Table Of ContentConfidence Intervals for Population Allele Frequencies:
The General Case of Sampling from a Finite Diploid
Population of Any Size
Tak Fung1,2*, Kevin Keenan3
1NationalUniversityofSingapore,DepartmentofBiologicalSciences,Singapore,Singapore,2Queen’sUniversityBelfast,SchoolofBiologicalSciences,Belfast,Northern
Ireland,UnitedKingdom,3Queen’sUniversityBelfast,InstituteforGlobalFoodSecurity,SchoolofBiologicalSciences,Belfast,NorthernIreland,UnitedKingdom
Abstract
The estimation of population allele frequencies using sample data forms a central component of studies in population
genetics. These estimates can be used to test hypotheses on the evolutionary processes governing changes in genetic
variation among populations. However, existing studies frequently do not account for sampling uncertainty in these
estimates,thuscompromisingtheirutility.Incorporationofthisuncertaintyhasbeenhinderedbythelackofamethodfor
constructingconfidenceintervalscontainingthepopulationallelefrequencies,forthegeneralcaseofsamplingfromafinite
diploid population of any size. In this study, we address this important knowledge gap by presenting a rigorous
mathematicalmethodtoconstructsuchconfidenceintervals.Forarangeofscenarios,themethodisusedtodemonstrate
that for a particular allele, in order to obtain accurate estimates within 0.05 of the population allele frequency with high
probability(§95%),asamplesizeofw30isoftenrequired.Thisanalysisisaugmentedbyanapplicationofthemethodto
empirical sample allele frequency data for two populations of the checkerspot butterfly (Melitaea cinxia L.), occupying
meadows in Finland. For each population, the method is used to derive §98:3% confidence intervals for the population
frequenciesofthreealleles.Theseintervalsarethenusedtoconstructtwojoint§95%confidenceregions,onefortheset
ofthreefrequenciesforeachpopulation.Theseregionsarethenusedtoderivea§95%confidenceintervalforJost’sD,a
measureofgeneticdifferentiationbetweenthetwopopulations.Overall,theresultsdemonstratethepracticalutilityofthe
method with respect to informing sampling design and accounting for sampling uncertainty in studies of population
genetics,important forscientific hypothesis-testingand also forrisk-based natural resource management.
Citation:FungT,KeenanK(2014)ConfidenceIntervalsforPopulationAlleleFrequencies:TheGeneralCaseofSamplingfromaFiniteDiploidPopulationofAny
Size.PLoSONE9(1):e85925.doi:10.1371/journal.pone.0085925
Editor:GuyBrock,UniversityofLouisville,UnitedStatesofAmerica
ReceivedJune23,2013;AcceptedDecember4,2013;PublishedJanuary21,2014
Copyright:(cid:2)2014Fung,Keenan.Thisisanopen-accessarticledistributedunderthetermsoftheCreativeCommonsAttributionLicense,whichpermits
unrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalauthorandsourcearecredited.
Funding:TFisbeingsupportedbytheNationalUniversityofSingaporestart-upgrantWBSR-154-000-551-133.Inaddition,TFacknowledgespastfunding
supportbyaPh.D.studentshipfromtheBeaufortMarineResearchAwardinEcosystemApproachtoFisheriesManagement,carriedoutundertheSeaChange
StrategyandtheStrategyforScienceTechnologyandInnovation(2006–2013),withthesupportoftheMarineInstitute,fundedundertheMarineResearchSub-
ProgrammeoftheIrishNationalDevelopmentPlan2007–2013(http://www.marine.ie/home/research/MIFunded/BeaufortAwards/).KKisbeingsupportedbya
Ph.D.studentshipfromtheBeaufortMarineResearchAwardinFishPopulationGenetics,fundedbytheIrishGovernmentundertheSeaChangeStrategyas
above.Thefundershadnoroleinstudydesign,datacollectionandanalysis,decisiontopublish,orpreparationofthemanuscript.
CompetingInterests:Theauthorshavedeclaredthatnocompetinginterestsexist.
*E-mail:[email protected]
Introduction populationallelefrequencieshavetobeestimatedusingasubsetof
sampledindividuals.Fordiploidorganismsinalargepopulation,it
Spatiotemporalpatternsofgeneticvariationamongpopulations
has been established that the frequency of an allele in a sample
are often used to test hypotheses about processes underlying the
provides an unbiased estimateof thefrequency in thepopulation
patterns, such as selection, migration and genetic drift (e.g.,
as a whole [10]. Thus, if samples of a given size are repeatedly
[1,2,3,4,5]). This genetic variation captures differences in genetic
taken from a large population, then the mean frequency of an
structure among populations, where the genetic structure of a
allele ina sample converges to thepopulation allele frequency as
population is determined by the distribution of alleles among
thenumberofsamplesincreases.However,manystudiesonlytake
individuals in the population. The allele distribution of a asinglesamplefromapopulationandpresentorusetheresulting
populationistheresultofarangeofbiologicalandenvironmental
frequencies of alleles in the sample, without accounting for
processes acting on the population and also on surrounding
sampling uncertainty (e.g., [11,12,13,14,15,16,17,18]). Therefore,
populations within the same geographical region, which result in
these studies implicitly assume that the sample allele frequencies
non-random mixing of gametes among individuals in all popula- areclosetothepopulationallelefrequencies,whichisbynomeans
tions. Patterns in allele distributions among populations are guaranteed. Interpretation of findings from these studies is
generally assessed by consideration of variations in allele therefore complicated by the potential for large sampling
frequencies (e.g.,[6,7,8,9]). uncertainty.Ideally,inthissinglesamplecase,uncertaintybounds
Logistic and ethical constraints mean that in practice, a forthepopulationallelefrequencieswouldbequantified,basedon
population is unlikely to be sampled in its entirety, such that the thesample allele frequencies.
PLOSONE | www.plosone.org 1 January2014 | Volume 9 | Issue 1 | e85925
ConfidenceIntervalsforAlleleFrequencies
ArecentstudybyHaleetal.[19]usedcomputersimulationsto a function of their population allele frequencies [9]. The data
draw samples from four diploid populations, with allele frequen- comes from a study [13] where it is unclear that the populations
ciesbasedonfourempiricaldatasets.Subsequently,theyusedthe can be assumed to be of infinite sizes or at HWE. This example
sample data to derive the means, variances and ranges of allele illustrates how the mathematical theory underlying our method
frequencies in samples of varying size, as well as the means, can be used to quantify sampling uncertainty not only in
variances and ranges of indicators that are a function of allele population allele frequencies, but also in parameters that are
frequencies (specifically, heterozygosity and F ). Using these functionsofthesefrequencies,importantforhypothesis-testingand
ST
results, Hale et al. [19] concluded that a sample size of 30 is also fornatural resource management.
sufficienttoaccuratelyestimatepopulationallelefrequencieswhen Overall, this study provides a rigorous mathematical quantifi-
using microsatellites. However, the authors did not quantify cation of sampling uncertainty in population allele frequencies,
uncertainty bounds for the population allele frequencies using without the typically unrealistic constraints of assuming that a
their sample data, such that their assessment of accuracy lacks a population has an infinite size and is at HWE. Thus, the results
rigorous quantitative basis. Furthermore, for each sample size, can beappliedtoa wide range ofstudies inpopulation genetics.
sample frequencies were calculated using only 100 replicate
samples, which may not give a good approximation of the true Methods
dispersioninthesamplefrequencies.Thishighlightsaweaknessof
using a computational approach that lacks an underlying Inordertoconstructconfidenceintervalsforthegeneralcaseof
mathematicaltheory.Moreover,Haleetal.[19]didnotconsider taking samples of any size from a diploid population of any size
the situation identified earlier, where one sample is taken and (larger than or equal to the sample size) and with any degree of
there is a need to quantify uncertainty of population allele deviation from HWE, the sampling distribution of the allele
frequencies using just the sample allele frequencies. For this frequencies in this general case is first exactly specified. This
situation,earlierstudieshavederivedconfidenceintervalsinorder distribution is then used to derive formulae specifying confidence
to capture uncertainty in the population allele frequencies. If a intervals that contain the population allele frequencies with
diploid population is of infinite size and is at Hardy-Weinberg probability above a known threshold. Using these formulae,
equilibrium (HWE), then the frequency of an allele in a sample confidence intervals are derived for a range of archetypal
followsabinomialdistribution[10].Thus,Gillespie[20]proposed scenarios, which are then used to calculate sample sizes that
the use of the Wald confidence interval [21]. However, this permitaccurateestimatesofpopulationallelefrequenciesinthese
intervalonlyperformswellforsufficientlylargesamplesizes–with scenarios.Inaddition,theformulaeareappliedtoarealscenario
small sample sizes, it is too short [22]. For the binomial where samples are taken from two butterfly populations [13], to
distribution, other confidence intervals have been derived that constructconfidenceintervalsforthepopulationallelefrequencies
do perform well for small sample sizes [22,23], notably the of these two populations. The intervals are then used to derive a
Clopper-Pearsonintervalthatwasderivedeightdecadesago[24]. corresponding confidence interval forJost’s D [9].
