Copyright Ó 2007 by the Genetics Society of America
DOI: 10.1534/genetics.107.075481
Unbiased Estimator for Genetic Drift and Effective Population Size
Per Erik Jorde*,1 and Nils Ryman†
*Department of Biology, Centre for Ecological and Evolutionary Synthesis (CEES), University of Oslo, N-0316 Oslo, Norway and †Department of
Zoology, Division of Population Genetics, Stockholm University, S-106 91 Stockholm, Sweden
Manuscript received May 4, 2007
Accepted for publication July 31, 2007
ABSTRACT
Amounts of genetic drift and the effective size of populations can be estimated from observed temporal
shifts in sample allele frequencies. Bias in this so-called temporal method has been noted in cases of small
sample sizes and when allele frequencies are highly skewed. We characterize bias in commonly applied
estimators under different sampling plans and propose an alternative estimator for genetic drift and
effective size that weights alleles differently. Numerical evaluations of exact probability distributions and
computer simulations verify that this new estimator yields unbiased estimates also when based on a modest
number of alleles and loci. At the cost of a larger standard deviation, it thus eliminates the bias associated
with earlier estimators. The new estimator should be particularly useful for microsatellite loci and panels
of SNPs, representing a large number of alleles, many of which will occur at low frequencies.
R
ECENT years have seen an increased interest in
using temporal shifts in allele frequencies to
estimate the genetically effective size of natural and
managed populations. This so-called temporal method
relies on sampling individuals from a population at two
or more times and estimating the amount of genetic
drift in the interim. Typically, two samples, of nx and ny
individuals, respectively, are drawn at random from the
population one (or more) generations apart. Screening
the samples for a number of polymorphic loci, the most
commonly used measures of allele frequency change
are those of Nei and Tajima (1981),
Fc ¼
a
1X
ðxi yi Þ2
a i¼1 zi xi yi
ð1Þ
F9 ¼ F 1
1
1
¼F ;
2nx 2ny
ñ
a
1 X
ðxi yi Þ2
;
a 1 i¼1
zi
ð2Þ
where a is the number of alleles at the locus and xi and yi
are the observed frequencies of the ith allele in the two
samples, respectively, with a (unweighted) mean of zi.
The expected values of the measures defined above
depend on how the samples were drawn from the population, and two different sample plans are recognized
(Nei and Tajima 1981; Waples 1989). Under plan I,
individuals are sampled after they have reproduced, or
they are sampled nondestructively and subsequently returned to the population before reproduction occurs.
Under plan II, in contrast, individuals are sampled be1
Corresponding author: Institute of Marine Research, Nye Flødevigveien
20, N-4817 His, Norway. E-mail: p.e.jorde@bio.uio.no
Genetics 177: 927–935 (October 2007)
ð3Þ
where ñ denotes the harmonic mean sample size, and F
represents Fc or Fk. When sampling follows plan I the
actual number of individuals (N) in the population
when the first sample was drawn is also a parameter,
1 1
F9 ¼ F 1 :
ñ N
and Pollak (1983),
Fk ¼
fore they reproduce and are not returned to the population. In the latter case, an estimator of genetic drift
over the t generations separating the samples is obtained
from the observed temporal allele frequency change, by
subtracting the expected contribution from sampling
from the quantities computed in Equations 1 or 2,
resulting in
ð4Þ
In practical applications of the temporal method several polymorphic loci are typically scored and an average estimate of drift over loci is calculated. If the
number of generations (t) between the two samples is
not too large (cf. Luikart et al. 1999), an estimate of effective population size is obtained from the average F 9 as
N̂e ¼
t
:
2F 9
ð5Þ
The procedure outlined above should yield approximately correct estimates of genetic drift and effective
population size, provided that the various assumptions
underlining the method are met and that a sufficiently
large number of loci are used. While computer simulations have typically verified this expectation, bias has
been noted in specific applications. Waples (1989)
928
P. E. Jorde and N. Ryman
noted that expressions (1) and (2) tended to be downward biased (resulting in an upward bias of Ne) when
allele frequencies are close to 0 (zero) or 1 (unity). This
problem was addressed in detail by Turner et al. (2001),
who found that bias is a complex function of population
allele frequency, true effective size, sample sizes, and
number of generations between the samples. A bias for
small sample sizes is also apparent in the computer
simulations presented by Nei and Tajima (1981) and
Waples and Yakota (2007). A downward bias in F, and
subsequent overestimate of Ne, was observed in severely
bottlenecked (Ne # 6) populations by Richards and
Leberg (1996) and Luikart et al. (1999).
