Wright—Phylogeny of early to middle Paleozoic crinoids


younger taxa are considered. For example, Ordovician cyathocrine cladids are typically recovered as a clade (Ausich, 1998; Ausich et al., 2015) but are sometimes nested within amore inclusive clade of cyathocrines and hybocrinids when hybocrinids are sampled in the same analysis. Testing whether the other cyathocrine cladids belong to this clade remains an open question and requires sampling younger species. Similarly, Ordovician cladids placed within the Dendrocrinida (sensu Moore and Laudon, 1943) are paraphyletic, but there may nevertheless be latent phylogenetic structure present among subsets of post-Ordovician dendrocrines that could inform taxonomic revisions. The analyses conducted herein build on and further test


recently proposed phylogenetic hypotheses that sample only Ordovician crinoids (Guensburg, 2012; Ausich et al., 2015). This analysis includes a broad sample of early to middle Paleozoic (Ordovician–Devonian) non-camerate crinoids and primarily focuses on resolving relationships among the problem- atic Cladida.


Bayesian phylogenetics and the fossilized birth–death process


Bayesian phylogenetic methods combine a likelihood model of evolution with a set of prior probabilities to generate a posterior probability distribution of phylogenetic trees and their asso- ciated parameters. The Bayesian framework used herein is adapted from Gavryushkina et al. (2015). This model uses time- stamped morphologic character data to simultaneously estimate a posterior distribution of phylogenetic trees, divergence times, and other macroevolutionary and sampling parameters. Let Ψ be a phylogeny (i.e., tree topology with branch


lengths in units of time), δ be a vector of parameters describing morphologic evolution, and π be the tree prior (and its asso- ciated hyper parameters). Using Bayes theorem, the posterior probability distribution is


f Ψ; π; δjX; d


½=f ½XjΨ; δ f ½djΨ f Ψ jπ ½f ½π f ½δ


fX; d ½


where X is a character by taxon matrix of morphologic character data and d is a vector of age ranges for each fossil taxon. The numerator on the right-hand side of the equation can be


separated into the tree likelihood function, f ½XjΨ; δ, and the remaining terms, which comprises the prior. Equations for


below) and f ½djΨ is the density of obtaining fossil occurrence ranges given Ψ (this term is treated herein as a constant,


calculating f ½XjΨ; δ are well described in literature, and therefore a disquisition on tree likelihoods is not presented here. Interested readers are advised to see summaries in Swofford et al. (1996), Lewis (2001), Felsenstein (2004), and Yang (2014). The density f Ψj π


½ describes the tree prior (see


see Gavryushkina et al., 2015). The denominator f [X, d]is a normalizing constant and is equal to themarginal probability of the data. Given the necessary inputs, the posterior distribution of trees is estimated using a numerical technique called Markov chain Monte Carlo (MCMC) that eliminates the need to calculate f [X, d] when estimating the posterior distribution of trees. Agreat strength of the Bayesian paradigm is that sources of


uncertainty can be explicitly incorporated into the evolutionary model via the use of prior distributions on pertinent parameters


; (1)


801


(Heath and Moore, 2014). For example, numerous factors can influence and potentially distort the accuracy of reconstructed evolutionary trees—arguably the most important parameter in phylogenetic inference. Even in cases where these other factors are not of primary interest, acknowledging and estimating ‘nuisance parameters’ is nevertheless important because it reduces the chance that any particular incorrect assumption will lead to the recovery of specious tree topologies (Huelsenbeck et al., 2002; Wagner and Marcot, 2010; Gavryushkina et al., 2014). Potential biasing factors may include variation in rates of morphologic evolution, taxonomic diversification rates, ancestor–descendant relationships, and (incompletely) sampling taxa over time rather than from a single time slice (Smith, 1994; Wagner, 2000b, 2000c; Wagner and Marcot, 2010; Bapst, 2012). Variability in evolutionary rates can be modeled with prior distributions to describe rate variation among characters. Similarly, rate variation among lineages can be modeled using uncorrelated ‘relaxed clock’ models where branch-specific rates are independently drawn from the same underlying parametric distribution (Lepage et al., 2007; Heath and Moore, 2014). Bayesian inference weights the likelihood of a tree by its


prior probability. The fossilized birth–death (FBD) process (Stadler, 2010; Didier et al., 2012; Heath et al., 2014) is an extension of the constant rate birth–death models commonly used in paleontology (e.g., Raup et al., 1973; Raup, 1985) and considers fossil preservation in addition to diversification dynamics. In the following, I briefly describe the FBD process as a tree prior and argue it is well suited to accommodate these additional sources of concern.


Tree prior.—The fossilized birth–death (FBD) process is a stochastic branching model for describing macroevolutionary dynamics, fossil preservation, and sampling (Stadler, 2010;Heath et al., 2014). The FBD process begins at some time to > 0in the past and endswhen t = 0.As timemoves forward (i.e., decreasing toward the ‘present’), each lineage may probabilistically undergo one of three process-based events, each according to a distinct constant rate Poisson process: branching (i.e., lineage splitting via speciation) with rate p, extinction with rate q, or fossil preser- vation and samplingwith rate r (Stadler, 2010;Heath et al., 2014). Lineages alive at the end of the process are sampled with probability ε corresponding to the sampling fraction of ‘extant’ taxa. It is important to note that the start and end times for the FBD process are arbitrary, and time can therefore be shifted to accommodate temporal frameworks more commonly used in paleontology. For example, paleontological systematists working on entirely extinct groups (e.g., trilobites) or sampling taxa froma restricted temporal interval (e.g., Paleozoic crinoids) can shift time such that t = 0 corresponds to the age of the youngest species sampled. The FBD process represents a major advance over other birth–death models in paleontology (e.g., Raup, 1985) because fossil preservation and sampling issues aremodeled in addition to clade diversification. In the implementation ofGavryushkina et al. (2014), a lineage may be sampled more than once, thereby producing an internal node connected to only two (rather than three) branches.Atwo-degree internal node in a phylogenetic tree implies a hypothesized ancestor–descendent relationship, via direct or indirect ancestry (Fig. 1) (Foote, 1996; Gavryushkina et al., 2014, 2015).


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