Comparative phylogeographic summary statistics for testing simultaneous vicariance.

Abstract

Testing for simultaneous vicariance across comparative phylogeographic data sets is a notoriously difficult problem hindered by mutational variance, the coalescent variance, and variability across pairs of sister taxa in parameters that affect genetic divergence. We simulate vicariance to characterize the behaviour of several commonly used summary statistics across a range of divergence times, and to characterize this behaviour in comparative phylogeographic datasets having multiple taxon-pairs. We found Tajima's D to be relatively uncorrelated with other summary statistics across divergence times, and using simple hypothesis testing of simultaneous vicariance given variable population sizes, we counter-intuitively found that the variance across taxon pairs in Nei and Li's net nucleotide divergence (pi(net)), a common measure of population divergence, is often inferior to using the variance in Tajima's D across taxon pairs as a test statistic to distinguish ancient simultaneous vicariance from variable vicariance histories. The opposite and more intuitive pattern is found for testing more recent simultaneous vicariance, and overall we found that depending on the timing of vicariance, one of these two test statistics can achieve high statistical power for rejecting simultaneous vicariance, given a reasonable number of intron loci (> 5 loci, 400 bp) and a range of conditions. These results suggest that components of these two composite summary statistics should be used in future simulation-based methods which can simultaneously use a pool of summary statistics to test comparative the phylogeographic hypotheses we consider here.

Full text

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Molecular Ecology (2005)doi: 10.1111/j.1365-294X.2005.02718.x
© 2005 Blackwell Publishing Ltd
Blackwell Publishing, Ltd.
Comparative phylogeographic summary statistics for testing
simultaneous vicariance
M. J. HICKERSON,
*
Museum of Vertebrate Zoology, University of California, 3101 Valley Life Sciences Building, Berkeley, California 94720-3160, USA,
†
Department of Zoology and Entomology, University of Queensland, St Lucia, 4072, Australia
* G. DOLMAN† and C. MORITZ*
Abstract
Testing for simultaneous vicariance across comparative phylogeographic data sets is
a notoriously difficult problem hindered by mutational variance, the coalescent variance,
and variability across pairs of sister taxa in parameters that affect genetic divergence.
We simulate vicariance to characterize the behaviour of several commonly used summary
statistics across a range of divergence times, and to characterize this behaviour in com-
parative phylogeographic datasets having multiple taxon-pairs. We found Tajima’s
relatively uncorrelated with other summary statistics across divergence times, and
using simple hypothesis testing of simultaneous vicariance given variable population
sizes, we counter-intuitively found that the variance across taxon pairs in Nei and Li’s net
nucleotide divergence (π π π π
net
), a common measure of population divergence, is often inferior
to using the variance in Tajima’s
D
across taxon pairs as a test statistic to distinguish
ancient simultaneous vicariance from variable vicariance histories. The opposite and more
intuitive pattern is found for testing more recent simultaneous vicariance, and overall we
found that depending on the timing of vicariance, one of these two test statistics can achieve
high statistical power for rejecting simultaneous vicariance, given a reasonable number of
intron loci (> 5 loci, 400 bp) and a range of conditions. These results suggest that compo-
nents of these two composite summary statistics should be used in future simulation-based
methods which can simultaneously use a pool of summary statistics to test comparative the
phylogeographic hypotheses we consider here.
D
to be
Keywords
summary statistic
: coalescent, comparative phylogeography, simultaneous vicariance, statistical power,
Received 20 March 2005; revision received 24 June 2005; accepted 25 July 2005
Introduction
One of the most central yet challenging puzzles in biogeo-
graphy is whether codistributed taxon pairs share a common
history of simultaneous vicariance (Darwin 1859; Nelson
& Platnick 1981; Ricklefs & Schluter 1993; Avise
Johns & Avise 1998; Knowlton & Weigt 1998; Schneider
et al
. 1998; Riddle
et al
. 2000; Johnson & Cicero 2004). While
conceptually simple, testing for simultaneous divergence
with genetic data is obscured by differences among taxon
pairs in parameters that affect genetic divergence including
mutation rates, ancestral population sizes, postdivergence
migration (admixture), and ancestral subdivision. Even when
et al
. 1998;
there is equality in such parameters across taxon pairs,
substantial variation in genetic divergence will arise from
mutational and coalescent variance, especially in taxa that
are heavily subdivided or have large and stable effective
population sizes (Arbogast
et al
In addition to these difficulties, incorrect sister species status
from undetected extinction events can also hinder inference
(Johnson & Cicero 2004).
Maximum-likelihood and Bayesian methods can in
principle tease apart simultaneous and variable divergence
histories by utilizing the full information content from the
data (Bahlo & Griffiths 2000; Edwards & Beerli 2000; Nielsen
& Wakeley 2001). However, extending these methods to
simultaneously analyse multiple phylogeographic data sets
involving complex models and idiosyncratic biogeographic
histories is less tractable than simulation-based methods
. 2002; Hickerson
et al
. 2003).
Correspondence: Michael J. Hickerson, Fax: (510) 643-8238;
E-mail: mhick@berkeley.edu

