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Molecular Ecology (2001) 10, 2361–2373
Statistical power when testing for genetic differentiation
Blackwell Science, Ltd
N . RY M A N * and P. E . J O R D E †
*Division of Population Genetics, Stockholm University, S-106 91 Stockholm, Sweden, †Division of Zoology, Department of Biology,
University of Oslo, PO Box 1050 Blindern, N-0316 Oslo, Norway
Abstract
A variety of statistical procedures are commonly employed when testing for genetic differentiation. In a typical situation two or more samples of individuals have been genotyped
at several gene loci by molecular or biochemical means, and in a first step a statistical test
for allele frequency homogeneity is performed at each locus separately, using, e.g. the contingency chi-square test, Fisher’s exact test, or some modification thereof. In a second step
the results from the separate tests are combined for evaluation of the joint null hypothesis
that there is no allele frequency difference at any locus, corresponding to the important case
where the samples would be regarded as drawn from the same statistical and, hence, biological population. Presently, there are two conceptually different strategies in use for
testing the joint null hypothesis of no difference at any locus. One approach is based on the
summation of chi-square statistics over loci. Another method is employed by investigators
applying the Bonferroni technique (adjusting the P-value required for rejection to account
for the elevated alpha errors when performing multiple tests simultaneously) to test if the
heterogeneity observed at any particular locus can be regarded significant when considered
separately. Under this approach the joint null hypothesis is rejected if one or more of the
component single locus tests is considered significant under the Bonferroni criterion. We
used computer simulations to evaluate the statistical power and realized alpha errors of
these strategies when evaluating the joint hypothesis after scoring multiple loci. We find
that the ‘extended’ Bonferroni approach generally is associated with low statistical power
and should not be applied in the current setting. Further, and contrary to what might be
expected, we find that ‘exact’ tests typically behave poorly when combined in existing procedures for joint hypothesis testing. Thus, while exact tests are generally to be preferred
over approximate ones when testing each particular locus, approximate tests such as the
traditional chi-square seem preferable when addressing the joint hypothesis.
Keywords: allele frequency, Bonferroni, chi-square, contingency table, Fisher’s exact test, statistical
analysis
Received 8 January 2001; revision received 9 April 2001; accepted 20 April 2001
Introduction
An increasingly common question in conservation and
evolutionary biology is whether a set of samples are
likely to represent the same gene pool. Several statistical
techniques are being applied when addressing this type of
problem, but there has been little discussion about their
relative merits for detection of genetic heterogeneity. This
lack is particularly obvious for methods used to combine
the information from multiple loci.
Correspondence: Nils Ryman. Fax: +46 8 154041; E-mail:
Nils.Ryman@popgen.su.se
© 2001 Blackwell Science Ltd
In a typical situation an investigator has collected tissue
samples from two or more groups of individuals that are
separated in space or time. Application of some biochemical or molecular techniques provides genotypes of the
sampled individuals at one or more nuclear loci or at the
mitochondrial genome, and each sample is described in
terms of its size and allele (or haplotype) frequencies. The
specific scientific questions may vary from study to study,
but a very basic one, which frequently determines how to
proceed with the analysis is the following one: Are the
allele frequency differences observed among samples large
enough to suggest that all the samples are not drawn from
the same population (gene pool)? It appears that in most
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2362 N . RY M A N and P. E . J O R D E
cases the underlying evolutionary model is one of ‘selective neutrality — isolation — genetic drift’, which implies that
all polymorphic loci examined are potentially informative
with respect to the question of overall genetic heterogeneity.
The general statistical approach most frequently used —
and the one dealt with in this paper — is first to conduct a
contingency test for allele frequency homogeneity for each
locus separately, and in a second step to evaluate the simultaneous, or joint, information from all loci examined. The test
procedure applied to each individual locus (contingency
table) implies assessment of the probability of obtaining —
if the null hypothesis (H0 ) of equal allele frequencies is
true — an outcome that is as likely as, or less likely than,
the observed one. This probability (P-value) can either be
calculated exactly (Fisher 1950; Mehta & Patel 1983), iterated
or simulated to a desired degree of precision (Roff &
Bentzen 1989; Raymond & Rousset 1995a,b), or approximated by means of some test statistic expected to follow a
chi-square distribution (Fisher 1950; Everitt 1977; Sokal &
Rohlf 1981). In the second step the results from the separate
tests are combined for evaluation of the joint null hypothesis (H0,J ) that there is no allele frequency difference at any
locus (i.e. H0,J is true when all the separate H0s are true).
Presently there appears to be two conceptually different
strategies in use for testing H0,J. One technique is based on
the summation of chi-square statistics and utilizes the fact
that the sum of a series of chi-square distributed variables
also follows a chi-square distribution (e.g. Everitt 1977;
Sokal & Rohlf 1981).
Another approach is used by investigators applying the
Bonferroni technique to test if the heterogeneity observed
at any particular locus can be regarded significant when
considered separately. The general idea behind the Bonferroni method is to account for the increased probability of
obtaining, by pure chance when the null hypothesis is true,
a significant result at one or more loci when performing
multiple tests (e.g. Rice 1989). Under this approach the joint
null hypothesis (H0,J ) is rejected if one or more of the component contingency tests is considered significant, i.e. at
least one ‘Bonferroni significance’ is required for rejecting
the joint null hypothesis of equal allele frequencies at all
loci. This strategy for testing H0,J is conceptually adequate
in the present context, although it has been noted that it
may be quite conservative resulting in too few rejections
(Legendre & Legendre 1998). We are aware of no study,
however, that evaluates the two approaches with respect to
their ability to detect true heterogeneity.
This paper compares the power of the above statistical
methods — summation of chi-square vs. application of the
Bonferroni method to determine if any one of the separate
locus tests can be considered significant — for detecting
genetic heterogeneity when multiple loci have been scored.
The results show that the efficiency may differ dramatically between the two approaches and, contrary to what
might be expected, this difference may become enhanced
as the number of loci increases.
