We can fit an OU model to data in a similar way to how we fit BM models in the previous chapters. For any given parameters ($$\bar{z}_0$$, σ2, α, and θ) and a phylogenetic tree with branch lengths, one can calculate an expected vector of species means and a species variance-covariance matrix. One then uses the likelihood equation for a multivariate normal distribution to calculate the likelihood of this model. This likelihood can then be used for parameter estimation in either a ML or a Bayesian framework.
We can illustrate how this works by fitting an OU model to the mammal body size data that we have been discussing. Using ML, we obtain parameter estimates $$\hat{\bar{z}}_0 = 4.60$$, $$\hat{\sigma}^2 = 0.10$$, $$\hat{\alpha} = 0.0082$$, and $$\hat{\theta} = 4.60$$. This model has a lnL of -77.6, a little higher than BM, but an AICc score of 161.2, worse than BM. We still prefer Brownian motion for these data. Over many datasets, though, OU models fit better than Brownian motion (see Harmon et al. 2010; Pennell and Harmon 2013).