MEC676.fm Page 1171 Wednesday, June 30, 1999 3:02 PM
Molecular Ecology (1999) 8, 1171–1178
Estimating genetic drift and effective population size from
temporal shifts in dominant gene marker frequencies
Blackwell Science, Ltd
P. E . J O R D E , S . P A L M and N . RY M A N
Division of Population Genetics, Stockholm University, S-10691 Stockholm, Sweden
Abstract
Measurement of temporal change in allele frequencies represents an indirect method for
estimating the genetically effective size of populations. When allele frequencies are estimated for gene markers that display dominant gene expression, such as, e.g. random amplified polymorphic DNA (RAPD) and amplified fragment length polymorphism (AFLP)
markers, the estimates can be seriously biased. We quantify bias for previous allele frequency estimators and present a new expression that is generally less biased and provides
a more precise assessment of temporal allele frequency change. We further develop an
estimator for effective population size that is appropriate when dealing with dominant
gene markers. Comparison with estimates based on codominantly expressed genes,
such as allozymes or microsatellites, indicates that about twice as many loci or sampled
individuals are required when using dominant markers to achieve the same precision.
Keywords: allele frequencies, effective population size, genetic drift, RAPD, AFLP, temporal
method
Received 7 October 1998; revision received 23 February 1999; accepted 23 February 1999
Introduction
Recent developments in molecular genetic techniques
have provided several types of genetic markers that
are suitable for routine screening of genetic variability
in populations of virtually any organism. Most of these
techniques utilize PCR-based amplification of DNA
fragments and can be applied to even minute amounts
of material, including DNA extracted from old collections.
With direct comparisons of the genetic characteristics
of old samples with recent ones comes the possibility
of detailed analyses of temporal genetic change within
populations (see, e.g. Nielsen et al. 1997). One application
of measurements of temporal change of allele frequencies
is to estimate the genetically effective population size of
natural or captive populations, a quantity of considerable
interest within the fields of conservation biology and
population management (e.g. Allendorf & Ryman 1987;
Lande & Barrowclough 1987). This so-called ‘temporal
method’ has been applied to estimate effective population
sizes in a number of species, using allozyme markers
Correspondence: P. E. Jorde. (On leave from the Department of
Biology, University of Oslo, PO Box 1050 Blindern, N-0316 Oslo,
Norway.) E-mail: p.e.jorde@bio.uio.no
© 1999 Blackwell Science Ltd
(Krimbas & Tsakas 1971; Begon et al. 1980; Hedgecock &
Sly 1990; Waples 1990; Hedgecock et al. 1992; Jorde &
Ryman 1996), microsatellites (Miller & Kapuscinski 1997),
minisatellites (Scribner et al. 1997), and mitochondrial
DNA haplotypes (Laikre et al. 1998).
To date, estimation of genetic drift and effective size
has been based on genes with codominant expression, for
which individual allelic variants are observed directly.
Many recent techniques, however, as well as some older
ones based on blood groups or certain allozymes, do not
allow for direct observations of genotypes. For instance,
random amplified polymorphic DNA (RAPD: Williams
et al. 1990) and amplified fragment length polymorphism
fingerprinting (AFLP: Vos et al. 1995) both amplify DNA
fragments that include certain sequence(s) that are recognized by the selected primer. As a result, only individuals that carry the specific sequence will display the
fragment, whereas others will not, giving the phenotypes
‘presence’ or ‘absence’ of the fragment, respectively. It is
typically difficult or impossible to tell whether the fragment is present in one or two copies, i.e. whether it occurs
in heterozygous or in homozygous condition, and the
amplified fragment is therefore regarded as dominant to
the recessive condition of absence of the fragment (Lynch
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1172 P . E . J O R D E , S . P A L M and N . R Y M A N
& Milligan 1994). Because dominant markers have different sampling properties from codominantly expressed ones
(e.g. Jorde & Ryman 1990; Lynch & Milligan 1994) it is
presently unclear how to apply them to estimate genetic
drift and effective size, and how dominant gene markers
compare to codominant ones with respect to bias and precision of the resulting estimates. Here, we address these questions and present new estimators for allele frequencies
and for temporal allele frequency shifts that are appropriate
when dealing with dominant gene markers of any kind.