However, these only apply in the limited cases when the
population is at HWE. Weir [10] proposed a confidence interval Derivation of sampling distribution of allele frequencies
that allows for deviations from HWE, analogous to the Wald Consider a population of M diploid individuals, from which a
confidence interval. However, like the latter, it is expected to sampleofNindividualsisrandomlydrawn(M§N).Atthelocus
perform well only for sufficiently large sample sizes [10]. This is ofinterest,therearenw1alleles,denotedbyAi,i[f1,2, ...,ng.
problematic because the accuracy at a particular sample size is Let the population allele frequencies be denoted by pi, with the
unknown, so ‘‘sufficiently large’’ cannot be rigorously quantified. corresponding sample allele frequencies being denoted by pi,N.
Moreover, all the confidence intervals considered only apply to Also,letPij bethefrequencyofindividualsinthepopulationwith
caseswhenthepopulationcanbeassumedtobeofinfinitesize,i.e. alleles A and A (i,j[f1,2, ...,ng), such that MP is the
i j ij
whenthepopulationsizeismuchlargerthanthesamplesize.This numberofcorrespondingindividualsinthebasepopulation.p is
i
does not reflect the range of scenarios encountered in empirical related to P by the formula p~P zPn (cid:2)P =2(cid:3). Here,
ij i ii j~1,j=i ij
research (e.g., [13,25,26,27]). P is a measure of the homozygosity of individuals in the
ii
In this study, we build on previous work by constructing population with respect to allele A. P ~p{Pn (cid:2)P =2(cid:3),
confidence intervals for population allele frequencies for the i ii i j~1,j=i ij
andthusP ƒp.If0ƒpƒ0:5,thentheminimumvalueofP is
generalcasewhere(i)thepopulationisdiploid,finiteandcanbeof ii i i ii
0, since all copies of allele A can be distributed among
anysize;(ii)thesamplecantakeanysizelessthanorequaltothat i
heterozygotes of allele A. However, if pw0:5, then this is not
ofthepopulation;and(iii)thepopulationcandeviatefromHWE i i
possible – there must be at least one homozygote of allele A. In
to any extent. These confidence intervals are guaranteed to i
this case, theminimum number of homozygotes is realized when
containthepopulationallelefrequencywithaprobabilityabovea
all heterozygotes have a copy of allele A, i.e. when
known threshold. The method derived for constructing these P zPn P ~1. Rearranging this for Pni P and
intervalsisthenusedtocalculatesamplesizesrequiredtoachieve ii j~1,j=i ij j~1,j=i ij
accurateestimatesofpopulationallelefrequencies,underarange substituting into Pii~pi{Pnj~1; j=i(cid:2)Pij=2(cid:3) gives the minimum
of scenarios. Here, accuracy is measured as the length of the value ofPii as2pi{1. Thus,overall, Maxf0,2pi{1gƒPiiƒpi.
confidenceintervals.Thesamplesizesderivedserveasaguidefor The frequency of allele Ai in the sample of size N is given by
determinationofsuitablesamplesizesinfuturepopulationgenetic pi,N~Yi,N=ð2NÞ,whereYi,N isthenumberofcopiesofalleleAi
studies. In particular, we show that a sample size of 30 does not inthesample.Thus,theprobabilitydistributionofpi,N isthesame
necessarilygiveaccurateestimates,thusrefiningtheconclusionof astheprobabilitydistributionofYi,N exceptwiththex-axisscaled
Haleetal.[19].Lastly,weprovideanexampleofhowthemethod byafactor1=ð2NÞ.Denotetheprobabilitymassfunction(pmf)for
can be applied to microsatellite data for two populations of the Yi,N by PðYi,N~yi,NÞ. The range of Yi,N is ½0,2N(cid:2), so
checkerspotbutterfly(MelitaeacinxiaL.)[13],toderiveconfidence PðYi,N~yi,NÞ~0 for yi,N outside this range. Thus, for the
intervals for population allele frequencies and also for Jost’s D, a following calculations, only yi,N [½0,2N(cid:2) are considered. Now,
measureofgeneticdifferentiationbetweentwopopulationsthatis Yi,N~2XiizXi:,whereXi:~Pnj~1,j=iXij andXij isthenumber
PLOSONE | www.plosone.org 2 January2014 | Volume 9 | Issue 1 | e85925
ConfidenceIntervalsforAlleleFrequencies
of individuals in the sample with alleles Ai and Aj. Xii, Xi: and (inequality (6)) ensures that xiiƒyi,N=2ƒN, as required. Denote
Zi~N{Xii{Xi: thusrepresentthenumberofindividualsinthe the upper and lower bounds in (8) by Lðyi,NÞ and Uðyi,NÞ
sample with two copies, one copy and no copies of allele Ai, respectively. ThenPðYi,N~yi,NÞcan bewrittenas:
respectively. Xii,Xi: andZi followa multivariatehypergeometric
distribution with pmf givenby: PðY ~y Þ
i,N i,N
P~ðX iiM~xPxiiiiii,!X0B@i:~Mxji,:Px,jn=iZ:iiP~ijz1CAiÞ0B@M{MPii{ziMj,Pjn=iPij1CA, ð1Þ ~~xii~fflloocooerril½½iXXUUngðð½yyLiið,,NNyiÞÞ,N(cid:2)(cid:2)Þ(cid:2)PPð XXii~ii~xixi,iiX,Xi:~i:~xiy:,i,NZ{i~2zxiiÞi,!, ð9Þ
M! Z~Nzx {y
xii~ceiling½Lðyi,NÞ(cid:2) i ii i,N
N
where zi has been rewritten using zi~N{xii{xi: and
where terms on the right-hand side in brackets are binomial xi:~yi,N{2xii, ceiling½a(cid:2) is the smallest integer larger than a,
coefficients. Since the number of individuals in the sample must andfloor½a(cid:2)isthelargestintegersmallerthana.Theceilingand
equal N, xiizxi:zzi~N. PðYi,N~yi,NÞ is the sum of floor functions are introduced to ensure that xii is an integer,
PðXii~xii,Xi:~xi:,Zi~ziÞ for all those biologically feasible whichitmustbetohaveabiologicalinterpretation.PðYi,N~yi,NÞ
combinations of xii, xi: and zi satisfying yi,N~2xiizxi:. It will isequaltoPðpi,N~yi,N=ð2NÞÞ,thepmfforpi,N,andthusdefines
nowbeshownthatthissummationcanbesimplifiedtothesumof the probability distribution of pi,N. Equation (9) can be used to
an expression that depends only on xii and yi,N, over all xii define the probability distribution of pi,N given only four
between lower and upper bounds that only depend on yi,N. This parameters: M,pi,Pii and N.
considerably eases calculation of PðY ~y Þ. Firstly, the
i,N i,N
number of individuals in the sample with two copies, one copy Derivation of confidence intervals for sample allele
ornocopiesofalleleAicannotexceedthecorrespondingnumbers frequencies
in thesampled population. Thisgives rise tothreeinequalities: For given population size (M), population allele frequency (p),
i
population frequency of homozygotes with allele A (P ) and
i ii
x ƒMP , ð2Þ sample size (N), the probability distribution for the sample allele
ii ii
frequency(p )canbecalculatedexactlyusingequation(9).The
i,N
mean value ofthisdistribution is:
n
X
xi:ƒMj,j=iPij, ð3Þ E½pi,N(cid:2)~E(cid:4)Y2Ni,N(cid:5)~2E½Xii2(cid:2)zNE½Xi:(cid:2)
2NP zN Pn P ð10Þ
N{xii{xi:ƒM{MPii{M Xn Pij: ð4Þ ~ ii 2Nj~1,j=i ij~Piiz Xn P2ij~pi,
j,j=i j~1,j=i
Secondly,thenumberofindividualswithtwocopies,onecopyor asinthecaseofsamplingfromaninfinitediploidpopulation[10].
