How reliably can we infer diversity-dependent diversification from phylogenies?

Slowdowns in lineage accumulation in phylogenies suggest that speciation rates decline as diversity increases. Likelihood methods have been developed to detect such diversity dependence. However, a thorough test of whether such approaches correctly infer diversity dependence is lacking. Here, we simulate phylogenetic branching under linear negative diversity-dependent and diversity-independent models and estimate from the simulated phylogenies the maximum-likelihood parameters for three different conditionings - on survival of the birth-death process given the crown age, on tree size (N) and on tree size given the crown age. We report the accuracy of recovering the simulation parameters and the reliability of the model selection based on the (2) likelihood ratio test. Parameter estimate accuracy: Conditioning on survival given the crown age yields a severe bias of the carrying capacity K towards N and an upward bias of the speciation rate, particularly in clades where diversity-dependent feedbacks are still weak (N