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2
M. J. HICKERSON, G. DOLMAN and C. MORITZ
© 2005 Blackwell Publishing Ltd,
Molecular Ecology
, 10.1111/j.1365-294X.2005.02718.x
such as approximate likelihood and approximate Bayesian
computation (ABC). Although such simulation-based approx-
imate methods utilize less information from the data using
summary statistics, they allow greater flexibility and can
better handle complicated and highly parameterized models
because the likelihood function, P(data|
to be calculated explicitly. Furthermore, simulation-based
methods can easily incorporate a priori idiosyncratic
biological realism or uncertainty in nuisance parameters
(Pritchard
et al
. 1999; Estoup
et al
Beaumont 2004). Such methods have been developed for
testing demographic histories of a single taxon (Tavaré
et al
. 1997; Weiss & von Haeseler 1998; Estoup
Tallmon
et al
. 2004; Excoffier
questions (Plagnol & Tavare 2002), but histories involving
sets of geographically codistributed taxa have not yet been
considered.
However, such simulation-based approximate methods
work best when using summary statistics that are relatively
unbiased and contain relevant information regarding a
parameter that is to be estimated, such as using the average
Φ
) does not have
. 2001; Beaumont
et al
. 2002;
et al
. 2004;
et al
. 2005) and phylogenetic
number of pairwise differences between individuals to
estimate
θ
(4 * the effective population size * mutation
rate) under neutrality and panmixia (Tajima 1983). When
testing more complex hypotheses such as estimating the
distribution of divergence times across taxon pairs, a single
summary statistic might not contain enough information
for robust inference across parameter space. In such cases,
a simulation-based method would benefit from simultane-
ously using a set of summary statistics that together capture
the essential features in the data.
Therefore, before embarking on a full ABC or approxi-
mate likelihood method for comparative phylogeographic
parameter estimation and hypothesis testing, we must
identify a pool of minimally correlated summary statistics
that independently demonstrate some statistical power
regarding comparative phylogeographic hypothesis test-
ing. To this end we employ simulations to investigate the
behaviour of various summary statistics under various single
and multiple taxon-pair vicariance models. Specifically, we
use simulations to investigate: (i) the general properties of
several commonly used summary statistics (Table 1) including
Table 1 Summary statistics of single taxon-pair divergence histories. The proposed comparative phylogeographic summary statistics
involve calculating variance of a subset of these summary statistics across 10 taxon pairs or covariance of pairs of a subset of these summary
statistics. All summary statistics are averaged across loci within each taxon pair
(a)
NotationSummary statisticReference
Expectation and variance
derived under vicarianceFormula and/or description
π
Average pairwise differences Tajima 1983Yes
; πij = differences
between the ith and jth sequence;
n is the number of DNA sequences
sampled
π restricted to between population
comparisons
π restricted to within population
comparisons
πb − πw
πb
Average pairwise differences
between populations
Average pairwise differences
within populations
Net average pairwise
differences between
populations
Raw number of segregating
sites
Takahata &
Nei 1985
Takahata &
Nei 1985
Takahata
& Nei 1985
Yes
πw
Yes
πnet
Yes
S
Watterson
1975
Expectation: Wakeley & Hey 1997
Variance under extreme vicariance:
Hudson, Kreitman & Aguadé 1987
Number of Polymorphic sites
θW
Watterson’s theta (number of
segregating sites normalized
for sample size)
Tajima’s D
Watterson
1975
No; n is number of DNA
sequences sampled
D
Tajima 1989No

e1 and e2 are coefficients defined in Tajima
(1989) n is number of DNA sequences
sampled
πij
i j
n
2
<∑∑










