O R I G I NA L A RT I C L E
doi:10.1111/j.1558-5646.2012.01775.x
SIMULATION-BASED LIKELIHOOD APPROACH
FOR EVOLUTIONARY MODELS OF PHENOTYPIC
TRAITS ON PHYLOGENY
Nobuyuki Kutsukake1,2,3 and Hideki Innan1,2
1
Department of Evolutionary Studies of Biosystems and Hayama Center for Advanced Studies, The Graduate University for
Advanced Studies, Hayama, Kanagawa 240-0193, Japan
2
PRESTO, Japan Science and Technology Agency, 4-1-8 Honcho Kawaguchi, Saitama, 332-0012, Japan
3
E-mail: [email protected]
Received November 27, 2011
Accepted July 26, 2012
Phylogenetic comparative methods (PCMs) have been used to test evolutionary hypotheses at phenotypic levels. The evolutionary
modes commonly included in PCMs are Brownian motion (genetic drift) and the Ornstein–Uhlenbeck process (stabilizing selection),
whose likelihood functions are mathematically tractable. More complicated models of evolutionary modes, such as branch-specific
directional selection, have not been used because calculations of likelihood and parameter estimates in the maximum-likelihood
framework are not straightforward. To solve this problem, we introduced a population genetics framework into a PCM, and here,
we present a flexible and comprehensive framework for estimating evolutionary parameters through simulation-based likelihood
computations. The method does not require analytic likelihood computations, and evolutionary models can be used as long as
simulation is possible. Our approach has many advantages: it incorporates different evolutionary modes for phenotypes into
phylogeny, it takes intraspecific variation into account, it evaluates full likelihood instead of using summary statistics, and it can
be used to estimate ancestral traits. We present a successful application of the method to the evolution of brain size in primates.
Our method can be easily implemented in more computationally effective frameworks such as approximate Bayesian computation
(ABC), which will enhance the use of computationally intensive methods in the study of phenotypic evolution.
KEY WORDS:
Approximate Bayesian computation (ABC), phenotypic evolution, phylogenetic comparative methods,
simulation-based likelihood computation.
Interspecific (or interpopulation) variation in phenotypes provides
useful insight for understanding the patterns and processes of
adaptive evolution. Because phylogenetically close species commonly exhibit similar phenotypic traits, consideration of the effect
of phylogeny is indispensable in the analysis of interspecific phenotypic variation. Because the first formal attempt to consider the
effect of the nonindependence of species in phylogeny, phylogenetic comparative methods (PCMs) have been developed and used
in a wide range of studies in comparative biology and evolutionary ecology (Felsenstein 1985; Harvey and Pagel 1991; Martins
1996; Freckleton 2009; Nunn 2011).
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355
In PCMs, most of the methods and conceptual frameworks
used to analyze continuous phenotypic traits are based on a simple
evolutionary mode, Brownian motion (BM) (Felsenstein 1985).
BM corresponds to neutral evolution or the process of genetic
drift (although selection toward a continuously fluctuating optimum can also be plugged in). The validity of using such a simple
model for phenotypic evolution has often been questioned (reviewed in Losos 2011). For example, the BM model assumes that
the evolutionary rate is constant across phylogeny, which may
be an unrealistic assumption in nature. Therefore, the BM model
has been extended to allow greater flexibility. Modifications have
C 2012 The Society for the Study of Evolution.
2012 The Author(s). Evolution Evolution 67-2: 355–367
N . K U T S U K A K E A N D H . I N NA N
included allowing different rates of BM (rate parameter or variance of BM) within a phylogeny (McPeek 1995; O’Meara et al.
2006; Thomas et al. 2006, 2009; Revell 2008; Eastman et al.
2012; Revell et al. 2012; Slater et al. 2012), applying BM with
accelerating or decelerating rates (Blomberg et al. 2003; Harmon
et al. 2010), and adjusting BM with punctuated evolution (Bokma
2008), although those models do not explicitly incorporate selection. One study modeled BM with a phenotypic mean that had an
evolutionary trend corresponding to directional selection (Hunt,
2006). Other extensions incorporated the effect of stabilizing selection for a single adaptive optimum in a particular branch or for
multiple adaptive optima in different branches or clades within
a phylogeny by using the Ornstein–Uhlenbeck (OU) process
(Martins 1994; Hansen 1997; Butler and King 2004). However,
actual phenotypic evolution does not always fit the BM or OU
models (Estes and Arnold 2007). Based on discrepancies between real processes of phenotypic evolution and the existing
evolutionary models, more flexible evolutionary models need to
be developed in the framework of PCM.
The primary reason for using BM or the OU process to model
phenotypic evolution is their mathematical tractability in the likelihood framework (Hansen 1997; Pagel 1999; O’Meara et al.
2006; Thomas et al. 2006, 2009). By using maximum-likelihood
(ML) methods, it is possible to estimate evolutionary parameters
of interest. However, difficulties in analytical likelihood computations have precluded the use of more complex and realistic
evolutionary modes in PCMs. To overcome those limitations, we
developed a new and, to the best of our knowledge, very flexible and comprehensive model to infer different modes and rates
of phenotypic evolution within a single phylogeny. In this model,
likelihood calculations are not straightforward; therefore, we took
advantage of computer simulations. Recent increases in computational power allowed us to simulate a huge number of replications
of phenotypic evolution given any mode and rate of evolution,
which made it possible to use simulation-based evaluations of
likelihood (reviewed in Marjoram and Tavare 2006). Furthermore,
we extended the method so that the approximate Bayesian computation (ABC) approach can be used (see Slater et al. 2012 for
an application of ABC to PCMs).
