Markov Maximum Likelihood ancestral states inference
We inferred ancestral states using the rayDISC function in the R package corHMM (Beaulieu, O’Meara, & Donoghue, 2013), with node.states=”marginal”, state.recon=”subsequently” and root.p=”maddfitz”. This algorithm allows multistate tip input, and infers marginal likelihoods (probabilities) for each state at each node including tips, the latter being in effect an implied state of common ancestor of the multistate tip (Felsenstein 2004, p. 255). Options for interpreting this post-analysis tip state marginal are to i) use these values or ii) to recode as equal probabilities to partially account for terminal dispersals within the tip lineage, or iii) explicitly count all implied terminal dispersals. In our eight-region scheme 86% of species are endemic to a single region, hence the moderate proportion (14%) of multistate species is a tolerable level for this type of ancestral state method to accommodate.
In order to select an optimal ML model structure, we applied an approximation intended to achieve a practical fair evaluation of what can be a prohibitively large number of possible rate models (Huelsenbecket al. , 2003; Felsenstein, 2004). Using the single consensus tree, an average of ER, SYM and ARD model rates (in order to buffer against potentially miss-specified extreme values) were rank ordered into a serially increasing number of rate category models from the simplest single rate (ER), evaluated by BIC (Table S5). We selected a four-rate model structure representing a balance of complexity and support, which was then used for all subsequent analyses (Table S6).
Rather than produce a single best ancestral state result, we focussed on broad biogeographical inferences summarizing both model ancestral state and phylogenetic uncertainty. To do this we applied stochastic node state mapping using 200 re-samplings of node state marginal likelihoods, to each of 100 BEAST tree samples. Randomly sampling node states according to marginal likelihoods selects an explicit node state – multiple resampling then fairly represents the marginal likelihood (Huelsenbeck et al. , 2003; Revell, 2012; O’Hara et al. , 2019). Doing this for a set of tree samples then includes phylogenetic uncertainty in the whole inference.
Transition events are summarized according to parent to daughter node (including tip) states; cases where the state remains the same are referred to as endemic cladogenesis. These results can be divided up by divergence age into time-bins, assigned by daughter node age. For counting state lineages, branches spanning a time point can be assigned to closest parent or daughter node state. These procedures for summarizing an explicitly resolved set of ancestral states are straightforward (Revell, 2012) and can be applied to any ancestral state method including DEC models (Bribiesca-Contreras et al. , 2019).
This entire procedure can be repeated on multiple trees and the entire set of transition events and state lineages per time-bin combined into a final matrix integrating the tree and model variation (Bribiesca-Contreras et al. , 2019). Such summary results of events per time-bin are best interpreted as a summary of the relative frequency or probability density of such events rather than an explicit count. Results can further be integrated by post-hoc combining several individual states and changes into larger summary categories, such as continental versus islands. By summarizing the results across the whole marginal likelihood the method accounts for multiple solutions and hence in effect to some degree accounts for multistate solutions, particularly for combined region summaries.
Analyses were executed, and summarized into 2 million year time-bin profile plots, in R using a custom script ASSMR2. For phylogenetic visualizations onto the single MCC consensus tree, we averaged marginal likelihoods across all 100 sample trees plus consensus tree, for all nodes in common with the consensus tree (identical taxon bipartitions) and the linked parent node. Nodes (hence branches) with maximum state probability <67% were then marked as indecisive. Except for inclusion of the parent node, these are often used procedures (Matzke, 2013; O’Hara et al. , 2019).