a novel comparative method for identifying shifts in the rate of
TRANSCRIPT
ORIGINAL ARTICLE
doi:10.1111/j.1558-5646.2011.01401.x
A NOVEL COMPARATIVE METHOD FORIDENTIFYING SHIFTS IN THE RATE OFCHARACTER EVOLUTION ON TREESJonathan M. Eastman,1,2 Michael E. Alfaro,3,4 Paul Joyce,5,6 Andrew L. Hipp,7,8,9 and Luke J. Harmon10,11
1Department of Biological Sciences, Institute for Bioinformatics and Evolutionary Studies (IBEST), University of Idaho,
Moscow, Idaho 83843-30512E-mail: [email protected]
3Department of Ecology and Evolutionary Biology, University of California, Los Angeles, Los Angeles, California 900954E-mail: [email protected]
5Departments of Statistics and Mathematics, University of Idaho, Moscow, Idaho 83843-30516E-mail: [email protected]
7The Morton Arboretum, 4100 Illinois Route 53, Lisle, Illinois 605328The Field Museum of Natural History, 1400 South Lake Shore Drive, Chicago, Illinois 60605
9E-mail: [email protected] of Biological Sciences, University of Idaho, Moscow, Idaho 83843-3051
11E-mail: [email protected]
Received February 11, 2011
Accepted June 19, 2011
Evolutionary biologists since Darwin have been fascinated by differences in the rate of trait-evolutionary change across lineages.
Despite this continued interest, we still lack methods for identifying shifts in evolutionary rates on the growing tree of life while
accommodating uncertainty in the evolutionary process. Here we introduce a Bayesian approach for identifying complex patterns
in the evolution of continuous traits. The method (auteur) uses reversible-jump Markov chain Monte Carlo sampling to more fully
characterize the complexity of trait evolution, considering models that range in complexity from those with a single global rate
to potentially ones in which each branch in the tree has its own independent rate. This newly introduced approach performs well
in recovering simulated rate shifts and simulated rates for datasets nearing the size typical for comparative phylogenetic study
(i.e., ≥64 tips). Analysis of two large empirical datasets of vertebrate body size reveal overwhelming support for multiple-rate
models of evolution, and we observe exceptionally high rates of body-size evolution in a group of emydid turtles relative to their
evolutionary background. auteur will facilitate identification of exceptional evolutionary dynamics, essential to the study of both
adaptive radiation and stasis.
KEY WORDS: Bayesian inference, Chelonia, comparative methods, primates,rate heterogeneity, reversible-jump MCMC, trait
evolution.
Explaining and identifying variation in rates of evolution among
lineages has long been a focus for evolutionary biologists (e.g.,
Darwin 1859; Simpson 1953; Gingerich 1983; Eldredge and Stan-
ley 1984; Estes and Arnold 2007). Simpson (1944, 1953) coined
a set of terms for the very purpose of distinguishing exceptional
rates of evolution in particular lineages (i.e., tachytely, bradtely,
horotely). Current approaches require that shifts in the evolution-
ary process are identified a priori (e.g., Butler and King 2004;
O’Meara et al. 2006; Revell and Collar 2009). What these and
related methods (Harmon et al. 2003; Freckleton and Jetz 2009)
1C© 2011 The Author(s).Evolution
J. M. EASTMAN ET AL.
lack is the acknowledgment and accommodation of uncertainty in
the evolutionary processes that give rise to trait values observed
in extant taxa (but see Revell et al., in revision). The quality of
inference is bounded by the quality of the models that we apply in
the comparative framework: even the best among a set of inade-
quate models is still inadequate. Relaxing the assumption of static
points in the tree where rates have shifted should diminish bias
in branchwise estimates of evolutionary rates. Most ideal would
be methods that limit influence of a priori expectations, while
fairly comparing fit among the largest possible set of models and
allowing robust inference of rate variation.
Drawing on recent progress in modeling molecular evolu-
tion (e.g., Huelsenbeck et al. 2004; Pagel and Meade 2008;
Drummond and Suchard 2010), we model trait evolution by al-
lowing phylogenetically localized shifts in the rate of a random-
walk process of trait change. In a novel Bayesian approach, here
referred to as AUTEUR, we allow the data to directly inform uncer-
tainty in the estimated evolutionary process. AUTEUR (Accommo-
dating Uncertainty in Trait Evolution Using R) samples across a
broad set of possible trait evolution scenarios, considering mod-
els differing in number and topological position of local rate
shifts.
We model trait evolution as a Brownian-motion process,
which describes a broad set of neutral and nonneutral models
of phenotypic evolution (Felsenstein 1973), and may be a reason-
ably adequate approximation of the evolutionary process in some
lineages (Harmon et al. 2010). Under the modeled random-walk
process, the trajectory of trait evolution (magnitude and direction-
ality) is independent of the current state of the character. Whereas
the expected value at the end of a random walk is simply the start-
ing value, variance in traits accumulates in proportion to both the
extent of independent evolution in lineages and the evolutionary
rate of the character (see Felsenstein 1973; O’Meara et al. 2006;
Revell et al. 2008). We thus expect little trait variance between
sister species who have just diverged, especially for a slowly
evolving trait. Its mathematical tractability makes Brownian mo-
tion an ideal framework in which to develop the model-fitting
approach described here.
AUTEUR applies a Bayesian approach to modeling rate het-
erogeneity on a phylogenetic tree using reversible-jump Markov
Chain Monte Carlo (Metropolis et al. 1953; Hastings 1970). This
reversible-jump approach is implemented to assess fit of models
of differing complexity, which in this context is the number of rate
shifts in the tree (Green 1995; Huelsenbeck et al. 2004; Drum-
mond and Suchard 2010). We construct a Markov chain such that,
upon convergence, distinct Brownian-motion models are sampled
according to their posterior probability (Bartolucci et al. 2006).
Of main interest, and inherently tied to the set of sampled models,
are the marginalized distributions of relative rates for each branch
in the tree.
General Approach: BayesianSampling of a MultirateBrownian-Motion ProcessAUTEUR takes as input a phylogenetic tree and character states
for sampled species. Trait data are assumed to have evolved by
Brownian motion, and evolutionary rates are presumed to be phy-
logenetically heritable. That is, the ancestral value for the evolu-
tionary rate of the trait is inherited in each descendant unless the
data provide evidence for a rate shift (see below). AUTEUR allows
exploration from the simplest Brownian-motion process (a global
evolutionary rate) to a highly complex model where each branch
evolves at a rate independent from all others (i.e., the “free model”
of Mooers et al. 1999).
In the following paragraphs, we detail the mechanics of
our Bayesian approach. Consider an ultrametric phylogenetic
tree with N species. Excluding the stem, this tree will have
2(N − 1) branch lengths, which we will refer to as �b =[b1, b2, . . . , b2(N−1)], listed in some arbitrary order. We will
call the vector of tip phenotypes �y = [y1, y2, . . . , yN ]. Under
a Brownian-motion model, these tip phenotypes come from a
multivariate normal distribution (Felsenstein 1973). In the case
where rates across every branch in the phylogeny are equal, this
distribution will have mean α and variance–covariance matrix
S = [si j ] = σ2[ci j ], where α is the state at the root of the tree and
σ2 is the rate parameter for the Brownian-motion process. Each
cij is the shared phylogenetic path length for species i and j. We
note that each sij (i.e., an element within the variance-covariance
matrix) can be written as the rate-scaled shared path length for
species i and j. Thus, si j = ∑2(N−1)k=1 σ2akbk , where ak = 1 if
branch k is shared by both i and j in the path from the root to each
species and 0 otherwise; bk is simply the length of the kth branch
as indicated above.
We will consider a set of models where characters evolve un-
der a Brownian-motion model, but the rate of evolution might vary
on each branch of the tree. As this model is highly dimensional,
with 2(N − 1) − N = N − 2 more parameters than trait val-
ues, we use reversible-jump Markov chain Monte Carlo sampling
for efficient searching of this space, and our prior is constructed
so to favor models with few distinct rate classes. We will spec-
ify branch-specific relative rates as �σ2 = [σ21, σ
22, . . . , σ
22(N−1)], or-
dered as the vector of branch lengths, �b. In this case, characters will
still follow a multivariate normal distribution, but the elements of
the variance–covariance matrix, S, will be si j = ∑2(N−1)k=1 σ2
kakbk ,
using each branch-specific σ2k to scale the path lengths. The set of
evolutionary rates scalars, �σ2, thus bears on the stochastic variance
of trait values expected to accumulate within the tree.
