I used NMDS axes of an ecological community as proxys for the community similarity in the different samples. I would like to "quantify" the importance of the different NMDS axis according to the stressreduction. Now my question is: If I calculate an NMDS with one dimension would the NMDS axis of that be the same as the first axis of a NMDS calculated with two dimensions? Of course, I tried that myself using the metaMDS function in vegan and the axis are not exactly same but correlate with a coefficient >0.9. How can that be explained?
Assuming the dimensions are relatively independent (e.g., orthogonal), then the results should be comparable. If not, then the first dimension of an analysis with 2 or more dimensions may not be the same as the single-dimension analysis. Though I do not know of a statistical test for this comparison (i.e., one that would give a P-value to assess the difference), a visual inspection of the one dimension from the two solutions would be a reasonable assessment of the argument that a one dimensional solution may suffice.
Hope this helps.