Exemplifying the problem
As an example of the statistical test options, Table 1
presents sample allele frequency data for 12 codominant
and di-allelic allozyme loci from two consecutive yearclasses of brown trout (Salmo trutta) collected from Lake
Blanktjärnen in central Sweden (see Jorde & Ryman 1996
for details). Are the observed allele frequency differences
large enough to suggest that there are true genetic
differences between year-classes?
The allele frequency difference at each individual locus
was tested using Fisher’s exact test, the conventional chisquare 2 × 2 contingency statistic ( X2, degrees of freedom,
d.f. = 1), and the chi-square statistic with Yates’ continuity
correction (X C2, d.f. = 1; chi-square test statistics are denoted
by X2 to distinguish them from values of a theoretical χ2
distribution). Both chi-square approximations provide Pvalues that are reasonably similar to the exact ones, but
those from the conventional X2 are generally smaller (less
conservative) whereas X C2 tends to produce larger ones. All
methods yield significant results (P < 0.05) for the same
two loci (sAAT-4 and bGALA-2).
When testing the joint null hypothesis (H0,J) that there
are no allele frequency differences at any locus, we note
that the sum of a set of χ2 distributed variables also follows
a χ2 distribution with d.f. equal to the sum of d.f. of the contributing variables. This summation is straightforward for
X2 and X C2, which are both expected to follow approximately a χ2 distribution under the null hypothesis. With
respect to the P-values obtained in the exact tests, Fisher
(1950) has shown that when the null hypothesis is true the
quantity –2ln(P) is expected to follow asymptotically a χ2
distribution with d.f. = 2. Thus, summing the negative of
twice the natural logarithm of the 12 P-values results in a
X2 statistic that is to be evaluated against a χ2 distribution
with d.f. = 24 (Table 1; this technique of summing –2ln(P)
is sometimes referred to as ‘Fisher’s method’, but it should
not be confused with Fisher’s exact test). Under each of the
three approaches the summed chi-square statistic is significant (P < 0.05) and results in rejection of the joint null
hypothesis (H0,J ) although the level of significance differs
among them (X2 and X C2 yielding the smallest and largest
summation P-value, respectively).
In contrast to the above ‘summation’ approaches, application of the Bonferroni method results in a different conclusion. The Bonferroni logic implies that no individual test in a
series of k-tests is to be judged significant unless the Pvalue is smaller than α/k, where α is the preassigned significance level for rejecting the null hypothesis (Rice 1989
and references therein; see below). In the present case with
k = 12 (12 loci) and α = 0.05 a P-value less than 0.05/12 =
© 2001 Blackwell Science Ltd, Molecular Ecology, 10, 2361– 2373
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D E T E C T I N G G E N E T I C D I F F E R E N T I AT I O N 2363
Table 1 Allele frequencies (the 100 allele) and 2 × 2 contingency test statistics for 12 di-allelic allozyme loci in two cohorts (1992 and 1993)
of brown trout from Lake Blanktjärnen, Sweden. The number of fish is 43 and 27 in 1992 and 1993, respectively
Allele frequency (100 allele)
Cohort
Fisher’s exact test
Chi-square test
X2
Locus
1992
1993
P
sAAT-4
DIA
bGALA-2
bGLUA
G3PDH-2
sIDHP-1
LDH-5
aMAN
sMDH-2
ME
MPI
PEPLT
0.6395
0.7791
0.7791
0.8256
0.7558
0.4302
0.7209
0.9884
0.9767
0.8953
0.1047
0.6395
0.4074
0.8148
0.9444
0.7407
0.6296
0.5741
0.6852
0.9259
0.9815
0.8704
0.2222
0.6667
0.009
0.673
0.009
0.285
0.129
0.119
0.705
0.073
1.000
0.786
0.087
0.856
Sum
Sum of d.f.
P
0.0042 is thus required to be regarded significant. No individual (single locus) P-value in Table 1 is that small, and consequently no locus can be considered displaying significant
heterogeneity when applying the Bonferroni technique. The
joint null hypothesis (H0,J ) is therefore also accepted, regardless of whether the basic contingency P-values were obtained
by means of chi-square approximation or exact calculation.
Thus, in this example the ‘summation’ method indicates
that the joint null hypothesis should be rejected for each
of the tests statistics applied, whereas application of the
Bonferroni technique consistently suggests acceptance when
using the same set of individual P-values. The difference is
crucial, but on the basis of the data available an investigator cannot determine what decision is most appropriate. It
appears that many scientific journals would accept either
approach without requiring additional data analysis.
In the remainder of this paper we present results of
computer simulations aimed at evaluating the probability
of detecting a true genetic difference (statistical power) and
the probability of falsely rejecting a true null hypothesis (α
or type I error) when addressing the joint H0,J (Bonferroni
vs. ‘summation’). We recognize that the present problem
can also be addressed by means of exact binomial or multinomial calculations, but we chose the simulation approach
for practical reasons.
Simulations
Random sampling of 2n genes (n diploid individuals) from
populations with known allele frequencies was simulated
© 2001 Blackwell Science Ltd, Molecular Ecology, 10, 2361–2373
–2ln(P)
9.421
0.793
9.489
2.512
4.097
4.261
0.700
5.240
0.000
0.482
4.881
0.311
7.222
0.258
6.849
1.454
2.550
2.748
0.205
3.756
0.036
0.205
3.596
0.107
42.19
24
0.012
28.98
12
0.004
Chi-square test
Yates’ correction
P
X C2
P
0.007
0.611
0.009
0.228
0.110
0.097
0.651
0.053
0.851
0.651
0.058
0.743
6.331
0.087
6.298
0.969
1.954
2.207
0.068
2.099
0.165
0.032
2.662
0.021
0.012
0.768
0.012
0.325
0.162
0.137
0.794
0.147
0.685
0.858
0.103
0.885
22.89
12
0.029
by means of pseudo random number generation. At each
locus the hypothesis of equal allele frequencies was tested
by various r × c contingency tests (r = number of samples
and c = number of alleles) using both Fisher’s exact method
and chi-square tests with d.f. = (r – 1) (c – 1). The number of
replicates (runs) of each simulation was typically in the
interval 5000–10 000, and the frequency of replicates
resulting in rejection and acceptance of a null hypothesis
provided estimates of the statistical power (probability of
rejecting a false hypothesis) or the α (type I) error (probability of rejecting a true hypothesis), respectively. The
intended α level was consistently kept at 0.05, rejecting
the null hypothesis for P < 0.05. A replicate was discarded
if the random sampling of genes resulted in less than
c alleles being observed in the combined material from
the r samples, and a new set of r samples was drawn in
those cases.