no copies of allele Ai in the sample must be non-negative. This In equation (10), the expectation E(cid:6)Xij(cid:7)~NPij has been used,
gives rise tothree moreinequalities: whichfollowsfromthefactthattheX ’s,withi,j[f1,2, ...,ng,
ij
followamultivariatehypergeometricdistributionwithparameters
xii§0, ð5Þ M, N and MPij [28]. In addition, it can be proved that the
varianceof p is
i,N
xi:§0, ð6Þ s2½p (cid:2)~ðM{NÞ(cid:2)pi{2p2izPii(cid:3) ð11Þ
i,N 2ðM{1ÞN
N{xii{xi:§0: ð7Þ (csoererecFtiiloenSfa1c).torThðMe {vaNriaÞn=cðeM{thu1sÞ tihnacltudteensdsatosta1ndasarMd ?fin?ite,
SPinnj~ce1,j=pii~PiPj~iiz2ðpPi{nj~P1i,ijÞ=ain(cid:2)dPixj=i:2~(cid:3)yia,Nn{d 2xyiii,N.T~h2uxs,iizthexii:n,eqtuhaeln- sas2½pi,N(cid:2)?(cid:2)rpeiq{ui2repd2iz. Pii(cid:3)=ð2TNhÞe,retfhoerev,ariance ains the casMe o?f?an,
infinitepopulationsize[10].Thecumulativedistributionfunction
ities (2)–(7) canberearranged toobtain thedouble inequality:
(cdf) for p isspecifiedby:
i,N
ny o
Max i,N{MpzMP ,y {N,0 ƒx ƒ (cid:8) y (cid:9)
2 i ii i,N ii Pðp ƒwÞ~P p ƒ i,N
ð8Þ i,N i,N 2N
n y o
Min MPii, i2,N,M{Nzyi,N{2MpizMPii : Xyi,N ð12Þ
~PðY ƒy Þ~ PðY ~kÞ,
i,N i,N i,N
It is noted that xi:~yi,N{2xii, which means that xi:§0 k~0
PLOSONE | www.plosone.org 3 January2014 | Volume 9 | Issue 1 | e85925
ConfidenceIntervalsforAlleleFrequencies
where y is an integer in the interval ½0,2N(cid:2). This cdf can be ofcopiesofallotherallelesinthepopulationmustbeatleastthe
i,N
calculatedusingequation(9)andisafunctionofp.Toconstructa number found in the sample, 2N{^yy ; this constrains the
i i,N
CIforp givenM,NandP ,considertestingthenullhypothesis maximum value of p according to
i ii i,0
H0 :pi~pi,0 against the alternative H1:pi=pi,0 at significance pi,0ƒ(cid:6)2M{(cid:2)2N{^yyi,N(cid:3)(cid:7)(cid:12)ð2MÞ~1{(cid:6)(cid:2)2N{^yyi,N(cid:3)(cid:12)ð2MÞ(cid:7). For
level a for an observed value of Yi,N, denoted by ^yyi,N. The null agivenvalueofpi,0,Maxf0,2pi,0{1gƒPii,0ƒpi,0,asdetermined
ahcycpeoptthaensciseisrengiootnredjeefcitneeddifbyusBinag~p(cid:6)iy~i,Np,ia,0=2,,^yyyii,,NN,1f{alðlas=2wÞ(cid:7)i,thwinhearne ienacrlluiedre. I^yyfi,PNi,Hfo’0r,aa~t1lea,stthoennethpeaiarccoefpvtaanluceesroegfiopni,0isanexdtePnidi,e0,dbtoy
yi,N,a=2isthelargestintegerforwhichP(cid:2)Yi,Nvyi,N,a=2(cid:3)ƒa=2and decreasing a.
yi,N,1{ða=2Þ is the smallest integer for which CI’sspecifiedbyequation(13)canbecalculatedgivenM,N,Pii
P(cid:2)Yi,Nwyi,N,1{ða=2Þ(cid:3)ƒa=2. Ba can be calculated using equation anda,whereasthosespecifiedbyequations(14a)and(14b)canbe
(12).Onemethodofconstructinga§100ð1{aÞ%CIforp isto calculated given just M, N and a. In this paper, computation of
i
determinethesetofvaluesofp forwhichthenullhypothesisis any CI’s using these equations, incorporating the underlying
i,0
not rejected at significance level a, denoted by formulae specified by equations (1), (8), (9) and (12), was carried
P ~(cid:10)p :^yy [B (cid:11), anddefiningtheCI as out using the software package Mathematica v5.0 [31]. Supporting
i,H0,a i,0 i,N a Webpage1providesMathematicacodeforcomputationoftheCI’s
and can be viewed at http://rpubs.com/kkeenan02/Fung-
h (cid:8) (cid:9) (cid:8) (cid:9)i
½L ,U (cid:2)~ Min P ,Max P ð13Þ Keenan-Mathematica/. However, other software packages such
a a i,H0,a i,H0,a
asMATLAB[32]andR[33]couldalsobeusedtoimplementthe
formulae; indeed, an R version of the code is provided on
[29]. This hypothesis-testing approach corresponds to the ‘‘test-
Supporting Webpage 2 and can be viewed at http://rpubs.com/
method’’ described by Talens [30], who applied it to construct
kkeenan02/Fung-Keenan-R/.Allsourcecodeforthecomposition
CI’sforparametersoftheunivariatehypergeometricdistribution.
IfP ~1,theemptyset,thentheacceptanceregionneedsto of Supporting Webpages 1 and 2, including the raw Mathematica
i,H0,a and R code used, can be accessed at https://github.com/
beextendedtoinclude^yy foratleastonevalueofp .Thus,in
i,N i,0 kkeenan02/Fung-Keenan2013/.
thiscase,aisdecreaseduntil^yy isintheacceptanceregionforat
i,N
least onep value.
i,0
The CI specified by equation(13) isnot an exact 100ð1{aÞ% Determining minimum sample sizes for accurate
CIbecause thedistribution forY isdiscreteandbecause there estimation of population allele frequencies
i,N
may be some p values between MinðP Þ and MaxðP Þ In studies of population genetics, it is desirable to obtain a
i,0 i,H0,a i,H0,a
forwhich^yy =[B ,sincePðY ƒy Þisnotguaranteedtobea sample allele frequency close to the population allele frequency
i,N a i,N i,N
monotonic function of p (see equations (9) and (12)). It is noted with high probability [19], i.e. a short CI of high probability.
