Using metric MDS with non-metric distances and assessing the fit quality

Question

I'm going to perform MDS by means of cmdscale function of standard R library. I spent several hours googling it and finally have a number of questions (some of them more general, some could be more specific to its implementations in R ).

1. I have a matrix of non-metric distances, e.g. based on Sorensen-Dice coefficient. Can I use the matrix for cmdscale function? As I understand, the function will perform "metric MDS"; so, metric MDS for non-metric distances.
2. How can I estimate the quality of my model? I've found the stress term and GOF component of output of cmdscale function, but I am confused with their interpretation. Could you clarify it for me please? Is there anything else?
3. In PCA one often calculates the percentage of explained variance. Is this term applicable in MDS as well? How can I interpret it?

Answer 1

• I can't say about cmdscale being not R user (and won't be reading its documentation right now) so can't answer your pt 2.

• You may do metric MDS for whatever data you wish (although for distances far from being euclidean or metric nonmetric MDS will usually give better map). Word "Metric" MDS has not quite that meaning as in "metric distances", so the term is a bit misleading. Actually, it refers to linear transformation of Dissimilarities to disparities. This transformation is a form of the so called optimal scaling task. Metric MDS imposes the constraint of linearity on it. Nonmetric MDS is more liberal-flexible and allows just monotonicity (thence it is "successful" more often).

• Torgerson's MDS based on PCA is a bit special form of metric MDS. (Note that there exist iterative metric MDS different from it!) It does not do the task of preserving the pairwise distances on a low-dim. map directly. Being based on PCA, it preserves variances. And here comes the answer to your pt 3: that percentage of explained variance isn't of much use or interest in MDS. Interest is for some form of stress which is about the reproduction of pairwise distances (more precise, of the disparities).

• Dice matching coefficient, when it is computed between objects based on dummy variables (e.g. a nominal variable recoded into such the set of binary indicators) is equal to Ochiai coefficient, which is the cosine. And cosine is directly convertible to euclidean distance. And euclidean distance is a distance for which metric MDS is quite naturally applicable (for it then just becomes the task to low the rank of the configuration). So, if it is your case - you have a reason to prefer it to nonmetric one.