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Background  Phylogenetic comparative methods allow us to test evolutionary hypotheses without the benefit of an extensive fossil record. These methods, however, make simplifying assumptions, among them that clades are always increasing or stable in diversity, an assumption we know to be false. This study simulates hypothetical clades to test whether the Binary State Speciation and Extinction (BiSSE) method can be used to correctly detect relative differences in diversification rate between ancestral and derived character states even as net diversification rates are declining overall. We simulate clades with declining but positive diversification rates, as well those in which speciation rates decline below extinction rates so that they are losing richness for part of their history. We run these analyses both with simulated symmetric and asymmetric speciation rates to test whether BiSSE can be used to detect them correctly.    Results  For simulations with a neutral character, the fit for a BiSSE model with a neutral character is better than alternative models so long as net diversification rates remain positive. Once net diversification rates become negative, the BiSSE model with the greatest likelihood often has a non-neutral character, even though there is no such character in the simulation. BiSSE’s usefulness in detecting real asymmetry in speciation rates improves with clade age, even well after net diversification rates have become negative.    Conclusions  BiSSE is most useful in analyzing clades of intermediate age, before they have reached peak diversity and gone into decline. After this point, users of BiSSE risk incorrectly inferring differential evolutionary rates when none exist. Fortunately, most studies using BiSSE and similar models focus on rapid, recent diversifications, and are less likely to encounter the biases BiSSE models are subject to for older clades. For extant groups that were once more diverse than now, however, caution should be taken in inferring past diversification patterns without fossil data.    Electronic supplementary material  The online version of this article (10.1186/s12862-018-1174-5) contains supplementary material, which is available to authorized users.

Binary-state speciation and extinction method is conditionally robust to realistic violations of its assumptions