Type I error rates for testing genetic drift with phenotypic covariance matrices: A simulation study

Abstract

Studies of evolutionary divergence using quantitative genetic methods are centered on the additive genetic variance-covariance matrix (G) of correlated traits. However, estimating G properly requires large samples and complicated experimental designs. Multivariate tests for neutral evolution commonly replace average G by the pooled phenotypic within-group variance-covariance matrix (W) for evolutionary inferences, but this approach has been criticized due to the lack of exact proportionality between genetic and phenotypic matrices. In this study, we examined the consequence, in terms of type I error rates, of replacing average G by W in a test of neutral evolution that measures the regression slope between among-population variances and within-population eigenvalues (the Ackermann and Cheverud [AC] test) using a simulation approach to generate random observations under genetic drift. Our results indicate that the type I error rates for the genetic drift test are acceptable when using W instead of average G when the matrix correlation between the ancestral G and P is higher than 0.6, the average character heritability is above 0.7, and the matrices share principal components. For less-similar G and P matrices, the type I error rates would still be acceptable if the ratio between the number of generations since divergence and the effective population size (t/N(e) ) is smaller than 0.01 (large populations that diverged recently). When G is not known in real data, a simulation approach to estimate expected slopes for the AC test under genetic drift is discussed.

Full text

ORIGINAL ARTICLE
doi:10.1111/j.1558-5646.2012.01746.x
TYPE I ERROR RATES FOR TESTING GENETIC
DRIFT WITH PHENOTYPIC COVARIANCE
MATRICES: A SIMULATION STUDY
Miguel Pr ˆ
oa,1,2Paul O’Higgins,1and Leandro R. Monteiro1,3
1Centre for Anatomical and Human Sciences, The Hull York Medical School, The University of York, Heslington, York,
YO10 5DD, United Kingdom
2E-mail: miguel.proa@hyms.ac.uk
3Laborat ´
orio de Ciˆ
encias Ambientais, Universidade Estadual do Norte Fluminense, Av. Alberto Lamego 2000, cep
28013-620, Campos dos Goytacazes, RJ, Brazil
Received November 22, 2010
Accepted July 3, 2012
Studies of evolutionary divergence using quantitative genetic methods are centered on the additive genetic variance–covariance
matrix (G) of correlated traits. However, estimating G properly requires large samples and complicated experimental designs.
Multivariate tests for neutral evolution commonly replace average G by the pooled phenotypic within-group variance–covariance
matrix (W) for evolutionary inferences, but this approach has been criticized due to the lack of exact proportionality between
genetic and phenotypic matrices. In this study, we examined the consequence, in terms of type I error rates, of replacing average
G by W in a test of neutral evolution that measures the regression slope between among-population variances and within-
population eigenvalues (the Ackermann and Cheverud [AC] test) using a simulation approach to generate random observations
under genetic drift. Our results indicate that the type I error rates for the genetic drift test are acceptable when using W instead
of average G when the matrix correlation between the ancestral G and P is higher than 0.6, the average character heritability is
above 0.7, and the matrices share principal components. For less-similar G and P matrices, the type I error rates would still be
acceptable if the ratio between the number of generations since divergence and the effective population size (t/Ne) is smaller than
0.01 (large populations that diverged recently). When G is not known in real data, a simulation approach to estimate expected
slopes for the AC test under genetic drift is discussed.
KEY WORDS: Cheverud’s conjecture, divergence rates, G-matrix, genetic drift test, multivariate evolution.
Quantitative genetic methods provide inferences of evolutionary
processes via the study of evolutionary divergence patterns and
their relationship to intrapopulation adult variation (Lande 1979;
Ackermann and Cheverud 2002, 2004; Marroig and Cheverud
2004; Monteiro and Gomes-Jr 2005; Perez and Monteiro 2009).
The connection between neutral microevolutionary processes and
macroevolutionary patterns is centered around the additive genetic
variance–covariance matrix (G) (Lande 1980; Arnold et al. 2001;
Jones et al. 2003; B´
egin and Roff 2004), which is thought to
determine both the response to selection and the pattern of neutral
divergence, at least among populations over a small time scale
(Lande 1980; Felsenstein 1988; Zeng 1988).
The expected pattern of phenotypic divergence among pop-
ulations caused by random genetic drift in correlated traits can be
used as a null hypothesis to test for neutral evolution (Lande 1979,
1980). The sampling distribution of the change in trait means in
one generation (¯
z) has a mean of 0 and variance–covariance ma-
trix G/Ne, the genetic covariance matrix in a population divided
by the effective population size (Lande 1979). If the average phe-
notype of a population ais represented by a column vector ¯
zaof
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2012 The Author(s). Evolution C
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MIGUEL PR ˆ
OA ET AL.
polygenic traits with additive genetic and environmental compo-
nents following multivariate normal distributions (Lande 1980),
the probability distribution after tgenerations will be
(¯
za,t)∼N[¯
z0,G(t/Ne)],(1)
which is a normal distribution with a mean equal to that of the
initial population and variance–covariance matrix G(t/Ne) (Lande
1979). If a number of populations are evolving independently (i.e.,
without gene flow), the expected among-population phenotypic
variance–covariance matrix (B) is a function of the genetic covari-
ance matrix (G), effective population size (Ne), and the number
of generations (t):
B=G(t/Ne).(2)
As a result, the comparison of among-population (Bpheno-
typic) and within-group (Ggenetic) variance–covariance matrices
can be used as a means to determine whether genetic drift as a
null model explains the pattern of divergence observed (Lofsvold
1986, 1988; Roff et al. 1999; Ackermann and Cheverud 2002;
B´
egin and Roff 2004).
Because phenotypic covariances are much easier to estimate
than their genetic counterparts, replacing average Gwith the
pooled phenotypic within-group covariance matrix (W), provided
that the phenotypic covariance matrices for diverging populations
remain similar, has been a widely used approach to study the evo-
lutionary mechanisms of divergence (Ackermann and Cheverud
2002, 2004; Marroig and Cheverud 2004; Perez and Monteiro
2009). Cheverud (1988) investigated the relationship between ge-
netic and phenotypic correlation matrices using data taken from
the literature and concluded that phenotypic correlations were
reasonable estimates (and generally proportional, although per-
haps not in a strict mathematical sense) of the respective ge-
netic correlations. A second conclusion from these data was that
phenotypic covariances Westimated with large samples might
approach Gmore accurately than genetic covariances estimated
from small effective sample sizes, at least for morphometric data
(Cheverud 1988; Revell et al. 2010). A number of meta-analyses
from literature reviews and empirical results have to some degree
corroborated Cheverud’s findings (Roff 1995; 1996; Koots and
Gibson 1996; Roff et al. 1999; Waitt and Levin 1998). Nonethe-
less, this approach has been criticized on several grounds (Willis
et al. 1991), but mostly because Wis not mathematically pro-
portional (i.e., having a constant ratio) to average G. Apart from
the issue of similarity and proportionality between matrices, more
specific consideration of the actual consequences of using Was
a surrogate of average Gin empirical studies (B´
egin and Roff
2004; Klingenberg et al. 2010) should prove fruitful and one such
aspect, the impact in terms of type I error rates, is the focus of the
present study.