While different sources of bias appear to affect the
temporal method, no systematic analysis of this problem
has been published and various suggestions to reduce
bias have been put forward. Here, we characterize bias
in the expressions above and propose an alternative
estimator for drift and effective size that is also unbiased
for small samples and highly skewed allele frequencies.
was calculated as Bin(Q2, 2Ne, q1). The second sampling
event is analogous to the first, except that the allele frequency is now q2, and the sample size ny may be different. The expected value of F is thus calculated as the
value, F(X, Y), obtained from Equations 1 or 2, when X
and Y copies of the allele are observed in the first and
second sample, respectively, times the probability of observing those numbers. Thus, under sample plan I
(assuming Ne ¼ N ),
EðF Þ ¼ F ðX ; Y Þ
3
2nx
X
HyperðX ; 2nx ; 2N ; q1 Þ
X ¼0
3
2ny
2Ne X
X
½BinðQ2 ;2Ne ;q1Þ3HyperðY; 2ny ; 2N ;q2 Þ;
Q2 ¼0 Y ¼0
ð6Þ
and, under plan II,
EðF Þ ¼ F ðX ; Y Þ
THEORETICAL CONSIDERATIONS
Expectation of F: To quantify bias in F, as defined in
Equations 1 and 2, we calculated exact expected values
for those expressions over a range of effective sizes (Ne),
initial population allele frequencies (q1), and sample
sizes (nx and ny) and compared these expected values
with the true amount of drift (1/2Ne per generation).
The calculations were carried out assuming a diploid
organism by summing E(F ) over all combinations of
sample- and population-allele frequencies, weighing by
the probabilities of those combinations. Briefly, sampling
of individuals under plan I was considered equivalent to
random drawing of 2n genes without replacement from
a pool of 2N genes, where N is the actual number of
individuals in the population and equals the effective
size under the Wright–Fisher model. The first sample of
nx individuals was taken when the population allele
frequency was q1. The probability of observing X copies
of the allele in this sample is given by the hypergeometric probability density function, Hyper(X, 2nx, 2N,
q1). Under plan II, the sampled individuals are permanently removed from the population and, as shown by
Waples (1989), sampling can then be regarded as being
carried out on the conceptually infinite gene pool preceding the generation. Sampling is thus independent of
population size and the probability of X was calculated
from the binomial probability density function, Bin(X,
2nx, q1). Assuming one generation between sampling
events, we considered a Wright–Fisher model and calculated probabilities of new population allele frequencies,
q2, in the next generation when the second sample is to
be drawn. Reproduction in this model is equivalent to
random sampling of 2Ne genes from an infinite gene
pool with allele frequency q1, and the probability of obtaining Q2 copies of the allele in the second generation
3
2nx
X
BinðX ; 2nx ; q1 Þ
X ¼0
3
2ny
2Ne X
X
½BinðQ2 ; 2Ne ; q1 Þ 3 BinðY ; 2ny ; q2 Þ:
Q2 ¼0 Y ¼0
ð7Þ
We exclude cases when both samples are fixed for the
same allele (i.e., X ¼ Y ¼ 0 or X ¼ 2nx and Y ¼ 2ny), as
these situations lead to undefined Fc and Fk. The expectations (6) and (7) are adjusted accordingly, by dividing
with the probability of no joint fixation in the two samples.
Variation in Fc and Fk over the range of sample sizes
(here, nx ¼ ny, ranging from 20 to 100) and initial allele
frequencies (q1, ranging from 0.005 to 0.995) are
apparent from the plots of E(F ) 1/n 1 1/N (sample
plan I) and E(F ) 1/n (plan II), depicted in Figure 1.
Because F should equal 1/(2Ne) regardless of sample
size and population allele frequency, this variation
demonstrates that Fc and Fk are not generally unbiased.
When samples are small, as compared to effective size, Fc
tends to be downward biased yielding estimates of Ne
that are too large, whereas Fk is upward biased so that N̂e
becomes too small when this estimator is used. Because
of the inverse relationship between F and Ne, bias in F
can translate to a substantial bias in N̂e . This source of
bias, being in opposite directions for Fc and Fk, was
noted in recent computer simulations by Waples and
Yakota (2007). This bias is apparent only when samples
are quite small, and Fc and Fk yield similar results when
samples are larger. Another source of bias is apparent
from Figure 1 when the locus is close to fixation, i.e.,
with q1 approaching 0 or 1. For such skewed allele
frequencies both Fc and Fk are downward biased for all
sample sizes under sample plan II (Figure 1, B and D), as
Estimating Effective Population Size
929
Figure 1.—Expected values of Fc (Equation 1) and Fk (Equation 2), after correction for sampling by subtracting 1/n (plan II) or
1/n 1/N (plan I). Expected values were calculated from Equations 6 (plan I) and 7 (plan II) for a population of effective size Ne
¼ 100, assuming one generation between samplings, and should equal 1/(2Ne) ¼ 0.005 over the entire parameter space for an
unbiased estimator of genetic drift. n is sample size and q1 is the initial population allele frequency.
noted previously by Waples (1989), Turneret al. (2001),
and others. Under sample plan I, on the other hand, the
bias is in the opposite direction and declines as a larger
fraction of the population is sampled and largely
disappears with exhaustive sampling (n ¼ N; Figure 1,
A and C). This behavior of Fc and Fk under sample plan I
has not been reported before and sample plan I has
generally received little attention in the literature.
The different types of bias described above demonstrate that the sampling adjustments to F made in expressions (3) and (4) are inadequate when samples are
small or allele frequencies are skewed. Those expressions were originally derived by approximating the
expected value of F by the ratio of the expected values
of its numerator and denominator separately (Nei and
Tajima 1981; Waples 1989). For Fk (Equation 2), which
can be rewritten Fk ¼ (x y)2/z(1 z) in the diallelic
case, the expected value has thus been approximated as
E(Fk) E½(x y)2/E½z(1 z). As noted by Turner et al.
