Categorical Ranks and Evolutionary Taxa in Numerical Taxonomy

The merits of various measures of divergence within a taxon are discussed, and it is concluded that the difference between the two most different OTU's in a taxon is the most reliable measure of divergence. This "maximum difference" criterion of divergence is shown to be related to the categorical rank of a taxon by an equation of the form cd( R) = k2R, where R is the rank of the taxon, and cd ( R) is the greatest amount of divergence permitted within a taxon designated to be of rank R. The rank criterion is related to a modification of the intensional "difference-from-type" form of taxon definition. It is noted that, if evolution does not proceed by saltation, certain intensional definitions of taxa imply extensional differences between monotypic overlapping taxa of different ranks, so that Gregg's Paradox is resolved. A classification of ferns is generated as an example of the use of the techniques proposed. It is concluded that the continuity of evolution invalidates the use of "gaps" between groups as criteria of rank. Continuity also implies that very similar OTU's may be placed in distinct taxa of high rank. One of the barriers to constructing formal classifications through numerical analysis has been the lack of an objective means to convert the continuous scale of numerical differences to the discrete scale of categorical ranks. For taxonomic statistics, such as the correlation coefficient or the matching coefficient (Sokal and Sneath, 1963), which vary between fixed limits, it was possible partly to resolve the difficulty by choosing standard "phenon" levels of similarity as criteria of rank (Sokal and Sneath, 1963). But the phenon levels remained arbitrary, even in relation to each other. Further, for some measures of relationship, it is difficult meaningfully to establish a priori phenon levels. I have shown (Farris, 1967) that the appropriate statistic with which to estimate overall patristic difference is a sum of estimated patristic unit character differences. In evolutionary studies, one typically wishes to weight characters to take their relative conservatism into account. I have suggested doing this by rescaling each unit character so that it has unit variance within OTU's (Farris, 1966). If the sum of unit character differences is used as the measure of overall difference, and if the scales of the unit characters are set a priori in this way, it is not possible to place a prior upper limit on the values of overall difference to be encountered in the study. While it is true that overall differences might be rescaled after the difference in the study would be unity, this does not seem to be the best procedure for evolutionary studies. The need for general evolutionary taxonomic studies is to establish relative ranks of taxa not only within a single study, but also to establish consistent rankings over a variety of diverse studies. This cannot be accomplished merely on the basis of relative ranks based on differences constrained to have a particular upper limit in each individual study. I shall present a general method for converting overall patristic differences into categorical ranks. Since a valid evolutionary taxon must be monophyletic, I shall limit my attention to taxa that are monophyletic in estimated evolutionary relationships. A taxon is estimated to be monophyletic if and only if it forms a connected group on an estimated cladogram. I shall refer to the estimated overall patristic difference between two OTU's, A and B, as d(A, B). MEASURES OF DIVERGENCE We must first decide on a measure of divergence within a set of OTU's forming a monophyletic group. Various statistics