Quantitative genetic theory predicts phenotypic covariances
within a single population (P) to be the sum of the genetic co-
variation (G) and the environmental covariances (E), P=G+
E(Falconer and Mackay 1996). A part-whole correlation is ex-
pected between phenotypic and genetic covariances; therefore,
phenotypic covariances can be considered an estimate of genetic
covariances with added error due to environmental covariances,
even if not mathematically proportional.
Most of the discussion on the surrogacy of average Gby
Wrevolves around the similarities and differences between phe-
notypic and genetic covariances in single populations or from
literature reviews, and the differences in empirical comparative
results obtained when using one kind of estimate or the other. The
latter are rare, due to the difficulty in estimating genetic param-
eters for a large number of species at the same time (B´
egin and
Roff 2004). Considering that Lande’s (1979, 1980) model expects
the among-population covariance matrix Bto be proportional to
the average Gwhen genetic drift is the sole evolutionary mecha-
nism, for the purpose of evolutionary divergence tests of neutral
evolution, the relevant discussion is not whether Gand Pare ex-
actly proportional in single populations, but whether using the
phenotypic pooled within-group covariance matrix Winstead of
the average Gwill add enough error (caused by the environmen-
tal covariances) to lead into erroneous conclusions. The tests that
have been used in the comparison of among-species phenotypic
covariances and genetic covariances (Lofsvold 1988; Ackermann
and Cheverud 2002, 2004) do not test for exact proportionality
between Band average G, but for similarity in different matrix
features, such as the correlation of principal components and the
distribution of eigenvalues. The expectation of proportionality
rests on a number of assumptions (Lande 1979) that are probably
violated in most natural populations (Lofsvold 1988), for exam-
ple, through the lack of large effective population sizes (Lofsvold
1988), or because of differences in the starting times of lineages
(Revell 2007). Furthermore, error in the estimation of the average
Gmight lead to unpredictable deviations from the expectation.
Lofsvold (1988) has suggested that the acceptance of genetic drift
as a null hypothesis will be more robust to the breaking of the
model’s assumptions than the rejection (so type I error rates are
of more concern than the power), and in real studies it might be
hard to determine the actual cause of rejection, natural selection
being one of the possible explanations. One might expect that
a consequence of using pooled within-group phenotypic instead
of genetic covariances would be to increase the probability of
rejecting (type I error rate) a true null hypothesis of genetic drift.
In this study, we examined the consequences of using pooled
within-group phenotypic instead of average genetic covariance
matrices in the Ackermann and Cheverud (2002) test of ge-
netic drift (referred to as the AC test from here on) in terms
of type I error rates using a simulation of phenotypic evolution in
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TYPE I ERROR RATES FOR GENETIC DRIFT TEST
diverging populations. We identified the most relevant parameters
and discuss a number of recommendations.
Material and Methods
SIMULATION MODELS: GENERAL DESCRIPTION
The simulations were performed using the quantitative genetic
theory from Lande (1979, 1980). Starting from an ancestral pop-
ulation with genetic covariance matrix Gand mean vector ¯
z0,
a number (15 or 30) of descendant population mean vectors ¯
za
were generated using the t-fold convolution in equation (1) for
a range of t/Neratios (0.000001–100 in increments of 1 in log10
scale). This approach is equivalent to a random walk in multi-
variate space where each descendant population is evolving at a
rate equivalent to G/Ne. Instead of generating the intermediate
phenotypes for each step (generation) of the random walk, the
convolution allows for a direct generation of the end points with
the same results and in a computationally efficient way.
The descendant populations from the ancestral distribution
N[¯
z0,G] were sampled ntimes (sample sizes 10–100, in incre-
ments of 10) according to the multivariate normal distribution
N[¯
za,P], for each population. The first step in the simulations
required an ancestral genetic variance–covariance matrix (G)to
generate species means and the second step required a phenotypic
within-population variance–covariance matrix (P) to generate
individual specimens for each population. The same Pwas
used for all populations (the pooled within-group phenotypic
covariance matrix Wis an estimate of the original P). Different
simulation models were used, either generating random Gand
Pmatrices as starting parameters (fully stochastic), or using
predetermined matrices obtained from real datasets. The fully
stochastic sets of simulations required the generation of random
positive definite covariance matrices (where all eigenvalues are
>0) that could be used as parameters in the generation of random
multivariate normal numbers representing individuals sampled,
as described below.
Throughout the paper, we used correlations of lower tri-
angular covariance matrices excluding diagonals (variances) as
one measure of structural similarity (alongside Common Princi-
pal Components—CPCs, Phillips and Arnold 1999). Note that
permutations of the matrix elements were not used for testing
significance of correlations. This procedure is not indicated for
testing similarity in covariance matrices if variables have differ-
ences in scale (Cheverud and Marroig 2007), but was the most
appropriate choice in our simulations. This is because the algo-
rithm for the generation of random positive definite matrices (see
details below) yielded matrices where covariances were small rel-
ative to variances. As a result, even when two covariance matrices
were independently generated, they presented high positive ma-
trix correlations when the diagonal was included (or when using
comparison methods such as random skewers), because the vari-
ances and covariances would systematically form two groups of
values in the matrix scatterplot. Only when diagonals were ex-
cluded, was the expected correlation for independent matrices 0.
This particular structure in the random matrices (high variances,
small covariances) is a consequence of generating positive defi-
nite matrices, because matrices with high covariances relative to
variances are likely to be nonpositive definite. Therefore, the most
accurate description of matrix similarities in our simulations was
derived from matrix correlations, using the lower triangular ele-
ments, excluding the diagonals. This is equivalent to comparing
correlation matrices derived from the covariance matrix, as the
information regarding variances is disregarded. In real datasets,
there would be no justification to exclude the variances from the
structural comparisons, as differences in scale of variances and
covariances are a relevant part of the structure.
