9.3: Pagel’s λ, δ, and κ
The three Pagel models discussed in Chapter 6 (Pagel 1999a,b) can also be applied to discrete characters. We do not create a phylogenetic variance-covariance matrix for species under an Mk model, so these three models can, in this case, only be interpreted in terms of transformations of the tree’s branch lengths. However, the meaning of each parameter is the same as in the continuous case:
- λ scales the tree from its original form to a “star” phylogeny, and thus quantifies whether the data fits a tree-based model or one where all species are independent;
- δ captures changes in the rate of trait evolution through time; and
- κ scales branch lengths between their original values and one, and mimics a speciational model of evolution (but only if all species are sampled and there has been no extinction).
Just as with discrete characters, the three Pagel models can be evaluated in either an ML / A I C c framework or using Bayesian analysis. One might expect these models to behave differently when applied to discrete rather than continuous characters, though. The main reason for this is that discrete characters, when they evolve rapidly, lose historical information surprisingly quickly. That means that models with high rates of character transitions will be quite similar to both models with low “phylogenetic signal” (i.e. λ = 0) and with rates that accelerate through time (i.e. δ > 0). This indicates potential problems with model identifiability, and warns us that we might not have good power to differentiate one model from another.
We can apply these three models to data on frog reproductive modes. But first, we should try the Mk and extended-Mk models. Doing so, we find the following results:
| Model | lnL | A I C c | Δ A I C c | AIC Weight |
|---|---|---|---|---|
| ER | -316.0 | 633.9 | 38.0 | 0.00 |
| SYM | -296.6 | 599.2 | 3.2 | 0.17 |
| ARD | -291.9 | 596.0 | 0.0 | 0.83 |
We can interpret this as strong evidence against the ER model, with ARD as the best, and weak support in favor of ARD over SYM. We can then try the three Pagel parameters. Since the support for SYM and ARD were similar, we will add the extra parameters to each of them. Doing so, we obtain:
| Model | Extra parameter | lnL | A I C c | Δ A I C c | AIC weight |
|---|---|---|---|---|---|
| ER | -316.0 | 633.9 | 38.0 | 0.00 | |
| SYM | -296.6 | 599.2 | 5.2 | 0.02 | |
| ARD | -291.9 | 596.0 | 0 | 0.37 | |
| SYM | λ | -296.6 | 601.2 | 5.2 | 0.02 |
| SYM | κ | -296.6 | 601.2 | 5.2 | 0.02 |
| SYM | δ | -295.6 | 599.2 | 3.2 | 0.07 |
| ARD | λ | -292.1 | 598.3 | 2.3 | 0.11 |
| ARD | κ | -291.3 | 596.9 | 0.9 | 0.24 |
| ARD | δ | -292.4 | 599.0 | 3.0 | 0.08 |
Notice that our results are somewhat ambiguous, with AIC weights spread fairly evenly across the three Pagel models. Interestingly, the overall lowest AIC score (and the most AIC weight, though only just more than 1/3 of the total) is on the ARD model with no additional Pagel parameters. I interpret this to mean that, for these data, the standard ARD model with no alterations is probably a reasonable fit to the data compared to the Pagel-style alternatives considered above, especially given the additional complexity of interpreting tree transformations in terms of evolutionary processes.