E(q2) = [E(q)]2 + V(q) = q2 + V(q)
(2)
where E denotes the expected value operator, so that
E(q) = q, and V(q) is the sample variance of q. For large
samples the variance is (e.g. Crow & Kimura 1970; p. 512):
1 – q2
V(q) = ------------4n
(3)
Substituting x for q2 in eqns 2 and 3 leads to the following estimator for the frequency of the recessive (‘absence’)
allele:
Methods
Allele frequency estimates
A problem specific to the use of gene markers with dominant gene expression concerns bias in estimates of allele
frequencies and derived quantities. The standard method
for estimating allele frequencies for dominant markers
utilizes the expected relationship between gene and genotype frequencies in a randomly mating population (i.e.
Hardy–Weinberg proportions) and estimates the frequency
of the recessive allele (q) from the observed number of
recessive homozygote individuals (X) in a sample of n
individuals:
q = X/n = x
of that quantity (Elandt-Johnson 1971; p. 104). In the present
context of q and q2:
(1)
where x = X/n is the observed proportion of recessive
homozygotes, i.e. the proportion of individuals that do
not display the dominant marker phenotype.
In eqn 1 and elsewhere, we assume that the genotypes
occur in Hardy–Weinberg proportions so that x represents an unbiased estimate of the population frequency of
recessive homozygotes (q2). The assumption of Hardy–
Weinberg genotype proportions is necessary in order to
estimate allele frequencies at dominant loci. In this
respect dominant markers have a disadvantage relative
to codominant ones because supplementary information
is needed to check whether this assumption is justified.
In the context of estimating effective size the assumption
of Hardy–Weinberg proportions should be reasonable,
however, because the application of the temporal method
typically rests on the assumption of a single, randomly
mating population with no evolutionary forces other than
genetic drift acting on the marker genes. Under these conditions, large deviation from Hardy–Weinberg proportions
is unlikely to occur and allele frequencies at dominant
markers can be estimated from the phenotype proportions.
However, even when genotypes do occur in Hardy–
Weinberg proportions, the estimator (eqn 1) often provides
biased estimates of allele frequency (Jorde & Ryman
1990). This bias arises because, generally, the expectation
of a squared quantity is larger than the squared expectation
1– x
q = x + ---------4n
(4)
as an alternative to the standard formula (1). Lynch &
Milligan (1994), also noting the bias in (1), suggested
another estimator
x
q = -------------------------x(1 – x)
1 – ----------------8nx2
(5)
This last expression (5) cannot be applied when no recessive homozygotes are found in the sample, i.e. when x = 0,
and q is set to zero in those cases.
Figure 1 compares bias in the three estimators (1) (4),
and (5) over a wide range of sample sizes and population
allele frequencies. For each estimator the expected value,
E(q), was calculated as the average estimate over all possible numbers of recessive homozygotes (X) observed,
weighted by the probability of obtaining that number in
a sample of n individuals when the true proportion of
recessive homozygotes in the population is q 2, i.e. the
n
binomial probability Bin(X; n; q2) =   (q2)X(1 − q2)n−X.
X
For estimator (5) we used an estimate of q = 0 whenever
X = 0 occurred in the calculations. It is obvious from the
results depicted in Fig. 1 that none of the three estimators
is unbiased over all combinations of allele frequency and
sample size. Bias is primarily a concern when samples are
small and the recessive allele is rare, often resulting in
quite erroneous allele frequency estimates. Comparing
the three estimators we note that the standard one (1) and
that of Lynch & Milligan (1994) (5) behave similarly with
respect to sample size and population allele frequency,
with maximum bias expected for small samples and
intermediate to low allele frequency. Overall, the latter
estimator represents only a minor improvement over
the standard formula with respect to bias. Our suggested
estimator (4) behaves quite differently from the other two
with significant bias expected primarily for very low q
values. In particular, for q = 0 (i.e. when the recessive
© 1999 Blackwell Science Ltd, Molecular Ecology, 8, 1171–1178
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E S T I M A T I N G E F F E C T I V E S I Z E 1173
1.0
a
frequency change because such measurements must in any
case be limited to polymorphic marker genes with q > 0.
Furthermore, reasonably accurate estimates of temporal
change cannot be expected unless fairly large samples are
used (below) and it is not clear from Fig. 1 which estimator to prefer under those circumstances.