SIMULATION MODEL 1
In this set of simulations, we used real matrices rather than ran-
domly generated ones. The matrices for the main simulations were
obtained from a honey bee (Apis mellifera) wing shape dataset
with 16 shape variables (partial warps), modified (used only
landmarks 11–20) from Monteiro et al. (2002), and a gastropod
shell shape dataset (Physa heterostropha) with 14 shape variables
(DeWitt 1996, 1998). The average heritability of bee wing vari-
ables (calculated as 1(IoG)(IoP)−11m−1,where1is a row vector
of ones, Iis the identity matrix, o is a Hadamard [element-wise]
product, and mis the number of variables) was 0.217, and the
effective sample size, given that 21 bee colonies were used, was
4.6. The effective sample size was calculated as the product of
heritability and the number of families, as described in Cheverud
(1988). For the shell shape dataset, the average heritability was
0.607, and the effective sample size was 11.5 (19 families were
used). These two datasets present important differences in the
structural similarity of Gand P. For the bee wing data, the matrix
correlation between Gand Pwas 0.804, whereas for the shell
dataset the correlation was 0.442. Although the average heritabil-
ity was smaller for the bee wing data, their genetic and phenotypic
matrices were more similar than in the shell dataset. This is not
unexpected, as these average heritabilities do not measure matrix
similarity, only the relative magnitudes of the genetic and phe-
notypic variances. A comparison of these genetic and phenotypic
covariance matrices via CPC indicated that the bee dataset matri-
ces shared the full set of principal components (full CPC model
supported by the jump-up approach) and the shell dataset matrices
shared no principal components (Unrelated model supported).
SIMULATION MODEL 2
In this model, Pand Gwere exactly proportional and dif-
fered only by a scalar multiplication. Gwas defined first as a
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MIGUEL PR ˆ
OA ET AL.
random positive definite covariance matrix using the eigenvec-
tor method from Marsaglia and Olkin (1984) and Joe (2006).
The eigenvector method first generates random eigenvalues
(λ1,...,λm) from a uniform distribution (the diagonal matrix L).
A lower bound of eigenvalues was set to 1 and an eigenvalue
ratio (between upper and lower bound) set to 10. The algorithm
then generates a random orthogonal matrix of eigenvectors Q(via
QR-decomposition) and constructs the genetic covariance matrix
Gas QLQT. The phenotypic covariance matrix was defined by
the scalar multiplication P=kG,wherekis a uniform random
number from 1 to 10. This approach generates a random uniform
distribution of covariance matrices in the space of positive defi-
nite covariance matrices (Joe 2006). In this set of simulations, as
in all other fully stochastic models, Gand Phad 15 dimensions.
SIMULATION MODEL 3
In this model, Gand Pare defined as independent random positive
definite covariance matrices using the uniform correlation matrix
method (Joe, 2006), where a random correlation matrix (R)is
first generated from a uniform distribution of partial correlation
coefficients. The variances are generated separately as a diagonal
matrix S=diag(σ12,...,σm2) with elements obtained from a uni-
form distribution ranging from 1 to 10. The random covariance
matrices are constructed as SRS.Gand Pwere independently
derived in this model, with the restriction that the variances in P
are always larger than the respective variances in G. To achieve
this, the variances (diagonal) of Pwere random multiples of the
respective variances in G. This procedure ensures that the vari-
ances of Pwere always larger, but Pand Gwere independent.
A series of 1000 simulations using this model yielded a distri-
bution of Gand Pmatrix correlations with a 95% confidence
interval (CI) (using 2.5 and 97.5 quantiles) of –0.194 to 0.201,
and a median of 0.0004. A further structural comparison of model
3 matrices was performed by CPC of 100 simulated GPpairs.
We compared estimated covariance matrices after generating 300
random observations from a multivariate normal distribution with
a mean vector of zeros and random Gand P(defined as above)
as parametric covariance matrices. Because the model fitting in
CPC depends on sample sizes, we have maintained a standard
n=300 for all other comparisons as well. The results indicated
the Unrelated model (no shared principal components) in all com-
parisons. Although the covariance structure is independently gen-
erated, all matrices generated by this method have variances on a
much larger scale than the covariances. Therefore, there is some
structural similarity because all matrices have clearly two groups
of elements (covariances and variances), and the variances are
always much larger than the covariances. This model is not bio-
logically reasonable because Gand Pindependence is unlikely
(even if not proportional) due to a part-whole relationship. The
model is included as a control, as the opposite to the mathemat-
ical proportionality of simulation model 2, allowing for a check
that the simulations behaved as expected at extremes of Gand P
similarity and independence.
SIMULATION MODEL 4
Simulation model 4 was designed to generate correlated Gand
Pmatrices, but without a CPC structure. To achieve this, we
have used the quantitative genetic relation P=G+E.Inthese
models, Gand Ewere defined first and independently. Pwas
then defined as a random matrix with expected value Gand a
random perturbation E(Marsaglia and Olkin 1984). Eand Gwere
generated by the uniform correlation matrix method described in
simulation model 3, where Ghas a range of variances between
1andσ2maxG (we used a maximum of 10), and Ehas a range
of variances between 0 and σ2maxE ,whereσ2maxG and σ2maxE
are the function parameters determining the upper limits of the
range of variances in Gand E, respectively. The expected value
of the average heritability of the variables in the simulations is the
ratio (σ2maxG – 1)/([σ2maxG –1]+σ2maxE ). This method generated
correlated Pand G, but without a common PC structure. This
pattern is ensured because the variables with larger variances in E
will be generally different than the variables with larger variances
in Gso that Pis less likely to inherit principal components from
G(H. Joe, pers. comm.). Of course, as the variances in Ebecome
smaller than variances in G(σ2maxE << σ2maxG), CPCs between P
and Gappear. The distribution of matrix correlations from 1000
model 4 simulations (using σ2maxG =10 and σ2maxE =9fora
similar range of variances) presented a 95% CI of 0.560–0.897,
and a median of 0.776. To check for common eigenstructure, we
performed a CPC analysis of 100 model 4 simulations of Gand P.
The simulations showed strong support for the Unrelated model
(no CPCs) in 65% of the cases, using the jump-up approach. The
remaining simulations supported 1 (26%) or 2 (9%) CPCs. In
simulation model 4, the perturbation of expected value Gby E
included random rotations of its eigenstructure, even if matrix
correlations were high.