Standard proposal mechanisms, described elsewhere (see,
e.g., Green 1995), are used to update parameters in the model. We
use multiplier and sliding-window proposals to update the vector
2 EVOLUTION 2011
IDENTIFYING RATE HETEROGENEITY IN CHARACTER EVOLUTION
of relative rates; sliding-window proposals are used to update the
root state. A critical proposal mechanism in our reversible-jump
Markov chain is the splitting or merging of relative rates across
the tree (see also Huelsenbeck et al. 2004). Steps in the chain that
involve this “split-or-merge” operation randomly choose a single
branch. If the relative rate of that branch is currently equivalent
to the relative rate of its immediate ancestor, a new local rate is
proposed for the descendant branches of the selected branch. This
new rate cascades through the descendants of that branch until
descendant tips are reached or other local rates, different from the
new rate, are encountered. Merge operations perform the reverse:
if in the current state of the Markov chain the immediate ancestral
relative rate differs from that of the randomly chosen branch, these
rates may be combined into one by the merge move. Both merge
and split operations are conducted such that the mean of relative
rates is preserved (Green 1995; Huelsenbeck et al. 2004).
The Markov chain is a succession of outcomes of model
comparisons, where the adoption of a particular model in each
state of the chain is driven by the posterior odds of that model
in light of the observed data. A newly proposed model, θ′, is
formulated for each state by updating some facet of the current
state (e.g., number of shifts, or values for relative rates), which is
compared to the current state of the chain, θ(t). In broadest terms,
the acceptance probability of the newly proposed state given the
current state is,
θ(t+1) =
⎧⎪⎨⎪⎩
θ′ with probability: min
{1,
P(θ′ | �y)
P(θ(t) | �y)
Q(θ(t) | θ′)Q(θ′ | θ(t))
},
θ(t) otherwise.(1)
Proposed models are thus adopted in inverse proportion to
the poorness of the model fit. This not only ensures that proposed
models with higher posterior odds will always be accepted, but
also enables the chain to adequately explore model space with-
out being confined to local optima of posterior support. The first
term of the acceptance probability involves the ratio of posterior
densities of the proposed and current states; the second term de-
scribes the transition kernel between adjacent states of the chain.
Beginning with the first term of the acceptance probability from
above,
P(θ′ | �y)
P(θ(t) | �y)= P(θ′)
P(θ(t))· P(�y | θ′)
P(�y | θ(t)). (2)
This term comprises the ratio of prior odds, P(θ′)P(θ(t)) multiplied by
the likelihood ratio. An important factor in Bayesian analysis is
the set of priors used in model comparisons. We describe details
of formulating the ratio of prior probabilities in the section to
follow.
The second term of the acceptance probability involves the
Hastings ratio, which is the ratio of probabilities of traversing
either direction between proposed and current states. If state-
traversal asymmetries exist in the proposal kernel, we thus need
to correct for the potential of biased sampling of the chain
(Hastings 1970). For instance, if in the current state of the chain
a single rate governs evolution across the tree, only proposals
that split the global rate into two independent rates are permis-
sible. This constraint would serve to bias the transition probabil-
ity toward higher dimension models. Analogously, if the current
state involves a unique rate for each branch in the tree, no fur-
ther rate shifts would be possible. We implement Green’s (1995)
method for calculating the correction factor (i.e., Hastings ratio),
which is
Q(θ(t) | θ′)Q(θ′ | θ(t))
= Q(K | K ′)Q(K ′ | K )
=
⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
K + 1
2n − 2 − Kif K ′ = K + 1,
2n − 2 − K + 1
Kif K ′ = K − 1,
1 if K ′ = K .
(3)
As shown, the proposal kernel is rewritten in terms of the number
of parameters in the models, K and K′, which describe dimension-
ality, or number of relative-rate shifts, in each model. Where com-
pared models do not differ in number of parameters and where
transitions between current and proposed states are symmetric,
the Hastings ratio reduces to one and does not contribute to the
posterior odds of the proposed model.
PROPOSALS AND PRIORS
Using the sliding-window proposal mechanism, proposal den-
sities for σ2k and on the state at the root, y0, are symmetric
between current and proposed values. Proposed values that ex-
ceed bounds (e.g., a negative relative-rate parameter) are simply
reflected back into the allowable range. The proposal kernels
are: σ2′k ∼ σ2
k + U (−ν, ν) and y′0 ∼ y0 + U( − ν, ν), where ν
is a proposal width and U is a uniform distribution. Priors on
these parameters are uninformative and span the following ranges:
σ2k ∼ U[0, ∞) and y0 ∼ U( − ∞, ∞) (see Schluter et al. 1997).
A multiplier proposal mechanism is also used to update σ2k and
may be asymmetric; this asymmetry is accommodated by the
Hastings ratio. The multiplier proposal draws new rates from
σ2′k ∼ σ2
k · expλ(u−0.5) , where u ∼ U(0, 1) and where λ = 2log (δ)
and δ is either the proposal width from above (if ν ≥ 1) or its re-
ciprocal (if ν < 1). The Hastings ratio for this multiplier proposal
is simply u.
We place high prior weight on few shifts in the evolutionary
process. Using uninformative priors for the other parameters, we
only need to consider the ratio of prior probabilities of having
K and K′ rate shifts in the phylogeny, computed using a Poisson
distribution that is truncated beyond the maximum number of rate
EVOLUTION 2011 3
J. M. EASTMAN ET AL.
shifts for a given tree (Kmax = 2n − 2; Drummond and Suchard
2010). For chain states that have at least one rate shift in the tree,
we introduce a proposal mechanism that locally moves the shift
one branch, with equiprobable chances of the shift moving tip-
ward or rootward. Because in a binary tree, there are always two
descendants at each node, this proposal is as well asymmetric: our
proposal mechanism disfavors moving toward the more tipward
branch. When used, we account for the asymmetric probabili-
ties of moving to and from a particular branch by the Hastings
ratio.
PROPOSAL-WIDTH CALIBRATION
To achieve reasonable rates of mixing when Markov chain sam-
pling occurs, an initial sampling period is used to calibrate the
proposal width (ν from above). An initial geometric progression
of widths (from 18 , 1
4 , . . . , 8) is used in these calibrations. Using
the maximum-likelihood estimate for a global-rate model as a
starting point, we compute a weighted average of proposal widths
to initiate sampling where weights are based on acceptance rates
for 1000 proposals under each proposal width. Thereafter, one-
tenth of the total run length is used to adjust the initially chosen
proposal width in the geometric progression. Updates to the pro-
posal width occur at progressively longer intervals in the recali-
bration period (e.g., at generation 101, 102..., 105 for a total run
length of 106 generations). For each recalibration, 1000 proposals
are again considered for a pair of adjusted proposal widths, which
are one-half and twice the current proposal width. As previously,
proposal width is readjusted using acceptance rates as weights
for sampling conducted with the pair of newly proposed widths.
When Markov chain sampling begins, the proposal width is con-
strained to the calibrated value, and samples from the calibration
period are obligatorily excluded from our estimate of the posterior
distribution of models.
MethodsEVALUATING STATISTICAL PROPERTIES BY SIMULATION
We use two general classes of simulations to investigate the prop-
erties of this method: under a global rate and under a heteroge-
neous process of evolution in which a single shift in relative rate
occurs. Within each class of simulation, we assess efficiency of
the method by comparing true to estimated parameters under a
range of tree sizes (16, 32, 64, and 128 taxa); second, we mea-
sure sensitivity by using a range of evolutionary rates across trees
(8-, 16-, 32-, and 64-fold reductions of a unit rate). Trees were
generated by the TREESIM package (version 1.3, Stadler 2010)
under a pure-birth model of diversification and with a stochastic
speciation rate of one new lineage per time unit. For multiple-
rate simulations, we confined tree space to those trees that bore
at least one clade with half the number of total tips in the tree
(i.e., clades with 8, 16, 32, or 64 tips). We positioned the single
rate shift to occur at the base of a selected subclade of appropri-
ate size: the ancestral rate (σ2A) was 1.0 and the rate descending
from the shifted node drew from one of four factor reductions
of this ancestral rate: σ2S ∈ (
18 , 1
16 , 132 , 1
64
). The sequence of tree
sizes and this latter set of reduced rates were used for simulations
conducted under a constant and global rate. Code from GEIGER
(version 1.0, Harmon et al. 2008) was used to simulate trait data
for all simulations. For simulations investigating the effect of
clade size, a shifted rate (σ2S) of 1
16 was used. We used a tree size
of 64 species for simulations assessing the influence of magnitude
of rate differences. Chain sampling in our simulations occurred
over 106 generations, of which the first half was discarded as a
burn-in.