S
i
i
n
∑
1
1
1
=
−






D
S
i
e S
1
e S S
2
i
n
∑

=

−

+
( );
−








=
−
π
1
1
1
1

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PHYLOGEOGRAPHIC SUMMARY STATISTICS
3
© 2005 Blackwell Publishing Ltd,
Molecular Ecology
, 10.1111/j.1365-294X.2005.02718.x
their expectations, variances, and pairwise correlations
throughout a range of divergence times; (ii) the variances
and covariances across taxon pairs in a subset of these
summary statistics (Table 1) under various simultaneous
and multiple vicariance histories (Figs 1 and 2); and (iii)
how statistical power to reject simultaneous vicariance
improves with collecting more nuclear loci under a simple
hypothesis testing framework in which the variance of a
summary statistic across taxon pairs or covariance of pairs
of summary statistics across taxon pairs are independently
used as test statistics (Excoffier
Maddison 2002; Hickerson & Cunningham 2005).
Results of this study will not only have general relevance
to comparative phylogeographic studies, but will addition-
ally suggest a useful pool of summary statistics to use in
the full ABC framework we are concurrently developing
for comparative phylogeographic studies (Beaumont
2002).
et al
. 2000; Knowles &
et al
.
Materials and methods
This study is partially motivated by the comparative phylo-
geographic data set that is emerging from the Australian
Wet Tropics (AWT) in which multiple intron loci are being
Fig. 1 A depiction of the model involving one ancestral popula-
tion splitting into two-daughter populations at time τ (4N generations)
before the present. The population mutation parameters are θA, θ1,
and θ2, where each θ is 4 * N * µ, such that µ is the per-gene per-
generation DNA mutation rate and where NA, N1, and N2 are effective
population sizes of the ancestral, and two-daughter populations,
respectively. Under the demographic expansion model, the two-
daughter populations are of sizes NBott1 and NBott2, respectively,
for times τBott1 and τBott2, respectively.
Fig. 2 Density curves depicting prior distributions that define the
10 different multiple taxon-pair vicariance hypotheses. Under
each hypothesis, divergence time varies among taxon-pairs within
a data set by randomly drawing from each respective prior
distribution. The density curves in (a) and (b) are simulated from
a gamma distribution using different means and variances for τ.
The density curve depicting Huniform in (c) is a uniform distribution
with values of τ ranging between 0 and 5.5 million years. The
density curve depicting Hmixed in (c) is a mixture of gamma
distributions H0.025A, H0.25A, H2.5A, and H5.0A with equal probab-
ility. Divergence time is alternatively given in units of 4N generations
and years.

