# Does a PCoA (or MDS) assume normality of the variables behind distances?

More precisely, if I conduct a cmdscale (classical multidimensional scaling) on an Euclidean distance matrix by considering $$n$$ observations of $$p$$ variables i.e. $$D_{ij}=\sqrt{ \sum_p (x_{ip} - x_{jp})^2}$$ with $$i$$ and $$j \in n$$:

1. Must the distributions of $$x_p$$ be normal (like in a PCA) ?
2. And actually, if $$p=1$$, does it then mean something to conduct the PCoA and then do a clustering analysis?
• Re 1: As far as I know, PCA does not require normality. – Richard Hardy May 4 '19 at 16:16
• What is a "cmdscale"? – whuber May 4 '19 at 16:17
• @RichardHardy Ok thx. So it means that I just have to find the appropriate transformation so that outliers do not outweight the PCoA, right ? – Romain May 4 '19 at 16:21
• @RichardHardy Wait, sorry I just realized. I think a PCA supposes normality when working out the covariances, but for the PCoA I don't know... – Romain May 4 '19 at 16:23
• I think PCA is a purely algebraic procedure with no assumptions, bar those that make PCA algebraically feasible. Assumptions like normality are needed for proving properties of statistical procedures, not algebraic ones. – Richard Hardy May 4 '19 at 17:15