In our approach, evolutionary change in a phenotype is
simulated within a population genetics framework. By using a
simulation-based likelihood approach, the following advantages
may be gained. (1) Our method allows a larger number of parameters to be incorporated compared with previous models, such as
branch- or clade-specific evolutionary parameters. This is particularly useful for testing for the presence of selection that occurs
locally within the phylogeny. (2) By simultaneously estimating
evolutionary parameters, the model allows us to infer the ancestral states of internal nodes. Methods to estimate ancestral states
have been developed in previous studies of PCM (reviewed in
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Nunn 2011), but the estimation has been difficult when phenotypic evolution does not follow BM (Oakely and Cunningham
2000; Polly 2001; Finarelli and Flynn 2006; Alberts et al. 2009).
Our framework provides a method for inferring ancestral states
under a model of phenotypic evolution that does not necessarily follow BM. (3) Intraspecific variation can be incorporated
when evaluating likelihood. Recently, several methods have been
developed to cope with intraspecific variation (e.g., Ives et al.
2007; Felsenstein 2008; Hadfield and Nakagawa 2010). (4) Our
method computes likelihoods of phenotypic data directly rather
than through summary statistics. By doing so, it is possible to
avoid the controversial issue of which and how many summary
statistics should be used in ABC studies (Beaumont et al. 2002;
Csillery 2010; Leuenberger and Wegmann 2010). (5) This model
can be applied to evolution of both continuous and discrete traits.
Algorithms
SIMULATION-BASED EVALUATION OF LIKELIHOOD
We first introduce simulation-based algorithms for estimating
evolutionary parameters given a species tree (). The algorithms
essentially follow Marjoram et al. (2003) and Marjoram and
Tavare (2006) (also see references therein and Slater et al. 2012).
The species tree consists of information on tree shape and
the lengths of all of the branches in the topology, which are assumed to be known. The tree shape represents the topology, that
is, the branching pattern from the most recent common ancestor
(MRCA) to all descendant species that are being considered. The
number of species is denoted by n. Let τi be the length of the ith
branch on the tree in a given unit (e.g., time or genetic distance).
Let θ0 be the focal phenotype of the MRCA, and we can simulate its evolutionary changes to the n descendant species, =
{θ1 , θ2 , θ3 , . . . θn }. By simulating a number of patterns of phenotypic evolution given a certain parameter set, we can evaluate
the likelihood of the parameter set by comparing the simulation
results with observed data, = {1 , 2 , 3 , . . . n }. Suppose
that, given a parameter set (), a large number (say, J replications) of simulations are performed. Then, the likelihood of is
approximately given by
L(|) =
J
1
Pr(| j ),
J j=1
(1)
where j represents a simulated phenotype dataset for the jth
replication. Pr(|j ) is the probability of the observed data given
j , which can be computed as a joint probability:
Pr(| j ) = Pr(1 , 2 , 3 , . . . , n |θ1 , θ2 , θ3 , . . . , θn ), (2)
where θi is the simulated value of the ith species. This form is
rigorous and flexible enough to incorporate information regarding
A C O M P U TAT I O NA L F R A M E W O R K F O R P H E N OT Y P I C E VO L U T I O N
intraspecific variation. However, this computation may not be
feasible in many cases. Therefore, we propose a conventional
method that uses composite probability. That is, equation 2 is
replaced by
Pr(| j ) =
n
Pr(i |θi, j ),
(3)
i=1
where θi,j is the simulated value of the ith species in the jth replication. This equation should work well when the sampled species
are reasonably diverged. Then, we might be able to approximate
Pr(|θ) by a normal distribution with mean θ. Again, it is difficult
to determine the standard deviation (SD) of the normal distribution (σ), but it might be reasonable to base σ on the observed
distributions of phenotypic values for each species.
In our framework, Pr(|θ) does not need to be a normal
distribution. Any function can be used for Pr(|θ), which makes
it possible to apply this method to a trait that does not follow
a normal distribution. By using this algorithm with a substantial
number of simulation replicates, likelihoods can be computed for a
wide range of parameter sets, and ML estimates of the parameters
can be obtained.
ABC ALGORITHM
We also developed an ABC algorithm (Marjoram et al. 2003;
Marjoram and Tavare 2006; Slater et al. 2012) that is flexible
enough to take prior information into account. The parameter set
() can also be inferred in the ABC framework, and the process
can be implemented as follows.
(1) Determine prior distributions for all parameters in .
(2) For each parameter, generate a random value from the prior
distribution. A random set of the parameters is denoted by
.
(3) Simulate phenotype data ( ) using .
(4) Accept with a probability that is proportional to
Pr(| ).
(5) Go back to step (2) until a large number of accepted ’s has
accumulated.
The result includes posterior distributions for all of the parameters in as a list of accepted ’s.
EVOLUTIONARY PHENOTYPE MODELS
The algorithms introduced above are flexible enough to incorporate any evolutionary model, from the commonly used BM and
OU models to complicated models that one wishes to assume as
alternatives. Our framework does not specify the genetic background of phenotypic evolution, but it is possible to incorporate
both a single-locus system and a multilocus system by setting
mutational effects on phenotype when desired. The framework
can also be used when the genetic background of a target trait is
unknown, which is generally the case in most phenotype studies.
As a first example, we describe a method for simulating
phenotypic evolution in a species tree. This method essentially
follows the BM model, but it is more flexible because the effect
of selection can be incorporated. Changes in phenotypic values
on individual branches can be simulated as follows. We consider
this process within a population genetics framework. In brief,
mutations occur randomly at a certain rate (u), and each mutation changes the phenotypic value. We assume that changes in
phenotypic values occur because of single mutations that follow
an arbitrary distribution, φ. This can be any distribution, from a
discrete distribution (corresponding to a binary phenotypic trait)
to a continuous distribution, depending on the nature of the focal
phenotype. Mutations occur and either increase or decrease the
phenotypic value at the rates u+ and u− , respectively, with the
assumption that their absolute phenotypic effects follow φ+ and
φ− , respectively. In many cases, it would be biologically reasonable to assume that u+ = u− and φ+ = φ− because they simply
reflect the mutational process. Then, natural selection works to
determine whether these mutations fix in the species or go extinct.