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M. J. HICKERSON, G. DOLMAN and C. MORITZ
© 2005 Blackwell Publishing Ltd,
Molecular Ecology
, 10.1111/j.1365-294X.2005.02718.x
collected across taxon pairs that are codistributed across a
putative historical barrier to gene flow (the Black Mountain
barrier, Schneider
et al
. 1998). Therefore, we incorporate a
priori information on effective population sizes (Fig. 3),
mutation rates, generation times and divergence times that
are based on estimates from seven introns collected from
two
carlia
skink taxon pairs (Dolman & Phillips 2004; Dolman
& Moritz in review). The simulated single taxon-pair data
sets consisted of samples of 12 diploid individuals per sister
taxon and 400 bp introns per locus (Fig. 1). The simulated
comparative phylogeographic data sets were identical to
this, yet consisted of 10 taxon pairs per data set.
Summary statistics
We chose to explore summary statistics based on whether
they were likely to contain information relevant to the para-
meters in our model, such as effective population sizes,
divergence times, and other demographic processes (Table 1).
With this in mind we initially chose two compound
summary statistics (
π
net
and Tajima’s
components (
π
w
and
b
from the former and
from the latter). Although Tajima’s
usually quantities that are calculated from a single species
or population, here we calculate each of these summary
statistics once per taxon pair as if they were being collected
from a single population or species. On the other hand,
and its components (
π
w
and
sister taxon treated as a distinct population, such that
the average pairwise differences between a sister pair,
D
) as well as their
S
,
π
S
,
π
, and
π
, and
θ
θ
W
D
,
W
are
π
net
π
b
) are calculated with each
π
b
is
π
w
is the average pairwise differences within the two sister taxa,
and
π
net
=
b
–
w
(Table 1). With respect to divergence
time, it is known that
π
b
,
net
, Tajima’s
as divergence time increases (Takahata & Nei 1985;
Hudson
et al
. 1987; Tajima 1989; Simonsen
respect to population sizes, we can expect πw, S, π, and θW
to all contain information (Watterson 1975; Tajima 1983;
Fu 1995; Vasco et al. 2001), and we should expect Tajima’s
D to extract information regarding population expansion
(Simonsen et al. 1995).
There were some commonly used summary statistics
that we did not consider because they are known to contain
less information for our parameters of interest. For example,
we considered πnet instead of FST, a commonly used metric
in population genetic studies. Although these two metrics
are very similar in that they both measure the ratio of inter-
population and intrapopulation genetic variance, we initially
chose the former because it is not upwardly bounded with
greater divergence times, while the latter has a maximum
value of 1.0. Likewise we did not choose to explore haplotype
diversity, the number of haplotypes or the number of single-
tons, because these measures do not have strong bearing
on divergence times. We also did not explore Fu and Li’s D
or F statistic because these measures are known to behave
very similar to Tajima’s D (Fu & Li 1993), but are only more
powerful than Tajima’s D when an outgroup is known
(Simonsen et al. 1995), a condition we do not assume.
ππ
π
D
,
S
,
π
,
θ
W
all increase
et al
. 1995). With
Expectation, variance and correlation of summary
statistics
To select candidate summary statistics from our initial pool
(Table 1), we determined their mean and variance through-
out a range of divergence times and determined the correla-
tions between all pairs of these seven summary statistics
under the various divergence times. The three criteria for
selecting candidate summary statistics for use in the multiple
taxon-pair analyses were: (i) a strong linear relationship
between divergence time and the mean of the summary
statistic; (ii) low variance of the summary statistic; and (iii)
low correlation between a summary statistic and other
summary statistics satisfying 1 and 2 (through a range of τ).
When a pair of summary statistics was shown to be heavily
correlated, elimination was based on the degree to which
criteria 1 and 2 were satisfied by either summary statistic
and whether either of the two summary statistics were
correlated with other summary statistics.
The mean and variance of the seven summary statistics
(Table 1) were calculated from 100 000 single taxon-pair
data sets simulated under 10 divergence times discretely
ranging from 0.0 to 2.0 coalescent time units (i.e. zero to
approximately 5 million years ago (Ma) when scaled by
4N generations, generation time of 3 years and θ = 1.0). The
3-year generation time was taken from the two previously
Fig. 3 Density curve depicting the prior distribution defining
among-taxon-pair variation in θ (4Nµ) used in the 10 taxon-pair
vicariance simulations.