SIMULATION MODEL 5
In simulation model 5, Gwas defined as a random positive definite
covariance matrix using the eigenvector method from Marsaglia
and Olkin (1984) and Joe (2006), but where σ2maxG is the max/min
eigenvalue ratio (this parameter will have a different interpretation
than in model 4, but the average heritabilities expected are exactly
the same in models 4 and 5). Pwas defined as the sum G+E,
where Ewas generated by the uniform correlation method, with a
variance range of 0 to σ2maxE. In this model, Preadily inherits the
principal component structure of G,evenwhenσ2maxE ∼σ2maxG.
The matrix correlation in these simulated matrices (with σ2maxG =
10, and σ2maxE =9) was smaller than in model 4 (matrix correla-
tion distribution 95% CI =0.061–0.462, median =0.274), but the
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TYPE I ERROR RATES FOR GENETIC DRIFT TEST
Figure 1. Simulation of genetic drift in four populations. The means of each population were evolved from an ancestral multivariate
normal distribution with mean =(0,0) and covariance matrix =G(t/Ne). Each population was randomly sampled 30 times using the
respective average and covariance matrix P. Left panels correspond to simulation model 4, where P and G are correlated, but do not
share principal components. Right panels correspond to simulation model 5, where P and G share principal components but have low
correlation. The ancestral genetic covariance matrix is depicted as a dashed ellipse. The population phenotypic covariance matrices are
depicted as solid ellipses. Filled circles correspond to population means and open circles correspond to individual observations.
CPC analysis shows strong support for a shared latent structure,
where 13% of the simulations supported the Unrelated model (0
CPCs), and 60% of the simulations supported models with 3 or
more common PCs. The perturbation caused by Egenerates ran-
dom differences between Pand G, but not a random rotation of the
eigenstructure of G(when σ2maxG >σ2maxE). This pattern is caused
by a lambda ratio (σ2maxG) of 10 or larger, which will produce G
matrices with sharp elliptical contours (noticeable principal com-
ponents), ensuring that the principal component structure of Gis
inherited by P,evenwhenσ2maxG ∼σ2maxE (H. Joe, pers. comm.).
An illustrative bivariate example of the typical main differ-
ences between simulation models 4 and 5 is depicted in Figure 1.
We simulated for each model and t/Ne, four populations descend-
ing from an ancestor (0,0) with a random genetic covariance
matrix (shown as dashed lines in Fig. 1) and a random phenotypic
matrix. The same phenotypic covariance matrices were used to
generate 30 observations in each population and these are depicted
as distinct clusters around each descendant. In simulation model
4, the matrix Pis a random rotation of G, whereas in simulation
model 5, the main axes of Gare preserved in P.
GENETIC DRIFT TEST
Genetic drift as a neutral model for phenotypic divergence was
tested by comparing the among-population covariance matrix (B)
and the within-population phenotypic covariance matrix (W,as
a surrogate of the average G) for the simulated data using the
method of Ackermann and Cheverud (2002, 2004). This involved
extracting the eigenvectors (M) and eigenvalues (m)ofW,and
projecting each population phenotypic vector of means ¯
zon M,
Y=¯
zM. The vector of means for each population was the one
estimated from the simulated samples, not the parametric means
generated from the ancestral Gand ancestral vector of means.
Finally, we calculated the variances for each column of Yand
performed a regression of the variances of Yon m
ln (Yi)=ln (t/Ne)+βln (mi)(3)
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MIGUEL PR ˆ
OA ET AL.
Testing with a t-test whether the slope of the regression (β)is
different from 1 indicates whether the pattern is compatible with
genetic drift. The null hypothesis of genetic drift is rejected if
the slope deviates significantly from 1 (Ackermann and Cheverud
2002).
For each combination of parameters (ancestral G,P,
t/Neratio, sample size, number of descendant populations,
σ2maxE/σ2maxG ) in different models, we simulated 1000 datasets
to estimate type I error rates. In the simulated datasets, the only
mechanism producing phenotypic divergence among the descen-
dant populations was genetic drift. When using a significance
level of α=0.05, we expect that a true null hypothesis has
a 5% chance of being rejected (a type I error). If the use of
phenotypic covariances as proxies for genetic ones in the ge-
netic drift test does increase the type I error rates, we expect
to find that, using a significance level of 5%, the null model of
genetic drift will be rejected in more than 5% of the simulated
samples.
All the simulations and analyses were run in the R environ-
ment (R Development Core Team, 2010) using functions from the
packages MASS (Venables and Ripley 2002), clusterGeneration
(Qiu and Joe 2009), and vegan (Oksanen et al. 2010). The R code
(commented) used for the simulations is available as Supporting
information.
Results
For the simulation using the bee wing shape data (genetic and
phenotypic covariance matrices) as starting parameters, the type
I error rate decreased with increasing sample sizes for small t/Ne
ratios (between 0.01 and 0.000001) irrespective of the number of
populations (15, 30) used (Figs. 2A, S1A). The error rate increased
for larger sample sizes when t/Ne≥0.1. The correlation between
Gand Wremained stable over simulations for all t/Ne, with a
median matrix correlation of 0.788, and a 95% CI (based on 0.025
and 0.975 quantiles) from 0.741 to 0.830. The matrix correlation
for the ancestral (original) Pand Gwas 0.804.
For the simulation using the shell shape data as starting pa-
rameters, the type I error rates remain at acceptable levels for
sample sizes above 20 in t/Neratios equal to or below 0.001, and
both numbers of populations (15, 30) (Figs. 2B, S1B). For the
simulation with t/Ne=0.01, the error rates increase with sam-
ple size. This is a slightly worse result than in the simulations
with bee wing parameters, because in the latter, the simulation
with t/Ne=0.01 yielded acceptable error rates (Fig. 2A, B). The
correlation between average Gand Walso remained stable over
simulations using the shell dataset for all t/Ne, with a median
matrix correlation of 0.441, and a 95% CI (based on 0.025 and
0.975 quantiles) from 0.405 to 0.479. The matrix correlation for
the ancestral (original) Pand Gwas 0.442.
In the simulation model 2, where Gand Pdiffered only by
a random constant (Fig. 2C), the resulting pattern showed slight
fluctuations around the expected type I error rate (0.05) for any
value of t/Ne. This result was observed for sample sizes above 40
individuals per population regardless of the number of populations
(15 or 30; Figs. 2C, S1C).