i
that for the probability parameter in a binomial distribution, Thus, under three representative scenarios, we calculate the
Clopper and Pearson [24] derived §100ð1{aÞ% CI’s using an minimumsamplesizesrequiredtoobtain§95%CI’swithlengths
analogous method. Thus, for the case of a large population size ƒ0:2andƒ0:1acrossallpossiblevaluesoftheobservedsample
relative to the sample size (MwwN) and HWE, where pi,N allele frequency, ^ppi,N~^yyi,N(cid:12)ð2NÞ. These minimum sample sizes
approximately follows abinomial distribution, CI’sforpi derived aredenotedbyNƒ0:2andNƒ0:1respectively.Theyarecalculated
usingourmethodwouldbevirtuallythesameasClopper-Pearson by starting with N~10, computing CI lengths for all possible
CI’s.Thiscanbeverifiedbyexplicitlycalculatingandcomparing values of ^ppi,N and then taking the maximum value. This is
CI’s using the two methods – for example, if repeatedforincreasingNinincrementsof10untilthemaximum
M~1,000,000wwN~30and^yy ~10,thenbothmethodsgive CIlengthbecomesƒ0:1.TheNvaluewithmaximumCIlength
i,N
theCI ½0:083,0:285(cid:2) forp. closest to 0.2 is then chosen, and then increased or decreased as
i
In deriving equation (13), it was assumed that Pii is known. necessarytofindNƒ0:2.Nƒ0:1 isderivedanalogously. Maximum
However,P isgenerallyunknown.Inthiscase,a§100ð1{aÞ% CIlengthsof0.2and0.1representsmallmaximumabsoluteerrors
ii
confidence region (CR) can be derived for p and P in an intheestimatedallelefrequencyof0.1and0.05respectively,ifthe
i ii
analogous way, by considering the null hypothesis estimate is taken as the mid-point of the CI. The first scenario
H’ :p~p ,P ~P .The CR canbe derived bydetermining examined,Scenario1,issamplingfromapopulationofM~1,000
0 i i,0 ii ii,0
the set of vectors ðp ,P Þ for which the null hypothesis is not at HWE. M~1,000 is larger than or on the same order of
i,0 ii,0
rejected at significance level a. This set is denoted by magnitude as the upper bound of the range of size estimates for
P ~(cid:10)ðp ,P Þ:^yy [B (cid:11), where B is as defined before. 15/24(63%)speciespopulationscollatedbyFrankhametal.[25],
Uis,Hin’0g,atheCRi,,0§1ii0,00ð1{i,NaÞ%CaI’sforp anda P canbedefinedas covering mammals, birds, insects and plants. Thus, M~1,000 is
i ii
takentorepresentalargepopulation.Later,Misvariedovertwo
ordersofmagnitudetoconsiderpopulationsthatrangefromsmall
(cid:6)L’pi,a,U’pi,a(cid:7)~hMin(cid:8)Pi,H’0,a,1(cid:9),Max(cid:8)Pi,H’0,a,1(cid:9)i, ð14aÞ toverylarge.SincethereisaHWE,Pii~p2i.Also,inthesampled
population, thenumber of homozygotes with allele i,MP , must
ii
and beaninteger.Thisrestrictsthenumberofvaluesthatp cantake
i
to11equallyspacedvaluesintheinterval½0,1(cid:2)(startingfrom0).
(cid:6)L’Pii,a,U’Pii,a(cid:7)~hMin(cid:8)Pi,H’0,a,2(cid:9),Max(cid:8)Pi,H’0,a,2(cid:9)i, ð14bÞ hoSldc,ensuacrhiosth2ataPnidi=3pr2ie.pIrnesSecnetnsairtuioat2io,nPsiiwihsearsesuHmWedEtodotaekseniotst
minimumvalue,whichisMaxf0,2p{1g,whereasinScenario3,
i
whereP isthesetofvaluesconsistingofthejthelementsin
i,H’0,a,j Pii is assumed to take its maximum value, which is pi. In
the set of vectors Pi,H’0,a. In determining Pi,H’0,a, pi,0 and Pii,0 comparisonwithScenario 1,pi cannowtake alargernumber of
values over the biologically feasible ranges are tested. Since the values–itcantakethe2,001equallyspacedvaluesintheinterval
numberofcopiesofalleleAiinthepopulationmustbeatleastthe ½0,1(cid:2) in Scenario 2 and the 1,001 equally spaced values in the
number foundinthesample,^yy (cid:12)ð2MÞƒp .Also,thenumber interval ½0,1(cid:2) in Scenario 3. Since the variance of p is an
i,N i,0 i,N
PLOSONE | www.plosone.org 4 January2014 | Volume 9 | Issue 1 | e85925
ConfidenceIntervalsforAlleleFrequencies
increasing function of P (equation (11)), CI’s derived using p terminology is not used in our study for clarity. The subpopula-
ii i,N
areexpectedtoincreaseinlengthwithP aswell.Thismeansthat tions form part of a total of about 536 subpopulations on the
ii
CIlengthsandminimumsamplesizes(Nƒ0:2 andNƒ0:1)derived A˚landIslands,withanestimatedtotalsizerangingfrom35,000to
forScenarios2and3areexpectedtoencompasstheentireranges at least 200,000 [13]. Thus, the size of each subpopulation is
of possiblevalues. assumed to be ½(35,000z200,000)=2(cid:2)=536~219, such that the
Inthethreescenariosexamined,P isspecifiedasafunctionof Pra¨sto¨ and Finstro¨m populations are assumed to have a size of
ii
p, such that equation (13) can be used to calculate a CI for p M ~2|219~438 and M ~7|219~1533 respectively. The
i i 1 2
givenavalueof^ppi,N.CalculationofthisCIinvolvesconsideringall observedsampleallelefrequenciesinthetwopopulations,denoted
possiblevaluesofp anddeterminingunderwhichvalues^pp falls by^pp and^qq respectively,aretakendirectlyfrom[13].These
i i,N i,N1 i,N2
within the corresponding acceptance region (as described above). areusedtocalculatetheobservednumberofcopiesofalleleA in
i
An alternative scenario, Scenario 4, is that the relationship eachpopulation,denotedby^yy and^zz respectively,usingthe
i,N1 i,N2
betweenPii andpi isunknown.Inthiscase,equation(14a)hasto formulae ^yyi,N1~2N1^ppi,N1 and ^zzi,N2~2N2^qqi,N2. Due to rounding
be used to calculate a CI for pi given a value of ^ppi,N, which errorin^ppi,N1 and^qqi,N2 valuesfrom[13],^yyi,N1 and^zzi,N2 hadtobe
involvesconsideringallpossiblevaluesofpiandPii.Thisresultsin rounded to the nearest integer. Since the true homozygosity of
a considerable increase in computation time; for example, when eachalleleineachpopulationisunknown[13],conservativeCI’s
sampling N~30 individuals from a population of M~1,000, are calculatedassuming P takesitsmaximumvalueof p (thisis
ii i
2,001 values of pi need to be considered but 1,002,001 expectedtomaximizethelengthsoftheCI’s,asexplained inthe
combinations of pi and Pii need to be considered, representing previous section). In addition, a~0:025=3 is chosen, such that a
an increase in computational time by two orders of magnitude. §98:3% CI is derived for each of the three population allele
However,foragivenvalueofpi,themaximumvalueofPii gives frequencies in each of the two populations. The reason for this
thehighestvarianceforpi,N andisthusexpectedtomaximizethe choice of a is because for each population, using the Bonferroni
lengthoftheacceptanceregionwithinwhich^ppi,N couldfallwithin. Inequality[34],thecubicregiondefinedbythethreeCI’scanbe
Thus,CI’sforpi derivedunderScenario4areexpectedtoclosely taken as a §100ð1{2bÞ% confidence region (CR) for the three
match those derived under the case of maximum homozygosity population allele frequencies, where 3a§b. The choice of
(Scenario3).ThiscanbeverifiedbyexplicitcalculationoftheCI’s a~0:025=3 allows §95% CR’s to be derived for each set of
– for example, given M~100, N~30 and ^ppi,N~15, the CI’s three population allelefrequencies inthetwopopulations.
derived under Scenario 4 and Scenario 3 are ½0:14,0:4(cid:2) and Todemonstratehowthetheorydevelopedinthispapercanbe
½0:15,0:39(cid:2), representing a difference in length of only 0.02. usedtoderiveCI’sforgeneticindicatorsthatareafunctionofthe
Similar results hold for ^ppi,N~10 and 5. Therefore, results from population allele frequencies, the §95% CR’s are used to
Scenario3areusedtoapproximatethoseforScenario4,obviating calculate a §95% CI for Jost’s D for the CINX1 locus and the
the need for long computational runtimes to explore all possible Pra¨sto¨ andFinstro¨mbutterflypopulations.Jost’sDisameasureof
combinations of pi andPii. genetic distance between populations [9]. For the case of two
Lastly,fromequation(11),thevarianceofpi,N isanincreasing populations, itisgivenby
function of M. Thus, CI lengths for p are expected to increase
i
with M, which would result in increases in Nƒ0:2 and Nƒ0:1. To (cid:4) (cid:13)J (cid:14)(cid:5)
test this explicitly, for a population at HWE and with N~30, D ~2 1{ T , ð15Þ
maximumlengthsforthe§95%CIforp (acrossallvaluesof^pp ) Jost JS
i i,N
are calculated for M~100, 250, 500, 750, 1,000, 2,500, 5,000,
where,using thenotation inthisexample,
7,500and10,000.Thisrangecorrespondstopopulationsthatare
smalltovery large [25].
