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Journal of Paleontology 91(4):799–814 Fossil tip-dating requires a model characterizing the
distribution of evolutionary rates throughout the tree. The case of a strict-morphological clock assumes rates are constant among lineages. However, the assumption of the strict-clock can be ‘relaxed’ by allowing rates to vary among branches throughout the tree (Lepage et al., 2007; Heath and Moore, 2014). To test whether evolutionary rates vary among lineages, strict- and relaxed-clock analyses were conducted. The inde- pendent gamma rates (IGR) relaxed-clock model was applied to account for variation in rates among branches. The IGR model facilitates episodic ‘white noise’ variation in rates across the tree and is appropriate because large-scale morphologic evolution is a function of both waiting times and stochastic selective forces (Wagner, 2012; Heath and Moore, 2014). A lognormal dis- tribution was placed on the base rate of the clock using methods outlined by Ronquist et al. (2012). A key assumption of tip-dating is that evolutionary change
is a function of time. In other words, geologically younger species are expected to have undergone a greater amount of within-lineage evolution (i.e., anagenesis) than older species because more time has elapsed for changes to occur (Smith et al., 1992; Wagner, 2000a). To test whether this assumption holds (and therefore whether the tip-dating method is valid for these data), the parsimony-based root-to-tip path length of each species from a non-clock analysis was regressed against median age dates from the IGR analysis using both phylogenetically corrected and uncorrected methods (Lee et al., 2014).
The FBD process was used as a tree prior (i.e., ‘samples-
trat = random’ in MrBayes 3.2.5). The implementation of the FBD in MrBayes reparameterizes the FBD process in terms of net diversification (= p – q), turnover (= q/p), and sampling probability (= r/(q+r). I placed an Exp(1) prior on net diver- sification, a Beta(1,1) uniform prior on turnover, and a Beta(2,2) prior on the sampling probability. To account for uncertainty in divergence time estimation, age ranges for fossil species were given broad uniform distributions typically corresponding to the stratigraphic range of their higher taxon and were taken from an updated version of Webster’s (2003) index of Paleozoic crinoids (Supplemental Data 2). Because the age of the most recent common ancestor of all species in the analysis is well con- strained by fossil evidence to be near the base of the Ordovician, the tree age prior was fixed to correspond to the earliest Tremadocian. Tip-dating is a computationally demanding phylogenetic
method that requires a time-consuming exploration of parameter space. To assist the analysis, several topological constraints were applied to reduce MCMC exploration of very unlikely trees and to test more specific phylogenetic hypotheses (see Guillerme and Cooper, 2016). For example, the monophyly of disparids and flexibles are well supported by other studies (Brower, 1995; Ausich, 1998; Guensburg, 2012; Ausich et al., 2015) and preliminary analyses. Thus, their status as clades is not in question. However, the branching position of these clades within the larger crinoid tree remains an open question, and their phylogenetic placement is evaluated herein. A partial constraint was placed on Eustenocrinus, Iocrinus, Ibexocrinus, and Heviacrinus that allowed for either Merocrinus and/or Alphacrinus to be included within the disparid clade if
the data support that hypothesis. This was done because whether Merocrinus is closer to cladids or disparids requires additional testing (cf. Ausich, 1998; Guensburg, 2012). Alphacrinus is a stratigraphically old taxon with a combination of unique and disparid-like traits that may or may not be stemward to ingroup Disparida (Guensburg, 2010). A hard constraint was placed on a flexible clade comprising Homalocrinus, Icthyocrinus, Lecanocrinus, Protaxocrinus, and Sagenocrinites. Markov chain Monte Carlo analyses consisted of two
independent runs of four chains sampling every 4,000 generations for 40 million generations per run with a burn-in percentage of 35%. Convergence was assessed using multiple criteria: average standard deviation of split frequencies among chains were below 0.01 (<0.05 for some IGR analyses) (Gelman and Rubin, 1992), potential scale reduction factors of ~1.0 (Lakner et al., 2008), effective sample sizes greater than 300 (with many >1,000), and visual inspection of log-likelihood plots among runs using Tracer v.1.6 (Drummond and Rambaut, 2007). Finally, the analysis with the best fit parameter settings was repeated three times to ensure estimates of optimal tree topologies were robust across runs. Together, these diagnostics indicate convergence among tree topologies and parameter estimates. To summarize the posterior distribution of tree topologies,
I generated a maximum clade credibility (MCC) tree using TreeAnnotator (Rambaut and Drummond, 2015). Although there is no single agreed-upon method for summarizing Bayesian posterior distributions of phylogenetic trees (Heled and Bouckaert, 2013), posterior probability can be viewed as an optimality criterion in phylogenetic inference (Rannala and Yang, 1996; Huelsenbeck et al., 2002; Wheeler and Pickett, 2007; Wheeler, 2012; Rambaut, 2014). TheMCCtree is the tree in the posterior distribution with the maximum product of clade posterior probabilities and represents a Bayesian point estimate of phylogeny (Rambaut, 2014). I also ran a series of sensitivity analyses (n>5) to explore
the effects of choosing different priors, including the prior placed: (1) on the variance of the gamma distribution in the IGR model, and (2) on the FBD turnover (= q/p) parameter. In all cases, statistically indistinguishable median estimates were obtained for node ages, branch lengths, and FDB parameters. In addition, I calculated the pairwise Robinson-Foulds (RF) distance (Robinson and Foulds, 1981) within and among tree distributions from separate analyses and ordinated the resulting RF matrix using principal coordinate analysis. A visual inspection of RF distances in principal coordinate space reveals substantial overlap between distributions, with no obvious gradient or isolated ‘islands’ (analogous to Maddison, 1991) of trees. Thus, the analysis presented herein is considered robust across a range of possible prior configurations.
Results
The relationship between within-lineage morphologic evolution and IGR age estimates indicates early to middle Paleozoic crinoids conform well to the assumptions of tip-dating. Parsimony-based branch lengths from a non-clock analysis reveals geologically younger taxa have higher amounts of anagenesis compared to older taxa (p =0.017) (Fig. 3). This relationship holds even when
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