Population allele frequencies were generally chosen to
result in a ‘realistic’ proportion of simulated contingency
tables where at least one cell had a small expected value
(expectancy less than 5 or 1) at low or moderate sample
sizes. This was done in order not to provide an overly
optimistic view of the fit of the various chi-square approximations to the expected χ2 distribution.
When simulating situations where multiple loci (k > 1)
are scored, all loci within a population were assumed to
segregate at identical allele frequencies (for example, for
di-allelic loci the allele under consideration may occur
at the frequency 0.10 at all loci in population 1 and at
0.15 at all loci in population 2). Of course, in the real world
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2364 N . RY M A N and P. E . J O R D E
it is not very likely to encounter a situation where all
the loci examined in a population segregate at exactly
the same frequency (although the true allele frequency
distribution is unknown in most cases). This model is
appropriate for the present purpose of examining differences between test procedures, however. When dealing
with relative frequencies both the power and the α error
are dependent on the population frequencies, and varying
those frequencies might obscure the comparison of test
procedures.
A ‘regular’ chi-square test statistic (X2) was calculated
for each simulated contingency table. In addition, Yates’
continuity correction was applied to provide a corrected
chi-square (X C2) for all 2 × 2 tables, and for larger tables the
G-statistic with and without Williams’ correction was used
(Sokal & Rohlf 1981; 737 – 38). Fisher’s exact test for 2 × 2
tables was performed as described by Sokal & Rohlf (1981;
740 – 42), and the algorithm of Mehta & Patel (1983) was
applied for larger tables. Computational results from the
statistical routines developed for the simulation programs
were checked with outputs from softwares such as biom
(Rohlf 1987 ), statistica (StatSoft Inc. 1998), StatXact-Turbo
(CYTEL Software Corporation 1992), and genepop
(Raymond & Rousset 1995b), and the simulated power
estimates were checked against exact calculations and
those obtained using the ‘standard’ normal approximation
for power assessment (e.g. Zar 1984; 398). When combining
the information from multiple loci for evaluation of the
joint null hypothesis (H0,J ) of no difference at any locus the
approaches of Bonferroni and summation of chi-square
statistics were applied as exemplified in the preceding
section.
A considerable number of simulations have been
conducted during the course of this study using different
combinations of allele frequencies, sample sizes, number
of alleles, loci, and populations. In order not to burden
the presentation unnecessarily, however, we have tried to
choose as simple combinations as possible for illustration
of general trends and principles. Thus, most of the paper
focuses on situations like that in Table 1 where the basic
statistical tests refer to 2 × 2 tables (2 samples and 2 alleles
per locus).
2 × 2 contingency tables
Four examples of typical simulation results for 2 × 2
contingency tests are depicted in Fig. 1(a). In the most basic
case we consider a single locus (k = 1) with two alleles (A
and A′), and a random sample of 20 diploid individuals (40
genes) is drawn from each of two populations (1 and 2)
where the A allele occurs in the true frequency Q1 and Q2,
respectively. In the case of multiple loci (k > 1) all of them
have the same allele frequency (Q1) in population 1, and
within population 2 all frequencies are Q2. Every plate in
Fig. 1(a) represents a specific combination of Q1 and Q2,
and for each test procedure the proportion of replicates
resulting in rejection of the joint null hypothesis (y-axis) is
indicated for different number of loci examined (x-axis;
k = 1, 2 … 5, 10, 20 … 50). As in the introductory example
(Table 1) the joint null hypothesis (H0,J ) was rejected for
P < 0.05 (summation of chi-squares), and when using the
Bonferroni method rejection of H0,J required at least one
single locus P-value smaller than 0.05 /k. For k = 1 the
‘summation’ is over a single value only, and the Bonferroni
rejection criterion coincides with that of the basic test.
When Q1 is different from Q2 (H0 is false; upper plates) the
proportion of H0,J rejections estimates the power of the test,
and when Q1 = Q2 (H0 is true; lower plates) it estimates the
realized α error.
In a first step we focus on the situations where H0 is false
(Q1 ≠ Q2; Fig. 1a, upper plates). Considering a single locus
only (k = 1) the power estimates for the Q1/ Q2 combination
of 0.10/0.15 are 0.107, 0.041, and 0.052 when using X2, X C2,
and Fisher’s exact test, respectively, and for the 0.50/0.60
combination the corresponding estimates are 0.150, 0.103,
and 0.103. The difference between these simulated power
estimates and the values expected theoretically is noticeable in the third decimal place only, indicating that the
simulations provide reasonably accurate results. Also,
Fisher ’s exact and the X C2 tests are both expected to be
more conservative than the X2 test (e.g. Everitt 1977 ), and
they yield accordingly lower power estimates within each
combination of Q1/Q2.
In contrast to the power observed for a single locus, the
results for tests involving multiple loci are more surprising.
Here, one might expect a reliable test to detect the true
divergence between populations more frequently as
additional loci are examined. It is only the ΣX2 approach,
however, that behaves consistently in this way at both
combinations of Q1 and Q2. In the case of Q1/ Q2 = 0.10/
0.15 the probability of rejecting H0 actually tends to decrease
when the number of loci included in the test increases for
all methods except the ΣX2. When Q1 / Q2 = 0.50/ 0.60 all
the procedures provide at least some increase of power
when more loci are considered, but the ΣX2 approach is
consistently the most powerful one.