(equation 1)
0.8
0.6
0.4
Estimating temporal change
0.2
−0.06
−0.02
−0.04
−0.02
0.0
Population frequency of recessive allele
20
40
60
80
100
1.0
b (equation 5)
0.8
0.6
0.4
0.2
−0.06
−0.04
20
40
−0.02
−0.02
0.0
60
80
100
1.0
c
(equation 4)
Temporal shifts in allele frequencies are typically estimated from two or more samples of individuals taken at
different occasions. Various strategies for sampling have
been devised, viz. sampling before or after the individuals
have reproduced, sampling individuals with or without
replacement, and with one or more generations lapsing
between samples (Nei & Tajima 1981; Waples 1989). For
the purpose of evaluating the usefulness of dominant
gene markers to estimate effective size we consider the
simplest situation and assume that generation intervals
are discrete (nonoverlapping) and that there are two
samples, each consisting of n individuals, that are drawn
exactly one generation apart before reproduction (i.e.
sample plan II of Nei & Tajima 1981). The extension to
other situations can easily be accommodated within the
framework used here and is not discussed further (see
Nei & Tajima 1981; Pollak 1983; Waples 1989, 1990; Jorde
& Ryman 1995). One commonly used measure of temporal
allele frequency shifts between samples is provided by Pollak
(1983). For di-allelic loci this measure can be written as
( qx – q y )2
Fk = ---------------------qz ( 1 – qz )
0.8
0.6
0.4
0.2
0.02
0.0
0.10
20
0.06
0.08
40
60
80
0.04
100
Sample size
Fig. 1 Expected amount of bias, E(q) – q, in allele frequency estimates for dominant gene markers under various combinations of
sample size (n) and population frequency of the recessive allele
(q). (a) The standard estimator (1). (b) Lynch and Milligan’s
estimator (5). (c) The proposed estimator (4). Lines represent
areas of equal bias indicated by numbers (i.e. isoclines). Note
that estimator (4) tends to yield estimates of q that are too high
(i.e. positive bias), whereas the other two are biased downwards
and yield estimates that are too low.
allele is lacking) this estimator always yields a (biased)
value of √1/(4n) (cf. eqn 4). While this particular property of (4) may in some cases be undesirable, it need not
be of major concern when estimating temporal allele
© 1999 Blackwell Science Ltd, Molecular Ecology, 8, 1171–1178
(6)
where qx and qy is the estimated frequency of the recessive
allele in the first and in the second generation, respectively, and qz = (qx + qy)/2 is their mean.
In the case of codominantly expressed, selectively neutral
genes the expected value of Fk has a relatively simple relationship with the effective population size, Ne, namely (cf. Nei
& Tajima 1981; Waples 1989): E(Fk) ≈ 1/(2Ne) + 1/n, and estimation of Ne can be done on the basis of that relationship:
1
Ne = -----------------------2Fk – 1/ñ
(7)
where ñ is the harmonic mean of the two sample sizes.
The term 1/ñ may be viewed as a ‘sample-correction’ to
Fk, accounting for the fact that Fk (eqn 6) provides an
upward biased estimate of temporal allele frequency change.
With dominant gene expression the statistical properties
of the allele frequency estimates are more complicated,
as discussed above, and we need to find the expected
relationship between Fk and Ne in this situation. We
address this problem by finding the expected value of the
observed allele frequency shift between the two samples
under dominant gene expression. Allowing Eδ to designate
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1174 P . E . J O R D E , S . P A L M and N . R Y M A N
the expected value operator for the change in allele
frequency (i.e. genetic drift) during the generation lapsing
between the two samples, and Eσ that for sampling from
the population, we have:
E[(qx − qy)2] = E[(qx − qx + qx − qy)2] = E[(qx − qx) – (qy − qx)]2 =
Eσ[(qx − qx)2] − 2Eσ[(qx − qx)Eδ(qy − qx)] + EσEδ[(qy − qx)2].
Here, Eσ(qx − qx) = 0 and Eσ [(qx − qx)2] = (1 – q2x)/(4n) is
the sample variance of qx (eqn 3). Further, EσEδ[(qy − qx)2] =
Eσ [(qy − qy)2] + Eδ [(qy − qx)2], where the first term is the
sample variance of qy and the second term is the variance
of the temporal allele frequency change (i.e. genetic drift)
in the population over a generation, or qx(1 – qx)/(2Ne)
(cf. Crow & Kimura 1970; eqn 7.3.2).
