Syst. Biol. 52(1):48–54, 2003
DOI: 10.1080/10635150390132713
Testing a Molecular Clock without an Outgroup: Derivations of Induced Priors
on Branch-Length Restrictions in a Bayesian Framework
M ARC A. S UCHARD ,1 R OBERT E. WEISS ,2
AND J ANET
S. S INSHEIMER 1,2,3
1
Department of Biomathematics, David Geffen School of Medicine, University of California, Los Angeles, California 90095-1766, USA
2
Department of Biostatistics, School of Public Health, University of California, Los Angeles, California 90095-1772, USA
3
Department of Human Genetics, David Geffen School of Medicine, University of California, Los Angeles, California 90095-1766, USA
Abstract.—We propose a Bayesian method for testing molecular clock hypotheses for use with aligned sequence data from
multiple taxa. Our method utilizes a nonreversible nucleotide substitution model to avoid the necessity of specifying either
a known tree relating the taxa or an outgroup for rooting the tree. We employ reversible jump Markov chain Monte Carlo
to sample from the posterior distribution of the phylogenetic model parameters and conduct hypothesis testing using
Bayes factors, the ratio of the posterior to prior odds of competing models. Here, the Bayes factors reflect the relative
support of the sequence data for equal rates of evolutionary change between taxa versus unequal rates, averaged over all
possible phylogenetic parameters, including the tree and root position. As the molecular clock model is a restriction of
the more general unequal rates model, we use the Savage-Dickey ratio to estimate the Bayes factors. The Savage-Dickey
ratio provides a convenient approach to calculating Bayes factors in favor of sharp hypotheses. Critical to calculating the
Savage-Dickey ratio is a determination of the prior induced on the modeling restrictions. We demonstrate our method on
a well-studied mtDNA sequence data set consisting of nine primates. We find strong support against a global molecular
clock, but do find support for a local clock among the anthropoids. We provide mathematical derivations of the induced
priors on branch length restrictions assuming equally likely trees. These derivations also have more general applicability to
the examination of prior assumptions in Bayesian phylogenetics. [Bayesian; hypothesis testing; Markov chain Monte Carlo;
phylogeny.]
et al., 1996; Whelan and Goldman, 1999). Bayes factors
have fewer difficulties in discrete spaces, because probability mass functions can be naturally substituted for
continuous distributions, and do not rely on large sample
asymptotics. Most importantly, when utilizing increasingly large data sets, a frequentist-based approach to
model selection, such as the LRT, will almost always reject the restricted model in favor of a larger one, whereas
the Bayes factor remains more conservative.
We recently published an article on Bayesian inference
in phylogenetics that included tests of molecular clocks
(Suchard et al., 2001). A molecular clock assumes that
the rate of evolutionary change remains constant over
time and across taxa, such that the evolutionary distances
between any two taxa and their most recent common
ancestor (MRCA) are equal. Imposing this restriction facilitates the incorporation of fossil evidence into analyses and allows divergence times to be estimated. Like
many previous approaches to testing molecular clocks
(Takezaki et al., 1995; Huelsenbeck et al., 2000), our tests
give weight to only a subset of possible trees. Specifically,
we limit ourselves to trees in which a particular taxon is
assumed to have diverged first and then conduct the test
conditional on the modal tree assuming this outgroup.
Although in general using these tests may pose little bias,
there are situations where there is uncertainty in the outgroup, particularly among organisms where historical
information is sparse or absent, and incorrect inference
may result. As an example, we consider inference among
closely related organisms where the fossil record is incomplete, such as primates. To assume an outgroup in
these cases may imprecisely estimate the variability of
test statistics used for inference by limiting the parameter space and can lead to erroneous results (Taylor et al.,
1996).