Only fixed mutations contribute to long-term evolution, which is
our main interest. If a phenotypic change is completely neutral,
the fixation rate is 1/2N in a diploid species, where N is the population size. The fixation rate is higher than 1/2N for advantageous
mutations and lower for deleterious mutations. We are primarily
interested in the number of fixed mutations; therefore, we let μ+
and μ− be the rates of fixed mutations that increase and decrease
the phenotypic value, respectively. Thus, the effect of adaptive
(directional) selection is included in μ. If there is no selection on
the phenotype, we expect μ+ = μ− .
Under this framework, we need to set μi + , μi − , φi + , and φi −
to simulate phenotypic evolution on the ith branch. As mentioned
above, we assume φ+ = φ− to be constant throughout the study.
Neutral evolution can be designated by assuming that μi + = μi − ,
which is essentially identical to BM. We refer to the model where
μi + = μi − is assumed to be true on all branches as the neutral
random BM (NRBM) model.
The simulation of neutral phenotypic evolution on the ith
branch with length τi is implemented as follows:
(1) Determine si + and si − , the numbers of fixed mutations that
increase and decrease the phenotypic value, respectively.
si + (si − ) is a random integer from a Poisson distribution
with mean μi + τi (μi − τi ).
(2) Determine the effect of each mutation on the phenotype,
which is a random value from the distribution φi . Then,
by adding the effects of all of the mutations, compute the
phenotypic value at the bottom node of the branch.
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θ = −α(θ − β)dt,
(4)
where θ is the phenotypic value before the mutational change,
and α and ß represent the strength of stabilizing selection and the
adaptive optimum state, respectively. In the original description
of the OU model, the right-hand side of equation 4 had a second
term that added a random factor in the stochastic process. As an
alternative, we introduce a modified version of the OU model,
called the random OU (ROU) model. Our model is similar to the
OU model; the critical difference is that the random process in
our NRBM is employed instead of the original random factor. In
practice, we set the condition that θ is equal to the phenotypic
value before the mutation, which determines the expected change
in the phenotype as calculated by equation 4. Then, a random factor is added such that random changes follow φ+ with probability
μ+ /(μ+ +μ− ) and φ− with probability μ− /(μ+ +μ− ).
We incorporated ROU in our framework because the OU
model has been commonly used in previous studies using PCMs.
However, the estimation of α may be meaningless especially when
we set dt in equation 4 equals 1. In this case, it is reasonable to
consider that 0 < α ≤ 1 is the possible range of this parameter.
When α is equal to or close to 1 within this range, the phenotype
value is expected to instantaneously reach the adaptive optimum
(β) through one or a few phenotypic changes. Therefore, if the
model assumes a gradual approach toward the adaptive optima,
the condition α<<1 should be met. If α >1, the phenotypic value
will oscillate and gradually separate from the adaptive optimum.
Finally, as is illustrated in an imaginary example (Fig. 1), in our
framework, α does not affect the outcome of OU-like phenotypic
evolution toward the adaptive optimum unless α is extremely
small. This is because the phenotypic value that is obtained by the
OU process is largely determined by ß, and α only determines its
trajectory. Consequently, the likelihood values of models do not
differ within a wide range of α. Therefore, we did not quantify α
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β=1000
1000
0.3
phenotypic value
Using this procedure, we can simulate the evolution of the
phenotype on the entire tree, from the common ancestor (θ0 ) to
all of the descendant species, = {θ1 , θ2 , θ3 , . . . θn }. It should be
noted that because the effect of the mutation rate, u, is included
completely in the evolutionary rate, μ, together with the effect of
selection (see below), the system does not directly involve u.
The incorporation of selection is straightforward using this
system. In contrast to the NRBM model, models that include
selection on at least one branch are referred to as selection random
BM (SRBM) models. If selection favors mutations that increase
the phenotypic value, then we expect μi + > μi − (assuming φ+ =
φ− ); the reverse is also assumed to be true. It is also possible to
simulate changes in the phenotypic value using the OU model. In
this model, the expected evolutionary change in the phenotypic
trait (
θ) because of a single mutation is given by
0.2
800
0.1
0.05
600
0.025
400
0.01
0.005
200
0.001
0
0
50
100
elapsed evolutionary time
150
Figure 1. An imaginary example of an evolutionary trajectory
under ROU with different α values. The phenotypic value (y-axis)
reached an adaptive optimum (set to 1000) from a phenotypic
value at MRCA (set to 0) after a given evolutionary time (x-axis,
set to 150 time units) under different values of α (indicated on
each line).
intensively, as we did for other parameters, and only used α to test
for the presence of stabilizing selection against neutral evolution
(see below).
In addition to these commonly used models, one can simulate evolutionary changes in phenotypes using any model. In other
words, as long as simulations can be conducted, any model can
feed into the likelihood and ABC algorithms described herein.
This even allows us to use very complicated selection models
(e.g., a model that incorporates different modes and intensities of
selection that are assigned to different branches), but parameter
inference in models with large numbers of parameters is computationally intensive.
Application
We applied our method to examine the evolution of brain size in
four great ape species, including humans, whose brain size has
enlarged rapidly relative to other species (Fig. 2). We assumed
that the phylogenetic relationships among the four species and
branch lengths (actual time) were accurately given by the software
“10ktrees” (Arnold et al. 2010). We used brain size data reported
by Bauchot and Stephan (1969) after removing six extreme outliers (Fig. 2). It is a common practice to log transform this type
of data; therefore, we analyzed the data using both untransformed
and log-transformed data. Because we obtained essentially identical results, we only show the results obtained using untransformed
data to explain the algorithm.