Putting this together we obtain the expected squared
difference in observed allele frequencies for samples drawn
one generation apart and scored for a dominant locus:
qx ( 1 – qx ) 1 – q 2x 1 – q 2 y
E[(qx − qy)2] = ---------------------+ ---------------- + ---------------4n
4n
2N e
qx ( 1 – qx ) 1 – q 2x
- + ---------------≈ --------------------2N e
2ñ
(8)
where the approximation holds when the true shift in
allele frequency over one generation is not very large, so
that q2y ≈ q2x, as would be the case when Ne is not extremely small. Even when Ne is very small this simplifying
assumption is reasonable because then the term involving 1/(2Ne) in equation 8 is likely to dominate over terms
in 1/(4n) and the entire expression becomes rather insensitive to the numerators of the latter. Making this simplification, the expectation of the denominator of (6) becomes
approximately equal to qx(1 – qx) and, to the extent that
the expectation of a ratio can be taken as the ratio of the
expectations of its numerator and denominator, we have
the expectation for Fk for dominant gene markers:
1 – q 2x
1
E(Fk) ≈ --------- + ----------------------------2N e 2ñqx(1 – qx)
(9)
Ideally, we would like to have a measure of temporal
allele frequency change that is related directly to 1/(2Ne)
and that is independent of sample size (n) and allele
frequency (qx). Equation 9 suggests that the modified
measure Fk′ satisfies this criterion:
( 1 – q 2x )
Fk′ = Fk − ----------------------------2ñqx(1 – qx)
(10)
Here, the estimated frequency qx is substituted for the
unknown true frequency qx in (9). Note that, when sample
sizes are small and Ne is reasonably large, the average
allele frequency over the two samples, qz, may be a more
precise estimate of qx and we use this average in place of
qx in (10) when calculating Fk′ in the following.
In analogy with the codominant case (cf. eqn 7), the
expression subtracted from Fk in (10) represents the contribution to Fk that is expected because of sampling errors
in the allele frequency estimates. Hence, Fk′ should represent an unbiased estimate of genetic drift, assuming that the
gene markers are selectively neutral, and can be applied
directly to estimate the effective size of the population:
1
1
Ne = ---------- = -----------------------------------------------2F′k
( 1 – q 2z )
2 Fk – ----------------------------2ñqz(1 – qz)
(11)
where Fk is calculated from eqn 6 using allele frequencies
estimated from one of the three alternative estimators (1)
(4) or (5).
Bias
In order to check the amount of bias that can be expected
in the proposed estimator for Ne (eqn 11) we consider the
situation of an infinitely large population being sampled
at two occasions. With an infinite size there is no genetic
drift between the sampling events and Fk′ should be zero:
corresponding to an infinite estimate of Ne. We check this
when Fk′ is calculated from sample allele frequencies that
are estimated using either of the three estimators above.
With two samples, each consisting of n individuals, there
are (n + 1)2 possible combinations of the observed number of recessive homozygotes (X and Y, respectively)
in the two samples, with X and Y ranging from 0 to n.
Summing the values of Fk′ that result for each combination of X and Y and weighting by the probability,
Bin(X; n; q2) × Bin(Y; n; q2), of that observation yields the
expected value of Fk′. (In these calculations we ignore
cases with no observed variation, i.e. when X = Y = 0 or
X = Y = n, because these observations yield no information about genetic drift.) The resulting expected values of
Fk′ for various sample sizes (n) and true allele frequencies
(q) are presented in Fig. 2.
Recalling that Fk′ should be zero in the case of an infinite population, the values depicted in Fig. 2 indicate that
bias in Fk′ may be a problem for dominant markers. In
particular, bias can be enormous when allele frequencies are estimated using either eqn 1 or 5, with maximum
bias of 0.6 and 0.8, respectively, yielding an estimate of Ne
(eqn 11) that is less than one! In comparison, our proposed
estimator (4) has a maximum amount of bias of an order
of magnitude below that of the other two (Fig. 2c). Also,
bias arising from this estimator is generally negative, so
that Ne would be judged too high, if anything, rather than
too low. (Note that a negative estimate of Ne is typically
interpreted as an infinite estimate.)
