On the Mkv Model with Among-Character Rate Variation

Models used in likelihood-based morphological phylogenetics often adapt molecular phylogenetics models to the specificities of morphological data. Such is the case for the widely used Mkv model-which introduces an acquisition bias correction for sampling only characters that are observed to be variable-and for models of among-character rate variation (ACRV), routinely applied by researchers to relax the equal-rates assumption of Mkv. However, the interaction between variable character acquisition bias and ACRV has never been explored before. We demonstrate that there are two distinct approaches to condition the likelihood on variable characters when there is ACRV, and we call them joint and marginal acquisition bias. Far from being just a trivial mathematical detail, we show that the way in which the variable character conditional likelihood is calculated results in different assumptions about how rate variation is distributed in morphological data sets. Simulations demonstrate that tree length and amount of ACRV in the data are systematically biased when conditioning on variable characters differently from how the data were simulated. Moreover, an empirical case study with extant and extinct taxa reveals a potential impact not only on the estimation of branch lengths, but also of phylogenetic relationships. We recommend the use of the marginal acquisition bias approach for morphological data sets modeled with ACRV. Finally, we urge developers of phylogenetic software to clarify which acquisition bias correction is implemented for both estimation and simulation, and we discuss the implications of our findings on modeling variable characters for the future of morphological phylogenetics.