(2001), however, a better approximation is given by
EðFk Þ E½ðx yÞ2 Cov½ðx yÞ2 ; zð1 zÞ
E½zð1 zÞ
E 2 ½zð1 zÞ
1
E½ðx yÞ2 3 Var½zð1 zÞ
;
E 3 ½zð1 zÞ
ð8Þ
where E denotes the expected value operator and Var
and Cov are the variance and covariance, respectively.
Expressions equivalent to (8) were used by Turner
et al. (2001) to evaluate bias in the temporal method and
by Raufaste and Bonhomme (2000) to evaluate bias in
an estimator devised by Robertson and Hill (1984) for
spatial genetic differentiation, FST. Turner et al. (2001)
derived an expression for bias in Fc (their Equation
A13), but applying their expression as a means to
correct for bias does not generally yield accurate results.
As is evident from the numerical examples presented in
Figure 2A, this bias correction tends to overcompensate
for the negative bias in the unmodified Fc (cf. Figure
1B), resulting in a positive bias that is excessive for small
samples. Raufaste and Bonhomme (2000) took a different approach and developed a correction for bias by
making a number of simplifying assumptions (one of
which was that allele frequencies are not highly skewed).
Adopting their bias correction ½their expression (24),
which was originally intended for FST, to the temporal
estimator Fk yields only a marginal improvement in bias
of the latter when samples are reasonably large (Figure
2B). Both the Turner et al. (2001) and the Raufaste and
Bonhomme (2000) corrections apply to sample plan II
(explicitly or implicitly) and no expression has so far
been put forward to correct for bias under sample plan I.
930
P. E. Jorde and N. Ryman
Figure 2.—Expected values for two bias-corrected estimators of genetic drift. Effective population size is 100, sampling
follows plan II, and expected values were calculated using
Equation 7, with the following expressions used for F(X, Y):
(A) following Turner et al. (2001), Fc 1/n bias(n, Ne,
q1), where bias(n, Ne, q1) is calculated according to their expressions (A13)–(A15); (B) adopted
P from Raufaste and
Bonhomme (2000), F k9 1 ðF k9Þ2 =2ð ai¼1 z1i 3aÞ, where F k9 is
given by Equation 2 and corrected for sampling using Equation 3. a is the number of alleles at the locus and zi is the mean
sample frequency of the ith allele.
Numerical evaluation of expression (8) indicates that
the approximation inherent in this expression may be
poor when allele frequencies are skewed. As seen from
Table 1, which refers to sample plan II, the components
on the right-hand side (i.e., A B 1 C) do not quite add
up to the left-hand value of E(Fk) when allele frequencies are close to zero (and unity). This implies that
expression (8) is unsatisfactory as a starting point for
developing a correction for bias in Fk (or Fc) under those
conditions. Inspection of Table 1 indicates, however,
that the ratio A ¼ E½(x y)2/E½z(1 z) is indeed
independent of population allele frequency and depends only on sample size n, which is known, and Ne,
which is to be estimated.
A new measure of F: The above observations suggest
that an unbiased estimator for Ne is also possible for
extreme allele frequencies if E½(x y)2 and E½z(1 z)
are calculated separately. Thus, we propose the following measure of temporal change:
Pa
ðxi yi Þ2
P
:
ð9Þ
Fs ¼ ai¼1
i¼1 zi ð1 zi Þ
This measure differs from those considered previously in that the numerator and the denominator
are estimated separately by summing over all alleles
before carrying out the division. This approach is equivalent
Pa to weighing each allele i by wi ¼ zi ð1 zi Þ=
i¼1 zi ð1 zi Þ (Raufaste and Bonhomme 2000, p.
288), as was recommended by Reynolds et al. (1983)
and Weir and Cockerham (1984) when estimating
spatial genetic differentiation (FST). Conversely, the
original estimator Fk (Equation 2) weights each allele
by wi ¼ (1 zi)/(a 1), which corresponds to the
weighing scheme used by Robertson and Hill (1984).
Relating Fs to Ne: To find an unbiased estimator for
effective population size it remains to find exact expressions for the expectations of the numerator and the
denominator of Equation 9. In the derivations we
consider a diallelic locus and make the simplifying
assumption that generations are discrete (nonoverlapping) and that samples are drawn exactly one generation apart, following either sample plan I or II. The
expectation of the numerator of Equation 9 under these
assumptions has been found previously (Waples 1989)
and is for sample plan II
1
1
1
2
E½ðx yÞ ¼ q1ð1 q1Þ
1
;
11 1 2nx
2ny
2Ne
ð10Þ
where, as before, nx and ny are the number of individuals
in the first and in the second sample, respectively, and q1
is the population allele frequency in the generation
when the first sample was drawn. The corresponding
expression for sample plan I is similar but includes an
additional term, 1/N, i.e., the inverse of the actual
number of individuals in the population (see below).