We estimated the parameters in three different evolutionary
models: NRBM, SRBM, and ROU. We used NRBM as a null
model in which we assumed a constant evolutionary rate on all
A C O M P U TAT I O NA L F R A M E W O R K F O R P H E N OT Y P I C E VO L U T I O N
0.2
0.1
orang-utan
[6]
0
0.2
[5]
0.1
gorilla
0
0.2
[1]
[4]
0.1
chimpanzee
0
[2]
proportion of data
MRCA
0.2
[3]
5
0
15
10
branch length (mya)
human
0.1
0
0
500
1000
1500
2000
brain volume (cm 3 )
Inter- and intraspecific variation in brain volume in four species of great apes, including humans, together with their phylogeny.
Data are shown in the Nexus format: ((1:8.65,(2:6.18,3:6.18):2.48):6.48,4:15.13); [1] gorilla, Gorilla gorilla (mean = 467.12, SE = 7.28, n =
Figure 2.
103); [2] human, Homo sapiens (mean = 1321.27, SE = 13.76, n = 81); [3] chimpanzees, Pan troglodytes (mean = 348.08, SE = 3.46, n =
179); [4] orang-utan, Pongo pygmaeus (mean = 334.76, SE = 8.43, n = 71).
branches, that is, μi + = μi − = μ0 , where μ0 is the baseline evolutionary rate. The model also assumed that φi + = φi − = φ0 for all
branches, where φ0 is given by an exponential distribution with
mean δ = 1. The use of an exponential distribution corresponds to
a well-known pattern in which large mutational effects that affect
the phenotypes of quantitative traits are rare, and most changes are
small (e.g., Orr 2005; Yang 2006). Thus, under this simple setting,
the NRBM model had two parameters, the baseline evolutionary
rate (μ0 ) and the phenotypic value of the MRCA (θ0 ). For the two
alternative models, SRBM and ROU, because we had predicted
a priori that brain size had increased in the human lineage after
splitting from the common ancestor with chimpanzees, an accelerated rate of brain size evolution was assumed only in the external
branch that led to humans, and a constant rate of neutral evolution was assumed in all of the nonhuman branches (i.e., μi + =
μi − = μ0 ). The models also assumed that φi + = φi − = φ0 for
all nonhuman branches. For the SRBM model, directional selection was incorporated in the human lineage by introducing a new
parameter, k, that was defined such that the upper evolutionary
rate increased by a factor k (i.e., μi + = μ0 k) and the rate in the
other direction decreased by a factor k (i.e., μi − = μ0 / k). When
k = 1, the situation is identical to the neutral case. This version of
the SRBM model had three parameters, θ0 , μ0 , and k. The number of free parameters in the ROU model was four because it included two selection-related parameters (α and β) in addition to θ0
and μ0 .
We first used the simulation-based likelihood approach. Under the null NRBM model, wide ranges of the two free parameters
(θ0 and μ0 , ranges shown in Table 1) were considered. Loglikelihood values for all possible pairs of the two parameters were
computed with at least 100,000 replicate simulations for each parameter set. In evaluating Pr(|θ) in equation 2, we used a normal
distribution for the brain size in each species:
1
(θi − i )2
Pr(i |θi ) = √
exp −
,
2σi2
2πσi
where i and σi are the mean and SD of the observed phenotypic
value of the ith species (i = {1,2,3,4}, corresponding to the four
species), respectively. Figure 2 demonstrates that the intraspecific
variation in the four species can be approximated by a normal
distribution.
All of the programs were written in the C language (available upon request), and the computations were performed using a
MacPro (OS 10.6.7, 2 × 2.93 GHz Quad-Core Intel Xeon). Note
that the ML estimation took a very long time; it took approximately 4 (for μ0 = 1) to 150 min (for μ0 = 10,000) to calculate
the likelihood of one parameter set with 100,000 replications of
the simulation in the BM model.
The obtained log-likelihood surface is presented in Figure 3,
which shows that the maximum log-likelihood (MLL = −30.150)
was given when θ0 = 490 and μ0 = 4075 (also see Table 1). This
estimate is conditional on our arbitrary assumption that δ, the
mean of φ, equals 1. To investigate the effect of this arbitrary
choice, we repeated the ML computation with different values
of δ (0.5, 5, and 10). We found that δ2 and μ0 were strongly
confounded; values of δ2 μ0 were nearly identical, approximately
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Table 1.
Summary of the parameters used in this study.
Parameter
Common parameters in all models
Most recent common ancestor (MRCA)
Baseline evolutionary rate per million years
Parameters specific to selection models
Directional selection
Strength of stabilizing selection in OU process
Adaptive optimum in OU
Maximum log likelihood
Range (increment)
NRBM
SRBM
ROU
θ0
μ0
330–1330 (5)
1–10,000 (5)
490
4075
365
15
365
5
k
α
β
1.0–20.00 (0.1)
0–1.0 (0.1)
330–1500 (5)
–
–
–
−30.150
10.3
–
–
−22.287
–
0.8
1325
−21.857
values (0.5, 5, 10) indicated that in this selection model, although
δ strongly affected the estimates of μ0 and k, their product, δμ0 k,
remained almost constant (δμ0 k = 153.75, 157.5, and 142.0).
This is easy to understand if we notice that the most meaningful
quantity is the expected change in the phenotypic value on the
lineage under selection, which is denoted by sel . In our SRBM,
sel is given by
log likelihood -30
μ− −
sel = kμ δ −
δ τ.
k
θ0
330 0
-100
µ
1330
10000
Figure 3.
Two-dimensional log-likelihood surface in NRBM. The
arrow points to the parameter set with the maximum likelihood.
4025, 4750, and 4600 for the three different δ values. The product
of δ2 and μ0 should represent the extent of random fluctuations in
the phenotypic values. Therefore, unless we are directly interested
in μ0 , we can treat δ as a nuisance parameter, even when we do
not know δ, and an arbitrary choice of δ (unless biologically
unreasonable) might work reasonably well because its effect can
be cancelled out by μ0 .