We use several measures to evaluate performance and statis-
tical properties of the method, for both global-rate and multiple
rate simulations. We summarize simulations by evaluating poste-
rior probabilities of the true complexity of the generating model
as well as the posterior probability of a model at least as com-
plex as the simulated process. We consider error in estimated
relative rates in a branchwise fashion, where we weight the pro-
portion of the estimated rate to the true rate by the length of each
branch. The average of these weighted branchwise proportions is
used as a summary of error across the entire tree. For instance,
if each relative rate, σ2k , is twice the true rate at each correspond-
ing branch, the tree-wide summary of the proportion of the true
rate would also be two. For simulations involving a shift in evo-
lutionary rate, we further assess rate-shift error by determining
the inferred posterior probability of a shift occurring at the truly
rate-shifted branch. Retaining every hundredth sample from the
Markov chain, we use a thinned posterior sample in evaluating
each measure of performance.
EMPIRICAL EXAMPLES
We investigate support for a heterogenous process in evolution
for a set of reasonably large datasets: body size for turtles (see
Jaffe et al. 2011; 226 species) and female body mass for primates
(see Redding et al. 2010; complete sampling of 233 species).
We use log-transformed measures of body size for each dataset.
Straight-line carapace lengths in Chelonia span well over an or-
der of magnitude. If we linearize by the cube root, raw body
masses for extant primates similarly range just over a single order
of magnitude: from the mouse lemur (Microcebus rufus) to the
gorilla (Gorilla gorilla). We ask whether this trait variance is as-
cribable to a rate-homogenous process or if observed variation is
attributable to lineage-localized shifts in the evolutionary process.
For each analysis, we combine results from three independent
Markov chains, discarding the first quarter of 106 generations of
sampling as burn-in.
4 EVOLUTION 2011
IDENTIFYING RATE HETEROGENEITY IN CHARACTER EVOLUTION
ResultsOver the range of examined simulations, our introduced Bayesian
sampler reliably and efficiently estimates the generating process
of trait evolution. Simulation results suggest that despite some im-
precision in identifying branch-specific locations of rate shifts in
our smallest trees, marginalized rate estimates at branches are very
reasonable across the range of datasets used herein. Our empiri-
cal datasets appear to confirm this result: while it may be seldom
that a particular node will receive substantial posterior support for
a rate shift, phylogenetically local patterns in rate heterogeneity
may be readily distinguishable if the marginal posterior densities
of rate estimates are used for purposes of rates comparison.
The range of overall acceptance rates within simulations
exhibited little variance between the two classes of simulations
(global-rate simulations: [0.38, 0.49]; multiple-shift simulations:
[0.32, 0.53]). Split-or-merge operations exhibited the weakest
rates of acceptance (from 0.01 to 0.04), which is often charac-
teristic of reversible-jump Markov chains (see Green and Hastie
2009). Yet for a dataset that exhibits strong signal for models of a
particular complexity, we should expect very few split-or-merge
operations to be accepted upon convergence of the chain. Includ-
ing calibration periods for the proposal width, three independent
chains for each empirical dataset were completed in ca. 11.5 h on
a machine with dual quad-core 2.4 GHz Xeon processors with 8
GB RAM running OS 10.6.7.
GLOBAL RATE SIMULATIONS
Simulations under a global rate for trees differing in the num-
ber of species suggest that AUTEUR performs well in recovering
the generating model complexity (on average, PP ≥ 0.6), with a
modest increase in posterior probability with increasing tree size
(Fig. 1A). Posterior weight for the generating, global-rate model
slightly exceeded prior weight of ca. 0.5, yet inferred model com-
plexity was generally overparameterized (Fig. 1A,C). Regardless
of tree size, rate estimates across the tree did not appear to deviate
substantially from the generating rates of evolution (Fig. 1B). For
all global-rate simulations, tree-wide summaries of rate estimates
fell well within the band between one-half and twice the true rate
(Fig. 1B,D), and variance of these estimates appeared to decrease
with increasing tree size (Fig. 1B). Under these simulations, any
deviation of rate estimates from the true rates would be indicated
where the tree-wide proportion of the true rate lies on either side
of one. Recall that this statistic accommodates branch length, such
that longer branches whose relative rates are estimated with high
error contribute greater weight in the calculation of our tree-wide
measure of error (see Methods).
As we should expect, the magnitude of the constant rate
of evolution appeared to exert little influence on the ability to
recover either the true model complexity or true relative rates in
these global rate simulations (Fig. 1C,D). Even where posterior
models were overly complex, relative rate estimates were very
near the true values and appear unbiased (Fig. 1B,D).
MULTIPLE RATE SIMULATIONS
Posterior densities of relative rates at branches appear to be rea-
sonably well estimated despite relatively low performance in de-
tecting specific branches that experienced a rate shift in simula-
tion. Yet, the largest trees in our simulations (≥64 tips) achieved
very high posterior probabilities in recovering adequate model
complexity (Fig. 2A,D), precision in topological identification of
the simulated rate shift (Fig. 2B,E), and overall estimate accuracy
of the underlying evolutionary rates (Fig. 2C,F). Marked improve-
ment in regard to identification of the rate-shifted branch is ob-
served between simulations conducted with 32 versus 64 species,
where the latter simulations exhibited little variance around com-
plete posterior support for a shift occurring at the truly shifted
branch (Fig. 2B). A similar improvement in recovering position
of the truly shifted branch is evident between simulations in-
volving eightfold and 16-fold reductions of the ancestral rates
(Fig. 2E). Even where the posterior sample of model complexity
was underparameterized, rate estimates at branches were gener-
ally quite reasonable (e.g., compare panels from Fig. 2A with
2C). Substantial improvement in the accuracy of rate estimates is
apparent with increasing tree size (Fig. 2C).
For the smallest of trees in our simulations, we were un-
likely to recover proper model complexity (Fig. 2A), and elevated
variance in rate estimates is likely a result of this underparame-
terization (Fig. 2C). Accuracy of rate estimates appears strongly
related to ability in recovering adequate model complexity where
at least some branches are allowed independent relative rates
(Fig. 2A,C).
Disparities between the ancestral and shifted rates (hereafter,
“effect sizes”) appear strongly related to method performance. For
eightfold or larger reductions in the ancestral rate at the simulated
rate shift, the primary mass of posterior density for model com-
plexity was strongly centered on multiple-rate models (Fig. 2D).
Precision in inferred placement of the rate shift improved with
increasingly large effect size, and an eightfold rate difference
may lie at the threshold of sensitivity for this method with 64
taxa (Fig. 2E). Noting that the potential for rate-estimate error
increases along the abscissa in Fig. 2F, we find branchwise rate
estimates to exhibit little error and bias: all effect-size simulation-
replicates fall well within a band of error ranging between one-half
and twice the true relative rates (Fig. 2F).
EMPIRICAL EXAMPLES
Posterior odds overwhelmingly support multiple-rate models of
body-size evolution in both turtles and primates. As shown by
Bartolucci et al. (2006), posterior odds of one model against
EVOLUTION 2011 5
J. M. EASTMAN ET AL.
Figure 1. Panels summarize method performance for global-rate simulations in relationship to clade size (“clade size”; leftmost panels)
and evolutionary rate (“global rate”; rightmost panels). All simulations were conducted with a global rate: for “clade size” simulations,
evolutionary rate (σ2) was 116 , and tree size is indicated along the abscissa; for “global rate” simulations, σ2 was drawn from ( 1
8 , 116 ,
132 , 1
64 ) and trees each had 64 tips. Throughout, the large shaded circles correspond to medians of eight simulation replicates. Ability of
auteur to recover the generating model used in simulation is plotted as a function of tree size (A) and evolutionary rate (C). Values in (A)
and (C) are posterior probabilities, PP, of a global-rate model. Tree-wide error in rate estimates are shown in relationship to tree size (B)
and evolutionary rate (D). In (B) and (D), the axis of ordinates spans a range of error, from average branchwise proportions from 15 the
true rate to a fivefold average difference between estimates and true rates. Points represent averaged proportions of the true relative
rate across all branches in the tree (see Methods). We consider error on a semi-log plot inasmuch as estimates of rates that are half or
double the true value are equally (un)reasonable. The expectation for the method, if rates are estimated without error, is a proportion
of the true rate of one (indicated by dot-dashed line).
another can be efficiently estimated from reversible-jump Markov
chains by computing the ratio of frequencies in the chain sam-
pling M1 versus M2, where each M represents a model of par-
ticular complexity (e.g., a global-rate model of trait evolution).