An exact expression for the expected value of z(1 z)
under the stipulated conditions can be found as follows:
E½zð1 zÞ ¼ EðzÞ E½ðz q1 1 q1 Þ2 ¼ q1 q12 E½ððx 1
yÞ=2 q1 Þ2 ¼ q1 ð1 q1 Þ 14 E½ðx q1 Þ2 2E½ðx q1 Þ
ðy q1 Þ 1 E½ðy q1 Þ2 g ¼ q1 ð1 q1 Þ 14VarðxÞ 14VarðyÞ
1 12Covðx; yÞ, where Var(x) is the sample variance of the
allele frequency in the first sample, q1(1 q1)/2nx, and
Var(y) is that in the second sample, q1 ð1 q1 Þf1 ½1
1=ð2ny Þ½1 1=ð2Ne Þg. Cov(x, y) is the covariance between the sample frequencies x and y and equals q1(1 q1)/2N under sample plan I and zero under plan II
(Waples 1989). Putting this together, we have the expected value of the denominator of Fs under sample
plan II:
E½zð1 zÞ
1
1
1
1
¼ q1 ð1 q1 Þ 1
1
:
1 1 8nx 4
2ny
2Ne
ð11Þ
The expression under sample plan I is similar but
includes an additional term, 1/(4N), arising from the
covariance between x and y (cf. Equation 14 below).
Estimating Effective Population Size
931
TABLE 1
Expected values for Fk [E(Fk)] and its components [expression (8)]
Ne
n
q1
E(Fk)
A
B
C
AB1C
20
20
20
20
20
100
20
100
0.025
0.025
0.5
0.5
0.06696
0.03277
0.07606
0.03549
0.07578
0.03518
0.07578
0.03518
0.04040
0.01879
0.00021
0.00030
0.02729
0.01112
0.00005
0.00001
0.06267
0.02751
0.07604
0.03548
100
100
100
100
20
100
20
100
0.005
0.005
0.5
0.5
0.05331
0.01321
0.05564
0.01504
0.05564
0.01503
0.05564
0.01503
0.01390
0.00898
0.00001
0.00001
0.00974
0.00606
0.00002
0.00000
0.05148
0.01211
0.05565
0.01504
500
500
500
500
20
100
20
100
0.001
0.001
0.5
0.5
0.05114
0.01055
0.05163
0.01103
0.05163
0.01103
0.05163
0.01103
0.00322
0.00304
0.00002
0.00000
0.00227
0.00212
0.00002
0.00000
0.05068
0.01011
0.05163
0.01103
The expected values were calculated from expression (7), assuming sample plan II, over a range of effective
population sizes, Ne, sample sizes, n, and initial population allele frequencies, q1. The components are A ¼ E½(x –
y)2/E½z(1 – z), B ¼ Cov½(x – y)2, z(1 – z)/E 2½z(1 – z), and C ¼ E½(x – y)2 3 Var½z(1 – z)/E 3½z(1 – z).
Carrying out the division of expression (10) by (11)
eliminates the population allele frequency, q1 and yields
the expected value for Equation 9. Under sample plan II
EðFs Þ ¼
1=ð2Ne Þ 1 1=ð2nx Þ 1 1=ð2ny Þ 1=ð4Ne ny Þ
:
1 1=ð8Ne Þ 1=ð8nx Þ 1=ð8ny Þ 1 1=ð16Ne ny Þ
ð12Þ
The expression for plan I is similar, but includes the
additional terms 1/N and 1/(4N) in the numerator
and in the denominator, respectively (cf. Equation 14).
Substituting Fs for 1/(2Ne) in expression (12) and
rearranging yields an estimator for drift that should be
unbiased under sample plan II,
F s9 ¼
Fs ½1 1=ð4ñÞ 1=ñ
;
ð1 1 Fs =4Þ½1 1=ð2ny Þ
ð13Þ
where ñ is the harmonic mean of the sample sizes nx and
ny . This expression is similar to Equation 14 of Nei and
Tajima (1981), but includes the additional factor 1 1/
(2ny). Under plan I the actual census size (N) of the
population at the time of first sampling is also a factor, so
that
F s9 ¼
Fs ½1 1=ð4ñÞ 1 1=ð4N Þ 1=ñ 1 1=N
:
ð1 1 Fs =4Þ½1 1=ð2ny Þ
ð14Þ
COMPUTER SIMULATIONS
Methods: The expected value of Fs derived above
(Equation 12) is exact under a Wright–Fisher model for
the stipulated conditions of a diallelic locus sampled
twice, one generation apart. It should be valid also when
Ne 6¼ N, for multiallelic loci, and at least approximately
so for samples collected more than one generation
apart. We used computer simulations to check on these
conjectures. The simulations started with a population
consisting of 2N different genes in the first generation,
i.e., with a ¼ 2N different alleles, each with a frequency
of q1 ¼ 1/(2N). Both ideal (Ne ¼ N) and nonideal (Ne ,
N) populations were simulated. For the case of an ideal
(Wright–Fisher) population, all 2N genes were equally
likely to be drawn for the offspring generation (random
drawing with replacement), whereas for nonideal populations we first picked (without replacement) 2Ne out
of the 2N genes to constitute parents and subsequently
drew 2N offspring genes (with replacement) from this
parental gene pool. Sampling was carried out by random drawing of 2nx ¼ 2ny genes from the population
(2N genes) with replacement (plan II), or without
replacement but with the genes subsequently returned
to the population before reproduction (plan I). Fs
(Equation 9) and Fk (Equation 2) were calculated from
the sample allele frequencies and corrected for sampling using Equations 3, 4, 13, or 14, as appropriate for
the estimator and sample plan and used to estimate Ne
by applying Equation 5.