Bayesian approaches are gaining popularity in phylogenetics (Rannala and Yang, 1996; Li et al., 2000) in
part because of the ease with which one may compare
a growing number of competing evolutionary hypotheses using Bayesian model selection (Huelsenbeck et al.,
2001; Suchard et al., 2001, 2002). Bayesian approaches are
used to estimate the posterior distribution of unknown
model parameters given the observed phylogenetic data
using Bayes theorem. Given a statistical model M with
parameter space θ and data X, Bayes theorem states that
the posterior distribution of θ is proportional to the sampling density of the data given θ, commonly referred to
as the model likelihood, multiplied by the prior distribution of θ :
p(θ |X, M) =
f (X|θ, M)q (θ|M)
,
m(X|M)
(1)
where
the reciprocal proportionality constant m(X|M) =
R
f
(X|θ,
M)q (θ |M)dθ is the marginal likelihood of the
θ
data X given model M.
One may describe various evolutionary models by restricting one or more parameters to specific ranges of
values and then may compare these different models using a Bayes factor, the ratio of the posterior probability
of one model to that of another (the posterior odds) divided by the ratio of the prior probabilities (the prior
odds). The Bayes factor is a measure of the change in
support for one model versus another given the data
and is the Bayesian analogue of the likelihood ratio test
(LRT). LRTs have been used very effectively in phylogenetics (Huelsenbeck and Rannala, 1997) but can be remiss in that the data are sparse and the space of possible
evolutionary trees is discrete so that standard likelihood
asymptotics may not apply (Goldman, 1993; Sinsheimer
48
2003
SUCHARD ET AL.—BAYESIAN MOLECULAR CLOCK TESTS
Here, we use an irreversible model of evolutionary
change. This model allows for inference of the location of
the root, the MRCA for all taxa under study. Our primary
aim, however, is not to estimate this position because this
task cannot be accomplished with great precision (Yang,
1994; Huelsenbeck et al., 2002). Rather, we use the irreversible model to explicitly integrate over all possible
root positions, so that neither the tree nor an outgroup
need be specified in the prior when testing molecular
clocks. We then outline our prior assumptions, discuss
inference methods based on Bayes factors, and lay out the
necessary tools to calculate or estimate induced priors.
An induced prior describes the marginal prior distribution of a function of model parameters that results from
assuming a joint prior on the parameters. For example,
assume θ1 and θ2 are two model parameters and we are
interested in testing the sharp hypothesis that θ1 − θ2 = c
for some constant c. If we assume a joint prior q (θ1 , θ2 ),
then the relevant induced prior is q (θ1 − θ2 ). In complicated models, such as the graphical models in phylogenetics, determining the induced prior is not always a
straightforward activity. The induced prior is a necessary component of the Bayes factor between two models
related by a sharp hypothesis. We have employed these
tools with a data set commonly used to report on new
phylogenetic methodology and briefly discuss the general implications of our model.
S TATISTICAL M ODEL
Data and Notation
We used aligned DNA or RNA sequences to determine the evolutionary histories relating N organisms. We
assumed an irreversible continuous-time Markov chain
model for evolutionary change at equilibrium, characterized in terms of an unknown evolutionary history and an
infinitesimal rate matrix. Here, an evolutionary history
consists of a rooted, bifurcating tree τ N where root node
R is the MRCA of all N extant taxa and a vector of edge
weights (branch lengths) tb ∈ T for b = 1, 2, . . . , 2N − 2.
There exist
EN =
(2N − 3)!
2 N−2 (N − 2)!
(2)
different rooted trees connecting N taxa (Felsenstein,
1978).
Rootable Model
We used a restricted parameterization of the infinitesimal rate matrix Q introduced by Schadt et al. (1998)
(SSL98) that consists of eight free, nonnegative parameters and is given by


−
a
b/2 c/2
 d
− b/2 c/2 


(3)
QSSL98 = (q s0 ,s1 ) = 
,
g 
 e/2 f /2 −
e/2 f /2 h
−
where the minus sign represents minus the sum of
the remaining row elements and s0 , s1 ∈ (A,G,C,T/U)
49
represent the initial and final nucleotide states. The stationary distribution π = (πA , πG , πC , πT/U ) of SSL98 was
given by Schadt et al. (1998). Only the product tb × Q
enters the likelihood calculation, so without loss of generality, we constrain Trace(Q) = −1, enforcing a + b +
c + d + e + f + g + h = 1.
