The Triangle Inequality and Character Analysis ’

With the advent of both analytical and evolutionary models that specify complex character-transformation weights (or probabilities) and computer programs in which they are implemented, the use of multistate characters with elaborate state-transformation costs (matrix characters; Sankoff and Rousseau 1975; Sankoff and Cedergren 1983) is increasing. What has not yet appeared, however, is a general discussion of these characters as they are implemented in various analyses. Since the most prominent use of these characters comes in the phylogenetic analysis of molecular data, it is in this area that the most attention is required. In its simplest incarnation, a matrix character is one which possesses a matrix of values-each cell of which describes a unique cost for a possible character transformation. These costs are the number of steps required to transform one state into another. In a nonadditive or unordered Fitch-type analysis (Fitch 197 1; Farris 1988)) all transformations are equally costly-each constitutes a single step. Programs such as Swofford’s ( 1990) PAUP allow the definition of elaborate “step matrices” in which the number of steps required by any transformation can be specified. This matrix may be as simple as a binary character with different costs for forward changes and reversals or as complex as a multistate character with scores of specified transformations. As an example, character models such as Do110 parsimony (Farris 1977 ) and irreversibility (Camin and Sokal 1965) are, in essence, simple matrix characters, specifying asymmetries in transformation between two states. In both of these cases, changes in one direction are more costly than the reverse, yielding a quantitative polarity statement. Multistate characters allow more complex delineations. In the case of nucleic acid sequence data, there are four nucleotide bases and 12 transformations to be specified, although not all of these are necessarily unique [actually 16, but the identity transformations play no role in parsimony analysis (Wheeler 1990a)]. Most commonly, the matrix specifies only two types of transformation-transitions (purine to purine, A-G; and pyrimidine to pyrimidine, C-T/U) and transversions (purine to pyrimidine and the reverse, such as A-T) (Brown et al. 1982; Liu and Beckenbach 1992). Transition-transversion ratios are frequently specified in analyses, regardless of whether the chosen costs are internally consistent. Transformation values among character states are, in essence, distances. Distances between sequences are these character changes summed over the length of the sequences. When analyzed as such, they must conform to certain logical strictures even if the events they describe do not. Foremost among these is the triangle inequality invoked by Farris ( 198 1, 1985) and others (Swofford 198 1) in their criticisms of distances. The triangle inequality is a property of metric spaces. Distances that do not conform to this relation are nonmetric and, hence, are internally inconsistent because