Not a single sample from the posterior density of model com-
plexities had fewer than two rate parameters, rendering Bayes
factor comparisons impossible from our Markov samples for
these taxa. To evaluate convergence of the Markov chains, we
used several diagnostics from the CODA package (version 0.14-2,
Plummer et al. 2010). Effective sample sizes of the relative rate
6 EVOLUTION 2011
IDENTIFYING RATE HETEROGENEITY IN CHARACTER EVOLUTION
Figure 2. Panels summarize method performance for multiple-rate simulations in relationship to clade size (“clade size”; leftmost panels)
and disparity between ancestral and shifted evolutionary rates (“effect size”; rightmost panels). All simulations were conducted in which
a shifted rate (σ2S) differed from that at the root of the tree (σ2
A = 1) and where the shifted rate affected one-half of all branches in the
phylogeny. For “clade size” simulations, evolutionary rate at the shifted branch (σ2S) and descendants thereof was 1
16 , and tree size is
indicated along the abscissa; for “effect size” simulations, σ2S was drawn from ( 1
8 , 116 , 1
32 , 164 ), and trees each had 64 tips. Shown is the
ability of our method to recover adequate model complexity under a rate-shift scenario, as related to tree size (A) and factor difference
between σ2A and σ2
S (D). Median PP for two-rate models were 0.49, 0.58, 0.73, and 0.77 with increasing clade size; similarly, for increasing
effect size, median posterior probabilities (PP) of two-rate models were 0.69, 0.77, 0.71, and 0.72. Power to precisely recover the branch
at which the simulated shift occurred is shown as a function of tree size (B) and effect size (E). Plotted posterior probabilities in (B) and (E)
are conditioned on there being at least one rate shift in the evolutionary model. Shown is the relationship between branchwise accuracy
of relative-rate estimates in relation to tree size (C) and effect size (F). See Figure 1 for further detail.
EVOLUTION 2011 7
J. M. EASTMAN ET AL.
parameters were generally well in excess of 103, and samples
from the thinned chains exhibited little autocorrelation. Model
likelihoods appeared to reach stationarity within several thousand
generations of sampling postcalibration.
Moderate support for several rate shifts is apparent within
Graptemys (Emydidae) and Geochelone (Testudinidae; 0.42 <
PP < 0.60; Fig. 3). A pair of shifts receives similar support
for two geoemydid turtles of likely intergeneric hybrid origin,
Mauremys iversoni and Ocadia philippeni (0.40 < PP < 0.56;
Fig. 3). We focus model-averaged rate comparisons within the
three polytypic families of the Testudinoidea: Emydidae, Geoe-
mydidae, and Testudinidae (Rhodin et al. 2010). Although rates
of body-size evolution within the tortoises (Testudinidae) appear
to be in excess of those of the Geoemydidae (P = 0.013), emydid
body sizes appear to evolve at truly exceptional rates (Fig. 3).
Rates for the Emydidae are distinguishable from both the phy-
logenetic background (P = 0.014; Fig. 3) as well as from their
sister group, comprising the Testudinidae and the Geoemydidae
(P = 0.042).
Although inferred evolutionary rates in the New World mon-
keys (Platyrrhini) were below the estimated median rate for
primates, we find no evidence to conclude that rates of body-
size evolution within this group are distinct from other primates
(P = 0.42). In fact, we find insufficient evidence for exceptional
evolution in any of the primary groups within the Primates us-
ing model-averaged rate comparisons (0.21 < P < 0.78). These
results are reinforced by the lack of posterior support for branch-
associated shifts in rate. Within primates, a pair of downturns in
evolutionary rate are weakly supported for the Simiiformes and
Platyrrhini (0.20 < PP < 0.45), and an upturn within macaques
(Macaca) and in the lineage leading to the pygmy marmoset (Cal-
lithrix pygmaea) receive similar weak support (0.32 < PP < 0.33;
Fig. 3).
DiscussionIn general, the Bayesian sampler of evolutionary rates presented
here provides accurate posterior samples of the modes and tempo
of trait evolution across phylogenetic trees. Although the method
appears susceptible to over-fitting of model complexity (Figs. 1
and 2), relative rates that are marginalized across model com-
plexity are estimated with very little error (Fig. 2C). Inasmuch
as true evolutionary processes may be quite discrepant from the
saltational shifts in rates modeled here, we advocate an emphasis
on model-averaged branchwise estimates of relative rates rather
than positions of inferred shifts. Such a focus should be more
resistant to ad hoc interpretations of the posterior results. This
model-averaging framework still allows comparisons of evolu-
tionary hypotheses at specific points or intervals on the tree, yet
provides a more accurate representation of the evidence for rate
shifts along nonfocal branches. We focus our discussion on per-
formance and potential extensions of this method.
CONSIDERATIONS
Although we model the evolutionary process as inherently
saltational—rates change discretely between branching events—
we expect that this method may as well be capable of capturing
more gradualistic change in rate. Support for shifted rates that is
diffusely spread over the backbone of a subtree (e.g., Fig. 3) may
be suggestive of a more gradual process of change that must nec-
essarily be discretized by the method introduced here. The lack
of overwhelming posterior support for a rate shift concentrated at
a particular branch may not imply that evolutionary rates within
the lineage are not atypical. The exceptional rates of body-size
evolution in the Emydidae appear to provide an example of just
this sort, where the posterior mass of support for rate shifts is
quite diffuse (Fig. 3). The converse also appears to hold: strong
posterior support for a shift at a given branch may not provide
sufficient evidence for distinguishable rates, but rather should be
formally tested (e.g., see Fig. 3 caption).
Model-averaged rates from the posterior summary of the
reversible-jump sampler provide a natural foundation for hypoth-
esis testing. The evolutionary rate within a lineage in comparison
to the phylogenetic background is one such testable hypothesis
(e.g., Fig. 3), although similar tests could be quite varied and
need not involve comparisons between monophyletic or para-
phyletic groups. Even where there is weak support for a rate shift
at any particular branch, model-averaged rates may provide suf-
ficient signal in detecting localized and distinguishable patterns
of incongruous evolution in distinct areas of tree. As shown here
(Fig. 3), heterogeneity in the process of evolution may well be
correlated with phylogeny (see also O’Meara et al. 2006) but also
with traits extrinsic to those whose rates are being modeled (e.g.,
selective regimes, as in Butler and King 2004). Correlating pos-
terior rate estimates with the states of extrinsic characters would
require trustworthy estimates of states at each branch in the phy-
logeny (e.g., ecological niches, biogeographical regions, etc.), an
issue that requires careful consideration (e.g., Goldberg and Igic
2007; Li et al. 2008; Ekman et al. 2008).
