Abstract
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.
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Willson, S.J. Minimum evolution using ordinary least-squares is less robust than neighbor-joining. Bull. Math. Biol. 67, 261–279 (2005). https://doi.org/10.1016/j.bulm.2004.07.007
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DOI: https://doi.org/10.1016/j.bulm.2004.07.007