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

Abstract

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 automat...

Early cephalopod evolution clarified through Bayesian phylogenetic inference

“…We made extensive use of contingent (or reductive) coding, which means that characters are hierarchically dependent on each other. Despite potential problems, this is currently the best available option for this study [ 90 – 92 ]. Although there has been recent progress in the development of methods dealing with inapplicable characters, most of these are not yet available for Bayesian inference, as they require different algorithms or coding strategies [ 92 – 94 ].…”

“…Despite potential problems, this is currently the best available option for this study [ 90 – 92 ]. Although there has been recent progress in the development of methods dealing with inapplicable characters, most of these are not yet available for Bayesian inference, as they require different algorithms or coding strategies [ 92 – 94 ]. Another recently developed alternative method using structured Markov models is computationally intensive for large hierarchies and needs further testing in empirical datasets [ 95 ].…”

New phylogenetic Markov models for inapplicable morphological characters

“…The ED has often been coded using inapplicable characters, which are then analyzed under parsimony or likelihood-based methods. In parsimony, the inapplicable coding was shown to be inappropriate (Maddison, 1993), which encouraged the development of new parsimony-based methods to handle the ED (Brazeau et al, 2019; Hopkins and St. John, 2021; Goloboff et al, 2021).…”

“…Most Markov models in phylogenetics treat characters as independent entities and, thereby, cannot be applied to ED. The study of Goloboff et al (2021) has recognized this fact that possibly led the authors to conclude that probabilistic treatment of ED requires new algorithms for likelihood calculation. They suggested that ED cannot be described with infinitesimal rate matrices ( Q ) alone, used for specifying MC evolution.…”

“…Recent publications have revitalized the interest in the long-standing problem of modeling inapplicable characters for phylogenetic analysis with discrete morphological data (Maddison, 1993; Hawkins et al, 1997). They offer new approaches for parsimony (Brazeau et al, 2019; Hopkins and St. John, 2021; Goloboff et al, 2021) and probabilistic methods (Tarasov, 2019). In morphology, character inapplicability refers to the hierarchical dependency of traits due to the anatomical dependency of their respective body parts.…”