Computer simulations were further used to check on
the accuracy and precision of the proposed estimator
when applied to a finite number of loci, segregating for
various numbers of alleles and with different allele
frequency profiles. Both diallelic (a ¼ 2) and multiallelic (a ¼ 10) loci were simulated with skewed and
even allele frequencies. In the case of skewed frequencies, simulations were initiated with one (for diallelic
loci) or nine (for multiallelic loci) rare alleles, each with
a frequency of q1 ¼ 0.01 and with one common allele
(with a frequency of 0.99 or 0.91). In the case of even
frequencies, all alleles were initiated with equal frequencies (i.e., 0.5 for diallelic and 0.1 for multiallelic
loci). Sampling and genetic drift over one generation
932
P. E. Jorde and N. Ryman
TABLE 2
Results of computer simulations checking theoretical expectations for old and new estimators of genetic drift and effective size
t¼1
Old Fk9 (N̂e )
t ¼ 10
New Fs9 (N̂e )
Old Fk9 (N̂e )
New Fs9 (N̂e )
Ne
n
20
100
100
500
500
20
20
100
20
100
0.02678
0.00774
0.00529
0.00210
0.00145
(19)
(65)
(94)
(238)
(346)
Plan I (Ne ¼ N)
0.02562 (20)
0.00523 (96)
0.00503 (99)
0.00102 (491)
0.00101 (496)
0.07581
0.02370
0.01506
0.00626
0.00461
(66)
(211)
(332)
(799)
(1084)
0.22954
0.04933
0.04911
0.01002
0.00995
(22)
(101)
(102)
(499)
(503)
20
20
100
100
500
500
20
100
20
100
20
100
0.00940
0.01365
0.00410
0.00336
0.00133
0.00072
(53)
(37)
(122)
(149)
(375)
(696)
Plan I (Ne ¼ 0.2N)
0.02532 (20)
0.02512 (20)
0.00503 (99)
0.00501 (100)
0.00101 (495)
0.00100 (500)
0.04553
0.01874
0.01896
0.00891
0.00541
0.00366
(110)
(267)
(264)
(561)
(924)
(1365)
0.21419
0.22317
0.04880
0.04879
0.00992
0.00995
(23)
(22)
(102)
(102)
(504)
(503)
20
20
100
100
500
500
20
100
20
100
20
100
0.01662
0.02272
0.00327
0.00320
0.00112
0.00055
(30)
(22)
(153)
(156)
(448)
(913)
Plan II
0.02490 (20)
0.02503 (20)
0.00501 (100)
0.00501 (100)
0.00095 (527)
0.00099 (505)
0.06762
0.06721
0.01955
0.01330
0.00530
0.00379
(74)
(74)
(256)
(376)
(943)
(1321)
0.22386
0.22369
0.04885
0.04891
0.00997
0.00995
(22)
(22)
(102)
(102)
(502)
(502)
A constant population of N diploid individuals was simulated, with effective size Ne ¼ N or Ne ¼ 0.2N, starting with only unique
alleles ½i.e., there are a ¼ 2N alleles, each with an initial frequency of q1 ¼ 1/(2N). The population is sampled twice, t generations
apart, according to sample plan I or II, with equal sample sizes (nx ¼ ny ¼ n). Reported are estimated amounts of genetic drift
(Fk9and Fs9), averaged over 10,000 replicate computer runs, and effective sizes corresponding to these averages, N̂e ¼ t=ð2F 9Þ.
were simulated as described above, following sample
plans II and I (assuming Ne ¼ N). The results of the new
estimator Fs were compared to two alternative estimators, viz., the original estimator Fk and the maximumlikelihood estimator proposed by Wang (2001). The
latter was calculated using the MLNE (ver. 1.0) software,
described by Wang and Whitlock (2003) and distributed by J. Wang (http://www.zoo.cam.ac.uk/ioz/software.
htm#MLNE), by applying it to a single isolated population with a maximum Ne of 2000 (a larger maximum
number resulted in excessive computer time). We
report mean estimates of genetic drift and their standard deviations, i.e., for Fs9, Fk9, and the inverse of twice
the MLNE estimate of Ne, calculated over 10,000
replicate computer runs (1000 for MLNE). Because
the MLNE method is not adapted to sample plan I, we
used this method only when simulating sampling
according to plan II. (At present, MLNE returns
estimates of Ne that are pegged against the maximum
value supplied by the user when running the program
on simulated data under sample plan I.)
