Resolving ubiquitous model congruence in phylogenetics and its application for studying macroevolution

A recent study (Louca and Pennell, 2020) spotlighted the issue of model congruence, or asymptotic unidentifiability, in time-dependent birth-death models used for reconstructing species diversification histories on phylogenetic trees. The present work investigates this issue in state-dependent speciation and extinction (SSE) models, commonly used to study trait-dependent diversification. We found that model unidentifiability is universal due to hidden states, with every SSE belonging to an infinite congruence class. Notably, any trait-independent model is congruent with trait-dependent models, raising concerns for hypothesis testing. To address this, we propose an analytical solution that resolves model selection within a congruence class. Our findings show that this type of congruence is the only one possible, and with our solution in place, model unidentifiability in SSEs becomes absolutely harmless for inference. However, model selection across congruence classes remains challenging due to extremely high false positive rates. The discovered congruence offers a clear explanation of this issue and suggests potential ways forward.