J ~X4 (cid:8)pizqi(cid:9)2~1ðp zq Þ2z1X3 ðpzqÞ2
Application to an empirical data set for checkerspot T 2 4 4 4 4 i i
i~1 i~1
butterflies ð16Þ
To demonstrate how the theory developed can be used in ~1 2{X3 p{X3 q!2z1X3 ðpzqÞ2
practice, it is applied to microsatellite data for samples from two 4 i i 4 i i
populations of the checkerspot butterfly (Melitaea cinxia L.) i~1 i~1 i~1
occupying meadows on the A˚land Islands in Finland [13]. and
Specifically, for the CINX1 locus, §100ð1{2aÞ% CI’s are
calculated for the frequencies of alleles A, B and C for the Pra¨sto¨
4 4
and Finstro¨m populations, using the corresponding sample allele Pp2zPq2
frequencies(Table1ofPaloetal.[13]).Foreachpopulation,aCI J ~i~1 i i~1 i ~1(cid:2)p2zq2(cid:3)z1X3 (cid:2)p2zq2(cid:3)
is not calculated for the population frequency of the fourth and S 2 2 4 4 2 i i
finalalleleD,becausethisisfixedbythepopulationfrequenciesof i~1 ð17Þ
tuhseedfiirnstouthrrseteudayll,ehleesn.cTefoorathc,haiellveelescoAn,sBis,teCncayndwDithartehreefneorrtaedtiotno ~1224 1{X3 pi!2z 1{X3 qi!235z12X3 (cid:2)p2izq2i(cid:3):
asallelesA ,A ,A andA respectively.Thesamplesizeforthe i~1 i~1 i~1
1 2 3 4
Pra¨sto¨ population is N ~53, whereas that for the Finstro¨m
1
ptoopduelnaotitoentihseNs2a~mp7l4e[a1l3le]l.epfi,rNe1quanendcqiei,sN2o,fiA[fi1n,2t,h3e,4Pg,ra¨asrteo¨uasnedd mAinloimwiezringlimthite tfuoncatio§n9in5%(15C) Igivfeonr tDheJosctoncsatnrainbtes tdheartivtehde sbixy
i
Finstro¨mpopulationsrespectively.Similarly,p andq areusedto population allele frequencies are contained within the two
i i
denote the population allele frequencies of A in the two corresponding §95% CR’s, and the two constraints
i
populations, respectively. The two populations consist of two 1{P3i~1pi~p4§0 and 1{P3i~1qi~q4§0. This lower limit
and seven subpopulations respectively. Thus, technically, the two is denoted by L . Similarly, the upper limit to a §95% CI for
D
populations can be referred to as metapopulations, although this D canbederivedbymaximizingthefunctionin(15)underthe
Jost
PLOSONE | www.plosone.org 5 January2014 | Volume 9 | Issue 1 | e85925
ConfidenceIntervalsforAlleleFrequencies
same constraints. This is denoted by U . In this study, the
D
‘‘Minimize’’ and ‘‘Maximize’’ functions in Mathematica v5.0 [31]
areusedtocomputeL andU ,butcorrespondingfunctionsmay
D D
beusedinothersoftwarepackages,suchasthe‘‘solnp’’functionin
the ‘‘Rsolnp’’ R package. A §95% CI for D can then be
Jost
definedastheinterval½L ,U (cid:2).SupportingWebpage1provides
D D
Mathematica code that can be used to calculate ½L ,U (cid:2) for the
D D
butterfly case study examined (http://rpubs.com/kkeenan02/
Fung-Keenan-Mathematica/).CorrespondingRcodeispresented
on Supporting Webpage 2 (http://rpubs.com/kkeenan02/Fung-
Keenan-R/).
Results
Maximum length of §95%confidence interval with
increasing sample size
For Scenario 1, where the sampled diploid population of size
M~1,000 was at HWE, the maximum length of the §95% CI
for the population frequency of allele A, p, considering all
i i
possible observed values of the sample allele frequency, ^pp ,
i,N
decreased non-linearly with sample size N (Figure 1). The Figure2.Changeinlengthof§95%confidenceintervalacross
minimum N required to achieve a maximum length ƒ0:2, differentobservedvalues,underHardy-Weinbergequilibrium.
Nlenƒg0t:h2,ƒwa0s:12,2N,ƒw0h:1e,rweaasst4h9e(Fmiginuirmeu1m). N required to achieve a cHFooanrrdfaiyd-esWanmeciepnbilneetreogrfvesaiqlzu(eCilINi)bfrtoiaurkmteh,negfprraoopmphulasahtpoioownpinufrgleaqttiuhoeennloceyfngositfzheanoMfalt~lheel1e,§0A090(5pa%t)
ForgivenNand^pp ,theCIforp consistsofallpossiblevalues i i
i,N i acrossallpossibleobservedvaluesofthesampleallelefrequency(p ).
i,N
ofpi forwhich^ppi,N liesinthecorrespondingacceptanceregion,as Thethreecurvescorrespond toN~10,30and50.Equation(13)was
described in Methods. This acceptance region is expected to usedtocalculatethelengthofeachCI.
doi:10.1371/journal.pone.0085925.g002
increasewiththevarianceofp ,s2½p (cid:2).AtHWE,thefrequency
i,N i,N
ofhomozygotesofalleleA,P ,isequaltop2;thus,accordingto
i ii i
equation (11), s2½p (cid:2) takes its highest values at intermediate
i,N
valuesofp.Asaresult,intermediatevaluesof^pp arelikelytofall
i i,N
within the acceptance regions of more values of p, such that the
i
corresponding CI lengths are generally longer. Simulation results
for Scenario 1 were broadly in agreement with these theoretical
expectations (Figure2).
InScenario2,wherethepopulationwasnolongeratHWEbut
attaineditslowestP ,themaximumCIlengthalsodecreasednon-
ii
linearlywithincreasingN(Figure1).ComparedwithScenario1,
Nƒ0:2wasslightlylargerandNƒ0:1waslargerbyaboutafactorof
two, taking values of 26 and 94 respectively (Figure 1). This is
contrary to expectations that minimum homozygosity would give
smallerNƒ0:2andNƒ0:1(seeMethods).Thereasonisthatalthough
s2½p (cid:2) is smaller for given N and p compared with the case of
i,N i
HWE, which would be expected to result in fewer p values for
i
which a given ^pp falls within the corresponding acceptance
i,N
regionsandhenceashorterCI,therearemorepossiblevaluesof
Figure 1. Change in maximum length of $95% confidence p (2,001 compared with 11 – see Methods), which increased the
i
interval with increasing sample size. Graph showing how the
number ofp valuesforwhich ^pp falls withinthecorresponding
maximum length of the §95% confidence interval (CI) for the i i,N
acceptance regions. For Scenario 2, P ~Maxf0,2p{1g, such
population frequency of an allele A (p) changes with increasing ii i
i i
sample size N, when sampling from a diploid population of size thats2½pi,N(cid:2)attainsitshighestvaluesatpi valuescloseto0.25and
M~1,000. For a given N, the maximum CI length was derived by 0.75(equation(11)).Thus,forgivenN,valuesof^pp near0.25and
i,N
calculating CI lengths for all possible values of the observed sample 0.75 are expected to generally exhibit the longest CI lengths.
allelefrequencyandthentakingthemaximumlength.Thethreecurves
SimulationresultsforScenario2werebroadlyinagreementwith
correspond to three scenarios where the population is (1) at Hardy-
Weinbergequilibrium(HWE),(2)attainsitslowesthomozygosityvalue these theoretical expectations (Figure3).