For all three estimators, however, bias drops rapidly
with increasing sample size and increasing frequency of
© 1999 Blackwell Science Ltd, Molecular Ecology, 8, 1171–1178
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E S T I M A T I N G E F F E C T I V E S I Z E 1175
a
(equation 1)
0.005 (corresponding to the dotted lines in Fig. 2), an even
wider domain of ‘unbiasedness’ results, in particular for
estimator (4). Unless sample sizes are very small, most gene
markers should lie within this domain and yield approximately unbiased estimates as judged by this criterion.
E( F′k )
0.6
0.4
0.2
0
1
0.8
0.6
0
20
40
n
80
0.2
100
q
0
b
(equation 5)
E( F′
k)
0.4
60
0.8
0.6
0.4
0.2
0
1
0.8
0.6
0
20
40
n
60
80
0.2
100
q
0
c
(equation 4)
E( F′
k)
0.4
0.02
0
−0.02
−0.04
−0.06
−0.08
1
0.8
0
0.6
20
40
n
0.4
60
80
0.2
q
0
100
Fig. 2 Expected estimated amount of temporal allele frequency
shift (bias) when sampling from an infinite population and
scoring a dominant locus. The surfaces depict the expected value
of Fk′ (eqn 10) under various sample sizes (n) and population
frequencies (q) of the recessive allele. (a) Using the standard
expression (1) for allele frequency estimation. (b) Using estimator (5). (c) Using estimator (4). Note the difference in scale of the
graphs. The dashed and the dotted line along the figure base
represent the isoclines Fk′ = 0.001 and 0.005, respectively, and
describe areas of fairly low bias. (Fk′ for an infinite population
should be zero.)
the recessive allele, implying that most of the bias can be
eliminated if analyses of temporal change are restricted to
large samples and highly polymorphic marker loci. This
restricted range corresponds to the area in Figs 2a–c
described by the dashed line: above this line bias is less
than 0.001 and the effective size is estimated to be 500 or
larger, which should probably be considered acceptable.
Relaxing the criteria for acceptable bias to, for example,
© 1999 Blackwell Science Ltd, Molecular Ecology, 8, 1171–1178
Precision
An important consideration, in addition to bias, when
estimating Ne from temporal shift in allele frequency is
the precision of the estimate. Because genetic drift is a
stochastic process, a single observation of temporal allele
frequency shift is insufficient for achieving any degree of
reasonable accuracy in the estimated Ne. Instead, estimation
of Ne should be based on multiple observations, typically
from several independently segregating gene markers,
and an average Fk′ over observations (gene loci and/or
generations) should be used when estimating effective
size from Ne = 1/(2Fk′) (eqn 11).
We performed computer simulations to check on the
precision of the estimates for Ne under various conditions, using uniformly distributed random population
allele frequencies, 0.1 ≤ q ≤ 0.9, in the first generation.
Random binomial drawings of 2Ne genes for reproduction and n phenotypes (recessive and dominant types)
for sampling were performed over one generation, using
various true effective sizes (Ne = 20; 100; 500), sample sizes
(n = 20; 100), and number of independently segregating
dominant marker loci (k = 5; 20; 100). Average Fk′ (eqs 6
and 10), its standard deviation, sFk′, and the corresponding estimate for effective size (Ne; eqn 11) were calculated
over 100 000 computer runs using allele frequency estimates provided by eqns 1, 4, and 5. In all cases pairs of
samples that were both monomorphic for either phenotype were excluded from the calculations. For comparison,
we also performed an analogous set of simulations using
codominantly expressed di-allelic loci.
Table 1 shows the estimates Fk′ and Ne resulting from
the computer simulations. Comparing the estimates of Ne
with the true values (leftmost column) when based on
different estimators for allele frequencies reveals large
amounts of bias in many cases. In particular, estimates
of Ne based on allele frequencies calculated from (1) or (5)
are generally too low, often seriously so. The point
estimates of Ne based on these equations are also highly
sensitive to sample size and appear quite misleading
under almost all conditions. This finding is not unexpected in view of the large upward bias in Fk′ noted in
Fig. 2 for these estimators. In contrast, calculating allele
frequencies from estimator (4) yields estimates of Ne that,
while not necessarily unbiased, appear more reasonable.