Although we assume stationarity, SSL98 does not necessarily satisfy detailed balance, meaning there exists at
least one pair s0 6= s1 where
πs0 q s0 ,s1 6= πs1 q s1 ,s0 .
(4)
The SSL98 model is therefore irreversible and can be used
to infer the root.
Yang (1994) proposed a more general irreversible
model (Y94) than SSL98. Y94 contains 12 parameters in
its infinitesimal rate matrix QY94 , and Huelsenbeck et al.
(2002) employed this model in a Bayesian framework.
Under Y94, calculations of finite time transition probabilities require numerical determination of the possibly
complex eigensystem of QY94 . This determination must
be repeated for each new set of parameter values examined and can become computationally expensive. An
advantage of SSL98 is that analytic expressions for the
finite time transition probabilities have been determined
(Schadt et al., 1998). Use of these expressions eases the
computational burden.
The reversible model proposed by Tamura and Nei
(1993) (TN93) is nested within SSL98 by setting
a e = df, bg = ch.
(5)
The model proposed by Hasegawa et al. (1985) (HKY85)
is further nested within these models by additionally
setting
bd = eh.
(6)
TN93 and HKY85 at equilibrium satisfy detailed balance
and therefore may not be used to infer the location of root
without further restrictions among the branch lengths
(Felsenstein, 1981).
B AYESIAN M ETHODS
Model Prior
Let θ = (τ N , T, µ, 3) be the model parameter space under SSL98, where µ is a hyperparameter used to define
the prior for T and 3 = (a , b, c, d, e, f, g, h). We employ
a vague yet completely proper prior on θ by setting
τ N ∼ Uniform over all trees with N taxa,
tb | µ ∼ Exponential(µ),
µ ∼ Inverse-gamma(α, β), and
3 ∼ Dirichlet(1, 1, 1, 1, 1, 1, 1, 1),
(7)
where α = 2.1 and β = 1.1. The notation x ∼ Y denotes that the random variable x is drawn from distribution Y. In Bayesian model specification, a random
50
VOL. 52
SYSTEMATIC BIOLOGY
variable x can represent either an estimable model parameter, e.g. τ N , tb , µ, and 3, or the observed data
themselves. When x represents the data, the resulting expression specifies the model likelihood. However, when
x is a model parameter, the expression describes a prior
distribution.
The priors on τ N and 3 are flat. Our choice of α and
β results in the hyperparameter µ having expectation 1
and a rather large variance of 10, because we know little about its tendencies a priori. The resulting marginal
prior on tb is diffuse on (0, ∞) but remains integrable.
The usual uninformative Jeffrey’s prior on tb ∈ (0, ∞)
is 1/tb . This latter prior is improper, precluding calculation of the Bayes factors necessary to test molecular
clocks. The Inverse-gamma distribution allows computation of the reciprocal moments of µ that will be needed
later.
Inference
We employed Bayes factors to infer the appropriateness of modeling restrictions within 3 and among branch
lengths. A Bayes factor (B10 ) in favor of model 1 (M1 ) over
model 0 (M0 ) is also the ratio of the marginal likelihood of
M1 over the marginal likelihood of M0 (Kass and Raftery,
1995).
