9.2: Beyond the Mk model
In Chapter 8, we considered the evolution of discrete characters on phylogenetic trees. These models fall under the general category of continuous-time Markov models, which consider a process that can occupy two or more states. Transitions occur between those states in continuous time. The Markov property means that, at some time t , what happens next in the model depends only on the current state of the process and not on anything that came before.
In evolutionary biology, the most detailed work on continuous time Markov models has focused on DNA or protein sequence data. As mentioned earlier, an extremely large set of models are available for modeling and analyzing these molecular sequences. One can also elaborate on these models by adding rate heterogeneity across sites (e.g. the gamma parameter, as in GTR + Γ ), or other complications related to mechanisms of sequence evolution (for a review, see Liò and Goldman 1998) .
However, there are two important differences between models of sequence evolution and models of character change on trees that make our task distinct from the task of modeling DNA or amino acid sequences. First, when analyzing molecular sequences, one typically has data for many thousands (or millions) of characters. Data sets for other characters – like the phenotypic characters of species – are typically much smaller (and harder to collect). Second, sequence analysis very often assumes that each character evolves independently from all other characters, but that all characters (or at least certain large subsets of those characters) evolve under a shared model (Liò and Goldman 1998; Yang 2006) . This means that, for example, the frequency of transitions between A and C at one location in a gene sequence contribute information about the same transition in a different location in the sequence.
Unfortunately, when analyzing morphological character evolution, we are often interested in single characters, and the use of shared models across characters seems impossible to justify. There is usually no equivalence between different character states for different characters: an A is an A for sequences, but a “1” in a character matrix usually corresponds to the presence of two completely different characters. The consequence of this difference is reflected in the statistical property of multivariate data. For gene sequence problems, adding more data in the form of additional characters (sites) makes model-fitting easier, as each site adds information about the overall (shared) model across sites. With character data, additional characters do not make the problem any easier, because each character comes with its own model parameters. In fact, we will see that when considering character correlations using a generalized Mk model, adding characters actually makes the problem more and more difficult. Perhaps these issues partially explain the slow pace of model development for fitting discrete characters to trees. There are a few potential solutions, such as threshold models [ Felsenstein (2005) ; Felsenstein (2012) ; discussed below]. More work is desperately needed in this area.
In this chapter, we will first discuss extensions of Mk models that allow us to add complexity to this simple model. We also discuss threshold models, a relatively new approach in comparative methods that is distinct from Mk models and has some potential for future development.