Minimum evolution using ordinary least-squares is less robust than neighbor-joining

The method of minimum evolution reconstructs a phylogenetic tree T for n taxa given dissimilarity data d. In principle, for every tree W with these n leaves an estimate for the total length of W is made, and T is selected as the W that yields the minimum total length. Suppose that the ordinary least-squares formula S _W (d) is used to estimate the total length of W. A theorem of Rzhetsky and Nei shows that when d is positively additive on a completely resolved tree T, then for all W ≠ T it will be true that S _W (d) > S _T (d). The same will be true if d is merely sufficiently close to an additive dissimilarity function. This paper proves that as n grows large, even if the shortest branch length in the true tree T remains constant and d is additive on T, then the difference S _W (d)-S _T (d) can go to zero. It is also proved that, as n grows large, there is a tree T with n leaves, an additive distance function d _T on T with shortest edge ε, a distance function d, and a tree W with the same n leaves such that d differs from d _T by only approximately ε/4, yet minimum evolution incorrectly selects the tree W over the tree T. This result contrasts with the method of neighbor-joining, for which Atteson showed that incorrect selection of W required a deviation at least ε/2. It follows that, for large n, minimum evolution with ordinary least-squares can be only half as robust as neighbor-joining.