Say I have a MDS plot as shown below:
Now, say everything in the MDS plot looks exactly the same (the position of the colored dots does not change). But only one thing changes: the x-axis ("first dimension"):
1) If the x-axis limits goes from -200 to 100 say (instead of -100 to 50 as it currently is), and the dots all appear the same distances from each other, can we say the S:N ratio is better (i.e. the groups are better separated) because the x-axis limit is larger?
2) If in one case the x-axis explains 25% of the variation, but in another case the x-axis explains 75% of the variation, if the dots all appear the same distances from each other in both cases, can we say the S:N ratio is better in the second case since those distances between the dots (samples) can be explained by even more variation?
Thank you for offering your ideas to these questions!
I don't think you can interpret the axes like that. MDS is just a method of contorting points separated in 3 dimensional space into 2 dimensions. You can't determine what proportion each of the axes explain in MDS unlike other constrained methods (PCA ect.) Instead, you are provided an R statistic, and when that R statistics exceeds 0.20, it means your MDS poorly represents the differences between the points.
Sorry that I can't provide a more 'mathematical' answer.