When M0 is nested within M1 , so that the parameter
space of M1 is (ω, φ) and the parameter space of M0 is
(ω = ω0 , φ), where ω0 is a known constant and the assumed priors under M0 and M1 satisfy q (φ|M0 ) ∝ q (ω =
ω0 , φ|M1 ), then we may estimate B10 using the Savage–
Dickey ratio (Verdinelli and Wasserman, 1995),
p(ω = ω0 |X, M1 )
1
= B01 =
,
B10
q (ω = ω0 |M1 )
(8)
where q (ω = ω0 |M1 ) is the prior and p(ω = ω0 |X, M1 ) is
the posterior of ω under M1 , both evaluated at ω0 .
We sampled from model posteriors using reversiblejump Markov chain Monte Carlo (MCMC) computation
(Green, 1995). MCMC methods provide an algorithm to
generate a Markov chain that explores the parameter
space of a given statistical model and whose stationary distribution equals the model posterior (Metropolis et al., 1953; Hastings, 1970). Metropolis et al. (1953)
suggested generating this chain by breaking each step
into two stages. In the first stage, an application-specific
transition kernel proposes a new destination state by
stochastically perturbing the current chain state. In the
second stage, the sampler accepts or rejects the proposal
based on the Metropolis–Hastings criterion. If accepted,
the chain moves to the destination state in the parameter space; otherwise, the chain state remains in place
during this chain step. Reversible-jump MCMC extends
these methods to sample across a countable number
of nonnested parameter spaces. As an example, branch
lengths do not necessarily retain definition from tree
to tree, so jumps between trees move the chain from
one nonnested parameter space to another. From the
posterior sample, we estimated the posterior ordinate of
ω in Equation 8 using either parametric or nonparametric
kernel density estimators. Most of the details of the sampling and density estimation procedures were given by
Suchard et al. (2001). One key exception is how we propose new root locations. This proposal is accomplished
by randomly selecting a new target branch and randomly
locating the new root along this branch. We accepted
or rejected the proposal using the Metropolis–Hastings
criterion.
M ATHEMATICAL D EVELOPMENTS
Critical to Bayes factor estimation between competing
models is the calculation of the induced prior evaluated
at the nested restriction in Equation 8. Here, we develop
the induced prior for restrictions among branch lengths
given equally likely evolutionary trees. We first outline
an algorithm to draw a random tree from this set. Such
an algorithm is paramount to Monte Carlo estimation
of the induced priors. We then determine the induced
prior distribution of a branch length restriction useful in
testing molecular clocks.
Drawing a Random Tree Using Recursion
To draw a random evolutionary tree τ N with N ≥ 2
labeled external nodes drawn from the uniform distribution q (τ N ) = 1/E N , we base our algorithm on a recursive enumeration procedure (Felsenstein, 1978). We
start with a tree τ2 containing taxa 1 and 2 connected
together by a single internal node, the root. For steps
s = 3, 4, . . . , N, we draw a random node Is uniformly
from the set of 2s − 3 current internal nodes in τs−1 and
attach taxon s as the new sibling to Is , where the labeling of taxa is unimportant. Sprouting taxon s from
τs−1 to form τs involves the addition of both the external node representing taxon s and a new internal node
Ps that becomes the immediate parent to both taxon
s and Is . If Is is not the root in τs−1 , then the parent
of Is in τs−1 becomes the parent of Ps and grandparent of Is in τs . Otherwise, Ps becomes the new root
in τs .
We prove this algorithm draws from the uniform distribution by induction. For N = 2, there exists exactly
one tree τ2 . If we assume τ N ∼ q (τ N ) = 1/E N , then there
exist 2N − 1 possible new trees τ N+1 that we could create
from a single τ N , each with probability 1/(2N − 1) given
our algorithm. Therefore,
q (τ N+1 |τ N ) =
1
, and
2N − 1
q (τ N+1 ) = q (τ N+1 , τ N ) = q (τ N+1 |τ N )q (τ N )
=
1
q (τ N )
2N − 1
2 N−2 (N − 2)!
=
(2N − 1)(2N − 3)!
µ
2N − 2
2N − 2
¶
2003
51
SUCHARD ET AL.—BAYESIAN MOLECULAR CLOCK TESTS
=
=
2 N−1 (N − 1)!
(2N − 1)!
