Computer Note
FSTAT (Version 1.2): A
Computer Program to
Calculate F-Statistics
J.
Goudet
Computation of Wright's
(1943,
1951) fixa-
tion indices
(F
u
,
F
w
and
FJ)
is widespread
among population biologists to assess ge-
netic differentiation of populations. F
u
is a
measure of the within population hetero-
zygote deficit, F
u
is a measure of the
among population heterozygote deficit
(the Walhund effect), and F
lt
is a measure
of the global heterozygote deficit. These
indices could be defined as the correlation
between uniting gametes (Wright 1943,
1951),
as a function of gene diversity in
the total population (Nei 1973, 1977; Nei
and Chesser 1983), or as a function of vari-
ance components from a nested analysis
of variance (Cockerham 1969, 1973; Cock-
erham and Weir
1986;
Weir and Cockerham
1984).
Deriving unbiased estimators of
these quantities has been at the center of
an argument between Nei on the one hand
(Nei 1986; Nei and Chesser 1983) and Weir
and Cockerham (Cockerham 1973; Cock-
erham and Weir 1986, 1987) on the other.
An in-depth review of the properties of
both families of estimators can be found
in Chakraborty and Dander-Hopfe (1991).
Cockerham and Weir (1993) presented an
interpretation of the differences between
Nei's estimator of F^ G
w
and their own, 9,
in terms of the probabilities of identity by
descent.
The most often used package for analy-
sis of allele and genotype frequencies is
Swofford and Selander's (1981) BIOSYS-1.
Although very useful for calculation of ge-
netic distances and building of phyloge-
netic trees, its module on F statistics es-
timation is outdated, because it is based
on Nei's (1977) article, which does not
take sampling effects into account.
1
pro-
pose a PASCAL program that performs the
calculation of Weir and Cockerham's
(1984) estimators of
F
statistics, based on
the Fortran listing published in Weir's
(1990)
Genetic
Data
Analysis,
with several
new features.
The Program FSTAT
The program FSTAT performs the follow-
ing:
• Estimated frequency of alleles per sam-
ple and overall (from which indices
such as Nei's gene diversity can be di-
rectly calculated).
• Observed and expected heterozygosity
per allele and sample. Expected hetero-
zygosity is calculated using Levene's
(1949) correction for small samples.
• F
u
(f in Weir and Cockerham's notation)
estimated per sample over loci.
• F
u
(F), F
u
(9), and F
a
(/) estimated per
allele, locus, and globally over all sam-
ples.
A fourth index has been added, a
measure of Hamilton's (1971) related-
ness,
r = 2FJ(l + F^, using the esti-
mator given in Queller and Goodnight
(1988).
This measure is the average re-
latedness of individuals within samples
when compared to the whole. It is often
used in studies of social insects.
• Confidence intervals based on resam-
pling schemes are provided: (1) Mean
and variance of F statistics per locus,
estimated from jackknifing over sam-
ples.
(2) Mean and variance of F statis-
tics over loci, estimated from
jackknif-
ing over loci. (3) Bootstrap confidence
intervals of F statistics performed on
the loci.
• Calculation of F
u
per pair of samples.
The output is a matrix of F
u
values that
can be used to carry out Slatkin's (1993)
method to test for isolation by distance
because F
a
is closely related to a genet-
ic distance (Reynolds et al. 1983). This
matrix could also be used in Mantel
tests (e.g., Manly 1985).
• Test of the significance of F
a
, F^ and /•"„
per locus and over all loci using per-
mutations (Excoffier et al. 1992; Hudson
et al. 1992; Manly 1991). The aim is to
obtain the distribution of the null hy-
pothesis, namely F^ not >0, and to
compare this null distribution with the
observed F^ The probability of obtain-
ing by chance a value as large or larger
than the observed is given: (1) For F
a
,
alleles are permuted among individuals
within samples. (2) For F
in
alleles are
permuted among samples. (3) For F
m
two types of tests can be carried out,
depending on the results of the test on
F
u
.
If
F,,
is not significantly different from
zero,
it is valid to permute alleles
among samples to test F
w
because al-
leles can be considered as independent.
If
Ft,
is different from zero, however, al-
leles within individuals are not indepen-
dent anymore, and the appropriate per-
mutation units are the genotypes, to be
permuted among samples.
These tests were developed to avoid the
caveats of existing tests, such as those of
Workman and Niswander (1970), based on
X
2
and therefore relying on large samples
(expected classes larger than 5). Raymond
and Rousset (in press) generalized Fish-
er's exact test for Hardy-Weinberg equilib-
rium to among samples differentiation.
Some problems remain however with this
test, because combining information from
different loci is carried out using Fisher's
procedure (Fisher 1954; Sokal and Rohlf
1981),
which does not weight loci. Fur-
thermore, their test for between sample
differentiation is based on the assumption
that there is Hardy-Weinberg equilibrium
within samples. If there is departure from
it, then alleles within individuals are not
independent, and the exact test for differ-
entiation would lead to erroneous results.
Permutations eliminate those caveats.
Slatkin (1994) pointed out that, when
studying population differentiation, it may
485
be more appropriate to use F^ a statistic
arising naturally, as a test statistic.
The Random Number Generator pro-
posed by L'Ecuyer (1988) was chosen for
the Bootstrap and permutation proce-
dures.
It combines two of the best Multi-
plicative Linear Congruential Generators
known and has passed all the tests for ran-
dom number generators.
The format of the output file (tab sepa-
rators) allows direct reading into many
commercially available spreadsheets, fa-
cilitating printing and graphical represen-
tation of the data.
A real mode version of the program
runs on 80286 (and above) PC compati-
bles.
No coprocessor is required, but will
speed up calculations. A protected mode
version will run on 80386 (and above) PC
compatibles. Again, no coprocessor is re-
quired. This version uses all the available
extended memory, therefore allowing the
processing of larger data sets. The actual
limits are:
• Number of samples: 200
• Number of locus: 50
• Number of alleles at the most polymor-
phic locus: 99
• Maximum number of individuals:
5,000
• Maximum number of permutations:
15,000
The program is also suited for haploid
data and appropriately handles missing
data, such as a locus missing completely
from one sample. The program is distrib-
uted with no charges. It can be sent elec-
tronically in a Binhexed or unencoded for-
mat (requests should be sent to jero-
me.goudet@izea.unil.ch). Alternatively, it
can be retrieved from the ftp server ora-
cle.bangor.ac.uk after anonymous login, in
the directory pub/fstat.
From the School of Biological Sciences, University of
Wales, Bangor, U.K., and Instltut de Zoologle et d'Ecol-
ogle Anlmale, Bat Biologle, Unlverslte de Lausanne,
Lausanne CH-1015, Switzerland This work forms part
of a research program into the evolutionary effect of
gene flow and has been partly funded by the University
of Wales, Bangor, the Department of Environment, U.K.,
and the Swiss National Science Foundation. It Is a con-
tribution to the Biodiversity Module of the Swiss Pri-
ority Program on the environment. I am Indebted to
Thierry DeMeeus for thorough checking of the program
capabilities. Many thanks are due to Chris Gliddon,
Michel Raymond, and Francois Rousset
The Journal of Heredity 1995:86(6)
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Received August 30, 1994
Accepted March 14, 1995
Corresponding Editor: Stephen O'Brien
486 The Journal of Heredity 1995 86(6)