We know from analyses of tree balance that the tree of life is more imbalanced than birth-death models predict. We can explore this variation in diversification rates by allowing birth and death models to vary along branches in phylogenetic trees. The simplest scenario is when one has a particular prediction about diversification rates to test. For example, we might wonder if diversification rates in one clade (clade A in Figure 12.2) are higher than in the rest of the phylogenetic tree. We can test this hypothesis by fitting a multiple-rate birth-death model.
The simplest method to carry out this test is by using model selection in a ML framework (Rabosky et al. 2007). To do this, we first fit a constant-rates birth-death model to the entire tree, and calculate a likelihood. We can then fit variable-rates birth-death models to the data, comparing the fit of these models using either likelihood ratio tests or AICC. The simplest way to fit a variable-rates model is to adapt the likelihood formula in equation 11.18 (or eq. 11.24 if species are unsampled). We calculate the likelihood in two parts, one for the background part of the tree (with rates λB and μB) and one for the focal clade that may have different diversification dynamics (with rates λA and μA). We can then compare this model to one where speciation and extinction rates are constant through time.
Consider the example in Figure 12.2. We would like to know whether clade A has speciation and extinction rates, λA and μA, that differ from the background rates, λB and μB – we will call this a “variable rates” model. The alternative is a “constant rates” model where the entire clade has constant rate parameters λT and μT. These two models are nested, since the constant-rates model is a special case of the variable rates model where λT = λA = λB and μT = μA = μB. Calculating the likelihood for these two models is reasonably straightforward - we simply calculate the likelihood for each section of the tree using the relevant equation from Chapter 11, and then multiply the likelihoods from the two parts of the tree (or add the log-likelihoods) to get the overall likelihood.
For a real example, let’s look at the phylogenetic tree of amphibians and evaluate the hypothesis that the tailed and New Zealand frogs, sister clade to the rest of frogs, diversified at a slower rate than other amphibians (Figure 12.3). We can use the phylogenetic "backbone" tree from Jetz and Pyron (Jetz and Pyron 2018), assigning diversities based on the classification associated with that publication. We can then calculate likelihoods based on Equation 11.24.
We can calculate the likelihood of the constant rates model, with two parameters λT and μT, to a variable rates model with four parameters λliop, μliop, λother, and μother. For this example, we obtain the following results.
|Constant rates||λT = 0.30||-1053.9||2111.8|
|μT = 0.28|
|Variable rates||lambdaliop = 0.010||-1045.4||2101.1|
|μliop = 0.007|
|lambdaother = 0.29|
|μother = 0.27|
With a difference in AICc of more than 10, we see from these results that there is good reason to think that there is a difference in diversification rates in these "oddball" frogs compared to the rest of the amphibians.
Of course, more elaborate comparisons are possible. For example, one could compare the fit of four models, as follows: Model 1, constant rates; Model 2, speciation rate in clade A differs from the background; Model 3, extinction rate in clade A differs from the background; and Model 4, both speciation and extinction rates in clade A differ from the background. In this case, some of the pairs of models are nested – for example, Model 1 is a special case of Model 2, which is, in turn, a special case of Model 4 – but all four do not make a nested series. Here we benefit from using a model selection approach based on AICC. We can fit all four models and use their relative number of parameters to calculate AICC scores. We can then calculate AICC weights to evaluate the relative support for each of these four models. (As an aside, it might be difficult to differentiate among these four possibilities without a lot of data!)
But what if you do not have an a priori reason to predict differential diversification rates across clades? Or, what if the only reason you think one clade might have a different diversification rate than another is that it has more species? (Such reasoning is circular, and will wreak havoc with your analyses!) In such cases, we can use methods that allow us to fit general models where diversification rates are allowed to vary across clades in the tree. Available methods use stepwise AIC (MEDUSA, Alfaro et al. 2009; but see May and Moore 2016), or reversible-jump Bayesian MCMC (Rabosky 2014, 2017; but see Moore et al. 2016).
For example, running a stepwise-AIC algorithm on the amphibian data (Alfaro et al. 2009) results in a model with 11 different speciation and extinction regimes (Figure 12.4). This is good evidence that diversification rates have varied wildly through the history of amphibians.
One note: all current approaches fit a model where birth and death rates change at discrete times in the phylogenetic tree - that is, along certain branches in the tree leading to extant taxa. One might wish for an approach, then, that models such changes - using, for example, a Poisson process - and then locates the changes on the tree. However, we still lack the mathematics to solve for E(t) (e.g. Equation 11.19) under such a model (Moore et al. 2016). Given that, we can view current implementations of models where rates vary across clades as an approximation to the likelihood, and one that discounts the possibility of shifts in speciation and/or extinction rates among any clades that did not happen to survive until the present day (Rabosky 2017) - and we are stuck with that until a better alternative is developed!