A reconsideration of inapplicable characters, and an approximation with step-matrix recoding.

Evidence for phylogenetic analysis comes in the form of observed similarities, and trees are selected to minimize the number of similarities that cannot be accounted for by homology (homoplasies). Thus, the classical argument for parsimony directly links homoplasy with explanatory power. When characters are hierarchically related, a first character may represent a primary structure such as tail absence/presence and a secondary (subordinate) character may describe tail colour; this makes tail colour inapplicable when tail is absent. It has been proposed that such character hierarchies should be evaluated on the same logical basis as standard characters, maximizing the number of similarities accounted for by secondary homology, i.e. common ancestry. Previous evaluations of the homology of a given ancestral reconstruction contain the unintuitive quantity "subcharacters" (number of regions of applicability). Rather than counting subcharacters, this paper proposes an equivalent but more intuitive formulation, based on counting the number of changes into each separate state. In this formulation, x-transformations, the homoplasy for the reconstruction is simply the number of changes into the state beyond the first, summed over all states. There is thus no direct connection between homoplasy and number of steps, only between homoplasy and extra steps. The link between the two formulations is that, for any region of applicability of any character, a subcharacter can be interpreted as the change into the state that is plesiomorphic in that region. Although some authors have claimed that the equivalence between maximizing explanatory power and minimizing independent originations of similar features (i.e. the standard justification of parsimony) does not hold for inapplicable characters, evaluating homoplasy with x-transformations clearly connects the two sides of that equation. Furthermore, as the evaluation with x-transformations provides a direct count and a straightforward interpretation of homoplasy, it extends naturally into implied weighting, and sheds light on problems with additive, step-matrix or continuous characters. It also allows deriving transformation costs for recoding hierarchies as step-matrix characters (where recoded states correspond to permissible combinations of states in primary and secondary characters), so that homology of the original observations is properly measured. Those transformation costs set the cost of gaining the primary structure to the maximum difference between "present" states plus cost of loss, and difference between "present" states to the sum of user-defined transformation costs between secondary features. With such recoding, invoking multiple independent derivations of the structure and similar features will cost as many extra "steps" as the instances of similarities (in both original characters) that are not being homologized. The step-matrix recoding also can take into account nested dependences. We present a simple convention for naming characters, which TNT can use to automatically convert the original data into a step-matrix form and set the proper transformation costs. Finally, the basic elements for handling inapplicable characters in the context of maximum-likelihood inference are outlined, and some quantitative comparisons between different approaches to inapplicables are provided.