Phylogenetic Analysis of Characters with Dependencies under Maximum Likelihood.

The dependencies between characters used in phylogenetic analysis (e.g., inapplicabilities, functional dependencies) can be taken into account by using combinations of character states as possible ancestral morphotypes, and using appropriate rates of transformation between such morphotypes. As every morphotype represents a permissible combination of the original character states, this allows easily ruling out specific combinations of character states, and taking into account changes that are either less or more likely to co-occur, or to occur in certain contexts. For inapplicable characters, Goloboff et al. (2021) used morphotypes but proposed obtaining transition probabilities between morphotypes from products of transition probabilities of the original characters and factors to incorporate dependencies. The product of transition probabilities is shown here to be flawed (failing the time-continuity requirement of phylogenetic Markov models, essential for statistical consistency under the model). Tarasov (2023) used the same delimitation of morphotypes but proposed obtaining transition probabilities from rate matrices, synthesized in a stepwise fashion from the hierarchy of dependencies. This paper shows that the rate matrices can easily be created, instead of with a stepwise synthesis, from direct comparisons between legitimate morphotypes (as done by Goloboff and De Laet 2023 for parsimony). Based on a few simple rules, the resulting rate matrices are (for inapplicable characters) identical to those obtained by Tarasov (2023). Additionally, in the computer program TNT, biological dependencies beyond mere inapplicability can be specified by the user with a simple syntax for (combinations of) states in "parent" characters restricting the states that "child" characters can take, using AND and OR conjunctions for elaborate interactions. These researcher-defined rules are used to internally convert the original characters into morphotypes, discarding morphotypes made impossible by the rules. In the case of biological dependencies (where, depending on the parent characters, there can be restrictions in the states that dependent characters can take, instead of the character being inapplicable), the rates of transition between morphotypes cannot be calculated solely from comparisons of states differing in both morphotypes -consideration of the conditions of dependency is needed as well.