In brief, this estimator tends to yield estimates that are
too large when based on small samples and estimates that
are somewhat too small for large samples. This latter
Estimator (5)
Estimator (4)
Codominant loci
© 1999 Blackwell Science Ltd, Molecular Ecology, 8, 1171–1178
Ne
n
k
Fk′
sFk′
Ne
Fk′
sFk′
Ne
Fk′
sFk′
Ne
Fk′
sFk′
Ne
20
20
20
20
20
20
100
100
100
100
100
100
500
500
500
500
500
500
20
20
20
100
100
100
20
20
20
100
100
100
20
20
20
100
100
100
5
20
100
5
20
100
5
20
100
5
20
100
5
20
100
5
20
100
0.0922
0.0919
0.0916
0.0342
0.0341
0.0340
0.0684
0.0687
0.0687
0.0116
0.0116
0.0117
0.0636
0.0637
0.0637
0.0072
0.0072
0.0072
0.1058
0.0530
0.0236
0.0372
0.0186
0.0083
0.0963
0.0482
0.0215
0.0255
0.0128
0.0057
0.0944
0.0469
0.0211
0.0235
0.0118
0.0053
5
5
5
15
15
15
7
7
7
43
43
43
8
8
8
69
69
69
0.1045
0.1041
0.1037
0.0343
0.0342
0.0342
0.0808
0.0811
0.0811
0.0119
0.0119
0.0120
0.0761
0.0762
0.0762
0.0076
0.0076
0.0076
0.1139
0.0570
0.0254
0.0380
0.0190
0.0085
0.1055
0.0528
0.0236
0.0266
0.0133
0.0060
0.1036
0.0515
0.0232
0.0248
0.0124
0.0056
5
5
5
15
15
15
6
6
6
42
42
42
7
7
7
66
65
66
0.0211
0.0213
0.0211
0.0254
0.0253
0.0253
0.0035
0.0037
0.0037
0.0054
0.0054
0.0054
– 0.0001
0.0000
0.0000
0.0015
0.0015
0.0014
0.0618
0.0310
0.0138
0.0279
0.0140
0.0062
0.0518
0.0260
0.0116
0.0162
0.0081
0.0036
0.0500
0.0249
0.0111
0.0141
0.0071
0.0032
24
24
24
20
20
20
144
135
136
93
93
92
inf
inf
inf
344
338
341
0.0258
0.0257
0.0256
0.0256
0.0255
0.0256
0.0056
0.0055
0.0055
0.0050
0.0051
0.0050
0.0016
0.0015
0.0017
0.0010
0.0010
0.0010
0.0461
0.0231
0.0103
0.0222
0.0111
0.0050
0.0345
0.0171
0.0077
0.0094
0.0047
0.0022
0.0321
0.0160
0.0071
0.0069
0.0035
0.0014
19
19
19
20
20
20
90
91
90
100
99
99
305
324
297
506
495
489
MEC676.fm Page 1176 Wednesday, June 30, 1999 3:02 PM
Estimator (1)
1176 P . E . J O R D E , S . P A L M and N . R Y M A N
Table 1 Estimated amount of genetic drift over one generation (Fk′: eqn 10) and effective size (Ne: eqn 11) in computer simulations (each based on 100 000 replicates). Initial allele
frequencies (qx ) varied uniformly between 0.1 and 0.9 among simulations and we used three different estimators when assessing the frequencies of the recessive gene from samples of
individuals, viz. the standard estimator (eqn 1), Lynch and Milligan’s estimator (5), and our new estimator (4). Estimates based on codominant loci are included for comparison. Ne is
the true effective size, n is the sample size (number of diploid individuals) in each generation, and k is the number of loci
MEC676.fm Page 1177 Wednesday, June 30, 1999 3:02 PM
E S T I M A T I N G E F F E C T I V E S I Z E 1177
downward bias for large samples is not very pronounced,
however, and should in most cases be judged acceptable.
As a side issue, we note that when using codominant
markers (e.g. eqn 7) there is a certain downward bias in
Ne for small samples (cf. rightmost column in Table 1 for
n = 20).