withrespecttoA,and(3)attainsitshighesthomozygosityvaluewith For the last scenario, Scenario 3, the population attained its
i
respect to Ai. For each curve, the two filled circles represent the highestPii,representingtheoppositeextremetoScenario2.Asfor
minimumNvaluesrequiredforthemaximumCIlengthtoreachvalues the previous two scenarios, the maximum CI length decreased
ofƒ0:2andƒ0:1.Forvisualguidance,thetwodashedhorizontallines
markmaximumCIlengthsof0.2and0.1.TheexactvaluesofNtested non-linearlywithincreasingN(Figure1),butthistime,Nƒ0:2and
aredescribedinMethods,andequation(13)wasusedtocalculatetheCI Nƒ0:1 took values of 94 and 285 respectively. These values were
lengths. approximately four and six times as large as the corresponding
doi:10.1371/journal.pone.0085925.g001 values in Scenario 1 (Figure 1). In Scenario 3, Pii~pi, and
PLOSONE | www.plosone.org 6 January2014 | Volume 9 | Issue 1 | e85925
ConfidenceIntervalsforAlleleFrequencies
Figure3.Changeinlengthof§95%confidenceintervalacross Figure4.Changeinlengthof§95%confidenceintervalacross
differentobservedvalues,underminimumhomozygosity.Fora differentobservedvalues,undermaximumhomozygosity.For
sample of size N taken from a population of size M~1,000with the asampleofsizeNtakenfromapopulationofsizeM~1,000withthe
minimum homozygosity possible for an allele A, graph showing the maximum homozygosity possiblefor an allele A, graph showingthe
i i
length of the §95% confidence interval (CI) for the population length of the §95% confidence interval (CI) for the population
frequencyofA (p)acrossallpossibleobservedvaluesofthesample frequencyofA (p)acrossallpossibleobservedvaluesofthesample
i i i i
allelefrequency(p ).ThethreecurvescorrespondtoN~10,30and allele frequency (p ). The four curves correspond to N~10, 30, 100
i,N i,N
100.Equation(13)wasusedtocalculatethelengthofeachCI. and200.Equation(13)wasusedtocalculatethelengthofeachCI.
doi:10.1371/journal.pone.0085925.g003 doi:10.1371/journal.pone.0085925.g004
equation(11)showsthatintermediatevaluesofpi givethehighest ½0:598,0:872(cid:2) and ½0:114,0:381(cid:2) respectively. Using the same
valuesofs2½pi,N(cid:2).Therefore,asinScenario1,intermediatevalues methodology, for the Finstro¨m population, the §98:3% CI’s for
of^ppi,N areexpectedtogenerallyexhibitthelongestCIlengthsfor the population frequencies of A1, A2 and A3 were calculated
given N. Again, simulation results were consistent with these as½0:003,0:109(cid:2),½0:714,0:914(cid:2)and½0:003,0:109(cid:2) respectively.
expectations (Figure4).
Maximum length of §95%confidence interval with
increasing population size
WhentakingasampleofsizeN~30fromadiploidpopulation
atHWE,themaximumlengthofthe§95%CIforp,acrossall
i
^pp , increased with population size M (Figure 5). As M was
i,N
increased from a small value of 100 to 1,000, the maximum CI
length remained the same at 0.2 (this is possible because there is
nothingtopreventdifferentvaluesofMgivingthesamemaximum
CIlength,usingequation(13)).ThemaximumCIlengthincreased
when M was increased from 1,000 to a large value of 2,500, but
only by 0.04. Thereafter, the length remained constant up to a
very large value of M~7,500, and only increased by a small
amount of 0.02 with a further increase in M to 10,000. Thus,
simulation results confirm the theoretical expectation that CI
lengthincreaseswithM(asexplainedinMethods–thisexpectation
arises because s2½p (cid:2) increases with M, according to equation
i,N
(11)).However,theincreaseintheCIlengthwasmodestasMwas
increased overtwoorders ofmagnitude.
Figure 5. Change in maximum length of §95% confidence
§95%confidence interval forJost’sDbetween two
intervalwithincreasingpopulationsize.Graphshowinghowthe
checkerspot butterfly populations maximum length of the §95% confidence interval (CI) for the
For the Pra¨sto¨ checkerspot butterfly population, the §98:3% population frequency of an allele Ai (pi) changes with increasing
CI’s for the population frequencies forthree of thefour alleles at populationsizeM,whentakingsamplesofsizeN~30.Thepopulation
is at Hardy-Weinberg equilibrium. For a given M, the maximum CI
theCINX1locuswerecomputedusingequation(13)anddatafrom
lengthwasderivedbycalculatingCIlengthsforallpossiblevaluesof
Palo et al. [13], as described in Methods. The three alleles are
the observed sample allele frequency and then taking the maximum
denoted by A1, A2 and A3 respectively, with the population length. M values of 100, 250, 500, 750, 1,000, 2,500, 5,000, 7,500 and
frequencies denoted by p1, p2 and p3 respectively. The three 10,000weretested,asindicatedbythefilledcircles.
corresponding §98:3% CI’s derived were ½0:002,0:080(cid:2), doi:10.1371/journal.pone.0085925.g005
PLOSONE | www.plosone.org 7 January2014 | Volume 9 | Issue 1 | e85925
ConfidenceIntervalsforAlleleFrequencies
Thethree§98:3%CI’sforthePra¨sto¨ populationwereusedto least100ð1{2aÞ,forthemoregeneralcasewherethepopulation
formacubicregion,whichcorrespondstoa§95%CRforA ,A size cantake anyvalue largerthan or equaltothesample size.
1 2
and A3 [34]. In the same way, the three §98:3% CI’s for the The method we derived was used to show that the sampling
Finstro¨m population were used to form a §95% CR for A1, A2 uncertaintyinpi,measuredasthemaximumlengthofthe§95%
and A3. Within these two CR’s, the maximum and minimum CI for pi across all possible values of the observed sample allele
valuesofDJost(DJostisgivenbyequation(15))werecalculatedand frequency^ppi,N,decreasednon-linearlywithNwhensamplingfrom
usedtoderivea§95%CIforDJost,asdescribedinMethods.This a large population (M~1,000) under three archetypal scenarios.
CI for DJost was derived as (cid:6)2:34|10{5,0:186(cid:7). If DJost was Thesethreescenariosrepresentthecaseswherethepopulation(1)
calculatedsimplyusingthesampleallelefrequenciesandequation isatHardy-Weinbergequilibrium(HWE),(2)hasthelowestvalue
(15),thenonly onevaluewould havebeen obtained: 0.043. ofPiiand(3)hasthehighestvalueofPii.Asexpectedfromtheory,
for any given N, the maximum CI lengths for Scenario 3 were
Discussion always greater than corresponding lengths in Scenarios 1 and 2.
However,themaximumCIlengthsforScenario2wasunexpect-
In scientific studies that use sample data to estimate unknown edlygreaterthanthatinScenario1forsomevaluesofN,reflecting
population parameters, the sampling uncertainty in the estimates the greater possible number of values p can take in a finite
i
needs to be quantified in order to make reliable inferences on population with minimum homozygosity compared to one at
population processes captured by the parameters. This forms an
HWE.Thisillustrateshowthefinitesizeofapopulationcangive
essential part of scientific hypothesis-testing. Therefore, in studies
oppositetrendstothoseobtainedunderanassumptionofinfinite
of population genetics, it is essential to quantify the sampling
size. According to theory and simulations, Scenario 3 gives CI
uncertainty of key population parameters used to infer past and
lengths that closely approximate those in the case where P is
ii
present evolutionary processes. These include allele frequencies,
unknown(seeMethods).Thus,ifP isunknown,CIlengthsderived
ii
which are often used to quantify genetic variation among
underScenario 3should be used.This wastheapproach usedin
populations,therebyallowinghypothesesonprocessesdrivingthis
the application of our method to sample data for two butterfly
variationtobetested(e.g.,[1,2,3,4,5]).However,manystudiesdo
populations,discussedfurtherbelow.Ontheotherhand,ifthereis
not include sampling uncertainty for allele frequencies, instead
evidence that the population is at HWE, then the shorter CI’s
presenting and/or using single point estimates based on one
derived underScenario1 could beused.
sampleperpopulation(e.g.,[11,12,13,14,15,16,17,18]).Thus,itis
Underthethreescenariosexamined,thenon-lineardecreasesin
not possible to assess the accuracy of any inferences from these
samplinguncertaintywithincreasingNareconsistentwithresults
studies.Thishindersnotonlytheadvanceofscientificknowledge
fromthesimulationstudyofHaleetal.[19],whofoundthatthe
but also decision-making based on this knowledge, such as for
average difference between p and p also exhibited non-linear
sustainablemanagementandconservationofnaturalresources.In i,N i
decreases with N. However, Hale et al. [19] did not use their
thiscontext,theworkpresentedinthispaperisvaluableinthatit
simulationresultstoquantifysamplinguncertaintyfortherealistic
provides a method of quantifying sampling uncertainty in allele
situation where only one sample is taken from a population; this
frequencies for diploid populations, in the form of confidence
situationwasconsideredinourstudy.Furthermore,asmentioned
intervals(CI’s)containingtruevalueswithprobabilityequaltoor
in the Introduction, for given N, they only used 100 samples to
greater than a desired threshold.