We also performed the ML estimation using the two alternative models, the SRBM and ROU models. In addition to θ0 and μ0 ,
the former model includes one more parameter (k), and the latter
includes two more (α and β). In the SRBM model, k varied between very small values (i.e., k ∼ 0 so that μi − >> μi + ) and 20.0
(i.e., μi + = 20; μ0 is 400 times more likely to occur than μi − =
μ0 / 20). Note that the SRBM model in which k = 1 is the same
as the NRBM model. We found the MLL = −22.287 at (θ0 , μ0 ,
k) = (365, 15, 10.3) (Table 1). We again examined the effect of
our arbitrary choice of δ. Additional simulations with different δ
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+ +
With our conventional setting (μ+ = μ− and δ+ = δ− ), sel is
given by δμ(k − 1 / k)τ, and if k >> 1, sel is approximately given
by δμkτ. Therefore, it makes sense that δμk remained constant in
our simulations. For our estimates, this quantity, sel , is given by
1 × 15 × 10.3 × 6.18 (million years) = 954.81.
In the ROU model, MLL = −21.839 was found at (θ0 , μ0 ) =
(365, 5) with (α, β) = (0.8, 1325) (see Table 1), but likelihood
was very similar among models with different values of α (see
Fig. 1). As was mentioned above, sel is generally determined
by the difference in β, unless α is very small, and the expected
phenotypic value at the beginning of the lineage under selection,
which is denoted by θ . If θ is replaced by θ0 , sel is 1325 – 365 =
960, which is in good agreement with the SRBM model (sel =
954.81).
Overall, the MLL values in the two selection models were
much larger than in the neutral model, which supports the hypothesis that selection occurred specifically in the human lineage.
Model fit was tested by a likelihood-ratio test. When comparing
NRBM and SRBM, the difference in the ML (after log transformation) was 7.86, which is about two times larger than the value
at the 5% cut-off value by the χ2 distribution (df = 1). The observed difference was highly significant (P < 0.001), indicating
that the SRBM explained the data much better than the NRBM
did. A closer investigation of the simulation results indicated that
the 95% confidence interval for k under the null SRBM was as
high as k = 5.2 when μ0 = 15 was fixed (i.e., k × μ0 = 78),
indicating that our ML estimate of k (10.3) was much larger than
could be detected in our framework.
A C O M P U TAT I O NA L F R A M E W O R K F O R P H E N OT Y P I C E VO L U T I O N
A similar result was obtained from the comparison between
NRBM and ROU. The ML improved significantly in the ROU
model (χ2 df =2 = 16.60, P < 0.001). The likelihood of the ROU
model was not statistically different from that of the NRBM when
α was less than 0.018. This result has two implications: (1) phenotypic evolution toward β was too weak to be distinguished from
neutral phenotypic evolution when α was below this value, and
(2) values of α larger than 0.018 do not create a difference in
likelihood between ROU and NRBM (see Fig. 1).
The estimated evolutionary rate was much higher for the
NRBM (μ0 = 4075) than for the two selection models (μ0 = 15
and 5). This is probably because the human data, which exhibited
an extreme deviation from data for the other three species, had
to be accounted for by mutation alone in the NRBM. In contrast,
in the two alternative models, this exceptional increase in brain
size could be explained by selection; therefore, we obtained much
lower estimates of the baseline evolutionary rate. The effects of
selection were evaluated separately through the selection parameters k in the SRBM and α and β in the ROU model. For the same
reason, the estimates of θ0 in the two alternative models were
much lower (θ0 = 365 for both SRBM and ROU) than those in
the NRBM (θ0 = 490).
The difference between the two alternative models in the
estimated evolutionary rate for the human lineage was simply
due to the nature of the selection models. In the ROU model,
the phenotypic value quickly approached the optimum, whereas
the process involved an almost linear increase in the SRBM.
Therefore, for a very drastic increase in the phenotype value,
the ROU requires a smaller number of changes than the SRMB
does. This is why we proposed that the most important quantity
to represent the effect of selection was sel , which was consistent
between the two selection models.
Next, we applied our ABC algorithm to the same data. We
only used the alternative models because the NRBM is simply
a special case of the alternatives (the SRBM model with k =
1 or the ROU model with α = 0). We assumed that the prior
distributions of all of the parameters were uniform; therefore,
we expected to obtain results that were essentially identical to
those from the likelihood approach. As we expected, the peaks of
the posterior distributions roughly agreed with the ML estimates
(Figs. 4, 5), with the exception of k in the SRBM. The inferred
value of k in the ML approach was 10.3, but the mode of the
posterior distribution of k was 1.2 with the ABC method. This
discrepancy occurred because μ0 and k are strongly confounded.
As shown in Figure 4C, the posterior distribution of the μ0 –k pair
falls roughly along the curve μ0 k = 170 (dashed line in Fig. 3C),
and the most accepted parameter set was located at (μ0 , k) = (15,
10.8), which is almost identical to the ML estimate (15, 10.3). In
the ROU, the posterior distribution of α was almost flat, which
was similar to the likelihood curve in the ML approach. As was
mentioned earlier, estimates of α are quite meaningless in our new
framework (also see Fig. 1).
There are several ways to statistically compare different models and to judge whether the null model (NRBM) should be rejected in favor of alternative models in our ABC framework. One
method is to look at the posterior probability of support for the alternative model over the null model (see Slater et al. 2012). In the
case of the SRBM, all of the ks in the accepted data were greater
than 1, which suggests robust support for the SRBM over the
NRBM. To handle confounding effects, sel is more appropriate
for comparing different models in the ABC approach, rather than
looking at each parameter separately. This is particularly effective in the case of the ROU because it is not easy to calculate the
posterior probability of support based on α, which should always
be 0 or larger. Figure 6 shows the posterior distributions of sel
for the SRBM and ROU models. All of the accepted data for sel
in the SRBM and 99.99% of the data in the ROU were greater
than 0, which can be considered strong evidence for selection.
