Are you imagining them as predictor or response variables (or both)?

Erin Larson I was envisioning them as response variables in a linear mixed-effects model. The model would have environmental predictors. For example, NMDS1 (change in species composition among sites along NMDS axis 1) ~ env1 + env2 + ranef (a random intercept term for experimental block).

The short answer is no. Why? Because nMDS plots don't technically have axes. Plots can be enlarged, reduced, rotated, flipped arbitrarily, and all that matters are the relative distances among objects in the chosen number of dimension in which the ordination has been produced. OK, axes (and scores) are used to plot the result, but they have no interpretation. What you want, to do what you describe, is output from an ordination method that explicitly tries to fit axes in the full multivariate space. PCA does this, and the axes are interpretable. PCoA is another option which allows a wider range of resemblance measures to be used. Finally, mMDS (metric MDS) might be an option, as the axes are interpretable, but mMDS often has high stress owing to the way it works, so something like PCA/PCoA might be better.

I agree with what Paul responded with. Another thing to consider would be to use PERMANOVA - you can do this with the 'adonis' function in the vegan package. I think the 'strata' argument in the adonis function can deal with your experimental blocks.

Thanks, Paul & Erin for your responses.

Actually, rather than PERMANOVA (which tests for differences among groups of samples) a closer approach would be to use DISTLM (distance based linear modelling), a semi-parametric form of multivariate multiple linear regression.

J. Aaron Hogan According to Buttigieg & Ramette, there are two warnings on using NMDS scores as response variables:

1. "Do not treat the results of distance-matrix-based analyses as 'raw data' in further procedures or interpretation. As pairwise dissimilarities have been calculated, there is now interdependence in the data. See Legendre et al. (2005) for further discussion."

2. "Avoid relating environmental variables (or other explanatory variables) to the distances in an NMDS ordination. As an NMDS ordination is typically only an approximate representation of a distance matrix, consider using the distance matrix itself. Consider distance-based redundancy analysis or the Mantel test."

https://mb3is.megx.net/gustame/dissimilarity-based-methods/nmds

I think Paul Somerfield made reference to the second one. As @Erin Larson said, I think you can handle nested or hierarchical designs using the permutations argument in the anova.cca function from vegan library.

Saludos!

Francisco Mora is right that you should be wary of using raw distances as data, owing to a lack of independence (changing one distance inevitably alters others), but that doesn't mean that in certain circumstances and with care that they can never be used. There is a whole class of multivariate analyses that use distance (or ranked distance) measures in a whole host of routines, including ones that explore relationships between different types of data such as biota and environmental variables. For nested or hierarchical designs (if that's what you really want to analyse) you could use ANOSIM or PERMANOVA.

I do agree that using ordination scores rather than the actual distances (or their ranks) is not the thing to do, as an ordination (or any other representation such as a cluster dendrogram) is only a representation of the true multivariate information, which lies in the distance matrix.

Dear J. Aaron Hogan , I hope you have a nice day.

Here is a paper in which we used a technique for related environmental variables with biotic interactions. Both procedures were developed in R software.

Article Flower-mediated plant-butterfly interactions in an heterogen...

Regards,

Cristian