It is an important observation that the remarkably better
power obtained through summation of X2 is not associated
with an unduly large α error when H0,J is true. Rather, this
method is the only one that consistently appears to provide
an α error that is reasonably close to the intended one of
α = 0.05 (Fig. 1a, lower panels).
Simulation results for the considerably larger sample
sizes of 500 individuals (1000 genes) are shown in Fig. 1(b).
Here, the population allele frequencies are the same as
those in Fig. 1(a) when estimating the α error (lower
plates), but for the sake of illustration the assessment of
power (upper plates) is based on smaller differences
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D E T E C T I N G G E N E T I C D I F F E R E N T I AT I O N 2365
Fig. 1 Proportion of statistical significances
when testing the joint null hypothesis
(H0,J) of no difference at any locus after
simulated drawing of a sample of size (n)
from each of two populations where the
true allele frequency at di-allelic loci are
Q1 and Q2, respectively. The number of
loci examined is 1– 50. A test for allele
frequency homogeneity is conducted for
each locus separately using X2, XC2, and
Fisher’s exact test, and the resulting Pvalues are combined into a joint test by the
‘summation’ and Bonferroni approaches.
The intended α is 0.05, and each data point
is based on 10 000 (Fig. 1a) and 5000
(Fig. 1b) replicates, respectively. See text
for details. Figure 1(a): n = 20 individuals
(40 genes). Figure 1(b): n = 500 individuals
(1000 genes).
between Q1 and Q2 to account for generally higher power
associated with larger samples. (When sample sizes as
large as 500 individuals the power of all methods is close to
unity for the combinations of Q1 and Q2 used in Fig. 1a). At
sample sizes of 1000 genes all methods are reasonably
successful in keeping the α error close to the intended one
of 0.05, except for those representing summation of X C2
and –2ln(P). This latter observation most likely reflects a
somewhat slower approach to the limiting χ2 distribution
for these two test statistics as compared to X2 (see below).
When H0,J is false (upper panels, Fig. 1b) summation of X2
is the technique that consistently provides the highest
probability of rejecting H0,J. For all methods, however, the
power increases as the number of loci grows, although the
‘summation’ approach in all cases appears more powerful
regardless of the statistical procedure applied in the basic
© 2001 Blackwell Science Ltd, Molecular Ecology, 10, 2361–2373
contingency tests. This is true also when summing X C2 and
–2ln(P), in spite of the lower realized α observed for these
statistics.
Reasons for low power
For both the Bonferroni and the ‘summation’ method the
low power is associated with the approximations involved
when assessing P-values for the basic 2 × 2 tables. Both
methods are expected to work satisfactorily in theory
when samples are large and allele frequencies intermediate. In practice, however, the approximate nature of
the test statistics and the P-values produced by the primary
contingency tests may provide a realized α in the combined
test that is far below the intended one, and the reduced α
typically results in a small power.
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2366 N . RY M A N and P. E . J O R D E
Fig. 1 Continued
Bonferroni. As noted above, the objective behind the
Bonferroni technique is to avoid excessive numbers of false
significances when performing multiple tests through
adjusting the P-value at which a particular component test
is to be judged significant. The probability of observing
false significances increases rapidly as the number of tests
grows. When performing k-tests of a true null hypothesis at
α = 0.05 the expected probability of obtaining one or more
significances by pure chance is 1 – (1 – 0.05) k; for k = 10,
for example, this probability exceeds 40%.
In order to maintain the intended α level (here 0.05) of
a particular test when a series of k-tests have been performed, the idea behind the Bonferroni method is to adjust
the P-value for rejection such that the probability of observing one or more significances among the k-tests remains at
α. This goal is met approximately if rejecting any particular
H0 at P < α/k (rather than at P < α). The rationale for this
criterion is based on the relationship
1 – (1 – α/k) k ≈ α.
The above arguments form the basis for the ‘extended’
application of the Bonferroni correction in the context of
testing the joint null hypothesis H0,J that all the components H0s are true when multiple tests have been
performed. That is, if all the component (single locus) H0s
are true (i.e. H0,J is true), the probability is α to observe one
or more ‘Bonferroni significances’ (at the α/k level). Thus,
H0,J is rejected when obtaining a single locus P-value less
than α/k, otherwise it is accepted.
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D E T E C T I N G G E N E T I C D I F F E R E N T I AT I O N 2367
If the Bonferroni technique is to work in practice,
however, two basic conditions must be met which are not
satisfied in many realistic situations. First, it is necessary
that each single locus test can produce a P-value that is
smaller than α/k. However, this is not possible for many
combinations of sample size (n) and population allele
frequencies (Q1/Q2) because of the restricted number of
potential outcomes of the sampling process. If basing the
evaluation of H0,J on a suite of such contingency tests there
is a substantial risk that the Bonferroni method yields a
realized α that is considerably smaller than the intended
one, and thereby a correspondingly reduced power.
To clarify the point we may consider the extreme example
of sampling two individuals (four genes) from each of
two populations that are fixed for different alleles. The
counts of the two types of alleles will be 4; 0 and 0; 4 for the
two samples, respectively, which represents the most
extreme outcome possible under the null hypothesis of
equal allele frequencies. Fisher’s exact test yields a (twosided) P-value of 0.029, which is significant at the 5% level,
and this is the smallest P-value that can be obtained with
the present sample sizes. Analysis of, say, five alternately
fixed loci would result in five P-values of 0.029, and intuition would justifiably make the investigator suspect that
the populations are genetically divergent. A Bonferroni
evaluation would dismiss such an interpretation, however,
because no P-value is smaller than 0.01 (0.05/5), a P-value
which is impossible to obtain with the present sample
sizes. Thus, in this situation the Bonferroni correction
results in a realized α level of zero, and the power is
thereby also reduced to zero. In the present simplified
example it is easy to see how the Bonferroni method, as an
effect of a discrete number of possible experimental outcomes, may result in a reduction of the realized α far below
the intended one, and thereby in a corresponding decrease
of power. The phenomenon is a general one, however, and
the magnitude of the effect depends on the sample sizes,
the number of loci (tests), and the true allele frequency
differences (Fig. 1).