RESULTS
Simulation results (Table 2) verify the theoretical
predictions presented in Figure 1 and demonstrate that
the original estimator Fk9 does yield biased estimates of
genetic drift and effective size when applied to loci with
skewed allele frequencies or small samples. With one
generation between the samples the true amount of
genetic drift is per definition 1/(2Ne). As predicted
above, however, the simulated Fk9’s are typically larger
than this value under sample plan I (with Ne ¼ N; top
rows), thus yielding estimates of Ne that are biased downward, approaching the true values only when sample
sizes are close to the effective size, i.e., when sampling is
exhaustive. When Ne , N (middle rows) and under
sample plan II (bottom rows), the bias in Fk9 is in the
opposite direction, resulting in estimates of Ne that are
usually too large. This behavior of Fk9 is in accordance
with the numerical results predicted for extreme allele
frequencies in Figure 1, C and D.
With several generations passing between samplings
the true amount of drift accumulates according to the
multiplicative expression 1 ½1 1/(2Ne)t ½Crow and
Kimura 1970, expression (7.3.4), and equals 0.22367,
0.04889, and 0.00996 for effective sizes 20, 100, and 500,
respectively, after t ¼ 10 generations. As seen from the
right-hand side of Table 2, Fk9 consistently fails to provide
accurate estimates of drift in these simulations, yielding
estimates that are too small (and N̂e too large) for all
sample sizes and sample plans. In summary, there
appear to be at least three causes of bias in the original
Estimating Effective Population Size
933
TABLE 3
Comparison of precision and accuracy in the new and previous estimators for genetic drift
Diallelic loci (a ¼ 2)
Estimator
10 loci
100 loci
Multiallelic loci (a ¼ 10)
10 loci
100 loci
q1 ¼ 9 3
q1 ¼ 0.01/0.99
Original Fk9
0.00379 (0.0047) 0.00382 (0.0015) 0.00387 (0.0016)
MLNE 1/(2N̂e ) 0.00332 (0.0027) 0.00303 (0.0010) 0.00318 (0.0011)
New Fs9
0.00473 (0.0067) 0.00497 (0.0023) 0.00501 (0.0038)
Mixed no. of alleles
10 loci
100 loci
0.01/0.91
q1 ¼ mixed extreme
0.00387 (0.0005) 0.00389 (0.0021) 0.00387 (0.0007)
0.00319 (0.0003) 0.00308 (0.0014) 0.00316 (0.0005)
0.00500 (0.0012) 0.00501 (0.0050) 0.00500 (0.0016)
q1 ¼ 2 3 0.5
q1 ¼ 10 3 0.1
q1 ¼ mixed even
Original Fk9
0.00496 (0.0066) 0.00501 (0.0021) 0.00499 (0.0022) 0.00498 (0.0007) 0.00493 (0.0029) 0.00498 (0.0009)
MLNE 1/(2N̂e ) 0.00494 (0.0052) 0.00443 (0.0020) 0.00447 (0.0020) 0.00454 (0.0007) 0.00451 (0.0026) 0.00450 (0.0009)
New Fs9
0.00490 (0.0065) 0.00497 (0.0021) 0.00501 (0.0022) 0.00500 (0.0007) 0.00490 (0.0039) 0.00501 (0.0012)
q1 ¼ mixed
q1 ¼ mixed
q1 ¼ mixed
Original Fk9
0.00444 (0.0058) 0.00446 (0.0018) 0.00443 (0.0019) 0.00444 (0.0006) 0.00441 (0.0028) 0.00443 (0.0009)
MLNE 1/(2N̂e ) 0.00368 (0.0033) 0.00347 (0.0012) 0.00355 (0.0013) 0.00360 (0.0004) 0.00347 (0.0019) 0.00368 (0.0006)
New Fs9
0.00505 (0.0090) 0.00503 (0.0029) 0.00497 (0.0028) 0.00500 (0.0009) 0.00500 (0.0056) 0.00500 (0.0017)
Estimates were calculated from computer simulated allele frequencies for a population with effective size Ne ¼ 100 that was
sampled twice for nx ¼ ny ¼ 100 individuals one generation apart, according to sample plan II. Replicate computer runs represent
a small (10) or a large (100) set of loci used for measuring genetic drift. ½For the lower right section, representing mixed number
of alleles and mixed allele frequency profiles (i.e., four different combinations), the number of loci are 8 and 80 instead. Both
diallelic (a ¼ 2) and multiallelic (a ¼ 10) loci, as well as a mixture of equal proportions of diallelic and multiallelic loci, were
simulated. The loci were initiated with skewed allele frequencies (rare allele, q1 ¼ 0.01), with even allele frequencies (q1 ¼ 1/a),
and with equal proportions of loci with skewed and with even frequencies. The reported estimates of genetic drift, and their standard deviations (in parentheses), were averaged over 10,000 replicate simulations (1000 for MLNE) and should equal 1/(2Ne) ¼
0.005 for an unbiased estimator.
estimator Fk9, viz., bias emerging with skewed allele
frequencies, small samples (n , , Ne), and long sample
intervals. In contrast to the original one, the new
estimator Fs9 proposed here appears unbiased or nearly
so for all sample sizes under both sample plans, even
with long sampling intervals.
