A Bayesian approach for evaluating the impact of historical events on rates of diversification

Abstract

Evolutionary biologists often wish to explore the impact of a particular historical event (e.g., the origin of a novel morphological trait, an episode of biogeographic dispersal, or the onset of an ecological association) on rates of diversification (speciation minus extinction). We describe a Bayesian approach for evaluating the correlation between such events and differential rates of diversification that relies on cross-validation predictive densities. This approach exploits estimates of the marginal posterior probability for the rate of diversification (in the unaffected part of the tree) and the marginal probability for the timing of the event to generate a predictive distribution of species diversity that would be expected had the event not occurred. The realized species diversity can then be compared to this predictive diversity distribution to assess whether rates of diversification associated with the event are significantly higher or lower than expected. Although simple, this Bayesian approach provides a robust inference framework that accommodates various sources of uncertainty, including error associated with estimates of divergence times, diversification-rate parameters, and event history. Furthermore, the proposed approach is relatively flexible, allowing exploration of various types of events (including changes in discrete morphological traits, episodes of biogeographic movement, etc.) under both hypothesis-testing and data-exploration inference scenarios. Importantly, the cross-validation predictive densities approach facilitates evaluation of both replicated and unique historical events. We demonstrate this approach with empirical examples concerning the impact of morphological and biogeographic events on rates of diversification in Adoxaceae and Lupinus, respectively.

Fig. 1.

Evaluating correlates of differential diversification rates using cross-validation predictive densities. Step 1 involves estimating the joint posterior probability density of the phylogeny and divergence times from the entire data set of nucleotide sequences. In step 2, the history of the event is estimated from the resulting posterior probability distribution of trees. Step 3 entails constructing the training data partition by removing the subset of species associated with the event and then subjecting this reduced data set to a second round of phylogeny/divergence-time estimation. Finally, we draw from the marginal posterior densities of t_k and λ_(k) (estimated in steps 2 and 3, respectively) to generate the cross-validation predictive diversity density (using Eq. 3), which permits us to assess whether the realized species diversity is significantly higher or lower than predicted (using Eq. 4).

Fig. 2.

Dated phylogeny for Adoxaceae based on a Bayesian analysis of the combined data set. Uncertainty in the tree topology and divergence times are indicated by bars on internal nodes: their length corresponds to the 95% highest posterior density (HPD) of node ages, and their shading reflects the posterior probabilities of nodes (shaded bars, nodal posterior probabilities ≥0.90; open bars, those <0.90). Circles adjacent to the tips are shaded to indicate the observed fruit type in the respective species, and branches of the tree have been shaded to reflect the posterior probability estimates for ancestral states. Numbered internal nodes correspond to the fossil constraints used to estimate divergence times (Table S2) and/or to internal nodes for which the posterior probabilities of the ancestral fruit morphologies were inferred. The inset histograms depict the estimated posterior probabilities of the ancestral trait values for the corresponding labeled nodes, which indicate that single-seeded fruits arose once along the branch subtending Sinadoxa corydalifolia and once along the branch subtending node 2.

Fig. 3.

Estimated correlation of fruit morphology with diversification rates in Adoxaceae. The graphs summarize the inferred correlation between diversification rate and the two instances of single-seeded fruits: each histogram depicts the predictive distribution of single-seeded species diversity, and the vertical line indicates the observed species diversity, with the corresponding posterior-predictive probability. These tests indicate that single-seeded fruits are correlated with significantly increased rates of diversification in Viburnum relative to background rates of diversification in Adoxaceae, but not in Sinadoxa corydalifolia.

Fig. 4.

Dated phylogeny for Lupinus based on a Bayesian analysis of the combined data set. Uncertainty in the tree topology and divergence times are indicated by bars on internal nodes: their length corresponds to the 95% highest posterior density (HPD) of node ages, and their shading reflects the posterior probabilities of nodes (shaded bars, nodal posterior probabilities ≥0.90; open bars, those <0.90). The numbered internal node corresponds to the fossil constraint used to estimate divergence times (Table S2). Observed species ranges are indicated across the tips of the tree, and the diagrams adjacent to internal nodes indicate the inferred locations of biogeographic dispersal events, including 2 into South America.

Fig. 5.

Estimated correlation between biogeographic dispersal into South America and rates of diversification in Lupinus. Each histogram depicts the predictive distribution of South American species diversity, and the vertical line indicates the observed species diversity, with the corresponding posterior-predictive probability. These results indicate that the Andean clade experienced significantly increased rates of diversification relative to background rates of diversification in Lupinus, whereas the other South American lineage did not.