The other requirement for the Bonferroni method to work
satisfactorily is that the realized α error of each of the basic,
single locus, contingency tests is reasonably close to the
intended one. In other words, when the component H0 is
true the probability of obtaining a P-value of 0.05 or less
should be close to 0.05, that of obtaining P ≤ 0.01 should be
close to 0.01, etc. It appears that this characteristic of the
sampling distribution is frequently taken for granted, in
spite of the fact that substantial deviations are quite common.
To exemplify the difference between intended and realized α errors in basic contingency tests we may consider
the occurrence of various P-values when drawing two independent samples of 20 individuals (40 genes) from a population with the allele frequency Q = 0.10. For illustration,
all the 2 × 2 tables possible when drawing two samples of
© 2001 Blackwell Science Ltd, Molecular Ecology, 10, 2361–2373
40 items were created, P-values based on regular chisquare (X2) and Fisher’s test were computed, the exact frequency of occurrence of each table and its associated P-values
was derived binomially, and the cumulative frequency of
occurrence of P ≤ 0.05 was depicted graphically (Fig. 2,
upper plate).
As seen in Fig. 2 (upper plate) the realized α may be considerably smaller than the intended one, and particularly
so for Fisher’s exact test. With Fisher’s exact test, for example, values of P ≤ 0.05 only occur in a frequency of 0.017
(less than half the ‘ideal’ rate), and the discrepancy is even
more pronounced for smaller P-values.
The corresponding cumulative distributions of P-values
when sampling 50 and 500 individuals (100 and 1000
genes) are depicted in the central and lower plates of Fig. 2,
respectively. For X2 the correspondence between intended
and realized α is fairly good at sample sizes of 50 individuals or more. In contrast, at the present population allele
frequency samples in the order of hundreds of individuals
are required for this to occur with Fisher’s exact test.
When realized α of the separate (single locus) contingency tests are smaller than intended, the Bonferroni
approach to testing the joint H0,J of no difference at any
locus may result in an overall realized α that is far below
the anticipated one. To exemplify, Table 2 gives exact realized values of α for 1–50 loci when applying Fisher’s exact
test to two independent samples of n = 10 (or n = 20)
diploids from a population with the true allele frequency
Q = 0.10. For n = 10 and k = 10 loci (tests), for instance, the
Bonferroni method implies that H0,J should be rejected
when observing at least one contingency P ≤ 0.005 (0.05/
10). With two samples of n1 = n2 = 10 diploids the largest
Fisher P-value meeting the criterion P ≤ 0.005 is 0.00385
(due to the restricted number of possible 2 × 2 tables at
these sample sizes), and P-values this small or smaller
occur at a frequency of 0.000107 (exact cumulative binomial probability). Thus, the realized α of the Bonferroni
approach to testing H0,J corresponds to the probability of
observing one or more single locus P-values of P ≤ 0.005,
which is α = 1 – (1 – 0.000107)10 = 0.0011. Clearly, this is
dramatically smaller than the intended α of 0.05, and with
such a small realized α the chance of detecting anything
but very large allele frequency differences is minor.
Chi-square summation. As noted above, the logic of the
summation approach is based on the fact that the sum of
two or more χ2 distributed variables will also follow a χ2
distribution with d.f. equal to the summed d.f. of the component variables. Under the null hypothesis, the test statistic computed for each particular locus (contingency table)
is expected to be asymptotically χ2 distributed with
d.f. = (r – 1)(c – 1) where r is the number of samples (rows)
and c is the number of alleles (columns). The fit may be
quite poor, however, particularly for small sample sizes
and skewed allele frequencies, and the sum of several such
MEC1345.fm Page 2368 Friday, September 21, 2001 3:36 PM
2368 N . RY M A N and P. E . J O R D E
Fig. 2 Exact cumulative frequency of occurrence of possible P-values when drawing
two independent samples of n = 20, 50, or
100 diploids (40, 100 or 1000 genes) from a
population where the true allele frequency
at a di-allelic locus is 0.1. The null hypothesis
of allele frequency homogeneity is tested
using regular chi-square (X2) and Fisher’s
exact test. Only the left-most part (P ≤ 0.05)
of the distribution is shown.
Table 2 Exact realized α error when applying the Bonferroni method to a series of k 2 × 2 tables. Each table represents two independent
samples of n diploid individuals (2n genes) from a population where the true gene frequency at di-allelic loci is 0.10, and where the H0 of
no gene frequency difference has been tested by Fisher’s exact method. Intended α = 0.05. Frequency of occurrence represents the
cumulative binomial probability of obtaining the realized P-value or a smaller one. See text for details
n = 10
n = 20
k = no. of
tested loci
Intended
P-value for
rejection
Realized P-value
for rejecting a
separate 2 × 2 table
Frequency of
occurrence
1
2
5
10
20
50
0.05
0.025
0.01
0.005
0.0025
0.001
0.04837
0.02484
0.00953
0.00385
0.00220
0.00077
0.012002
0.003010
0.000622
0.000107
0.000015
0.000002
Realized α
Realized P-value
for rejecting a
separate 2 × 2 table
Frequency of
occurrence
Realized α
0.012002
0.006010
0.003107
0.001065
0.000304
0.000090
0.04817
0.02470
0.00931
0.00483
0.00240
0.00096
0.016770
0.005559
0.002022
0.000678
0.000514
0.000056
0.016770
0.011087
0.010067
0.006761
0.010225
0.002802
© 2001 Blackwell Science Ltd, Molecular Ecology, 10, 2361– 2373
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D E T E C T I N G G E N E T I C D I F F E R E N T I AT I O N 2369
‘poorly fitted’ variables may deviate dramatically from the
expected χ2 distribution.
