8.2: Fitting Mk models to Comparative Data
The equations in Chapter 7 give us enough information to calculate the likelihood for comparative data on a tree. To understand how this is done, we can first consider the simplest case, where we know the beginning state of a character, the branch length, and the end state. We can then apply the method across an entire tree using a pruning algorithm, which will allow calculation of the likelihood of the data given the model and phylogenetic tree.
Imagine that a two-state character changes from a state of 0 to a state of 1 sometime over a time interval of t = 3. What is the likelihood of these data under the Mk model? As we did in equation 7.17, we can set a rate parameter q = 0.5 to calculate a probability matrix:
\[ \mathbf{P}(t) = e^{\mathbf{Q} t} = exp( \begin{bmatrix} -0.5 & 0.5 \\ 0.5 & -0.5 \\ \end{bmatrix} \cdot 3) = \begin{bmatrix} 0.525 & 0.475 \\ 0.475 & 0.525 \\ \end{bmatrix} \label{8.1}\]
For this simple example, we started with state 0, so we look at the first row. Along this branch, we ended at state 1, so we should look specifically at p 12 ( t ): the probability of starting with state 0 and ending with state 1 over time t . This value is the probability of obtaining the data given the model (i.e. the likelihood): L = 0.475.
This likelihood applies to the evolutionary process along this single branch.
When we have comparative data the situation is more complex. If we knew the ancestral character states and states at every node in the tree, then calculation of the overall likelihood would be straightforward – we could just apply the approach above many times, once for each branch of the tree. However, there are two problems. First, we don’t know the starting state of the character at the root of the tree, and must treat that as an unknown. Second, we are modeling a process that is happening independently on many branches in a phylogenetic tree, and only observe the states at the end of these branches. All of the character states at internal nodes of the tree are unknown. The likelihood that we want to calculate has to be summed across all of these unknown character state possibilities on the internal branches of the tree.
Thankfully, Felsenstein (1973) provides an elegant algorithm for calculating the likelihoods for discrete characters on a tree. This algorithm, called Felsenstein’s pruning algorithm, is described with an example in the appendix to this chapter. Felsenstein’s pruning algorithm was important in the history of phylogenetics because it allowed scientists to efficiently calculate the likelihoods of comparative data given a tree and a model. One can then maximize that likelihood by changing model parameters (and perhaps also the topology and branch lengths of the tree; see Felsenstein 2004) .
Pruning also gives some insight into how we can calculate probabilities on trees; many other problems in comparative methods can be approached using different pruning algorithms.
Felsenstein’s pruning algorithm proceeds backwards in time from the tips to the root of the tree (see appendix, section 8.8). At the root, we must specify the probabilities of each character state in the common ancestor of the species in the clade. As mentioned in Chapter 7, there are at least three possible methods for doing this. First, one can assume that each state can occur at the root with equal probability. Second, one can assume that the states are drawn from their stationary distribution, as given by the model. The stationary distribution is a stable probability distribution of states that is reached by the model after a long amount of time. Third, one might have some information about the root state – perhaps from fossils, or information about character states in a set of outgroup taxa – that can be used to assign probabilities to the states. In practice, the first two of these methods are more common. In the case discussed above – an Mk model with all transition rates equal – the stationary distribution is one where all states are equally probable, so the first two methods are identical. In general, though, these three methods can give different results.