To facilitate comparisons among models differing in com-
plexity, we allow for Bayes factor comparisons in the provided
software (Jeffreys 1935; Kass and Raftery 1995). Within a single
chain (or from pooled chains), the Bayes factor provides a poste-
rior measure of model support while integrating over uncertainty
in the precise placement of rate shifts in the tree, if the model
so allows. We further allow sampling to be parametrically con-
strained (e.g., considering only k-rate models, rather than the full
range of model complexities) for alternative statistical compar-
isons between models with differing complexities. For instance,
sampling can be constrained to maximal model complexity to
8 EVOLUTION 2011
IDENTIFYING RATE HETEROGENEITY IN CHARACTER EVOLUTION
Els
eya
la
tiste
rnu
m
Ch
elo
din
a lo
ng
ico
llis
Ph
ryn
op
s g
ibb
us
Aca
nth
och
ely
s r
ad
iola
ta
Aca
nth
och
ely
s m
acro
ce
ph
ala
Aca
nth
och
ely
s p
alli
dip
ecto
ris
Aca
nth
och
ely
s s
pix
ii
Pla
tem
ys p
laty
ce
ph
ala
Ch
elu
s f
imb
ria
tus
Pe
lom
ed
usa
su
bru
fa
Pe
lusio
s w
illia
msi
Ery
mn
och
ely
s m
ad
ag
asca
rie
nsis
Pe
lto
ce
ph
alu
s d
um
erilia
na
Po
do
cn
em
is v
og
li
Po
do
cn
em
is e
xp
an
sa
Po
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cn
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is s
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be
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lata
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cn
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ryth
roce
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ala
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do
cn
em
is le
wya
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Ch
ely
dra
se
rpe
ntin
a
Ma
cro
ch
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s t
em
min
ckii
Pla
tyste
rno
n m
eg
ace
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alu
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ote
stu
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fo
rste
nii
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stu
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issin
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laco
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us t
orn
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rsfie
ldii
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stu
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i b
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mo
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s f
em
ora
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ara
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nic
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da
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so
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olo
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so
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ussu
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dra
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ep
ta
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och
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ch
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ecki
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ph
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ium
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ha
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en
sis
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och
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ra a
bin
gd
on
i
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och
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ne
nig
ra h
oo
de
nsis
Ge
och
elo
ne
pla
tyn
ota
Ge
och
elo
ne
ele
ga
ns
Ge
och
elo
ne
su
lca
ta
Ge
och
elo
ne
de
nticu
lata
Ge
och
elo
ne
ca
rbo
na
ria
Kin
ixys e
rosa
Kin
ixys h
om
ea
na
Kin
ixys s
pe
kii
Kin
ixys b
elli
an
a
Kin
ixys n
ata
len
sis
Ch
ers
ine
an
gu
lata
Ho
mo
pu
s s
ign
atu
s c
afe
r
Ho
mo
pu
s b
ou
len
ge
ri
Ho
mo
pu
s b
erg
eri
Ho
mo
pu
s a
ero
latu
s
Psa
mm
ob
ate
s o
cu
life
r
Psa
mm
ob
ate
s g
eo
me
tric
us
Psa
mm
ob
ate
s t
en
toriu
s
Astr
och
ely
s y
nip
ho
ra
Psa
mm
ob
ate
s p
ard
alis
Ma
no
uria
im
pre
ssa
Ma
no
uria
em
ys
Go
ph
eru
s p
oly
ph
em
us
Go
ph
eru
s f
lavo
ma
rgin
atu
s
Go
ph
eru
s a
ga
ssiz
ii
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ph
eru
s b
erla
nd
eri
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ino
cle
mm
ys n
asu
ta
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ino
cle
mm
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ulc
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a in
cis
a
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ino
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mm
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ino
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reo
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ino
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mm
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un
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ela
no
ste
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ino
cle
mm
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un
ctu
laria
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laria
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ino
cle
mm
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ub
ida
Ge
ocle
mys h
am
ilto
nii
Mo
ren
ia o
ce
llata
Ha
rde
lla t
hu
rjii
ind
i
Ka
ch
ug
a s
ylh
ete
nsis
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ch
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a t
ecta
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ch
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a t
en
toria
circu
md
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ch
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ii sm
ith
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ch
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ng
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llag
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bo
rne
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nsis
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ch
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rivitta
ta
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tag
ur
ba
ska
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tag
ur
aff
inis
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tag
ur
ka
ch
ug
a
Ma
laye
mys s
ub
triju
ga
Orlitia
bo
rne
en
sis
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oe
myd
a ja
po
nic
a
Ge
oe
myd
a s
pe
ng
leri
Sa
ca
lia p
se
ud
oce
llata
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be
nro
ckie
lla le
yte
nsis
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be
nro
ckie
lla c
rassic
olli
s
Me
lan
och
ely
s t
riju
ga
triju
ga
Ma
ure
mys le
pro
sa
Ma
ure
mys c
asp
ica
Ma
ure
mys r
ivu
lata
Ma
ure
mys s
ine
nsis
Ma
ure
mys ja
po
nic
a
Ch
ine
mys n
igrica
ns
Ma
ure
mys r
ee
ve
sii
Ma
ure
mys m
eg
alo
ce
ph
ala
Ma
ure
mys a
nn
am
en
sis
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dia
gly
ph
isto
ma
Ma
ure
mys m
utica
Oca
dia
ph
ilip
pe
ni
Ma
ure
mys ive
rso
ni
Cu
ora
zh
ou
i
Cu
ora
trifa
scia
ta
Cu
ora
au
roca
pita
ta
Cu
ora
pa
ni
Cu
ora
mcco
rdi
Cu
ora
fla
vo
ma
rgin
ata
eve
lyn
ae
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ora
ga
lbin
ifro
ns
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ora
pic
tura
ta
Cu
ora
am
bo
ine
nsis
am
bo
ine
nsis
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ea
mo
uh
otii
He
ose
mys s
pin
osa
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ose
mys g
ran
dis
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mys d
ep
ressa
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ose
mys a
nn
an
da
lii
Cycle
mys s
p.
fusca
Cycle
mys s
p.
ge
me
li
Cycle
mys o
ldh
am
ii
Cycle
mys t
ch
ep
on
en
sis
Cycle
mys s
ha
ne
nsis
Cycle
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en
tata
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mys s
p.
en
igm
atica
Cycle
mys p
ulc
hristr
iata
Cycle
mys a
trip
on
s
Le
uco
ce
ph
alo
n y
uw
on
oi
No
toch
ely
s p
laty
no
ta
Sa
ca
lia q
ua
drio
ce
llata
Sa
ca
lia b
ea
lei
Em
ys t
rin
acris
Em
ys o
rbic
ula
ris c
olc
hic
a
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yd
oid
ea
bla
nd
ing
ii
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em
ys m
arm
ora
ta
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pte
mys m
uh
len
be
rgii
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pte
mys in
scu
lpta
Te
rra
pe
ne
ne
lso
ni
Te
rra
pe
ne
orn
ata
Te
rra
pe
ne
co
ah
uila
Te
rra
pe
ne
ca
rolin
a t
riu
ng
uis
Cle
mm
ys g
utt
ata
Pse
ud
em
ys n
els
on
i
Pse
ud
em
ys a
lab
am
en
sis
Pse
ud
em
ys c
on
cin
na
Pse
ud
em
ys t
exa
na
Pse
ud
em
ys r
ub
rive
ntr
is
Pse
ud
em
ys p
en
insu
laris
Pse
ud
em
ys c
on
cin
na
flo
rid
an
a
Pse
ud
em
ys s
uw
an
nie
nsis
Pse
ud
em
ys g
orz
ug
i
Ch
ryse
mys p
icta
Gra
pte
mys g
eo
gra
ph
ica
Tra
ch
em
ys g
aig
ea
e
Gra
pte
mys c
ag
lei
Gra
pte
mys p
se
ud
og
eo
gra
ph
ica
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pte
mys n
igrin
od
a
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pte
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se
ud
og
eo
gra
ph
ica
ko
hn
ii
Gra
pte
mys o
cu
life
ra
Gra
pte
mys p
ulc
hra
Gra
pte
mys o
ua
ch
ite
nsis
Gra
pte
mys f
lavim
acu
lata
Gra
pte
mys v
ers
a
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pte
mys b
arb
ou
ri
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pte
mys g
ibb
on
si
Tra
ch
em
ys s
crip
ta t
aylo
ri
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ch
em
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tejn
eg
eri
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ch
em
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crip
ta e
leg
an
s
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lacle
mys t
err
ap
in
De
iro
ch
ely
s r
eticu
laria
Ch
elo
nia
myd
as
Na
tato
r d
ep
ressa
Le
pid
och
ely
s k
em
pii
Le
pid
och
ely
s o
liva
ce
a
Ca
rett
a c
are
tta
Ere
tmo
ch
ely
s im
brica
ta
De
rmo
ch
ely
s c
oria
ce
a
Ste
rno
the
rus o
do
ratu
s
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uro
typ
us t
rip
orc
atu
s
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rma
tem
ys m
aw
ii
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itra
in
dic
a
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itra
ch
itra
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itra
va
nd
ijki
Pe
loch
ely
s c
an
torii
Pe
loch
ely
s b
ibro
ni
Trio
nyx t
riu
ng
uis
Ra
fetu
s e
up
hra
ticu
s
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alo
ne
sp
inife
ra a
sp
era
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alo
ne
fe
rox
Ap
alo
ne
mu
tica
Pe
lod
iscu
s s
ine
nsis
Do
ga
nia
su
bp
lan
a
Asp
ide
rete
s le
ith
ii
Asp
ide
rete
s g
an
ge
ticu
s
Asp
ide
rete
s h
uru
m
Asp
ide
rete
s n
igrica
ns
Nils
so
nia
fo
rmo
sa
Pa
lea
ste
ind
ach
ne
ri
Am
yd
a c
art
ilag
ine
a
Cyclo
de
rma
au
bry
i
Cyclo
de
rma
fre
na
tum
Cycla
no
rbis
ele
ga
ns
Cycla
no
rbis
se
ne
ga
len
sis
Lis
se
mys s
cu
tata
Lis
se
mys p
un
cta
ta
Ca
rett
och
ely
s in
scu
lpta
1 2 3 4 5 6 7 8 9
0.0
0.2
0.4
0.6
0.8
1.0
prior
posterior
rates
pro
babili
ty
marginal rates
0.4349
0.1551
0.0553
0.0197
0.0070
0.0025
0.0009
0.0003
0.0001
0.005 0.010 0.020 0.050 0.100
0100
200
300
400
rate estimates
density
Testudinidae
Geoemydidae
0.005 0.010 0.020 0.050 0.100 0.200
0100
200
300
400
rate estimates
density
Emydidae
background
Em
Te
Ge
A B
Figure 3. Posterior comparisons of rate of body-size evolution in turtles using model-averaging of rate estimates. Shown are posterior
densities of rates comparing (A) those between the Emydidae (red; Em) and the phylogenetic background (gray); and (B) rates between
the Geoemydidae (gray; Ge) and the Testudinidae (red; Te). Bars represent the highest posterior density (0.95) of rate estimates for each
group. Rates are computed as a weighted average of posterior rate estimates, where weighting is determined by branch length (see
Methods); note that rate densities are plotted on a log-scale. To test whether rates were distinguishable between lineages, we conducted
104 Monte Carlo sampling iterations, each of which was a comparison between randomly sampled draws from the posterior distributions
of lineage-specific relative rates. Expecting the sign of these comparisons to be random if the two posterior distributions were truly
identical, we interpret the proportion of comparisons in a particular direction to be an approximate probability value in a one-tailed test.