1
E N+1
.
(9)
N taxa. Then, trivially, P1,1,2 = 1, as there is only one tree
connecting A and B, who are both at level 1 away from
their MRCA, the root. Further, PnA ,nB , N = 0 if nA = 0, nB =
0 or nA + nB > N. Summing the conditional probabilities
yields the recurrence relationship
¶
µ
nA + nB
PnA ,nB , N = PnA ,nB , N−1 1 −
2N − 3
µ
¶
nA − 1
+PnA −1,nB , N−1
2N − 3
¶
µ
nB − 1
.
+PnA ,nB −1, N−1
2N − 3
Removing the final root location also generates an unrooted evolutionary tree drawn uniformly from all unrooted trees of size N.
Induced Prior Distribution of the Distance of Two Taxa
from Their MRCA
Assuming equally likely trees, we derive the prior
marginal density of the height difference 1i j evaluated at
0 for any two arbitrary taxa i and j drawn from a random
tree of fixed size N. The height difference 1i j = h i − h j ,
where h i and h j are the sum of branch lengths measured
from the MRCA of taxa i and j to taxa i and j themselves. Under a molecular clock, 1i j = 0. Because we are
considering only two taxa, the MRCA of i and j is not
necessarily the root. Assuming tb ∼ Exponential (µ),
q (1i j = 0|ni , n j , µ)
¡
¢
µ ¶
(ni + n j − 2)!0 n j − 12
1
,
= (n −n +1) √
i
j
2
π(ni − 1)!(2n j − 2)! µ
Given an Inverse-gamma(α, β) prior on µ,
q (1i j = 0)
Z Z Z
=
q (1i j |ni , n j , µ)q (µ)q (ni , n j )dµdni dn j
ni
=
(10)
where ni is the level of taxon i and n j is the level of
taxon j from their MRCA in a given tree (Suchard et al.,
2001), and 0(·) is the gamma function. The level n` of an
arbitrary node ` from a given MRCA counts the number
of edges traversed from the node to the MRCA, such that
the level of the MRCA is 0 and the children of a node at
level k are at level k + 1.
Integrating Equation 10 against the prior q (ni , n j , µ) =
q (ni , n j )q (µ) returns the sought-after marginal density of
q (1i j ) evaluated at 0 for use in a pairwise molecular clock
Bayes factor test.
We determine q (ni , n j ) given our prior q (τ N ) using
a recursion argument. Without loss of generality, we
arbitrarily assume taxa i and j are A and B, the first two
taxa connected in τ2 and, starting recursively, calculate
the probability of randomly generating a tree from q (τ N )
with (nA , nB ) = (k, `). There exist three independent
ways of generating a tree with (nA , nB ) = (k, `) from q (τ N )
given a tree with one less taxon drawn from q (τ N−1 ).
First, we start with a tree where (nA , nB ) = (k, `) and add
the Nth taxon as nearest neighbor to any node in τ N−1
that is not along the path from A or B to their MRCA.
Given that there are 2N − 3 possible nearest neighbors,
this occurs with probability 1 − (nA + nB )/(2N − 3).
Second, we start with a tree where (nA , nB ) = (k − 1, `)
and add the Nth taxon as nearest neighbor to any node
along the path from A to the MRCA with probability
(nA − 1)/(2N − 3). Third and likewise, the proceeding
event occurs for taxon B.
Let PnA ,nB , N be the probability that A is at level nA and B
is at level nB away from their MRCA in a tree containing
(11)
α
β
nj
µ
ni +n
j ≤N
X
ni ,n j =1
(ni + n j − 2)!0(n j − 12 )
Pn ,n , N .