The mean and variance of a χ2 distribution is d.f. and
2d.f, respectively, and as an example of the fit to χ2 Table 3
gives the observed mean and variance of the test statistics
obtained when simulating the drawing of two independent samples from a population with an allele frequency of
0.1 (Q1 = Q2 = 0.1; n = 3 – 500). Both of X2 and X C2 have
d.f. = 1, and if the fit is perfect we expect these statistics to
yield means and variances of 1 and 2, respectively. With
respect to the P-value from Fisher’s exact test (exact P) the
quantity –2ln(exact P) should be asymptotically χ2 distributed with d.f. = 2 (see above), and with a perfect fit we
expect the mean and variance to be 2 and 4, respectively.
With respect to the traditional chi-square statistic (X2),
the fit of the observed sampling distribution to the theoretical χ2 is quite good, except for a reduced variance and
a slightly inflated mean at the smallest sample sizes. In
contrast, the approach to the limiting χ2 distribution is
markedly slower for X C2 and –2ln(exact P). The observed
sampling distributions tend to be located to the left of the
Mean
expected one, and this shift in location produces too few
false significances (realized α < 0.05), and thereby a low
power. The mean and variance of X2 are fairly close to their
expected values when n ≥ 50 individuals, but X C2 and
–2ln(exact P) both yield markedly smaller means and variances even at n = 500. The shift of location relative to χ2
may not seem alarming when plotting the test statistic
sample distribution, but a variable representing the sum of
several observations from such a distribution may deviate
dramatically from the one expected for the sum. To
exemplify, the simulated and expected distributions of
the 2 × 2 contingency X C2 (d.f. = 1) for n = 20 diploids and
Q1 = Q2 = 0.1 are shown in Fig. 3(a). The deviation from the
χ2 distribution with d.f. = 1 may not appear overly large,
but when summing 10 observations and comparing with
χ2 with d.f. = 10 the difference is dramatic (Fig. 3b). Clearly,
in a situation like this when X C2 for 10 loci is being summed,
the realized α error is far below the expected one, and the
probability of rejecting H0,J is reduced correspondingly.
In the case of 2 × 2 contingency tables we have focused
on the X2, X C2, and –2ln(exact P) test statistics, and it is
Variance
Test statistic
n
Observed
Expected
Observed
Expected
X2
X2
X2
X2
X2
X2
X C2
X C2
X C2
X C2
X C2
X C2
–2ln(exact P)
–2ln(exact P)
–2ln(exact P)
–2ln(exact P)
–2ln(exact P)
–2ln(exact P)
G
G
G
G
G
G
Williams’ G
Williams’ G
Williams’ G
Williams’ G
Williams’ G
Williams’ G
3
10
20
50
100
500
3
10
20
50
100
500
3
10
20
50
100
500
3
10
20
50
100
500
3
10
20
50
100
500
1.09
1.04
1.01
1.00
1.02
1.00
0.25
0.44
0.54
0.67
0.78
0.88
0.34
0.84
1.13
1.41
1.59
1.80
1.43
1.24
1.10
1.04
1.03
1.01
1.05
1.07
1.02
1.02
1.01
1.00
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
0.75
1.52
1.77
1.95
2.04
1.98
0.15
0.49
0.83
1.28
1.53
1.75
0.59
1.74
2.47
3.15
3.49
3.70
1.30
2.58
2.44
2.20
2.14
1.95
0.86
2.01
2.12
2.09
2.08
1.94
2
2
2
2
2
2
2
2
2
2
2
2
4
4
4
4
4
4
2
2
2
2
2
2
2
2
2
2
2
2
© 2001 Blackwell Science Ltd, Molecular Ecology, 10, 2361–2373
Table 3 Mean and variance of test statistics
from 2 × 2 contingency tables when simulating the drawing of two independent samples of equal size (n = 3 – 500 diploids) from
a population where the true allele frequency
at a di-allelic locus is 0.1. The number of
replicates (number of 2 × 2 tables) is 10 000
at each sample size. Expected values are
those for a χ2 distribution with d.f. = 1 [or
d.f. = 2 for –2ln(exactP) ]. See text for details
MEC1345.fm Page 2370 Friday, September 21, 2001 3:36 PM
2370 N . RY M A N and P. E . J O R D E
clear that the traditional chi-square test appears to provide
the statistic (X2) that is to be preferred when combining
information from multiple loci by summation (Fig. 1,
Table 3). It is especially interesting to note that the fit of
–2ln(exact P) is markedly poor when n is small, i.e. when
the use of an exact test is typically regarded most
warranted for comparisons of allele frequencies at each
particular locus considered separately.
As a comparison, Table 3 also gives simulated means
and variances for G and Williams’ G for the case of
Q1 = Q2 = 0.10. Here, Williams’ G performs as well as X2,
and sometimes even better. In contrast to the other statistics, however, the sampling distribution of G seems to be
shifted to the right of χ2 for many sample sizes (larger mean
and variance than expected). The G-test therefore appears
to produce an excess of false significances in the basic
contingency tests (realized α > 0.05), and this tendency is
expected to grow progressively stronger when testing H0,J
through summation of G-values from multiple loci.
two rows or columns (Everitt 1977), and when testing H0,J
the realized α is therefore expected to be in better agreement with the intended one for both the Bonferroni and the
‘summation’ methods. As an example, Table 4 gives the
results from simulated drawings of three samples (r = 3)
from populations segregating for two and five alleles,
respectively (c = 2 or 5), for the diploid sample sizes (n) of
10, 20, and 50.