We report two-tailed probability values under this resampling procedure, given that we proposed no prior expectation for the sign of
these differences. Comparisons in panels (A) and (B) both strongly indicate rate differences (P = 0.014 and P = 0.013, respectively). The
depicted turtle phylogeny (C) is from Jaffe et al. (2011). Hue and size of circles at branches denote posterior support for a rate shift at
the indicated branch. Larger and redder circles suggest higher posterior support for an upturn in evolutionary rate (see text for details).
Branches in the phylogeny are colored such that rates not deviant from the median are shaded gray; rates below (or above) the median
are shaded blue (or red). Rates corresponding to each hue are indicated in the legend.
examine posterior support for the “free model” (sensu Mooers
et al. 1999), where each branch evolves at a unique rate. Al-
though AUTEUR may be capable of sampling from the ’free model’
in unconstrained analyses, it is likely that few datasets will ap-
proach the number of rate categories as there are branches in the
tree.
We note a potential difficulty in distinguishing a heteroge-
neous process of evolution where a shift occurs on a rootmost
branch. If we were to simulate a downturn in evolutionary rate
at the base of the tree (on a particular branch), we might ex-
pect posterior support for rate shifts to be spread between this
and its sister branch (i.e., with some posterior weight supporting
an upturn on the sister branch and some posterior support for
a downturn on the opposite branch). In simulations conducted
on trees perfectly balanced at the rootmost split, we find this to
be the case: posterior support for rate shifts tends to be spread
EVOLUTION 2011 9
J. M. EASTMAN ET AL.
Alle
nopithecus n
igro
virid
is
Allo
cebus trichotis
Alo
uatta b
elz
ebul
Alo
uatta c
ara
ya
Alo
uatta c
oib
ensis
Alo
uatta fusca
Alo
uatta p
alli
ata
Alo
uatta p
igra
Alo
uatta s
ara
Alo
uatta s
enic
ulu
s
Aotu
s a
zara
i
Aotu
s b
rum
backi
Aotu
s h
ers
hkovitzi
Aotu
s infu
latu
s
Aotu
s lem
urinus
Aotu
s m
iconax
Aotu
s n
ancym
aae
Aotu
s n
igriceps
Aotu
s trivirgatu
s
Aotu
s v
ocifera
ns
Arc
tocebus a
ure
us
Arc
tocebus c
ala
bare
nsis
Ate
les b
elz
ebuth
Ate
les c
ham
ek
Ate
les fuscic
eps
Ate
les g
eoffro
yi
Ate
les m
arg
inatu
s
Ate
les p
anis
cus
Avahi la
nig
er
Bra
chyte
les a
rachnoid
es
Cacaja
o c
alv
us
Cacaja
o m
ela
nocephalu
s
Calli
cebus b
runneus
Calli
cebus c
alig
atu
s
Calli
cebus c
inera
scens
Calli
cebus c
upre
us
Calli
cebus d
onacophilu
s
Calli
cebus d
ubiu
s
Calli
cebus h
offm
annsi
Calli
cebus m
odestu
s
Calli
cebus m
olo
ch
Calli
cebus o
enanth
e
Calli
cebus o
lalla
e
Calli
cebus p
ers
onatu
s
Calli
cebus torq
uatu
s
Calli
mic
o g
oeld
ii
Calli
thrix a
rgenta
ta
Calli
thrix a
urita
Calli
thrix fla
vic
eps
Calli
thrix g
eoffro
yi
Calli
thrix h
um
era
lifera
Calli
thrix jacchus
Calli
thrix k
uhlii
Calli
thrix p
enic
illata
Calli
thrix p
ygm
aea
Cebus a
lbifro
ns
Cebus a
pella
Cebus c
apucin
us
Cebus o
livaceus
Cerc
ocebus a
gili
s
Cerc
ocebus g
ale
ritu
s
Cerc
ocebus torq
uatu
s
Cerc
opithecus a
scaniu
s
Cerc
opithecus c
am
pbelli
Cerc
opithecus c
ephus
Cerc
opithecus d
iana
Cerc
opithecus d
ryas
Cerc
opithecus e
ryth
rogaste
r
Cerc
opithecus e
ryth
rotis
Cerc
opithecus h
am
lyni
Cerc
opithecus lhoesti
Cerc
opithecus m
itis
Cerc
opithecus m
ona
Cerc
opithecus n
egle
ctu
s
Cerc
opithecus n
ictita
ns
Cerc
opithecus p
eta
urista
Cerc
opithecus p
ogonia
s
Cerc
opithecus p
reussi
Cerc
opithecus s
cla
teri
Cerc
opithecus s
ola
tus
Cerc
opithecus w
olfi
Cheirogale
us m
ajo
r
Cheirogale
us m
ediu
s
Chiropote
s a
lbin
asus
Chiropote
s s
ata
nas
Chlo
rocebus a
eth
iops
Colo
bus a
ngole
nsis
Colo
bus g
uere
za
Colo
bus p
oly
kom
os
Colo
bus s
ata
nas
Daubento
nia
madagascariensis
Ery
thro
cebus p
ata
s
Eule
mur
coro
natu
s
Eule
mur
fulv
us
Eule
mur
macaco
Eule
mur
mongoz
Eule
mur
rubrivente
r
Euoticus e
legantu
lus
Euoticus p
alli
dus
Gala
go a
lleni
Gala
go g
alla
rum
Gala
go m
ats
chie
i
Gala
go m
oholi
Gala
go s
enegale
nsis
Gala
goid
es d
em
idoff
Gala
goid
es z
anzib
aricus
Gorilla
gorilla
Hapale
mur
aure
us
Hapale
mur
griseus
Hapale
mur
sim
us
Hom
o s
apie
ns
Hylo
bate
s a
gili
s
Hylo
bate
s c
oncolo
r
Hylo
bate
s g
abriella
e
Hylo
bate
s h
oolo
ck
Hylo
bate
s k
lossii
Hylo
bate
s lar
Hylo
bate
s leucogenys
Hylo
bate
s m
olo
ch
Hylo
bate
s m
uelle
ri
Hylo
bate
s p
ileatu
s
Hylo
bate
s s
yndacty
lus
Indri indri
Lagoth
rix fla
vic
auda
Lagoth
rix lagotr
icha
Lem
ur
catta
Leonto
pithecus c
ais
sara
Leonto
pithecus c
hry
som
ela
Leonto
pithecus c
hry
sopygus
Leonto
pithecus r
osalia
Lepile
mur
dors
alis
Lepile
mur
edw
ard
si
Lepile
mur
leucopus
Lepile
mur
mic
rodon
Lepile
mur
muste
linus
Lepile
mur
ruficaudatu
s
Lepile
mur
septe
ntr
ionalis
Lophocebus a
lbig
ena
Loris tard
igra
dus
Macaca a
rcto
ides
Macaca a
ssam
ensis
Macaca c
yclo
pis
Macaca fascic
ula
ris
Macaca fuscata
Macaca m
aura
Macaca m
ula
tta
Macaca n
em
estr
ina
Macaca n
igra
Macaca o
chre
ata
Macaca r
adia
ta
Macaca s
ilenus
Macaca s
inic
a
Macaca s
ylv
anus
Macaca thib
eta
na
Macaca tonkeana
Mandrillu
s leucophaeus
Mandrillu
s s
phin
x
Mic
rocebus c
oquere
li
Mic
rocebus m
urinus
Mic
rocebus r
ufu
s
Mio
pithecus tala
poin
Nasalis
concolo
r
Nasalis
larv
atu
s
Nycticebus c
oucang
Nycticebus p
ygm
aeus
Oto
lem
ur
cra
ssic
audatu
s
Oto
lem
ur
garn
ettii
Pan p
anis
cus
Pan tro
glo
dyte
s
Papio
ham
adry
as
Pero
dic
ticus p
otto
Phaner
furc
ifer
Pithecia
aequato
rialis
Pithecia
alb
icans
Pithecia
irr
ora
ta
Pithecia
monachus
Pithecia
pithecia
Pongo p
ygm
aeus
Pre
sbytis c
om
ata
Pre
sbytis fem
ora
lis
Pre
sbytis fro
nta
ta
Pre
sbytis h
osei
Pre
sbytis m
ela
lophos
Pre
sbytis p
ote
nzia
ni
Pre
sbytis r
ubic
unda
Pre
sbytis thom
asi
Pro
colo
bus b
adiu
s
Pro
colo
bus p
ennantii
Pro
colo
bus p
reussi
Pro
colo
bus r
ufo
mitra
tus
Pro
colo
bus v
eru
s
Pro
pithecus d
iadem
a
Pro
pithecus tatters
alli
Pro
pithecus v
err
eauxi
Pygath
rix a
vunculu
s
Pygath
rix b
ieti
Pygath
rix b
relic
hi
Pygath
rix n
em
aeus
Pygath
rix r
oxella
na
Saguin
us b
icolo
r
Saguin
us fuscic
olli
s
Saguin
us g
eoffro
yi
Saguin
us im
pera
tor
Saguin
us inustu
s
Saguin
us labia
tus
Saguin
us leucopus
Saguin
us m
idas
Saguin
us m
ysta
x
Saguin
us n
igricolli
s
Saguin
us o
edip
us
Saguin
us tripart
itus
Saim
iri boliv
iensis
Saim
iri oers
tedii
Saim
iri sciu
reus
Saim
iri ustu
s
Saim
iri vanzolin
ii
Sem
nopithecus e
nte
llus
Tars
ius b
ancanus