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PHYLOGEOGRAPHIC SUMMARY STATISTICS 5
© 2005 Blackwell Publishing Ltd, Molecular Ecology, 10.1111/j.1365-294X.2005.02718.x
mentioned skink taxon pairs (Dolman & Moritz in review).
The population mutation parameter, θ (4Νµ), was fixed at
1.0, such that µ, the mutation rate was 1.2 × 10−6 per locus
per generation following an infinite sites model. Simulations
were conducted under a model without demographic
expansion and a model that included a bottleneck after
divergence such that the demographic expansion parameters
were NBott1 = 0.5N1, NBott2 = 0.5N1, τBott1 = 0.5τ, and τBott2 = 0.5τ
(Fig. 1; Table 2).
Although it is preferable to derive the expectation and
variance of these summary statistics rather than use
simulations, the range of models and summary statistics
we explore makes this endeavour beyond the scope of
this study. In cases where the first two moments have been
previously derived under vicariance models, we compare
the simulated expectations and variances to the derived
values as a check.
Multiple taxon-pair divergence model
For the multiple taxon-pair simulations, we implement
a hierarchy of parameters. The ‘higher level’ parameters
describe the mean and variance in divergence time (τ) and
θ within each simulated multiple taxon-pair data set, while
the ‘lower level’ parameters are the values of τ and θ for each
taxon pair within each simulated data set.
Each simulated data set consisted of 10 taxon pairs that
diverged at time τ, and samples from each sister taxon con-
tained 12 diploid individuals (Fig. 1). For each of the 10 000
data sets simulated under each of the 10 divergence time
hypotheses (Fig. 2), we calculated several comparative
phylogeographic summary statistics (Table 1). For eight
out of 10 of these divergence time hypotheses, τ varied
across taxon pairs by drawing it from a particular gamma
distribution that was constrained by a particular mean and
variance of τ (Fig. 2a, b). For the two other divergence time
hypotheses, τ was either drawn from a uniform prior dis-
tribution or a mixed gamma distribution with four peaks
corresponding to major events of climate change (Fig. 2c):
the last glacial maximum (20 000 years bp); the previous glacial
maximum (150 000 years bp); the Pliocene–Pleistocene
transition (2 million years ago (Ma)); and the Pliocene–
Miocene boundary (5 Ma).
For each simulated taxon pair, the population mutation
parameter (θ) of the ancestral (θA) and two daughter
populations (θ1 and θ2 at τ = 0) were held equal to each
other while θ1 and θ2 were equally rescaled during the
bottleneck phase (Fig. 1). These parameters (θA, θ1 and θ2)
varied among taxon pairs within each 10 taxon-pair data
set according to a gamma distribution with α = 3.5 and
β = 0.288, approximately corresponding to values of θ that
range between 0.0 and 3.0 (mean of 1.02 and SD = 0.55;
Fig. 3). This gamma distribution describes the range of
θ that is normally found for intron loci and the mean and
variance were specifically estimated from seven intron loci
collected from two taxon pairs of carlia skinks (Dolman
& Phillips 2004; Dolman & Moritz in review) using the
program im (Hey & Nielsen 2004). We explored multiple
taxon-pair divergence models with and without a bottleneck.
In the former case, the demographic expansion parameters
were fixed at NBott1 = 0.5N1, NBott2 = 0.5N1, τBott1 = 0.5τ, and
τBott2 = 0.5τ (Fig. 1; Table 2).
Table 2 Parameters used for simulation of vicariance histories depicted in Fig. 1

ParameterDescription
Values under single taxon-
pair vicariance histories
Prior distributions under 10 taxon-pair vicariance
histories
τ
Divergence time before the
present in units of 4N
generations
9 fixed values discretely
ranging between 0.0–2.0
(4N generations)
Gamma distributed under 11 different
hypotheses defined by various values of E(τ) and var(τ)
shown in Fig. 2a, b. Uniformly distributed under one
hypothesis with τ between 0.0 and 5.0 4N generations
(Fig. 2c), and a mixed gamma distribution under one
hypothesis with four means (Fig. 2c)
θ (4 * N * µ)4 * effective population size
* DNA mutation rate with
α = 3.50 and β = 0.288 (Fig. 3)
1.0 Gamma distributed priors (per locus per generation);
NA = N1 = N2 where N1 and N2 correspond to daughter
taxon 1 and 2, respectively, and NA corresponds to the
ancestral population (Fig. 1)
NBott1 and NBott2
Effective population size of
daughter taxa 1 and 2 during
bottleneck (Fig. 1)
No bottleneck or fixed at
0.5N1 and 0.5N2
No bottleneck or fixed at 0.5N1 and 0.5N2
τBott1 and τBott2
Longevity of bottleneck
subsequent to τ in daughter
taxon 1 and 2
No bottleneck or fixed at 0.5τ
No bottleneck or fixed at 0.5τ