Under the conditions simulated the difference between
the contingency test statistic sampling distribution and the
expected χ2 is in most cases minimal for the largest sample
size (n = 50). As a result, the ‘summation’ method for
addressing H0,J generally provides a realized error rate that
is reasonably close to the intended α at n = 50, although
that of –2ln(exact P) is still a bit low in the 3 × 2 tests
(Table 4). At all sample sizes the Bonferroni approach usually results in a realized α that is smaller than that obtained
by summation. At the smaller sample sizes (10 and 20) the
–2ln(exact P) statistic yields an α error that appears unduly
small in the 3 × 2 tests, but in the 3 × 5 tables the error is
close to the intended one. Most strikingly, however, summation of the G-statistic tends to produce unacceptably
high rates of false significances (10 – 37%) at the smallest
sample sizes, an effect of a markedly poor fit to the
expected χ2 distribution. The traditional chi-square (X2)
generally seems to behave in fairly good agreement with
expectation, and the ‘summation’ method appears to perform
better than Bonferroni over a wide range of sample sizes.
Although the differences between test statistics and
methods for evaluating H0,J are less pronounced than for
the 2 × 2 tables, the tendencies are similar. Combining exact
P-values by means of ‘Fisher’s method’ tends to result
in an unduly small α error more frequently than when
summing X2, particularly at small or moderate sample
sizes, and it seems that ‘summation’ should be preferred
before Bonferroni.
Finally, it should be noted that the present results indicating generally better ‘summation properties’ of X2 relative to the other test statistics is not caused by choosing
combinations of sample sizes (n) and allele frequencies (Q)
that produce unduly few tables with low expectancy cells.
Considering 2 × 2 contingency tables with n = 20 and
Q = 0.1 (Table 3), for example, over 70% of the simulated
tables have two (out of four) cells with an expectation less
than five. Similarly, at n = 20 nearly 80% of the simulated
3 × 2 contingency tables (Table 4) have three cells with an
expected value of less than five; almost all the 3 × 5 tables
have 6–12 such cells and over 10% have 3 – 6 cells with an
expectation of less than unity.
General r × c contingency tables
Concluding remarks and recommendations
The fit of the test statistic sample distributions to χ2 is
generally improved for contingency tables with more than
It is obvious from the above that the choice of statistical
method may be crucial for the probability of drawing
Fig. 3 Simulated sampling distribution of the X2C test statistic
(2 × 2 contingency chi-square with Yates’ correction) when
drawing two independent samples of 20 diploids (40 genes) from
a population where the true allele frequency at di-allelic loci
is 0.1. The corresponding χ2 distributions are those expected
asymptotically under large sample theory. (a) Examining a single
locus; expected χ2 has d.f. = 1. (b) Testing 10 loci separately and
summing the test statistic values; expected χ2 has d.f. = 10.
© 2001 Blackwell Science Ltd, Molecular Ecology, 10, 2361– 2373
MEC1345.fm Page 2371 Friday, September 21, 2001 3:36 PM
D E T E C T I N G G E N E T I C D I F F E R E N T I AT I O N 2371
Table 4 Mean and variance of test statistics from 3 × c contingency tables when simulating the drawing of three samples of equal size
(n = 10 – 50 diploids) from the same population. The number of alleles (c) is 2 and 5 occurring in the frequencies 0.1 and 0.9 (c = 2) and 0.7,
0.1, 0.1, 0.05, and 0.05 (c = 5). The number of replicates (number of 3 × c tables) at each sample size is 10 000 and 1000 for the 3 × 2 and 5 × 2
tables, respectively. Expected values are those for a chi-square distribution with d.f. = 2(c – 1) [d.f. = 2 for –2ln(exact P) ]. Realized α refers
to a situation where the information for 10 loci is combined by means of the summation or the Bonferroni approach and the intended α is
0.05. See text for details
Mean
Realized α (10 loci)
Variance
c
Test statistic
d.f.
n
Obs.
Exp.
Obs.
Exp.
Summation
Bonferroni
2
2
2
2
2
2
2
2
2
2
2
2
5
5
5
5
5
5
5
5
5
5
5
5
X2
X2
X2
–2ln(exact P)
–2ln(exact P)
–2ln(exact P)
G
G
G
Williams’ G
Williams’ G
Williams’ G
X2
X2
X2
–2ln(exact P)
–2ln(exact P)
–2ln(exact P)
G
G
G
Williams’ G
Williams’ G
Williams’ G
2
2
2
2
2
2
2
2
2
2
2
2
8
8
8
2
2
2
8
8
8
8
8
8
10
20
50
10
20
50
10
20
50
10
20
50
10
20
50
10
20
50
10
20
50
10
20
50
2.01
2.02
2.00
1.26
1.62
1.83
2.38
2.17
2.04
2.09
2.05
1.99
8.19
8.14
7.78
1.96
2.04
1.86
9.82
9.21
8.07
8.11
8.38
7.79
2
2
2
2
2
2
2
2
2
2
2
2
8
8
8
2
2
2
8
8
8
8
8
8
3.25
3.71
3.94
2.70
3.60
3.79
4.77
4.86
4.26
3.82
4.30
4.07
10.96
14.33
13.90
3.65
4.21
3.26
15.06
19.86
15.66
10.65
16.53
14.58
4
4
4
4
4
4
4
4
4
4
4
4
16
16
16
4
4
4
16
16
16
16
16
16
0.043
0.049
0.052
0.001
0.019
0.029
0.150
0.100
0.068
0.072
0.066
0.057
0.030
0.060
0.040
0.040
0.070
0.050
0.370
0.240
0.060
0.030
0.070
0.060
0.027
0.035
0.047
0.013
0.025
0.037
0.052
0.079
0.064
0.018
0.061
0.056
0.000
0.049
0.020
0.020
0.068
0.020
0.049
0.086
0.020
0.000
0.058
0.020
the right conclusion when testing for genetic differentiation. To date, it appears that the primary statistical interest
has been devoted to avoidance of false significances
(α errors), whereas considerably less attention has been
paid to the prospect of detecting true differences (power).