Tars
ius d
ianae
Tars
ius p
um
ilus
Tars
ius s
pectr
um
Tars
ius s
yrichta
Thero
pithecus g
ela
da
Tra
chypithecus a
ura
tus
Tra
chypithecus c
rista
tus
Tra
chypithecus fra
ncois
i
Tra
chypithecus g
eei
Tra
chypithecus johnii
Tra
chypithecus o
bscuru
s
Tra
chypithecus p
hayre
i
Tra
chypithecus p
ileatu
s
Tra
chypithecus v
etu
lus
Vare
cia
variegata
1 2 3 4 5 6 7 8
rates
pro
babili
ty0
.00
.20
.40
.60
.81
.0
prior
posterior 9.546
2.278
0.544
0.130
0.031
0.007
0.002
0.001
0.000
marginal rates
Alle
nopithecus n
igro
virid
is
Cerc
ocebus a
gili
s
Cerc
ocebus g
ale
ritu
s
Cerc
ocebus torq
uatu
s
Cerc
opithecus a
scaniu
s
Cerc
op
ithecus c
am
pbelli
Cerc
op
ithecus c
ephus
Cerc
op
ithecus d
ian
a
Cerc
opithecus d
rya
s
Cerc
opithecus e
ryth
rogaste
r
Cerc
opithecus e
ryth
rotis
Cerc
opithecus h
am
lyn
i
Cerc
opithecus lhoesti
Cerc
opithecus m
itis
Cerc
opithecus m
on
a
Cerc
opithecus n
egle
ctu
s
Cerc
opithecus n
ictita
ns
Cerc
opithecus p
eta
urista
Cerc
opithecus p
ogonia
s
Cerc
opithecus p
reussi
Cerc
opithecus s
cla
teri
Cerc
opithecus s
ola
tus
Cerc
opithecus w
olfi
Chlo
rocebus a
eth
iops
Colo
bus a
ngole
nsis
Colo
bus g
uere
za
Colo
bus p
oly
kom
os
Colo
bus s
ata
nas
Ery
thro
cebus p
ata
s
Gorilla
gorilla
Hom
o s
apie
ns
Hylo
bate
s a
gili
s
Hylo
bate
s c
oncolo
r
Hylo
bate
s g
ab
riella
e
Hylo
bate
s h
oo
lock
Hylo
bate
s k
lossii
Hylo
bate
s la
r
Hylo
bate
s leucogenys
Hylo
bate
s m
olo
ch
Hylo
bate
s m
uelle
ri
Hylo
bate
s p
ileatu
s
Hylo
bate
s s
yndacty
lus
Lophocebus a
lbig
en
a
Macaca a
rcto
ides
Macaca
assam
ensis
Macaca c
yclo
pis
Macaca fascic
ula
ris
Macaca fuscata
Macaca
maura
Macaca
mula
tta
Macaca
nem
estr
ina
Macaca n
igra
Macaca o
chre
ata
Macaca r
adia
ta
Macaca s
ilenus
Macaca
sin
ica
Macaca s
ylv
anu
s
Macaca thib
eta
na
Macaca tonkean
a
Mandrillu
s leuco
phaeus
Mandrillu
s s
phin
x
Mio
pithecus tala
poin
Nasalis
concolo
r
Nasalis
larv
atu
s
Pan p
anis
cus
Pan tro
glo
dyte
s
Papio
ham
adry
as
Pre
sbytis c
om
ata
Pre
sb
ytis fem
ora
lis
Pre
sbytis fro
nta
ta
Pre
sbytis h
osei
Pre
sbytis m
ela
lopho
s
Pre
sbytis p
ote
nzia
ni
Pre
sbytis r
ubic
unda
Pre
sbytis thom
asi
Pro
colo
bus b
adiu
s
Pro
colo
bus p
ennantii
Pro
colo
bus p
reussi
Pro
colo
bus r
ufo
mitra
tus
Pro
colo
bus v
eru
s
Pygath
rix a
vunculu
s
Pygath
rix b
ieti
Pygath
rix b
relic
hi
Pygath
rix n
em
aeus
Pygath
rix r
oxella
na
Sem
nopithecus e
nte
llus
Thero
pithecus g
ela
da
Tra
chypithecus a
ura
tus
Tra
chypithecus c
rista
tus
Tra
chypithecus fra
ncois
i
Tra
chypithecus g
eei
Tra
chypithecus johnii
Tra
chypithecus o
bscuru
s
Tra
chypithecus p
hayre
i
Tra
chypithecus p
ileatu
s
Tra
chypithecus v
etu
lusCatarrhini
Alo
uatta b
elz
ebul
Alo
uatta c
ara
ya
Alo
uatta c
oib
ensis
Alo
uatta fusca
Alo
uatta p
alli
ata
Alo
uatta p
igra
Alo
uatta s
ara
Alo
uatta s
enic
ulu
s
Aotu
s a
zara
i
Aotu
s b
rum
backi
Aotu
s h
ers
hko
vitzi
Aotu
s infu
latu
s
Aotu
s lem
urinus
Aotu
s m
iconax
Aotu
s n
ancym
aae
Aotu
s n
igriceps
Aotu
s trivirgatu
s
Aotu
s v
ocifera
ns
Ate
les b
elz
ebuth
Ate
les c
ham
ek
Ate
les fuscic
eps
Ate
les g
eoffro
yi
Ate
les m
arg
inatu
s
Ate
les p
anis
cu
s
Bra
chyte
les a
rachnoid
es
Cacaja
o c
alv
us
Cacaja
o m
ela
nocephalu
s
Calli
cebus b
runneu
s
Calli
cebus c
alig
atu
s
Calli
cebus c
inera
scen
s
Calli
cebus c
upre
us
Calli
cebus d
onacophilu
s
Calli
cebus d
ubiu
s
Calli
cebus h
offm
annsi
Calli
cebus m
odestu
s
Calli
cebus m
olo
ch
Calli
cebus o
enanth
e
Calli
cebus o
lalla
e
Calli
cebus p
ers
onatu
s
Calli
cebus torq
uatu
s
Calli
mic
o g
oeld
ii
Calli
thrix a
rgenta
ta
Calli
thrix a
urita
Calli
thrix fla
vic
eps
Calli
thrix g
eoffro
yi
Calli
thrix h
um
era
lifera
Calli
thrix jacchus
Calli
thrix k
uhlii
Calli
thrix p
enic
illata
Calli
thrix p
ygm
ae
a
Cebus a
lbifro
ns
Cebus a
pella
Cebus c
ap
ucin
us
Cebus o
livaceu
s
Chiropote
s a
lbin
asu
s
Lagoth
rix fla
vic
aud
a
Lagoth
rix lagotr
ich
a
Leonto
pithecus c
ais
sara
Leonto
pithecus c
hry
som
ela
Leonto
pithecus c
hry
sopygu
s
Leonto
pithecus r
osalia
Pithecia
aequato
rialis
Pith
ecia
alb
icans
Pithecia
irr
ora
ta
Pithecia
monachus
Pithecia
pithecia
Saguin
us b
icolo
r
g
Saguin
us g
eoffro
yi
Saguin
us im
pera
tor
Sag
uin
us inustu
s
Saguin
us labia
tus
Saguin
us leucopu
s
Saguin
us m
ida
s
Saguin
us m
ysta
x
Saguin
us n
igricolli
s
Saguin
us o
edip
us
Saguin
us tripart
itus
Saim
iri boliv
iensis
Saim
iri oers
tedii
Saim
iri sciu
reu
s
Saim
iri ustu
s
Saim
iri va
nzolin
ii
Chiropote
s s
ata
nas
Saguin
us fuscic
olli
s
Allo
cebus trichotis
Arc
tocebus a
ure
us
Arc
tocebus c
ala
bare
nsis
Avahi la
nig
er
Cheirogale
us m
ajo
r
Cheirogale
us m
ediu
s
Daubento
nia
madagascariensis
Eule
mur
coro
natu
s
Eule
mur
fulv
us
Eule
mur
macaco
Eule
mur
mongo
z
Eule
mur
rubrivente
r
Euoticus e
legantu
lus
Euoticus p
alli
dus
Gala
go a
lleni
Gala
go g
alla
rum
Gala
go m
oholi
Gala
go s
enegale
nsis
Gala
goid
es d
em
idoff
Gala
goid
es z
anzib
aricu
s
Hapale
mur
aure
us
Hapale
mur
griseu
s
Hapale
mur
sim
us
Indri indri
Lem
ur
catta
Lepile
mur
dors
alis
Lepile
mur
edw
ard
si
Lepile
mur
leuco
pus
Lepile
mur
mic
rodo
n
Lepile
mur
muste
linus
Lepile
mur
ruficaudatu
s
Lepile
mur
septe
ntr
ionalis
Loris tard
igra
dus
Mic
rocebus c
oquere
li
Mic
rocebus m
urin
us
Mic
rocebus r
ufu
s
Nycticebus c
oucang
Nycticebus p
ygm
aeus
Oto
lem
ur
cra
ssic
audatu
s
Oto
lem
ur
garn
ettii
Pero
dic
ticus p
otto
Pha
ner
furc
ifer
Pro
pithecus d
iadem
a
Pro
pithecus tatters
alli
Pro
pithecus v
err
eauxi
Vare
cia
variegata
Cheirogale
us
majo
r
Gala
go
mats
chie
i
Platyrrhini Strepsirrhini
Figure 4. Marginalized rate estimates of body-mass evolution in primates and posterior support for a multiple-rate model by comparison
of prior and posterior odds of varying model complexities. Three primary clades within the Primates are indicated at top (Catarrhini:
Old World monkeys and apes; Platyrrhini: New World monkeys; and Strepsirrhini: lemurs and allies). Although multiple-rate models are
clearly favored (inset), permutation tests did not indicate strong support for disparate rates of evolution between these major lineages.
The primate tree used here is from FitzJohn (2010), which is based primarily on work by Vos and Mooers (2006). See Figure 3 for further
detail on symbols and shading.
between each of the first descendant branches of the root (data
not shown). In excluding the root branch from the constructed
evolutionary models, the ability to resolve the nature of a shift
at the rootmost branches will be limited. We note, however, that
