Robustness of the approximate likelihood of the protracted speciation model

The protracted speciation model presents a realistic and parsimonious explanation for the observed slowdown in lineage accumulation through time, by accounting for the fact that speciation takes time. A method to compute the likelihood for this model given a phylogeny is available and allows estimation of its parameters (rate of initiation of speciation, rate of completion of speciation and extinction rate) and statistical comparison of this model to other proposed models of diversification. However, this likelihood computation method makes an approximation of the protracted speciation model to be mathematically tractable: it sometimes counts fewer species than one would do from a biological perspective. This approximation may have large consequences for likelihood-based inferences: it may render any conclusions based on this method completely irrelevant. Here, we study to what extent this approximation affects parameter estimations. We simulated phylogenies from which we reconstructed the tree of extant species according to the original, biologically meaningful protracted speciation model and according to the approximation. We then compared the resulting parameter estimates. We found that the differences were larger for high values of extinction rates and small values of speciation-completion rates. Indeed, a long speciation-completion time and a high extinction rate promote the appearance of cases to which the approximation applies. However, surprisingly, the deviation introduced is largely negligible over the parameter space explored, suggesting that this approximate likelihood can be applied reliably in practice to estimate biologically relevant parameters under the original protracted speciation model.