Even more conservatively, the posterior distribution of sel can
be compared to its null distribution under the NRBM (hereafter,
called null ). null was obtained through simulation with 1000
replications using the posterior distribution of μ0 . We found that
the overlap between sel and null was very small, only 0.02% for
SRBM and 4.4% for ROU, which again provided strong evidence
for selection.
Thus, the algorithm worked reasonably well with simple uniform prior distributions for all free parameters. However, the acceptance rate was very low, and therefore the simulation took a
very long time (13 and 37 h for the SRBM and ROU models,
respectively). The performance of the ABC algorithm could be
improved if appropriate prior distributions could be determined.
Figures 4 and 5 show that when uniform prior distributions were
used, we obtained wide posterior distributions of μ0 and θ0 . This
is most likely because simulated data that had a high evolutionary
rate in combination with larger values of θ0 and weak selection
parameters were occasionally accepted (see above). This problem
can be solved if we take advantage of our a priori prediction: the
effect of selection is negligible except in the human lineage. If this
holds true, we may be able to apply the NRBM model to the three
nonhuman species. This analysis would provide better inference
for θ0 and μ0 because none of the lineages would involve strong
selective forces. In this way, we obtained posterior distributions
of μ0 (mode: 25; 95% credible interval: 20.44–8820.87) and θ0
(mode: 350; 95% credible interval: 336.52–444.08) for the three
nonhuman species. These distributions were much narrower than
the ones obtained from the four-species models (Figs. 4 and 5).
These posterior distributions were then used as prior distributions
when using the ABC algorithm with the SRBM that included all
four species. With this approach, the computational time was reduced significantly, and as a result, the effect of selection was more
EVOLUTION FEBRUARY 2013
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A
20000
mode: 375
CI [336.52 - 491.14]
Frequency
10000
0
300
30000
8000
mode: 10
CI [7.47 - 500.73]
4000
15000
0
0
400
500
600
700
0
800
mode: 1.2
CI [1.16 - 18.78]
100
200
300
400
0
5
10
15
20
B
20000
mode: 365
CI [336.79 - 421.47]
10000
0
300
mode: 10
CI [7.15 - 197.36]
30000
4000
15000
0
400
500
600
700
800
mode: 2.4
CI [1.40 - 18.82]
8000
0
0
100
θ0
200
300
400
0
5
10
µ
C 1000
15
20
k
δ * k * µ = 169.6
δ*k*µ
160
140
120
100
µ 500
80
60
40
20
0
1
5
10
k
15
20
Figure 4. Posterior distributions of parameters in the SRBM obtained by the ABC method with 50,000 accepted parameter sets. (A) Results
with uniform prior distributions. (B) Results with nonuniform prior distributions. In each panel, the line shows the prior distribution of
each parameter. The mode of each estimated parameter value and its 95% credible interval are also shown. (C) Relationship between
the evolutionary rate (μ0 ) and the strength of directional selection (k) on the value of δμk.
pronounced in the human lineage (Fig. 4). In the ROU model,
the computational time was also substantially reduced (4.2 and
2.1 h for SRBM and ROU, respectively), although the posterior
distributions did not change dramatically. This is because, as was
mentioned above, this model could predict a very quick increase
in the phenotypic value with only a small number of substitutions,
even under weak stabilizing selection (α). Therefore, the effect of
selection was well evaluated regardless of the prior distribution
of μ0 . Again, 100% of sel for SRBM and 99.99% for ROU were
positive. Compared with null , the posterior probabilities of support for SRBM and ROU were 99.02% and 95.10%, respectively,
which were still sufficient to reject the NRBM model.
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EVOLUTION FEBRUARY 2013
THE POWER TO DETECT SELECTION
Here, we report the power and error rates for our frameworks. As it
is possible to calculate error rates and power in various settings, we
evaluated the power of our approach in a relatively simple setting
(Fig 7). We created symmetrical and balanced phylogenies with
different numbers of species (4, 8, 16, and 32). Then, using the
SRBM, we considered a case in which positive selection worked
on one external branch. We performed 100–1000 replicates of
simulations. For each replication, we applied both the SRBM and
the NRBM in the ML and ABC frameworks. We explored the
power of this approach using various levels of selection (i.e., k).
In the ML framework, the power was measured as the proportion
A C O M P U TAT I O NA L F R A M E W O R K F O R P H E N OT Y P I C E VO L U T I O N
A
20000
mode: 370
10000
CI [334.87 - 798.33]
mode: 10
CI [14.34 - 3645.64]
Frequency
10000
3000
20000
mode: 0.2
CI [0.03 - 0.97]
1500
0
600
0
1200
10000
0
0
400
2000
4000
mode: 1300
CI [1066.17 - 1480.68]
0
0
800 1000 1200 1400
1
B
10000
mode: 360
20000
CI [336.62 - 433.08]
mode: 10
3000
CI [4.30 - 1750.87]
10000
1500
0
0
300
400
500
600
0
0
1000
θ0
Figure 5.
2000
20000
mode: 0.6
CI [0.03 - 0.97]
mode: 1300
CI [1065.6 - 1480.86]
10000
0
0
1
800 1000 1200 1400
α
µ
β
Posterior distributions of parameters in the ROU obtained by the ABC method with 50,000 accepted parameter sets. (A) Results
with uniform prior distributions. (B) Results with nonuniform prior distributions.
Frequency
5000
5000
mode: 940
CI [637.34 - 1264.21]
SRBM
4000
4000
3000
3000
2000
2000
1000
1000
0
0
0
1000
2000
0
3000
sel
Figure 6.
ROU
mode: 900
CI [544.83 - 1189.88]
1000
2000
sel
Posterior distributions of sel in the SRBM and ROU models.
of simulation replications that rejected the null model at the 95%
level. In the ABC framework, we concluded that the SRBM was
supported more than the NRBM if the posterior distribution of
k did not overlap to 1. As a result, power increased with k in
both the ML and ABC frameworks (Fig. 7). Power was higher
with a larger number of species in the ML framework, but it was
generally comparable among models with different numbers of
species. We did not investigate the power of the ROU because it
is meaningless to investigate different α values when using our
approach (see above and Fig. 1).
