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How can we improve accuracy of macroevolutionary rate estimates?

Nee et al. (1994) presented likelihood equations for estimating speciation and extinction rates based on phylogenies of only extant species; in particular their method can infer extinction patterns without extinct species data. Meanwhile, even for the simplest model of speciation and extinction, namely, the constant rate birth–death process, a number of studies have been published using different likelihood equations (Thompson 1975; Rannala and Yang 1996; Yang and Rannala 1997; Gernhard 2008; Stadler 2009). The likelihood functions differ due to conditioning the likelihood on different quantities, like the age of the tree, survival of the tree, or the number of species in the tree. Which conditionings yield the most accurate speciation and extinction rate estimates? In order to answer this question, I present an overview of 7 likelihood functions (which have been published in previous articles), conditioning on different aspects of the tree. I investigate and discuss the impact of the different conditionings toward accuracy of the maximum-likelihood rate estimates by inferring rates based on simulated phylogenies. The second part of this article discusses a possible bias in speciation and extinction rate estimates when analyzing incomplete phylogenies, that is, phylogenies in which not all extant species are included. The analytic considerations reveal that we cannot estimate the fraction of nonsampled species, but have to know it, when estimating speciation and extinction rates. The conclusions reached in this article, assuming the simple constant rate birth–death model, will also apply when assuming the more realistic macroevolutionary models allowing for nonconstant rates (Rabosky 2007; Alfaro et al. 2009; FitzJohn et al. 2009; Morlon et al. 2011; Stadler 2011a; Silvestro et al. 2011; Etienne et al. 2012), as these general models all contain the constant rate birth– death model as a special case. This article ends with contrasting these different method implementations (Table 1) and providing some recommendations for end users in order to facilitate model comparison across packages. SEVEN TREE LIKELIHOOD FUNCTIONS