numerically construct the distribution for p , resulting in an
The method presented pertains to the general case of a locus i,N
incomplete distribution that may not closely reflect the true
withnalleles,withasampleofsizeNtakenfromapopulationof
distribution. This highlights a weakness of a simulation-based
size M and any degree of homozygosity with respect to the n
approachwithoutarigorousmathematicalunderpinning,whichis
alleles.Inthiscase,themethodallowsconstructionofaCIforthe
presentinourapproach.Haleetal.[19]concludedthatN=25–
populationfrequencyofeachallele,whichcanthenbecombined
30issufficienttogiveaccurateestimatesofp,butthisconclusion
tocreateajointconfidenceregion(CR)forallthepopulationallele i
has to be interpreted in light of the limitations identified. Our
frequenciesatagivenlocus.Itisnotedthatifmorethanonelocus
results refine this conclusion by showing that across the three
is considered simultaneously, then a joint CR for all population
scenariosexamined,N=49–285isrequiredtoensurethat,witha
allelefrequenciesatalllocicanbecalculatedbycombiningCI’sin
highprobabilityof§0:95,anestimateforp canbederivedfrom
an analogous way. For the subcase of an infinite population size i
(M??) and a large sample size of N§30, Weir [10] had any one sample that is within 0.05 of the true value; this
proposed an approximate 100ð1{2aÞ% CI for the population correspondstoaCIoflengthƒ0:1.Toensurethattheestimateis
within0.1ratherthan0.05,correspondingtoaCIoflengthƒ0:2,
allele frequency of an allele A, p. This is
½pi,N{z1{ass^½pi,N(cid:2),pi,N{zass^½pi,N(cid:2)(cid:2),wherepi,iN isthiesampleallele our results show that N=22–94 is required. Thus, N~30 is not
frequency; z satisfies Wðz Þ~a, with W being the cumulative guaranteed to give ‘‘accurate’’ estimates of pi under all or most
a a
scenarios, and N values up to 10 times larger could be required.
distributionfunction(cdf)ofthestandardNormaldistribution;and
ss^½p (cid:2)isanestimateofthestandarddeviationofp usingsample Decreasing the population size M from 1,000 would help to
i,N i,N
data.ss^½p (cid:2)isspecifiedbyequation(11)withp replacingp and decreasesamplinguncertainty,butresultsshowedthatdecreasing
i,N i,N i
P , the sample frequency of homozygotes with allele A, Movertwoordersofmagnitudefrom10,000to100onlyresulted
reipi,Nlacing P , the corresponding population frequency of homoi- inmodestdecreasesinthemaximumCIlengthofƒ0:06,withno
ii
zygotes. However, the accuracy of this CI depends both on how decrease when M was decreased from 1,000 to 100. Thus, the
close ss^½p (cid:2) istothetrue standard deviation s½p (cid:2) (specified by overallconclusionisthatN~30isofteninsufficienttoguarantee
i,N i,N
equation (11)) and how close the cdf of pi,N is to a Normal accurateestimatesofpi,inthesensethatpiiswithin0.05or0.1of
distributionwiththesamemeanandvariance.Thisaccuracyhas the estimate. Considering that alleles at highly polymorphic loci,
notbeenquantified[10]andthus,itisnotknownwhethertheCI such as microsatellites, often occur at population frequencies of
actually contains pi with a probability of at least 100ð1{2aÞ, v0:05[35],itmightbedesirabletoderiveCI’sforpi thatareof
renderingitsuseproblematic.Inthisstudy,wehaverectifiedthis lengthv0:1.Thus,samplesizesevenlargerthanthevaluesfound
problembyconstructingaCIforp withprobabilitycoverageofat inour simulations mightberequired under somecircumstances.
i
PLOSONE | www.plosone.org 8 January2014 | Volume 9 | Issue 1 | e85925
ConfidenceIntervalsforAlleleFrequencies
The application of our method to empirical data for two ofboththebinomialandunivariatehypergeometricdistributions,
populations of the checkerspot butterfly [13] demonstrated how when compared with a hypothesis-testing approach analogous to
the underlying theory can be applied to construct joint §95% the one used in this paper. If their method could be extended to
CR’s for the population frequencies of multiple alleles at a single the population allele frequency parameter for the more complex
locus. These CR’s were then used to construct a §95% CI for distribution used in this paper (equation (9)), based on a
Jost’s D, which measures genetic differentiation between the two multivariate hypergeometric distribution (equation (1)), then this
populations. This illustrates how our method can be used to mightresultintighterCI’sforpopulationallelefrequencieswhen
quantify sampling uncertainty in genetic indicators that are a samplingfrom afinite diploid population.
function of population allele frequencies, thus facilitating hypoth- In addition, the CI’s derived in our paper were designed to
esis-testing and also risk-based natural resource management. In quantifytheuncertaintyinpopulationallelefrequenciesthatarises
theexampleconsidered,thesinglepointestimateofJost’sDusing fromtakingarandomsampleofafinitediploidpopulation,which
thesampleallelefrequencieswasaboutfourtimeslowerthanthe can exhibit any degree of relatedness. The ‘‘random’’ refers to
upper bound of its §95% CI. Thus, use of the single point equalprobabilityofchoosingindividualsthatmaybeofanytype,
estimatewithoutaccountingforsamplinguncertaintycouldleadto whichdoesnotimplythatonceanindividualofaparticulartype
misleading conclusions. The effectiveness of any management hasbeensampled,thenextindividualsampledisequallylikelyto
measures based on such conclusions would be compromised, belong to any of the types (i.e., does not imply individuals in the
hinderingtheachievementofobjectivesrelatedtoconservationor population or sample are unrelated). Relatedness among individ-
sustainable use. Our example therefore highlights the practical ualsinthepopulationisimplicitlyincludedwithinthepopulation
utility of themethodthat wehavederived. frequencies of the different genotypes, and is thus accounted for
In conclusion, we have presented a rigorous mathematical when calculating the sampling distribution for the population
method for quantifying sampling uncertainty in estimates of frequency of an allele A, as specified by equation (1). However,
i
population allele frequencies, for a general case that has hitherto this representation does not give an explicit quantification of the
notbeenanalyzed.Inaddition,wehavedemonstrateditspractical degree of relatedness among individuals in the population, for
application in informing sampling design and determining exampleusingkinshipcoefficients.DeGiorgioandRosenberg[38]
uncertainty in genetic indicators. Thus, the method derived used such coefficients to derive an unbiased estimator of
advances both theory and practice, with broad implications for a heterozygosity (H~1{Pn p2, for a locus with n alleles) in
rangeofdisciplines,including:conservationgenetics,evolutionary i~1 i
thecaseofsamplingfromapopulationofdiploidindividualsthat
genetics, genetic epidemiology, genome wide association studies
couldberelated,andDeGiorgioetal.[39]extendedtheseresults
(GWAS),forensicsandmedicalgenetics.Inparticular,themethod
to the case of individuals with arbitrary ploidy. In their
provides exact answers to the question of how many individuals
calculations, the number of copies of allele A in each sampled
i
needtobesampledfromapopulationinordertoachieveagiven
individualkwastreatedasarandomvariableandthecovariances
levelofaccuracyinestimatesofpopulationallelefrequencies.This
ofthesevariableswerethenrelatedtothekinshipcoefficients.This
is a question that has rarely been studied before [19], despite its
is different to calculations in our paper, where the number of
important practical implications. Previous studies have derived
sampledindividualswithallelesA andA wastreatedasarandom
i j
samplesizesrequiredtosample,withhighprobability,atleastone
variable(seeMethods).Themethodinthispapermightberevised
copy of all alleles at a locus above a given frequency [35,36,37],
byconsideringthenumberofcopiesofalleleA foreachsampled
but these do not correspond to sample sizes required to achieve i
individual instead, following [38,39]. This could allow explicit
accurate estimates of population allele frequencies. Derivation of
quantificationofrelatednessinthecontextofderivingCI’sforthe
the latter requires explicit quantification of sampling uncertainty,
population allele frequencies.
as wehavedone inthisstudy.