The simulation results presented in Table 3 expand
on those above by including loci with different numbers
of alleles (2 and 10) and allele frequency profiles
(skewed and even), but keeping the true effective size
fixed at Ne ¼ 100. When applied to multiple loci, an
average Fk is calculated over all single-locus measures
(Equation 2), which are weighted by the number of
independent alleles (i.e., a 1) at each locus. However,
when calculating Fs for these simulations, the weighing
scheme for alleles used in Equation 9 is extended over
loci, by summing over all alleles and loci in the
numerator and denominator separately before carrying
out the division. The results (Table 3) validate this
procedure and demonstrate that unbiased estimates are
obtained with this new estimator also when applied to
combinations of loci with different numbers of alleles
and with very different allele frequency profiles (right
columns). Briefly, the new estimator is seen to provide
unbiased estimates of drift, not deviating from the true
value of 1/(2Ne) ¼ 0.00500 by .2% when applied to
multiallelic loci or to a sufficient number of diallelic
loci, also when allele frequencies are highly skewed, as
in the top rows of Table 3. The worst-case scenario
depicted in Table 3 refers to a small number (10) of
diallelic loci with skewed allele frequencies (top left),
where the residual bias is 5%. With increasing numbers of loci (100) the original Fk9 estimator remains
biased (by .20%), whereas the residual bias in Fs9
disappears. The reduction of bias in this case is entirely
due to the extension of the allele-weighing scheme in
expression (9) over loci. For loci segregating for
multiple alleles, bias is nearly absent from Fs9 already
when calculated for a single locus (10 alleles with
skewed frequencies, Fs9¼ 0.00478; data not shown).
Table 3 also includes results for the MLNE maximumlikelihood procedure. Estimates of genetic drift, 1=ð2N̂e Þ,
from this method are clearly biased when allele frequencies are skewed (top rows). A slight amount of bias is
apparent for multiallelic loci also when allele frequencies are even (middle rows) and bias is not reduced with
increasing numbers of loci. There appears to be a direct
inverse relationship between accuracy and precision of
the three estimators of genetic drift. As judged by their
standard deviations (Table 3), the least accurate (i.e., most
biased) estimator, MLNE, is also the most precise one,
having a lower standard deviation than the other estimators in all simulations involving skewed allele frequencies. Conversely, the new estimator (Fs9) has the largest
standard deviation of the three, except when allele frequencies are even. In that case the three estimators are
all largely unbiased and have similar standard deviations.
Under sample plan I with exhaustive sampling (i.e., all
individuals sampled nondestructively and returned to
the population before reproduction), bias is small for
934
P. E. Jorde and N. Ryman
either estimator (data not shown). The mean point
estimate of drift based on the original Fk9 estimator
ranges from 0.00502 for loci with even allele frequencies
to 0.00543 when the frequencies are highly skewed. The
worst case here represents ,10% bias from the true
value of 0.00500 (inverse of 2Ne) and may be considered
negligible in most practical applications. These observations verify the theoretical expectation (cf. Figure 1C)
that bias in Fk9 is reduced under sample plan I when a
large fraction of the population is sampled. The new
estimator Fs9 is again virtually unbiased for all allele
frequency profiles (mean point estimates range from
0.00499 to 0.00506). Both estimators have similar standard deviations, except for loci with many rare alleles
when SD for Fs9 approaches twice that for Fk9. However,
for all combinations of numbers of loci, numbers of
alleles, and allele frequency profiles, the standard deviations for both estimators are greatly reduced relative to
that observed for plan II, typically being one-half (for
Fk9) to one-third (for Fs9) of those reported in Table 3.
DISCUSSION
There is an obvious trade-off between accuracy and
precision when selecting estimators for genetic drift and
effective size with the temporal method. Unbiased
estimates are provided with the new estimator, Fs9(Equations 9 and 13 for sample plan II or 14 for plan I),
whereas other estimators are seen to yield quite biased
estimates in some situations.
The simulation results presented in Table 3 allow for
comparing alternative estimators with respect to both
accuracy and precision, under relatively realistic scenarios pertaining to sample plan II. Here, the results of
simulated mixtures of skewed and even allele frequencies for various number of loci are of practical interest.
For example, 10 diallelic loci with mixed allele frequency profiles may exemplify an isozyme study of
temporal allele frequency change (e.g., Jorde and
Ryman 1996), 8–10 multiallelic loci a typical microsatellite study (Miller and Kapuscinski 1997; Ovenden
et al. 2006), and 100 diallelic loci a realistic SNP study.
Under all three scenarios, the new estimator should
yield unbiased estimates of genetic drift and effective
size (Table 3, bottom row), whereas the original Fk9 estimator and the MLNE method do not.
The improved accuracy of the new estimator comes at
a cost of reduced precision, however, as is evident from
the larger standard deviation of Fs9 relative to Fk9. The
reduction in precision is probably caused by a more
aggressive allele weighing scheme in the new estimator.