relative rates in these balanced-tree simulations do tend to accu-
rately reflect the underlying known rates, which reemphasizes the
importance of using marginal distributions of branch-specific rate
estimates.
Some of the poorest estimates from this method occur where
data were few (Figs. 1B and 2C). Under multiple-rate trait evo-
lution, relative rates across the tree may be estimated with high
variance as a result of the inability to recover proper model com-
plexity; in simulation, this was most apparent for our smallest
datasets (i.e., 16 and 32 taxa; Fig. 2C). If error is due to un-
derparameterization of model complexity, we expect that as the
difference between σ2S and σ2
A becomes small (thereby converg-
ing toward a single rate), error in rate estimates will approach
zero (as is the case where the difference between σ2S and σ2
A is
large and thus detectable; Fig. 2C,F). There may, however, be a
range of disparities between σ2S and σ2
A within which a true rate
difference may be difficult to confirm and rate-estimate error may
accumulate in the tree.
EXTENSIONS
To improve mixing in the Bayesian sampler, we note the potential
benefit of Metropolis coupling of chains (e.g., Huelsenbeck and
Ronquist 2001). We have observed that chains may occasionally
become resistant to slight model perturbation in the topological
position of a rate shift, especially where a set of branches provides
strong signal for an exceptional rate. This can be problematic if
the chain becomes trapped by sampling models with suboptimal
placement of a rate shift. We suspect Metropolis coupling of hot
and cold chains will enable more precise estimation of the topo-
logical locations of rate shifts. At the very least, we stress the
importance of combining posterior estimates from multiple inde-
pendent chains, which we facilitate with the provided software.
Phylogenetic error, especially where error is nonrandom with re-
spect to the true phylogeny, is certain to contribute to erroneous
results in AUTEUR (e.g., see Martins and Hansen 1997 and Revell
et al. 2005). In fact, results that weakly support shifted rates for
two putative intergeneric hybrids (see Fig. 3 and Results) may
be attributable to violation of the assumption of a nonreticulating
phylogeny. A difficult scenario, distinct from hybridization, could
be where the estimated phylogeny does not contain the bipartition
whose stem has truly experienced a shift toward an exceptional
1 0 EVOLUTION 2011
IDENTIFYING RATE HETEROGENEITY IN CHARACTER EVOLUTION
rate of trait diversification. A reasonable approach in dealing with
phylogenetic error might be to integrate inferences from AUTEUR
over a sample of credible trees. Sensitivity of results to phylo-
genetic error could otherwise be investigated by simulation. We
further note that measurement error can be readily incorporated
into these analyses where the extent of error associated with each
trait value is known (Martins and Hansen 1997; Ives et al. 2007;
Harmon et al. 2010).
An extension of the Revell and Collar (2009) likelihood
method of testing evolutionary shifts in correlated evolution of a
pair of characters could be readily implemented in the reversible-
jump Bayesian framework. These authors ask whether the evo-
lution of two functional–morphological characters in centrarchid
fish is correlated, and if so, whether the relationship is uniform
across evolutionary history. Rather than (or in addition to) fit-
ting models where the rates of evolution of quantitative char-
acters changes at particular nodes in the tree, the evolutionary
matrix among characters would be the parameter of interest in
a reversible-jump approach. In the most highly complex model,
each branch in the phylogeny would have an independent evolu-
tionary matrix among characters. Reversible-jump Markov sam-
pling would provide a natural framework in which to simultane-
ously fit a broad range of character-correlation models and without
the limitation of needing to select the most appropriate model(s)
for comparison a priori.
The reversible-jump approach of AUTEUR is sufficiently flex-
ible as to allow for broader extensions into other commonly im-
plemented models of trait evolution and for which model likeli-
hoods are calculable (e.g., Ornstein–Uhlenbeck process, Hansen
1997; Early-burst evolution, Blomberg et al. 2003). We expect
development toward an implementation of AUTEUR under a gener-
alized Ornstein–Uhlenbeck model of constrained trait evolution.
In such a model, trait values are largely bounded to optima such
that values strongly deviant from the optimum are expected to
be drawn to fitted medial values (Hansen 1997; Butler and King
2004). Rather than comparing fit of potentially numerous a pri-
ori models with distinct histories of transitions between selective
regimes, one could foresee the utility in a reversible-jump sam-
pler of which one purpose is to “identify” these shifts between
selective regimes.
ConclusionsThe Bayesian sampler of phenotypic rates, presented here and
provided as an R software package (AUTEUR), greatly broadens
the field of possibilities in modeling continuous-trait evolution
and identifying exceptional rates of change in lineages. AUTEUR
can be used as a framework in which to test existing hypotheses
for rate heterogeneity or as a method for generating hypotheses of
rate variation. We emphasize the use of model-averaged rates in
posterior analyses, as estimate error across a wide range of simula-
tion conditions appears reasonably small. We find that the ability
to recover model complexity is highly related to the disparity in
rates across the tree and dataset size. Thus, where the potential
for large absolute error in rate estimates is most menacing, this
method appears consistently reliable.
ACKNOWLEDGMENTS
We are indebted to M. Pennell, J. Rosindell, D. Jochimsen, G.
Slater, two anonymous reviewers, and P. Lindenfors for edito-
rial and general comments on a previous draft of the present
manuscript. We thank A. Jaffe and G. Slater for their generosity
in enabling our use of the chelonian dataset. We thank J. Brown
for his comments on an earlier version of this paper and for his
advice leading us toward the analytic implementation adopted for
AUTEUR. Financial support for MEA and LJH was provided by
NSF DEB 0918748 and 0919499, respectively.
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