Supporting Information
Possible future extensions
The CI’s and CR’s constructed using our method are File S1 Derivation of the variance of the sample allele
conservative in the sense that they contain the true values with a frequency.
probability equal to or greater than a desired threshold. This (PDF)
conservative property is useful in hypothesis-testing if there is a
needtodecreasetheprobabilityofobtainingafalsepositivebelow Acknowledgments
a certain threshold. However, researchers would ideally like to
construct CI’s and CR’s containing true values with a known Wewouldliketothanktwoanonymousreviewersforhelpfulandinsightful
probability, not just with probability at or above a known comments,whichhaveledtogreatimprovementsinthismanuscript.
threshold. Therefore, future research could attempt to tighten
the intervals and regions that we have derived, ideally until they Author Contributions
coveraknownprobability.Forexample,CaiandKrishnamoorthy
Conceived and designed the experiments: TF KK. Performed the
[23] devised a method (using their ‘‘Combined Test’’ approach) experiments: TF. Analyzed the data: TF KK. Contributed reagents/
thatwasshowntogiveshorterCI’sfortheprobabilityparameters materials/analysistools:TFKK.Wrotethepaper:TFKK.
References
1. BowcockAM,KiddJR,MountainJL,HebertJM,CarotenutoL,etal.(1991) 3. LuikartG,EnglandPR,TallmonD,JordanS,TaberletP(2003)Thepowerand
Drift, admixture, and selection in human evolution: A study with DNA promiseofpopulationgenomics:fromgenotypingtogenometyping.Nature4:
polymorphisms.ProcNatlAcadSciUSA88:839–843. 981–994.
2. AkeyJM,ZhangG,ZhangK,JinL,ShriverMD(2002)Interrogatingahigh- 4. deKovelCGF(2006)Thepowerofallelefrequencycomparisonstodetectthe
densitySNPmapforsignaturesofnaturalselection.GenomeRes12:1805– footprintofselectioninnaturalandexperimentalsituations.GenetSelEvol38:
1814. 3–23.
PLOSONE | www.plosone.org 9 January2014 | Volume 9 | Issue 1 | e85925
ConfidenceIntervalsforAlleleFrequencies
5. FriedlanderSM,HerrmannAL,LowryDP,MephamER,LekM,etal.(2013) 21. LaplacePS(1812)The´orieanalytiquedesprobabilite´s.Paris,France,Courcier.
ACTN3 allele frequency in humans covaries with global latitudinal gradient. 464p.
PLoSONE8(1):e52282. 22. Agresti A, Coull BA (1998) Approximate is better than ‘‘exact’’ for interval
6. NeiM(1972)Geneticdistancebetweenpopulations.AmNat106:283–292. estimationofbinomialproportions.AmStat52:119–126.
7. NeiM,ChesserRK(1983)Estimationoffixationindicesandgenediversities. 23. Cai Y, Krishnamoorthy K (2005) A simple improved inferential method for
AnnHumGenet,47:253–259. somediscretedistributions.ComputStatDataAnal48:605–621.
8. Weir BS, Cockerham CC (1984) Estimating F-statistics for the analysis of 24. ClopperCJ,PearsonES(1934)Theuseofconfidenceorfiduciallimitsillustrated
populationstructure.Evolution38:1358–1370. inthecaseofthebinomial.Biometrika26:404–413.
9. JostL(2008)GSTanditsrelativesdonotmeasuredifferentiation.MolEcol17: 25. Frankham R (1996) Relationship of genetic variation to population size in
4015–4026 wildlife.ConservBiol10:1500–1508.
10. WeirBS(1996)GeneticdataanalysisII.Sunderland,USA,SinauerAssociates. 26. GreenDM(2003)Theecologyofextinction:populationfluctuationanddecline
445p. inamphibians.BiolConserv111:331–343.
11. EanesWF,KoehnRK(1978)AnanalysisofgeneticstructureintheMonarch 27. Vredenberg VT,Knapp RA,TunstallTS, BriggsCJ(2010)Dynamicsofan
butterfly,DanausplexippusL.Evolution32:784–797. emergingdiseasedrivelarge-scaleamphibianpopulationextinctions.ProcNatl
12. Forbes SH, Hogg JT, Buchanan FC, Crawford AM, Allendorf FW (1995) AcadSciUSA107:9689–9694.
Microsatelliteevolutionincongenericmammals:domesticandbighornsheep. 28. JohnsonNL,KotzS,BalakrishnanN(1997)Discretemultivariatedistributions.
MolBiolandEvol12:1106–1113. Hoboken,USA,Wiley-Blackwell.328p.
13. Palo J, Varvio S-L, Hanski I, Va¨ino¨la¨ R (1995) Developing microsatellite 29. CoxDR,HinkleyDV(1974)Theoreticalstatistics.London,UK,Chapmanand
markers for insect population structure: complex variation in a checkerspot Hall.511p.
butterfly.Hereditas123:295–300. 30. TalensE(2005)StatisticalauditingandtheAOQL-Method.Ridderkerk,The
14. LuikartG,AllendorfFW,CournuetJ-M,SherwinWB(1998)Distortionofallele Netherlands,LabyrintPublications.164p.
frequencydistributionsprovidesatestforrecentpopulationbottlenecks.JHered 31. WolframResearchInc.(2003)MathematicaEdition:Version5.0.Champaign,
89:238–247. Illinois,,USA,WolframResearchInc.
15. Rank NE, Dahlhoff EP (2002) Allele frequency shifts in response to climate 32. The MathWorks Inc. (2012) MATLAB version 8.0. Natick, Massachusetts, ,
change and physiological consequences of allozyme variation in a montane USA,TheMathWorksInc.
insect.Evolution56:2278–2289. 33. RDevelopmentCoreTeam(2010)R:alanguageandenvironmentforstatistical
16. Hugnet C, Bentjen SA, Mealey KL (2004) Frequency of the mutant MDR1 computing.Vienna,Austria,RFoundationforStatisticalComputing.1731p.
alleleassociatedwithmultidrugsensitivityinasampleofcolliesfromFrance. 34. Rice JA (1995) Mathematical statistics and data analysis, second edition.
JVetPharmacolTher27:227–229. Belmont,USA,DuxburyPress.651p.
17. SeiderT,FimmersR,BetzP,LedererT(2010)Allelefrequenciesofthefive 35. ChakrabortyR(1992)Samplesizerequirementsforaddressingthepopulation
miniSTR loci D1S1656, D2S441, D10S1248, D12S391 and D22S1045 in a geneticissuesofforensicuseofDNAtyping.HumBiol64:141–159.
Germanpopulationsample.ForensicSciIntGenet4:e159–e160. 36. GregoriusH-R(1980)Theprobabilityoflosinganallelewhendiploidgenotypes
18. Petrejcˇ´ıkova´E,Sota´kM,Bernasovska´J,Bernasovsky´I,Re˛bałaK,etal.(2011) aresampled.Biometrics36:643–652.
Allelefrequenciesandpopulationdatafor11Y-chromosomeSTRsinsamples 37. Sjo¨grenP,Wyo¨niP-I(1994)Conservationgeneticsanddetectionofrarealleles
fromEasternSlovakia.ForensicSciIntGenet5:e53–e62. infinitepopulations.ConservBiol8:267–270.
19. Hale ML, Burg TM, Steeves TE (2012) Sampling for microsatellite-based 38. DeGiorgioM,RosenbergNA(2009)Anunbiasedestimatorofgenediversityin
population genetic studies: 25 to 30 individuals per population is enough to samplescontainingrelatedindividuals.MolBiolEvol26:501–512.
accuratelyestimateallelefrequencies.PLoSONE7(9):e45170. 39. DeGiorgioM,JankovicI,RosenbergNA(2010)Unbiasedestimationofgene
20. Gillespie JH (2004) Population genetics: a concise guide, second edition. diversityinsamplescontainingrelatedindividuals:exactvarianceandarbitrary
Baltimore,USA,TheJohnHopkinsUniversityPress.217p. ploidy.Genetics186:1367–1387.
PLOSONE | www.plosone.org 10 January2014 | Volume 9 | Issue 1 | e85925