To investigate error rates, we performed power simulations
with the assumption that k = 1 (i.e., NRBM) but with different
evolutionary rates (μi + = μi − = 1, 5, 10, 20, 50). The calculation
of the error rate for the ROU in the ABC framework was not
EVOLUTION FEBRUARY 2013
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proportion of data (power)
A: SRBM (ML)
B: SRBM (ABC)
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
4 spp.
8 spp.
16 spp.
32 spp.
0.2
0
1.5
2
3
5
0
10
1.5
2
3
k
proportion of data (error rate)
C: SRBM (ML)
10
D: SRBM (ABC)
0.1
0.1
4 spp.
8 spp.
16 spp.
32 spp.
0.05
0.05
0
0
1
5
10
20
50
1
5
10
20
50
μ
μ
E: ROU (ML)
proportion of data (error rate)
5
k
F: ROU (ABC)
0.1
0.1
0.05
0.05
0
0
1
5
10
20
50
μ
1
5
10
20
50
μ
Figure 7. (A, B) Power analysis for the SRBM with different k values (shown on the x-axis) using the ML (A) and ABC (B) approaches.
The lengths of all of the branches were assumed to be constant (5 time units), and the SDs of all of the tip phenotypes were set to 5. μ0
for the simulated data was set to 10. We fixed MRCA to 0 because this is a baseline value for all of the phenotypic values; therefore, it is
less likely that different MRCA values will affect the statistical performance. (C–F) Error rates for SRBM (C: ML; D: ABC) and ROU (E: ML;
F: ABC) using an imaginary dataset that included different numbers of species (n = 4, 8, 16, 32). The influence of each parameter was
investigated by changing the value of the parameter while holding the other parameters fixed. μ0 is shown on the x-axis.
straightforward. Ideally, we should test whether a posterior distribution of α overlaps with 0 (i.e., no stabilizing selection). However, a negative value of α as a prior distribution is biologically
questionable; therefore, it is impossible to use this approach. Instead, we examined the posterior distribution of sel and deter-
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mined whether this distribution overlapped 0. We found that the
type-I error rate was less than 5% in both the ML and ABC
frameworks (Fig. 7), and the number of species in the model did
not affect this result. These results suggest that our framework is
conservative when rejecting the NRBM.
A C O M P U TAT I O NA L F R A M E W O R K F O R P H E N OT Y P I C E VO L U T I O N
We also investigated the error rates and power assuming
the phylogeny used for the primate brain analysis by conducting
additional simulations in the same way as reported above. As a
result, we found that the estimated parameter sets had low error
rates and high power (see Supporting information for details).
Discussion
This article introduces a new framework that can be used to infer evolutionary parameters under different models of phenotypic
evolution. The primary advantage of this framework lies in its
flexibility; it can incorporate any evolutionary model with a number of parameters of interest. This flexibility stems from a recent
dramatic increase in computational power, which makes it possible to apply simulation-based computations of likelihood. That
is, likelihood can be obtained without analytic expressions as
long as simulation is possible (reviewed in Marjoram et al. 2003
and Marjoram and Tavare 2006; Slater et al. 2012). This framework will be useful for analyses of complex phenotypic evolution
on phylogeny that cannot be analyzed using traditional PCMs.
Similar simulation-based likelihood approaches are commonly
used in studies of molecular evolution and population genetics
(Marjoram and Tavare 2006), but so far they have been less commonly used in PCMs of phenotypic evolution.
The greatest advantage of the simulation-based approach is
that we are able to incorporate very realistic models of phenotypic
evolution. Evolutionary changes in the phenotype are modeled
based on the framework of population genetics. Our approach
makes it possible to consider many parameters that are related
to evolutionary modes and tempos. Increases in the number of
parameters, which were confined to interactions among δ, μ, and
k in this study, may result in confounding interactions among parameters. In such cases, it will be important to test the evolutionary
hypothesis by considering such confounding interactions among
parameters, but not by treating each parameter separately. Our
framework can also include branch- or clade-specific BM evolution, which has been examined in previous studies (O’Meara
et al. 2006; Thomas et al. 2006, 2009; Venditti et al. 2011;
Eastman et al. 2012, Revell et al. 2012; Slater et al. 2012, see
Butler and King 2004 for OU). In addition, one important extension of our framework is that is allows us to incorporate branchor clade-specific parameters for directional selection. Previously,
Hunt (2006) modeled an occurrence of directional selection in
one lineage. Hunt (2006) used sequential fossil data, whereas
this study only used tip data for extant species, and the phenotypic value of the root in which directional selection was assumed
to have occurred was unknown. Therefore, our method is similar to Hunt’s (2006) but differs in that this framework attempts
to model the likelihood of phenotypic evolution in the entire
phylogeny.
As an example, our simulation-based algorithms were applied to the evolution of brain size in great apes. In this phylogeny,
increases in brain size are thought to have occurred specifically
in the human lineage after it split from the common ancestor
of humans and chimpanzees. We found that the selection models explained the data much better than the null model that did
not include selection (Table 1), which matched our a priori understanding. Our approach used the information of intraspecific
variation in the computation of likelihood and provided an estimate of the ancestral state of the phenotypic value in a model
that included a heterogeneous evolutionary pattern within a phylogeny. Although estimates of the ancestral state based on the
BM have been available in previous PCM methods, the inclusion
of branch-specific evolutionary modes (i.e., directional selection)
alters the estimate of the ancestral state (Oakely and Cunningham
2000; Polly 2001; Finarelli and Flynn 2006; Alberts et al. 2009).
We also introduced an ABC algorithm that was more flexible
and efficient than the likelihood approach (cf. Slater et al. 2012).
