM() analyse to another (except for the sampling issue outlined just above) because related to the sampled locations themselves. However, my DV matrix (intraspecific $$\beta$$-diversity) and one of my IVs matrices (pairwise genetic index) are species-specific.

What I want is to compare the standardized coefficients for each IVs between species, which means, compare the standardized coefficients from one MRM() analyse to another. For example :

MRM for Species A:

MRM(BetaDiv.A ~ Geog.dist.A + Connectivity.A + GeneticID.A, mrank=T, nperm=10000)
$coef BetaDiv.A pval Int 5.81657290 0.9950 Geog.dist.A 0.34923860 0.1224 Connectivity.A 0.27729070 0.2096 GeneticID.A 0.16573590 0.4709$r.squared
R2      pval
0.2697909 0.0574000

$F.test F F.pval 6.281003 0.057400  MRM for Species B: MRM(BetaDiv.B ~ Geog.dist.B + Connectivity.B + GeneticID.B, mrank=T, nperm=10000)$coef
Int             -6.5066862 0.9154
Geog.dist.B     0.28720590 0.1748
Connectivity.B  0.46442990 0.0023
GeneticID.B     0.53126360 0.0258

$r.squared R2 pval 0.7004525 0.0041000$F.test
F   F.pval
31.9577   0.0041


Here, I would like to tell that Connectivity has a stronger influence on BetaDiv for Species B than it has for Species A (e.g., 0.46442990 > 0.27729070), so does GeneticID, while the opposite goes for Geog.dist. But, I feel like I cannot actually say that, considering each MRM()was built with unequal number of observations between species, and with one species-specific IV matrix.

I have been reading things about more classical linear modelling, where some people assess that is possible to directly compare standardized beta coefficients and/or R² between two lm() models -eventhough these come with different DVs (i.e., outside of a model selection or model-averaging routine), while other people urge to rely on other approaches like multivariate regression models. I am really not sure how to perform in my case considering the use of matrices, and given my experimental design.

Any suggestion will be appreciated :-)

Cheers!

References:
Goslee & Urban 2007, The ecodist package for dissimilarity- based analysis of ecological data. Journal of Statistical Software, 22:1-19.

Legendre et al. 2014, Modeling Brain Evolution from Behavior: A Permutational Regression Approach, Evolution, 48(5):1487-1499. doi: 10.1111/j.1558-5646.1994.tb02191.x.