The simulation model 3, where Pand Gwere generated in-
dependently (Figs. 2D, S1D), presented acceptable type I error
rates only for t/Neratios equal to or below 0.00001, regardless of
the number of populations. The simulations with t/Ne>0.001 all
presented type I error rates above 0.8 and are not shown in the
Figure. Because in this model Gand Phave independent covari-
ances, the test would be expected to show significant deviations
from the unity slope for any combination of parameters. This
suggests that the power of the test must be small for such values
of t/Ne.
Simulation models 4 and 5 were designed to generate Gand
Pcorrelated matrices, where P=G+E. In simulation model 4,
the random matrix Eadds variation to the genetic covariances and
variances, including a random rotation of the eigenstructure when
Pis calculated, even if the range of variances in E(σ2maxE)isthe
same or a bit smaller than the range of variances in G(σ2maxG). In
simulation model 5, the Ematrix only causes differences in the
principal components of Gand Pwhen σ2maxE >σ2maxG.Thefirst
set of analyses was performed using the same range of variances
in Gand Efor both models. The simulation model 4 presented
acceptable error rates for sample sizes larger than 20 regardless
of t/Neratio and number of populations. The simulation model 5
presented acceptable type I error rates only for t/Neratios equal
to or below 0.001, regardless of the number of populations (Figs.
2E, F, S1E, S1F).
Exploring the simulations with a larger range of parameters,
we found that the ratio of upper limits of environmental and ge-
netic variance ranges (σ2maxE/σ2maxG ) also influences the type I
error rates of the test. One unexpected result was that in simulation
model 4, as σ2maxE gets smaller than σ2maxG, the type I error rates
increase. We performed the simulations again, with fixed sample
sizes (100), number of groups (15), and t/Ne(10) to assess the in-
fluence of σ2maxE/σ2maxG on the slope of the AC test (Fig. 3). In the
right panel of Figure 3, using simulation model 5 (where Preadily
inherits the eigenvectors of G), as the value of σ2maxE/σ2maxG gets
smaller, the slope of the test converges to 1, as expected under
genetic drift. On the other hand, in the simulation model 4 (left
panel of Fig. 3), the expected value of the slope under simulation
of drift is 1 only when σ2maxE ∼σ2maxG. As the ratio of variance
ranges get smaller, the expected slope converges to approximately
1.3, and this pattern explains why the type I error rates increase
when σ2maxE gets smaller than σ2maxG. The simulations using
the real matrices (model 1) and the same parameters described
above had expected slopes of 1.3 (bees) and 0.8 (Physa shells).
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TYPE I ERROR RATES FOR GENETIC DRIFT TEST
A
D
FE
C
B
Figure 2. Type I error rates for the simulated analyses with varying sample sizes and t/Neratios (drift intensities). The legends and line
types indicate the value of t/Neused (only when differences among lines are noticeable). The dashed horizontal straight line indicates
the expected type I error rate of 0.05. All simulations in this figure were performed with 15 populations. (A) Error rates for the bee wing
dataset. (B) Error rates for the shell dataset. (C) Stochastic simulations (model 2) where G was random and P was exactly proportional to
it P =kG (multiplication by a random scalar kdrawn from a uniform distribution between 1 and 10). (D) Stochastic simulation (model 3)
where both G and P were random and completely independent. (E) Stochastic simulation (model 4) where G and P were correlated (P =G
+E), but did not share a common latent structure (G and E with the same range of variances). (F) Stochastic simulation (model 5) where
G and P were correlated and shared a common latent structure (G and E with the same range of variances). See text for model details.
Slopes larger than 1 might be obtained when the variance among
population averages projected on the first eigenvectors of Wis
larger than the corresponding eigenvalues, whereas slopes smaller
than 1 are the result of less among-population variation than pre-
dicted by the eigenvalues of the first PCs of W.
Considering that, for simulation model 5, smaller variance
range ratios lead to the expected slope under genetic drift, we
explored the combination of simulation parameters that would
lead to acceptable type I error rates on the AC test (Table 1).
When we decrease the σ2maxE/σ2maxG ratio, the correlations
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MIGUEL PR ˆ
OA ET AL.
Figure 3. Slopes (β) of the Ackermann and Cheverud test in relation to the ratio of upper bounds of environmental (σ2maxE ) and genetic
(σ2maxG) variances in the simulations for models 4 (Sim4) and 5 (Sim5), using t/Ne=10, 15 dimensions in G, 15 populations, and 100
observations per population. Genetic variances ranged between 1 and σ2maxG =10 and the environmental variances ranged between 0
and σ2maxE =1–15. The solid lines show the expected (mean) value for the slope over 1000 simulations, whereas the dashed lines indicate
the upper and lower limits of 95% CIs. The dotted line indicates the unity slope, which is the theoretical expectation under genetic
drift.
Tab l e 1 . Type I error rates for the genetic drift test using simulation model 5 (1000 repetitions), with 15 variables, 15 groups, and 50
individuals per group (α=0.05), with varying t/Ne.σ2maxE/σ2maxG is the ratio of the upper bounds of variances in the environmental
and genetic matrices (see text), CI-h2is the 95% CI for the average heritability in each set of simulations, CI-MatCor is the 95% CI for G P
matrix correlations in each set of simulations, fCPC is the percentage of significant full CPC models for G and P in 100 simulations, CICPCs
are the 95% CI (percentiles) for the number of common principal components for G and P in 100 simulations.
GP matrix comparisons t/Ne
σ2maxE/σ2maxG CI-h2CI-MatCor fCPC CICPCs 100 10 1 0.1
0.1 0.86 – 0.94 0.808 – 0.970 100 14 0.058 0.062 0.056 0.048
0.2 0.76 – 0.88 0.543 – 0.891 100 14 0.048 0.057 0.058 0.043
0.3 0.70 – 0.84 0.385 – 0.812 79 7–14 0.097 0.084 0.102 0.077
0.4 0.64 – 0.80 0.277 – 0.729 67 4–14 0.139 0.159 0.188 0.132
between Pand Gincrease, as well as the number of CPCs. If
σ2maxE is around 20% of σ2maxG, the matrix correlations observed
are not particularly high, as compared to real Pand Gmatrices
estimated with large sample sizes, but they do share a common
eigenstructure, and for any value of t/Ne, the type I error rates ap-
proach acceptable values. Performing the same simulations with
more variables (m=30), the same results are obtained with larger
within-population sample sizes (n>100) (results not shown). It
is evident from these results that the combination of parameters
yielding acceptable type I error rates is sensitive to the models
under which the starting matrices were generated.