√
2(ni − n j + 1) π (ni − 1)!(2n j − 2)! i j
(12)
Testing Restrictions among Branch Lengths
The existence of a molecular clock between two taxa
i and j restricts 1i j to be exactly 0. To test the hypothesis that 1i j = 0 unconditional on tree and without assuming an outgroup, we estimated the posterior density p(1i j = 0|X) from a sample of 1i j drawn using
our rjMCMC algorithm. The posterior p(1i j |X) may be
nonnormal and multimodal with different modes corresponding to alternative root locations. We controled for
these factors during density estimation by first simulating p(τ R |X) and recalling that
p(1i j = 0|X) =
X
p(1i j = 0|τ R , X) p(τ R |X).
(13)
τR
We then exploited Occam’s window (Madigan and
Raftery, 1994). Occam’s window states that a reasonable
estimate of p(1i j = 0|X) can be obtained by averaging
over only the fraction of all possible τ R that possess nominal posterior probability. We independently estimated
p(1i j = 0|τ R∗ , X), for each τ R∗ such that p(τ R∗ |X) ≥ ², using
a kth nearest neighbor density estimator. A kth nearest
neighbor density estimator is a multivariate, nonparametric method that works by determining the minimum
volume V of a hypersphere necessary to contain the k
closest samples to the point of interest x. The estimated
density at x is k/(nV), where n is the total sample size. If
the true density at x is locally
linear, then this method is
√
unbiased. Usually, k ≈ n is chosen (Loftsgaarden and
Quesenberry, 1965). The conditional simulations insure
52
SYSTEMATIC BIOLOGY
VOL. 52
that there are sufficient posterior samples of 1i j for each
τ R∗ to estimate the density accurately. The marginal posterior density is
p(1i j = 0|X) ≈
X
τ R∗
p(1i j = 0|τ R∗ , X) p(τ R∗ |X).
(14)
Dividing Equation 14 by the prior given in Equation 12
yields the Bayes factor in favor of a molecular clock. Pairwise or multitaxon molecular clock comparisons may
also be used as a diagnostic to identify subsets of taxa
that do or do not support a local molecular clock.
A global molecular clock among N taxa makes N − 1
restrictions, h i = h j ∀ i, j = 1, 2, . . . , N, where h i is the
distance from the inferred root to taxon i. We reparameterized these equalities into a vector of N − 1 differences
H and estimate p(H = 0|X) by averaging the conditional
densities over trees, similar to Equation 14.
We estimated the prior ordinate q (H = 0) by draw( p)
ing τ N using the algorithm presented here. We drew
( p)
µ , T ( p) |µ( p) from our priors and calculated H ( p) for
p = 1, 2, . . . , P, via a kth nearest neighbor density estimator. The Bayes factor in favor of a molecular clock
is
B MC =
p(H = 0|X)
.
q (H = 0)
(15)
NUMERICAL EXAMPLE: PRIMATE M ITOCHONDRIAL DNA
To illustrate these tests, we analyzed a set of mitochondrial DNA sequences taken from nine primates (human,
chimpanzee, gorilla, orangutan, gibbon, macaque, squirrel monkey, tarsier, and lemur) (Brown et al., 1982). The
data set comprises the protein coding regions for subunits 4 and 5 of the enzyme NADH-dehydrogenase and
three tRNAs and contains 888 sites after the removal of
alignment gaps. We previously used these data in testing molecular clocks with the HKY85 model in which
we assumed the lemur was the outgroup (Suchard et al.,
2001).
Figure 1 depicts the modal rooted tree for the primate
data set under SSL98. Branch lengths in the figure are
the posterior means conditional on the modal tree and
are drawn to scale. The model properly clusters the apes
(human, chimpanzee, gorilla, orangutan, and gibbon),
monkeys (macaque and squirrel monkey), and prosimians (tarsier and lemur).