Estimates of temporal allele frequency shifts based on
(4) also have considerably lower standard deviation (sFk′)
than those obtained from the other two estimators for
dominant alleles, implying higher precision of the point
estimate. However, comparing the standard deviations to
their means, it is evident that the precision is not very
high in any of the estimates, regardless of which formula
is used to calculate allele frequencies. This fairly low
precision is, albeit to a lesser extent, also observed for
codominant genes (cf. Table 1), and is only partially
attributable to dominant gene expression. For estimates
calculated on the basis of eqn 4, using dominant marker
genes reduces precision (as measured by sFk′) by a factor
of about 2/3 relative to what can be obtained if the same
number of di-allelic codominant marker genes was used.
Discussion
Analysis of genetic drift from dominantly expressed gene
markers has a number of disadvantages relative to using
codominantly expressed ones. From the statistical point
of view, problems with dominant markers arise because
only phenotypes are observed, rather than the genes
directly. One consequence of this is that, with the same
number of diploid individuals screened, the number
of observations (phenotypes or genes) is only half that
for codominant markers, resulting in reduced precision.
Further, when the genotypes occur in Hardy–Weinberg
proportions (as assumed herein) the frequencies estimated
for dominant and recessive alleles are not necessarily
unbiased and neither are estimates of temporal allele frequency shift and effective population size. We have presented an estimator (eqn 4) for allele frequencies that,
while not always unbiased, does minimize bias in measures of temporal allele frequency change and effective
size. This estimator also yields more precise estimates of
these quantities (i.e. having a lower standard deviation)
than either the standard estimator (eqn 1) or the one proposed by Lynch & Milligan (1994) (eqn 5). We conclude
that for measurements of temporal change, allele frequencies should preferably be calculated from eqn 4.
While using the proposed estimator for allele frequencies should yield the least biased and most precise
estimates of effective size for dominant gene markers, it is
clear from the large standard deviations of Fk′ reported in
Table 1 that the precision is often quite poor: in many of
the simulations the standard deviation is larger than the
mean value, implying particularly uncertain point estimates
© 1999 Blackwell Science Ltd, Molecular Ecology, 8, 1171–1178
in those situations. However, it is also clear from the
table that precision can be increased by using larger
sample sizes and scoring more marker loci. The standard
deviation of Fk′ declines approximately inversely with
the square root of the number of loci or individuals
(cf. Table 1). This implies that in order to increase the
precision for dominant markers to match that for codominant ones, i.e. reducing sFk′ by approximately 2/3, about
twice the number of loci or individuals need to be
sampled [(3/2)2 = 2.25]. As when using codominant
markers, precision can be increased further by sampling
more than two generations (Nei & Tajima 1981; Waples
1989) or age classes (Waples 1990; Jorde & Ryman 1995).
With samples taken from multiple generations or age
classes the expected temporal shifts are larger than over a
single generation and, hence, more easily measured in the
presence of sampling errors.
Measures of temporal allele frequency change calculated from dominant loci can further be combined with
measures based on other kinds of gene markers, including
codominantly expressed allozymes or microsatellites.
Because these latter markers often segregate for multiple
alleles, Fk′ for each marker locus should be weighted by
the number of ‘independent’ alleles used for that locus,
or K − 1 for a locus with K alleles. This assures that F for
each allele is given equal weight in the average Fk′ and
implies that each di-allelic marker locus with dominance
is given a weight of 1 (one), whereas codominant loci
with multiple alleles receive a higher weight. At this point
it does not seem appropriate, without examining this
question specifically, to reduce the weight for dominant
markers further on account of their lower precision. This
is because giving unduly low (or high) weight to the temporal shift that happened to occur at one or a few alleles
may compromise rather than improve precision of the
average. For similar reasons, Fk′ values are generally not
weighted by sample size either.
Acknowledgements
This study was supported through grants to N.R. from the
Swedish Natural Science Research Council and the Swedish
research program on Sustainable Coastal Zone Management
(SUCOZOMA), founded by the Foundation for Strategic Environmental Research (MISTRA). P.E.J. was supported by a Marie
Curie postdoctoral fellowship from the European Commission.
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This study was conducted at the Division of Population
Genetics, Stockholm University, headed by Nils Ryman. The
Division’s research relates to the genetic structure of natural
populations, genetic conservation, and to the genetic effects of
human interference with natural populations. This paper will be
a part of Stefan Palm’s PhD thesis. Per Erik Jorde is presently a
postdoctoral fellow at the University of Oslo, working on genetic
monitoring.
© 1999 Blackwell Science Ltd, Molecular Ecology, 8, 1171–1178