Discussion
Testing diversification by genetic drift is a useful starting point in
the study of evolutionary variation (Lynch 1990; Ackermann and
Cheverud 2004; Weaver et al. 2007; Perez and Monteiro 2009).
Cheverud’s (1988) suggestion that genetic covariance matrices
could be safely replaced by phenotypic matrices for evolutionary
inferences was greeted with scepticism, and “Cheverud’s conjec-
ture” (Roff 1995) has been tested and discussed in a number of
papers (e.g., Roff 1995, 1996; Koots and Gibson, 1996; Waitt
and Levin 1998; Roff et al. 1999; B´
egin and Roff 2004; Hadfield
et al. 2006; Kruuk et al. 2007), usually by comparing the simi-
larity of genetic and phenotypic covariances, seldom by check-
ing the influence of matrix differences in the results of tests.
Thus, the evidence gathered has been equivocal and the most
relevant studies (large reviews of data) indicate a general agree-
ment with Cheverud (1988), but also recommend caution in the
interpretations of results because matrix comparisons among iso-
lated populations using genetic or phenotypic covariances might
192 EVOLUTION JANUARY 2013

TYPE I ERROR RATES FOR GENETIC DRIFT TEST
differ in important ways (Roff et al. 1999; B´
egin and Roff
2004).
Our results indicate that the type I error of Ackermann and
Cheverud’s (2002, 2004) test of proportionality between Band W
is influenced mainly by the structural similarity between the an-
cestral Gand P, the ratio of variance ranges (approximated by the
average heritability), and the ratio of time and effective population
size t/Ne. If the parametric genetic and phenotypic covariance ma-
trices are exactly proportional, as in the simulation model 2, the
type I error rates are acceptable for any t/Neratio (as expected).
On the other extreme (simulation model 3), where Gand Pwere
generated with an unrealistic minimum of structural similarity,
the type I error rate is unacceptable for most values of t/Ne.
The simulations showed that even if the ancestral Gand P
are not proportional but do share a large number of principal
components, have an average heritability around 0.5, and matrix
correlation above 0.7 over all variables (as in our simulation model
5), acceptable type I error rates will be obtained for any t/Neratio.
When Gand Pdo not share principal components but are highly
correlated (r>0.7) and have average heritabilities approaching
0.5, the type I error rates should be acceptable for any t/Neratio
(as in our simulation model 4). Average heritabilities different
from 0.5 will bias the expectation of the slope in the AC test due
to concentration of variation among projections of population
averages in the first eigenvectors of W. In these cases, type I error
rates will still be acceptable for t/Ne<0.01.
The combination of parameters laid out is not an unrealis-
tic expectation. The literature indicates that considerable agree-
ment between genetic and phenotypic correlations is often found
and that the correlations between Gand Pare usually above
0.6 for morphological data when effective sample sizes are large
(Cheverud 1988; Koots and Gibson, 1996; Roff 1996; Waitt and
Levin 1998; Begin and Roff 2004; Kruuk et al. 2007; de Oliveira
et al. 2009).
In a study where only phenotypic data are available, it might
be complicated or impossible to determine whether the relation-
ship between the ancestral Gand Pfits into the assumptions out-
lined above. These parameter values can, nevertheless, be used
as guidelines for comparisons among populations as indirect ev-
idence of ancestral Gand Psimilarity (de Oliveira et al. 2009),
or one might use the Monte Carlo simulation approach described
below to estimate a CI for the slope of the AC test under drift.
Our example datasets (simulation model 1) seem to behave
in a similar way to simulation model 4 for extremes of low and
high σ2maxE/σ2maxG . The expected slope for the simulations using
the bee matrices was 1.3, the similarity of Gand Pwas high,
they did share principal components, but the average heritability
was low (it should have been higher than 0.6 to fit the model
4 more closely). On the other hand, the simulations with shell
matrices had an expected slope of 0.8, Gand Psimilarity was
low, they did not share principal components, but the average
heritability was high (should have been lower than 0.3). Such
results would be observed if model 4 was changed to calculate
P=k(G+E), so that the average heritability would be decreased
or increased by the scalar kwithout influence in the correlation
or shared structure between Pand G. These results suggest that
Gand Pare related in complex ways, which can hardly be reduced
to scalar comparisons without considerable loss of information. If
some information about Gand Pis available, one might use this
simulation approach to estimate the expected slope of the AC test
and use this expectation in the test of the real data (instead of the
theoretical unity slope). For example, in the bee wing analyses,
we could have used a slope of 1.3 as parameter in the t-test of
the AC tests and the type I error rates would be acceptable for
any value of t/Ne. Alternatively, the 95% CI for the expected AC
test slope under genetic drift simulations ranged from 1.1 to 1.5,
and an observed slope could be compared with this interval for
evidence of departure from the neutral expectation. When genetic
data are not available, it might be possible to use the between-
population covariance matrix (B), estimated from phylogenetic
independent contrasts if possible (Revell 2007) and the within-
population phenotypic covariance matrix Was proxies for the
ancestral Gand P, respectively, in the simulations to estimate
the expected slope under drift. A simulation function provided as
Supporting information (simulationAC-slope.R) will calculate a
mean estimate and a 95% CI for AC test slopes under genetic drift
for any ancestral Gand P. Observed slopes can be compared to
the CI or the mean estimate can replace the parameter slope =1
in the ordinary t-test.
Within-population sample sizes influence the type I error
rates, but they need to be considered in conjunction with the num-
ber of populations and the dimensionality of the matrices. For our
fully stochastic simulations, all matrices had 15 dimensions and
most acceptable type I errors were observed for within-population
samples larger than 40. The number of populations used had a
slight but negligible effect.
It is possible that sampling error in the estimation of Gmight
lead to a similar pattern of type I errors as when average Gis re-
placed by W, because the parametric and estimated Gmatrices are
not likely to be exactly proportional as well. It is not clear whether
sampling error in the estimation of Gis comparable to the environ-
mental covariance matrix E, but a part of Cheverud’s conjecture
was that Wcould be a more reliable estimate of the parametric
Gthan a genetic matrix estimated from a small effective sample
size (Cheverud 1988), and phenotypic correlation estimates are
often within the CIs of genetic correlations (Koots and Gibson
1996; Roff 1996). The instability of covariance matrix and factor
estimation for small sample sizes is well known in multivariate
statistics (MacCallum et al. 1999; Krzanowski 2000), and genetic
covariance matrices can be particularly demanding with respect to
EVOLUTION JANUARY 2013 193

MIGUEL PR ˆ
OA ET AL.
samples sizes (Cheverud 1988). Patterns caused by sampling error
in the estimation of genetic covariance matrices, such as biases on
eigenvalues, are well known (Meyer and Kirkpatrick 2010) and
a considerably large statistical literature is devoted to such top-
ics. As long as the sampling error can be considered independent
from the parametric G, the simulation function provided as Sup-
porting information can be adjusted to address specific concerns
regarding the error in the estimation of G.