Because the two quantities are interrelated, excessive
focus on one of them may have undesirable effects on
the other.
It appears that the current trend in many fields of evolutionary biology is to score a steadily growing number of
loci in a quite restricted number of individuals, frequently
in the range, say, 5 – 30. The question of how to combine the
information from multiple loci is therefore becoming
increasingly significant. Our results indicate strongly that
summation of chi-square (X2) tends to perform better than
any of the alternatives examined, even at fairly small
sample sizes. This approach should typically be the method
of choice when testing the joint null hypothesis of no
difference at any locus.
The technique of summing twice the negative logarithm
of P-values from Fisher’s exact test [Σ – 2ln(exact P) ]
© 2001 Blackwell Science Ltd, Molecular Ecology, 10, 2361–2373
appears to be used increasingly often. It should be stressed,
though, that this approach (‘Fisher’s method’ applied to
Fisher’s exact P) may be associated with a strikingly small
power even when sample sizes are quite large, and particularly so for 2 × 2 tables. It may seem superficially
appealing to base the joint test on probabilities that
are computed ‘exactly’ for each contributing contingency
table. Nevertheless, the poor fit to the asymptotically expected χ2 distribution frequently makes this
method profusely conservative. Therefore, we suggest that
results from summation of –2ln(exact P) should generally be accompanied by the results from chi-square
summation.
It should be stressed that we by no means suggest that
exact contingency tests should be abandoned. Whenever
possible, exact calculation of P is to be preferred before
any approximation relying on large sample theory, and
particularly so when sample sizes are modest or small.
Although necessarily conservative, there is an obvious
advantage with exact tests in that the investigator is
guaranteed that the realized α will never exceed the
MEC1345.fm Page 2372 Friday, September 21, 2001 3:36 PM
2372 N . RY M A N and P. E . J O R D E
intended one. The problem we address arises when
combining the information from several exact tests by
means of an approximation such as ‘Fisher’s method’.
There is no contingency test that is universally ‘best’
from every perspective and under all circumstances, and
there are many situations where an investigator may have
valid concern regarding the appropriateness of the chisquare statistic, which necessarily represents an approximation. With small expected values in one or more cells
the risk of excessive rates of false significances cannot
be ignored, although several reports suggest (as do the
present simulation results) that the severity of this continuity problem may be overrated (e.g. Cochran 1954; Lewontin
& Felsenstein 1965; Everitt 1977). Exact tests, on the other
hand, may be overly conservative and fail to detect true
differences more often than anticipated. When evaluating
a single contingency table, however, exact calculation of
the P-value should be the primary method of choice. It
must be noted, though, that the observation of a nonsignificant P-value (P > 0.05) may be quite uninformative
in the absence of minimal information on realized α and
power of the test at the sample sizes at hand. When
combining the information from several contingency tables,
however, we recommend that the decision on overall
genetic divergence is made on the basis of summation of
chi-square (X2).
The Bonferroni correction is also being applied more
commonly as a tool for evaluating the joint null hypothesis
of no difference at any locus, and its frequently poor
performance in the present context may be perceived a bit
surprising. It should be noted, though, that the Bonferroni
correction was primarily designed to reduce the probability of obtaining false significances when performing
several independent tests. It was not aimed at combining
the information from multiple tests that address the same
null hypothesis. The Bonferroni method focuses exclusively
on the occurrence of (very) small P-values and largely ignores,
for example, tendencies of weak significances to be overrepresented. When testing for genetic heterogeneity under
a ‘selective neutrality — genetic drift’ model, however, any
indication of allele frequency differences at any polymorphic locus (regardless of direction) should ideally
contribute information that makes the joint null hypothesis
less likely. The ‘summation’ method is directly aimed at
picking up such tendencies, and the difference between
the two approaches becomes particularly obvious when the
underlying test statistic distributions are characterized
by marked discontinuities.
As with exact tests, we do not suggest that the Bonferroni method should be avoided in general. The Bonferroni
(or the sequential Bonferroni) approach represents a most
valuable tool for controlling the α error when an investigator, after conducting multiple tests, focuses on a particular
null hypothesis (Rice 1989). Rather, because of the
markedly low power we recommend against its ‘extended’
use when testing the joint null hypothesis (H0,J) that all
the component H0s are true.
This paper is focused on contingency tests for allele
frequency heterogeneity, but problems with statistical
power similar to those discussed here may also occur
in other testing situations. The Bonferroni approach,
for example, is often applied for evaluation of multiple
P-values obtained when testing for Hardy–Weinberg
proportions or linkage equilibrium between pairs of loci.
For the Bonferroni method to work properly in such cases,
it is also necessary that the contributing tests can produce
P-values as small as α/k and that those P-values occur
at frequency reasonably close to α/k when the null hypothesis is true. For instance, if it is impossible in practice
to obtain P < α/k in many or most of the contributing
tests, then the power of the joint test (the Bonferroni
evaluation) may be very close to zero. In such a situation,
any attempt to interpret an observed lack of significance in
biological terms would typically be meaningless and
potentially erroneous.
Acknowledgements
We thank Linda Laikre, Ole Christian Lingjærde, Stefan Palm,
Associate Editor Laurent Excoffier, and two anonymous reviewers
for comments on earlier versions of this paper. The study was
supported by grants to N.R. from the Swedish Natural Science
Research Council and from the Swedish research program on
Sustainable Costal Zone Management, SUCOZOMA, funded by
the Foundation for Strategic Environmental Research, MISTRA.
P.E.J. was supported by a grant from the National Research
Council of Norway.
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Nils Ryman is a professor of genetics and heads the Division of
Population Genetics at the Stockholm University. His research has
focused primarily on the genetic structure of natural populations,
the genetic effects of human exploitation of such populations, and
related conservation genetics issues. Per Erik Jorde graduated
in genetics in Stockholm and is presently a postdoctoral fellow at
the University of Oslo, working on microsatellite DNA analyses
of fishes.