Illustrated in Figure 1 is the modal position of the root,
located between the lemur and the MRCA of the remaining taxa. The inferred root position accounts for 93.2% of
the total posterior over all trees. Alternative rootings include those between the squirrel monkey and the remaining taxa (4.7% posterior density), between the prosimians (the MRCA of both the lemur and tarsier) and the
remaining taxa (0.7%), and between the tarsier and the
remaining taxa (0.6%). Each of the remaining possible
root positions account for <0.5% of the posterior. Based
on the fossil record and comparative biology, the MRCA
FIGURE 1. Modal rooted evolutionary tree relating the primates
under the SSL98 model. MRCA = inferred most recent common ancestor to all nine extant taxa, and its plotted location has 93.2% posterior
support. Branch lengths are the posterior means, and approximately
equal distances of the anthropoids from the root is suggestive of a local
molecular clock. Care should be taken in making this inference because
no measures of branch length uncertainty are depicted.
of all primates is assumed to lie in one of two positions.
The first position is along the branch that partitions the
lemur from the remaining taxa, and the second position
is along the branch that partitions both prosimians from
the remaining taxa (Schwartz, 1986). Thus, the estimated
posterior probability for the root position is consistent
with prior evidence.
Matrix 16 presents the posterior means and SDs of the
evolutionary parameters in 3:
Q SSL98


—
0.082(0.006) 0.074(0.006) 0.026(0.005)
 0.131(0.016)
—
0.074(0.006) 0.026(0.005) 

=
 0.071(0.004) 0.005(0.002)
—
0.241(0.012) 
0.071(0.004) 0.005(0.002) 0.193(0.012)
—
(16)
The posterior means and SDs of the reparameterization
of (a , b, c, d, e, f, g, h) into π are πA = 0.308(0.012), πG =
0.122(0.009), πC = 0.278(0.011), and πT = 0.292(0.012).
Approximately equal distances (heights) from the
MRCA for all anthropoids (N = 7) in Figure 1 offers an indication of support for a local molecular clock within this
group. However, caution is necessary, because branch
2003
53
SUCHARD ET AL.—BAYESIAN MOLECULAR CLOCK TESTS
length variability is not reported in the figure. In the
following paragraphs, we formally test the appropriateness of the SSL98 model and the validity of the existence
of a molecular clock while taking this variability into
consideration.
Testing the Appropriateness of SSL98 over HKY85
We tested the appropriateness of the SSL98 model for
these data by estimating the Bayes factor in favor of
SSL98 over HKY85. Unlike the irreversible model fit test
proposed by Yang (1994), the Bayes factor approach does
not require that we condition on a fixed tree or root position. Previously, we showed that the HKY85 model provides a better description of the data than do more reduced models (Suchard et al., 2001). We estimated the
numerator in the Savage–Dickey ratio using a multivariate normal approximation, because marginal posterior
histograms of these parameters are approximately normal in appearance (Fig. 2).
Using the three restrictions given in Equations 5
and 6, we estimated the log10 posterior density at
the restriction to be −9.22. Via Monte Carlo simulation of length P = 100,000 under the prior q (3) ∼
Dirichlet(1, 1, 1, 1, 1, 1, 1), we estimated the log10 induced prior density at the restriction to be 3.54. The log10
Bayes factor in favor of SSL98 against HKY85 was 12.76,
offering decisive support (Kass and Raftery, 1995) in favor of SSL98.
Pairwise Molecular Clock Comparisons
Based on the distribution of trees, we set ² = 0.01 and
calculated the log10 Bayes factors in favor of a molecular
clock two taxa at a time (Table 1). All log10 Bayes fac-
FIGURE 2. Marginal posterior and induced prior densities of parameter restrictions relating the SSL98 and HKY85 models. Parameters
a , b, c, d, e, f , g, and h are elements of the infinitesimal rate matrix under
SSL98. Histograms of the marginal posterior samples, plots of the multivariate normal approximation to the posterior (solid line), and plots
of the nonparametric estimation of the induced prior (dashed line) for
the three restrictions that nest HKY85 within SSL98 are shown. Under
HKY85, each of these restrictions equals 0 (dotted line). Joint density
ordinates at 0 are used in calculating the Savage–Dickey ratio.