In some of the simulations, particularly model 1 (with pre-
determined matrices) and the fully stochastic simulation where
Gand Pwhere random and completely independent (model 2), a
trend was observed where for higher values of t/Ne, the type I error
rates increase with within-population sample size (see Fig. 2A, B,
D). This counterintuitive result was also observed in simulation
model 4, when σ2maxE is smaller than σ2maxG (Fig. S2). Consider-
ing that, depending on this ratio, the differences between Gand P
caused the expected value of the slope of the AC test to be larger
than 1 (as shown in Fig. 3 due to more variation among popu-
lations than predicted by the eigenvalues of W), the type I error
rates increase with sample sizes because the CIs become narrower
(there is an expected increase in power) and a larger percentage
of simulated tests will show significant results. The type I error
converges to a value that depends on the magnitude of deviation
of the expected AC test slope from 1 and the size of the CI. For
smaller t/Neratios, there is a reduction in the contribution of the
Gmatrix to among-population variation (it will be proportional
to t/NeG). Because the simulations calculate among-group vari-
ation using averages estimated from the nobservations generated
by Pat each population (and not the parametric means generated
by G), when t/Nedecreases, most among-population variation is
generated and predicted by W, and the expected slope of the AC
test is 1. This also explains the effect in reverse, when σ2maxE >
σ2maxG, causing among-population variation to be smaller than
the eigenvalues of Wand the expected slope of the AC test to be
<1 (Fig. 3).
In summary, replacing Gwith Wwhen testing the null hy-
pothesis of divergence by genetic drift is not likely to increase
the type I error rates of the AC test, unless the ancestral Gand
Ware structurally dissimilar (mathematical proportionality is not
a required condition), the t/Neratio is large, and sample sizes
are small (<40 per group). A Monte Carlo simulation approach
might be used to estimate the expected slope of the AC test un-
der drift, taking into account the structural differences between
Gand P. A number of other methods have been proposed, which
were used to compare among-population and within-population
covariance matrices (Lofsvold 1988; B´
egin and Roff 2001; Revell
et al. 2007). Not all of these alternative methods will have their
type I error rates increased when average Gis replaced by W.
Type II errors are also a possibility, as genetic drift is the alterna-
tive hypothesis in some tests (Revell et al. 2007). The simulation
function provided as Supporting information can be modified to
account for other methods of matrix comparison. Alternatively,
model-based approaches (Butler and King 2004) should provide
reliable and possibly more informative tests of evolutionary pro-
cesses and scenarios. The simulation approaches, particularly the
more sophisticated individual-based models (Revell 2007), should
prove useful in further analyses, comparing methods and testing
evolutionary quantitative genetics models.
ACKNOWLEDGMENTS
The authors would like to thank T. DeWitt for allowing the use of the
Physa shape data as simulation parameters. Previous versions of the
manuscript were greatly improved by comments from G. Marroig and
anonymous reviewers. MP was funded by the Fundac¸˜
ao para a Ciˆ
encia e
a Tecnologia (Portugal), through the Ph.D. Programme in Computational
Biology, Instituto Gulbenkian de Ciˆ
encia (Portugal). LRM is funded by
the Conselho Nacional de Desenvolvimento Cient´
ıfico e Tecnol´
ogico
(CNPq, Brazil) and the Fundac¸˜
ao de Amparo `
a Pesquisa do Estado do
Rio de Janeiro (FAPERJ). The authors declare no conflict of interest.
LITERATURE CITED
Ackermann, R. R., and J. M. Cheverud. 2002. Discerning evolutionary pro-
cesses in patterns of tamarin (genus Saguinus) craniofacial variation.
Am. J. Phys. Anthropol. 117:260–271.
———. 2004. Detecting genetic drift versus selection in human evolution.
Proc. Natl. Acad. Sci. USA 101:17946–17951.
Arnold, S. J., M. E. Pfrender, and A. G. Jones. 2001. The adaptive landscape
as a conceptual bridge between micro- and macro-evolution. Genetica
112–113:9–32.
B´
egin, M., and D. A. Roff. 2001. An analysis of Gmatrix variation in two
closely related cricket species, Gryllus firmus and G. pennsylvanicus.J.
Evol. Biol. 14:1–13.
———. 2004. From micro- to macroevolution through quantitative genetic
variation: positive evidence from field crickets. Evolution 58:2287–
2304.
Butler, M. A., and A. A. King. 2004. Phylogenetic comparative analysis: a
modeling approach for adaptive evolution. Am. Nat. 164:683–695.
Cheverud, J. 1988. A comparison of genetic and phenotypic correlations.
Evolution 42:958–968.
Cheverud, J., and G. Marroig. 2007. Comparing covariance matrices: random
skewers method compared to the common principal components model.
Genet. Mol. Biol. 30:461–469.
de Oliveira, F. B., A. Porto, and G. Marroig. 2009. Covariance structure in the
skull of Catarrhini: a case of pattern stasis and magnitude evolution. J.
Hum. Evol. 56:417–430.
DeWitt, T. J. 1996. Functional tradeoffs and phenotypic plasticity in the fresh-
water snail Physa. Ph.D. diss, Binghamton University, Binghamton, New
Yor k .
———. 1998. Costs and limits of phenotypic plasticity: tests with predator-
induced morphology and life history in a freshwater snail. J. Evol. Biol.
11:465–480.
Falconer, D. S., and T. F. C. Mackay. 1996. Introduction to quantitative ge-
netics. 4th ed. Pearson Prentice Hall, Harlow, U. K.
Felsenstein, J. 1988. Phylogenies and quantitative characters. Annu. Rev. Ecol.
Syst. 19:445–471.
Hadfield, J. D., A. Nutall, D. Osorio, and I. P. F. Owens. 2006. Testing the
phenotypic gambit: phenotypic, genetic and environmental correlations
of colour. J. Evol. Biol. 20:549–557.