Because Fs weights alleles according to their heterozygosity, or z(1 z) (z being the allele’s average sample
frequency), alleles that occur in a low frequency in the
samples are given proportionally lower weight and
contribute less to the mean estimate. For example,
when calculating Fs an allele that occurs with an average
sample frequency of z ¼ 0.1 would receive a weight that
is more than nine times greater than would a rarer allele
with z ¼ 0.01, the relative weight being ½0.1 3 (1 0.1)/
½0.01 3 (1 0.01) ¼ 9.09. In the measure Fk, on the
other hand, there would be much less difference in
weight given to these two alleles; (1 0.1)/(1 0.01) ¼
0.91, or nearly equal weights. With intermediate allele
frequencies the differences in weight between the two
schemes are modest, and Fk9 and Fs9 have similar means
as well as standard deviations when allele frequencies
are even (cf. middle rows of Table 3). The allele
weighing scheme used in the new estimator is advantageous because it drastically reduces bias associated with
rare alleles (cf. Figure 1), allowing for an unbiased
mean. The lower weight given to rare alleles also largely
eliminates the bias caused by allele fixation when
sample intervals are long (e.g., t ¼ 10 generations, Table
2). On the other hand, the new weighing scheme is
disadvantageous in the sense that the mean estimate is
dominated by fewer alleles and therefore becomes more
susceptible to random errors raising its standard
deviation.
Discussion of the temporal method in the literature
has so far focused on the precision that can be expected
of the resulting estimates of genetic drift and effective
size. This is justified because genetic drift is a stochastic
process and estimates of drift are often strongly dominated by random errors when based on a limited
number of loci and on limited sample sizes. With
increased automation of the genotyping process, larger
numbers of individuals and numbers of loci are becoming realistic and systematic errors (bias) are then
becoming increasingly important. Thus, the better the
data, the more important the issue of bias is. For
example, with a large number of multiallelic loci with
skewed allele frequencies (cf. Table 3, 100 loci), the
original Fk9 estimator yields an estimate of Ne ¼ 1/(2 3
0.00387) ¼ 129 that is not only biased (by 29.2%), but
the 95% confidence interval for the estimate ½1/(2 3
(0.00387 6 1.96 3 0.0005)) ¼ 103 173 does not even
contain the true value of 100! Despite this bias, the
mean square error (MSE ¼ bias2 1 SD2) of the original
Fk9 estimator is only a little higher than the MSE for the
new, unbiased Fs9. This is because Fk9 has a lower SD.
Thus it yields a more precise, but biased estimate of
effective size. Clearly, unbiased estimators may sometimes be preferable even if they sacrifice some precision.
When precision is of major concern, the best strategy
is to sample a large fraction of the population according
to plan I. When such a strategy can be applied, both
random errors as well as bias associated with the original
estimators (Fc9 and Fk9) are greatly reduced. While
impractical for many organisms, such exhaustive sampling may nevertheless be realistic for reasonably small
populations that can be sampled nondestructively.
Nondestructive sampling is trivial for most plants and
Estimating Effective Population Size
may be adopted for a number of animals using, e.g., skin
or blood biopsy, fin clipping, or noninvasive sampling of
feathers, hairs, or feces. The latter has proved effective
in sampling otherwise elusive large mammals such as
brown bears (Taberlet et al. 1997) and wolverines
(Flagstad et al. 2004).
In recent years, various maximum-likelihood approaches for estimating effective size from temporal
allele frequency data have been developed and claimed
to be less biased than moment-based estimators such as
Fc9 or Fk9 (Williamson and Slatkin 1999; Anderson
et al. 2000; Berthier et al. 2002; Wang and Whitlock
2003). Contrary to these claims, we find that bias may
also be a problem for maximum-likelihood methods under certain conditions. For example, the MLNE method
yields downward-biased estimates of drift, and hence too
large N̂e , when calculated from loci with highly skewed
allele frequencies (Table 3, top rows). Further simulation results presented in Table 3 (middle rows), and
additional ones not shown here, indicate that the MLNE
method performs appropriately when applied to loci
with reasonably even allele frequencies. However, making a complete evaluation of the performance of this
and other maximum-likelihood estimation procedures
is outside the scope of the present article ½see Tallmon
et al. (2004), for some comparisons. Instead, we focus
on moment-based estimators because they are the most
widely applied in the empirical literature and because
they are more readily adopted to various sampling
scenarios (i.e., the two different sample plans referred
to herein) and population characteristics, including
semelparous (Waples 1990) and iteroparous ( Jorde
and Ryman 1995; Waples and Yakota 2007) breeding
schemes with age structure and overlapping generations and different ploidy (e.g., haploid mtDNA)
(Laikre et al. 1998; Turner et al. 2001). The new Fs9
estimator should therefore provide an unbiased alternative to other estimators of genetic drift and effective
size in a wide range of applications. A computer program implementing the new estimator can be downloaded from http://folk.uio.no/ejorde/ or http://www.
zoologi.su.se/ryman/.
We thank Jinliang Wang and Robin Waples and two anonymous
reviewers for valuable comments on a previous version of this
manuscript. This work was supported by the Research Council of
Norway (P.E.J.) and by the Swedish Research Council (N.R.). Parts of
this work were carried out when P.E.J. was on leave at the Division of
Population Genetics, Stockholm University, with a Marie Curie postdoctoral stipend from the European Science Foundation.
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Communicating editor: M. Veuille