In our example with the SRBM, we first confirmed that a posterior
distribution of the accepted parameter set (θ0 , μ0 , and k) in the
ABC with noninformative (flat) prior distributions agrees with
the result of the ML analysis. The effective incorporation of prior
distributions not only reduces the computational load but also
makes it possible to take the uncertainties of nuisance parameters
into account. This would work well, particularly when we have
reliable phenotypic values from fossil records or a strong evolutionary hypothesis because researchers can use an informative
prior distribution. The computational treatment for such special
cases will be published elsewhere.
Thus, by taking advantage of dramatic improvements in computational power, we can use a flexible and powerful approach
to infer models and parameters of phenotypic evolution. This
simulation-based likelihood approach is particularly useful for
complex data with many taxa, although full likelihood of simple
data like the brain evolution in four primate species might be obtained analytically. The described framework is very simple and
it could benefit from a number of potential improvements and
extensions, both biological and statistical. Biologically, the oversimplified assumptions that were used in this article can easily
be relaxed so that parameter inference can be performed under
more realistic conditions. For example, we can assume that different modes of selection with different strengths act on different
branches. It is even possible to model variation in the selection
mode and intensity, even within a single branch. This can be applied to situations where drastic environmental or climate changes
are known to have occurred during the period represented by a
single branch. Additionally, although we assumed that the species
tree (both topologies and branch lengths) was known, we can easily include uncertainty in the tree by incorporating it as a prior
distribution. To correctly interpret the results of this framework, it
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is strongly recommended that researchers examine the power and
error rates of the model in parallel with the simulation to test the
evolutionary hypothesis. Simulations should also be effectively
used to assess statistical differences between different models,
as was done in our analysis when comparing the null NRBM
model with alternative models that included branch-specific
selection.
An issue worthy of further consideration is the generation
time effect. It is quite common that species at the tip of a phylogeny have different life histories, including generation time,
which might cause substantial variation in the evolutionary rate
among branches. The need to consider generation time has been
discussed in previous PCM studies (e.g., Polly 2001). Given the
biological importance of generation time in PCM, it might be
useful to adjust μ0 proportionally with the species-specific generation time instead of using a constant baseline evolutionary rate,
μ0 , throughout the entire phylogeny. One would need to investigate whether such an adjustment would improve a model’s fit
to the data compared with a model without the adjustment. This
idea is analogous to previous models that applied plural BM with
different rate parameters to a phylogeny (O’Meara et al. 2006;
Thomas et al. 2006, 2009), although these previous models did
not incorporate selection.
Another issue that we should consider is the scaling of phenotypes. We used an arbitrary value of δ = 1 as a mean of φ+ and φ−
to simulate phenotypic data ranging from 330 to 1330 (Table 1).
We confirmed that the use of other δ values did not dramatically
change the results, but it is important to use a realistic value for
δ based on the extent of the observed phenotypic variation. If a
realistic value of δ makes the computational time too long, it may
be possible to use a larger value. However, values of δ that are
too large would cause unnecessarily large random fluctuations,
which would make the performance of the likelihood evaluation
worse. In any case, it is essential to check the robustness of the
results against different values of δ.
It is also possible to statistically extend our model in many
ways. In the case of ML, the null and alternative models were compared using a likelihood-ratio test, but it is also possible to perform
model fitting with a number of models, perhaps using the Akaike
Information Criteria (AIC). Although the computational load was
not enormous in our analyses of only four primate species with
simple settings, it will become computationally intensive with
increases in the amount of data and the number of parameters.
In such situations, the ABC inference process should be implemented in the Markov chain Monte Carlo (MCMC) framework
(Marjoram et al. 2003; Marjoram and Tavare 2006).
In summary, we believe that this simulation-based computational approach will be very useful, and that it has great potential
for future studies of phenotypic evolution. It allows us to understand the mode and intensity of selection in a very realistic
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situation, and inferred models and parameters can be tested statistically. We also point out that this framework can be used to
detect and quantify selection in PCMs. The initial motivation of
this work was largely on the pioneering work in molecular evolution by Yang and colleagues (Yang 2006). They established a
fundamental statistical framework that could be used to infer selection on amino acid substitutions in a phylogeny. In their framework, the rate of amino acid changing nucleotide substitutions
(i.e., nonsynonymous substitutions) is measured by standardizing the rate of synonymous substitutions, which are generally
considered to be neutral and constant among generations. This
relative ratio, called the dN/dS or Ka/Ks ratio, is a well-accepted
summary statistic for representing the direction and intensity of
selection in studies of molecular evolution (Li 1997). For studies
of phenotypic evolution, it would be beneficial to have such a
conventional expression, and we propose the use of sel for this
purpose. This index represents the total amount of inferred phenotype change in a specific branch. The basic idea of sel is similar
to the unit of evolutionary change, the darwin (Haldane 1949),
which is defined as the difference in (log transformed) phenotypic changes divided by the time span between two samples. Our
sel extends this concept in two ways. First, sel can be calculated within the PCM framework where the ancestral state of each
branch is unknown. Second, the darwin is a descriptive value that
does not allow inference about evolutionary processes, such as
testing whether selection explains observed phenotypic changes
better than neutral changes. On the other hand, sel can be used
for these purposes because null can be obtained by simulation.
ACKNOWLEDGMENTS
G. Slater provided important comments on the manuscript. This study was
financially supported by Precursory Research for Embryonic Science
and Technology (PRESTO) at Japan Science and Technology Agency
(JST) (to NK and HI, respectively) and the Center for the Promotion of
Integrated Sciences (CPIS) at Sokendai.
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Associate Editor: P. Lindenfors
Supporting Information
The following supporting information is available for this article:
Figure S1 (A) Power of the SRBM with different k values (shown on the x-axis) using the ML and ABC approaches.
Supporting Information may be found in the online version of this article.
Please note: Wiley-Blackwell is not responsible for the content or functionality of any supporting information supplied by the
authors. Any queries (other than missing material) should be directed to the corresponding author for the article.
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