194 EVOLUTION JANUARY 2013

TYPE I ERROR RATES FOR GENETIC DRIFT TEST
Joe, H. 2006. Generating random correlation matrices based on partial corre-
lations. J. Multivariate Anal. 97:2177–2189.
Jones, A. G., S. J. Arnold, and R. B¨
urger. 2003. Stability of the G-matrix in a
population experiencing pleiotropic mutation, stabilizing selection, and
genetic drift. Evolution 57:1747–1760.
Klingenberg, C. P., V. Debat, and D. A. Roff. 2010. Quantitative genetics
of shape in cricket wings: developmental integration in a functional
structure. Evolution 64:2935–2951.
Koots, K. R., and J. P. Gibson. 1996. Realized sampling variances of estimates
of genetic parameters and the difference between genetic and phenotypic
correlations. Genetics 143:1409–1416.
Kruuk, L. E. B., J. Slate, and A. J. Wilson. 2007. New answers for old ques-
tions: the evolutionary quantitative genetics of wild animal populations.
Annu. Rev. Ecol. Evol. Syst. 39:525–548.
Krzanowski, W. J. 2000. Principles of multivariate analysis: a user’s perspec-
tive. 2nd ed. Oxford Univ. Press, Oxford, U. K.
Lande, R. 1979. Quantitative genetic analysis of multivariate evolution, ap-
plied to brain:body size allometry. Evolution 33:402–416.
———. 1980. Genetic variation and phenotypic evolution during allopatric
speciation. Am. Nat. 116:463–479.
Lofsvold, D. 1986. Quantitative genetics of morphological differentiation in
Peromyscus. I. Tests of homogeneity of genetic covariance structure
among species and subspecies. Evolution 40:559–573.
———. 1988. Quantitative genetics of morphological differentiation in
Peromyscus. II. Analysis of selection and drift. Evolution 42:54–
67.
Lynch, M. 1990. The rate of morphological evolution in mammals from the
standpoint of the neutral expectation. Am. Nat. 136:727–741.
MacCallum, R. C., K. F. Widaman, S. Zhang, and S. Hong. 1999. Sample size
in factor analysis. Psychol. Methods 4:84–99.
Marroig, G., and J. M. Cheverud. 2004. Did natural selection or genetic drift
produce the cranial diversification of neotropical monkeys? Am. Nat.
163:417–428.
Marsaglia, G., and I. Olkin. 1984. Generating correlation matrices. SIAM J.
Sci. Statist. Comput. 5:470–475.
Meyer, K., and M. Kirkpatrick. 2010. Better estimates of genetic covariance
matrices by “bending” using penalized maximum likelihood. Genetics
185:1097–1110.
Monteiro, L. R., and J. L. Gomes, Jr. 2005. Morphological divergence rate tests
for natural selection: uncertainty of parameter estimation and robustness
of results. Genet. Mol. Biol. 28:345–355.
Monteiro, L. R., J. A. F. Diniz-Filho, S. F. dos Reis, and E. D. Ara´
ujo.
2002. Geometric estimates of heritability in biological shape. Evolution
56:563–572.
Oksanen, J., F. G. Blanchet, R. Kindt, P. Legendre, R. B. O’Hara, G. L.
Simpson, P. Solymos, M. H. H. Stevens, and H. Wagner. 2010. vegan:
Community Ecology Package. R package version 1.17–4. Available at
http://CRAN.R-project.org/package=vegan. Accessed April 30, 2010.
Perez, S. I., and L. R. Monteiro. 2009. Nonrandom factors in modern hu-
man morphological diversification: a study of craniofacial variation in
southern south american populations. Evolution 63:978–993.
Phillips, P., and S. J. Arnold. 1999. Hierarchical comparison of genetic
variance-covariance matrices I. Using the Flury hierarchy. Evolution
53:1506–1515.
Qiu, W., and H. Joe. 2009. clusterGeneration: random cluster generation (with
specified degree of separation). R package version 1.2.7. Available
at http://CRAN.R-project.org/package = clusterGeneration. Accessed
September 18, 2011.
R Development Core Team. 2010. R: a language and environment for statistical
computing. R Foundation for Statistical Computing. Vienna, Austria.
Available at http://www.R-project.org. Accessed April 30, 2010.
Revell, L. J. 2007. Testing the genetic constraint hypothesis in a phylogenetic
context: a simulation study. Evolution 61:2720–2727.
Revell, L. J., L. J. Harmon, R. B. Langerhans, and J. J. Kolbe. 2007. A
phylogenetic approach to determining the importance of constraint on
phenotypic evolution in the neotropical lizard Anolis cristatellus.Evol.
Ecol. Res. 9:261–282.
Revell, L. J., D. L. Mahler, J. R. Sweeney, M. Sobotka, V. E. Fancher, and
J. B. Losos. 2010. Nonlinear selection and the evolution of variances
and covariances for continuous characters in an anole. J. Evol. Biol.
23:407–421.
Roff, D. 1995. The estimation of genetic correlations from phenotypic corre-
lations: a test of Cheverud’s conjecture. Heredity 74:481–490.
———. 1996. The evolution of genetic correlations: an analysis of patterns.
Evolution 50:1392–1403.
Roff, D., T. Mousseau, and D. Howard. 1999. Variation in genetic architecture
of calling song among populations of Allonemobius socius,A. fasciatus,
and a hybrid population: drift or selection? Evolution 53:216–224.
Venables, W. N., and B. D. Ripley. 2002. Modern applied statistics with S.
4th ed. Springer, New York.
Waitt, D. E., and D. A. Levin. 1998. Genetic and phenotypic correlations in
plants: a botanical test of Cheverud’s conjecture. Heredity 80:310–319.
Weaver, T. D., C. C. Roseman, and C. B. Stringer. 2007. Were neandertal
and modern human cranial differences produced by natural selection or
genetic drift? J. Hum. Evol. 53:135–145.
Willis, J. H., J. A. Coyne, and M. Kirkpatrick. 1991. Can one predict the
evolution of quantitative characters without genetics? Evolution 45:
441–444.
Zeng, Z. B. 1988. Long-term correlated response, interpopulation covariation,
and interspecific allometry. Evolution 42:363–374.
Associate Editor: G. Marroig
Supporting Information
The following supplementary material is available for this article:
Figure S1. Type I error rates for the simulated analyses with varying sample sizes and t/Neratios (drift intensities).
Figure S2. Type I error rates for the simulated analyses under model 4 with varying σ2maxE /σ2maxG, sample sizes, and t/Neratios
(drift intensities).
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.
EVOLUTION JANUARY 2013 195