TABLE 1. Pairwise diagnostic tests for a molecular clock among
the primates. Each entry is the log10 Bayes factor in favor of a molecular clock between the two indexed taxa. Values >1 strongly support a molecular clock, and values <−1 strongly reject the clock.
HU = human; CH = chimp; GO = gorilla; OR = orangutan; GI = gibbon; MA = macaque; SQ = squirrel monkey; TA = tarsier; LE = lemur.
LE
TA
SQ
MA
GI
OR
GO
CH
HU
CH
GO
OR
GI
MA
SQ
TA
−1.69
−1.17
0.29
0.90
0.52
1.06
1.33
0.94
−1.72
−1.24
−0.13
0.62
−0.11
0.48
1.06
−1.69
−1.17
0.29
0.94
0.44
1.01
−1.66
−1.07
0.61
1.05
1.02
−1.63
−0.92
0.88
0.93
−1.63
−1.06
0.72
−1.70
−0.69
−1.38
tors for the lemur and any other taxa and all but two for
the tarsier and any other taxa are less than −1, providing strong evidence against a molecular clock within the
prosimians and between the prosimians and the remaining taxa. Mostly positive log10 Bayes factors between the
anthropoids suggests a local molecular clock; however,
these estimates are highly correlated and will be considered jointly.
Joint Molecular Clock Tests
Via Monte Carlo simulation of length P = 100, 000, we
estimated q (H = 0) to be 0.076 for N = 9 and 0.227 for
N = 7. The SDs between 10 independent runs were 0.004
for N = 9 and 0.032 for N = 7, showing small simulation
variability.
Among all primates, we estimated log10 p(H = 0|X) =
−5.73, resulting in a log10 Bayes factor of −4.61 in favor of a molecular clock. This calculation decisively rejects (Kass and Raftery, 1995) the molecular clock model
among all primates. Among the anthropoids, we estimated log10 p(H = 0|X) = 1.80. The log10 Bayes factor
was 2.44 and offers decisive support for a local molecular clock.
R EMARKS
The log10 Bayes factor of −4.61 is substantially stronger
than the −1.3 estimate of Suchard et al. (2001). This difference results from the better fit using the larger model
and from the fact that previously we were forced to ignore a large portion of the information from the lemur
data by fixing the lemur as the outgroup and calculating a
Bayes factor between the remaining eight taxa. Inclusion
of the lemur in the analysis adds substantially greater
weight against a molecular clock (Table 1). Additionally,
the nonreversible substitution model provides a significantly better description of the data than did the HKY85
model.
Although Bayesian approaches are gaining popularity in phylogenetics, little research has focused on how
prior choice affects the prior induced on parameter restrictions that nest competing phylogenetic models. The
54
SYSTEMATIC BIOLOGY
induced prior illustrates how the prior and modeling
assumptions affect inference about other modeling assumptions; the Bayes factor used to compare these models is highly dependent on the induced prior at the nested
restriction (Verginelli and Wasserman, 1995). In general,
researchers should pay more attention to prior choice
sensitivity when using Bayesian phylogenetics. A general comparison between Bayes factor and LRT molecular clock tests is in progress.
ACKNOWLEDGMENTS
This work was partially supported by a HHMI predoctoral
fellowship and NIH training grants GM08042 and GM08185 (M.A.S.)
and USPHS grant CA16042 (J.S.S.). Java source code used to calculate the finite time transition probabilities under the SSL98 irreversible model is available at http://www. biomath.medsch.ucla.edu/
msuchard/software.html. The primate data set is available at http://
www.biomath.medsch.ucla.edu/msuchard/datasets.html.
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First submitted 19 April 2002; reviews returned 6 August 2002;
final acceptance 27 August 2002
Associate Editor: Bruce Rannala