13.5: Potential Pitfalls and How to Avoid Them
Recently, a few papers have been published that are critical of state-dependent diversification models (Rabosky and Goldberg 2015, Maddison and FitzJohn (2015) ) . These papers raise substantive critiques that are important to address when applying the methods described in this chapter to empirical data. In this section I will attempt to describe the critiques and their potential remedies.
The most serious limitation of state-dependent models as currently implemented is that they consider only a relatively small set of possible models. In particular, the approach we describe above compares two models: first, a model where birth and death rates are constant and do not depend on the state of the character; and second, a model where birth and death rates depend only on the character state (Maddison et al. 2007) . But there is another possibility that might be (in general) more common than either of the models we consider: birth and death rates vary, but in a way that is not dependent on the particular character we have chosen to analyze. I say that this is probably a common pattern because we know that birth and death rates vary tremendously across lineages in the tree of life (Alfaro et al. 2009) , and it seems probable to me that many of our hypotheses about which characters might contribute to that variation are, at this point, stabs in the dark.
This issue is a normal one for statistical analyses – after all, there are always other models outside of our set of considered possibilities. However, in this case, the fact that state-dependent diversification models fail to consider the possibility outlined above causes a very particular – and peculiar – problem: if we apply the tests to empirical phylogenetic trees, even with made-up data, we almost always find statistically significant results (Rabosky and Goldberg 2015) . For example, Rabosky and Goldberg (2015) found that there is very often a statistically significant “signal” that the number of letters in a species name is significantly associated with speciation rates across a range of empirical datasets. This result might seem ridiculous and puzzling, as there is no way that species name length should be associated with the diversification processes. However, if we return to our alternative model above, then the results make sense. Rabosky and Goldberg (2015) simulated character evolution on real phylogenetic trees, and their results do not hold when the trees are simulated along with the characters (this is also why Rabosky and Goldberg’s (2015) results do not represent “type I errors,” contra their paper, because the data are not simulated under the null hypothesis). On these real trees, speciation and/or extinction rates vary across clades. Among the two models that the authors consider, both are wrong; speciation and extinction are independent of the character but not constant through time. Of the two alternatives, the state-dependent model tends to fit better because, from a statistical point of view, it is important for the model to capture some variation in birth and death rates across clades. Even a random character will pick up some of this variation, so that the alternative model tends to fit better than the null – even though, in this case, the character has nothing to do with diversification!
Fortunately, there are a number of ways to deal with this problem. First, one can compare the statistical support for the state-dependent model with the support that one obtains for random data. The random data could be simulated on the tree, or one could permute the tips or draw random data from a multinomial distribution (Rabosky and Goldberg 2015) . One can then compare, for example, the distribution of Δ A I C c scores obtained from these permutations to the Δ A I C c for the original data. There are also semi-parametric methods based on permutations that have similar statistical properties (Rabosky and Goldberg 2017) . Alternatively, we could explicitly consider the possibility that some unmeasured character is actually the thing that is influencing diversification rates (Beaulieu and O’Meara 2016) . This latter approach is the most elegant as we can directly add the model described in this section to our list of candidates (see Beaulieu and O’Meara 2016) .
A more general critique of state-dependent models of diversification was raised by Maddison and Fitzjohn (Maddison and FitzJohn 2015) . This paper pointed out that statistically significant results for these tests can be driven by an event on a single branch of a tree, and therefore be unreplicated. This is a good criticism that applies equally well to a range of comparative methods. We can deal with this critique, in part, by making sure the events we test are replicated in our data. Together, both of these critiques argue for a stronger set of model adequacy approaches in comparative methods.