Title: Bounds and normalization of the composite linkage disequilibrium coefficient.

Transcription

1 Ttle: ounds and normalzaton of the composte lnkage dsequlbrum coeffcent. (Genetc Epdemology 004 7:5-57 uthor: Dmtr V. Zaykn * * GlaxoSmthKlne Inc., Research Trangle Park, NC Contact detals: Dmtr Zaykn Natonal Insttute of Envronmental Health Scences Natonal Insttutes of Health Research Trangle Park, NC 7709 emal: zayknd@nehs.nh.gov Runnng ttle: Composte LD bounds

2 bstract The composte lnkage dsequlbrum (LD measure s often calculated for two-locus genotypc data, especally when couplng and repulson double heterozygotes cannot be dstngushed. Ths measure has been reported to have good statstcal propertes and was suggested for routne testng of LD regardless of Hardy-Wenberg equlbrum at ether of two loc [Wer, 979, Schad, 004]. However, the bounds for ths measure have not been yet reported. These bounds are derved here as functons of one-locus genotype or allele frequences. They provde standardzed measures of composte lnkage dsequlbrum defned as the proporton of ts mum attanable value gven observed allele or genotype frequences.

3 3 Introducton Calculaton of the composte dsequlbrum coeffcent,, for measurng assocaton between alleles and at two loc s a routne step n analyss of multlocus genotypc data [Wer 979, Wer and Cockerham, 979, Wer, 996]. Defnton of the composte coeffcent s P + P p p, where P s the frequency of gamete, P s the jont frequency of alleles and at two dfferent gametes, and p, p are the frequences of alleles and at two loc [Wer 996]. Ths coeffcent s drectly estmated from two-locus genotypc data and under random matng corresponds to the usual measure of lnkage dsequlbrum, D P p p. Ths s because the nongametc frequency reaches the equlbrum P p p after one generaton of random matng. Wer [996] suggests the defnton n whch the focus s on two alleles at a tme,,, and other alleles at these two loc are combned and collectvely referred to as a, b wth frequences p,( p. Wer [996] defnes the correlaton coeffcent, r (, whch s one possble normalzaton of, r ( p ( p + D ( p ( p + D where D, D are the estmates of Hardy-Wenberg dsequlbrum coeffcents and p, are sample allele frequences at loc,. It s most useful n that t drectly p relates to the asymptotc ch-square test statstc wth one degree of freedom for testng the hypothess 0. The relaton s X, where n s the number of ndvduals nr n the sample. Constrants for the frequences of genotypes and alleles at each locus

4 4 mpose bounds for the mnmum and mum values of r. s the result, the range of possble values of r s smaller than (- to. In certan stuatons, t mght be s useful to report values of as the proporton of ts mum possble value,. For example, Clark et al. [003] evaluated the dstrbuton of ths standardzed measure across 4,833 SNPs n the human genome, however they obtaned the bounds ( numercally. The purpose of ths report s to derve the bounds on. The standardzed measure s defned as and ranges between and. The standardzaton makes the new measure ndependent of sngle locus frequences, n the sense of the range that the coeffcent can take. The allele frequences are very much part of the usual normalzed LD defned n the same way, D D 988] and t would be mstaken to nterpret ether D or dependences on the allele or genotype frequences. D, [Hedrck, 987, Lewontn, as beng free of Statstcal Methods Usng the notaton of Wer [996], the composte LD coeffcent s estmated from dlocus counts and sample allele frequences as (n + nb + na + nab p p ( n where n s the number of ndvduals n the sample, and n,..., n ab denote d-locus sample genotype counts. One way to derve bounds for the sample value of s to use

5 5 the relaton C(, (n x y where C( x, y n n n n x y x y, and x,y are vectors of genotype values for two loc, re-coded as f genotype s x 0 f genotype s a f genotype s y f genotype s 0 f genotype s b f genotype s The correspondence between and C( x, y can be shown by wrtng the composte LD coeffcent n an alternatve form: ( + 4 n ( n n ( n n a n ( n + n n n ( p p + p p p p p p ab + n + n b n n a n n ( p p ( p p a ( n n ( n n ( b b b a Sums n C( x, y can be wrtten n terms of the genotype counts as x y n n ( ( + n n n ( ( + + n n (( + n (( x y ( n n ( n n Therefore, C x, y n( n n n + n ( n n ( n n. Dvdng ths ( by n, the relaton to the composte LD as defned n ( s C( x, y (n. Sample allele frequences are related to x, y as follows:

6 6 p p n + n n n + n n a b x + n n y + n n n x n y p p Therefore, by takng expectatons of these expressons, E( x p and E( y p. It can be further verfed that Cov( x, y E( x, y Cov( x, y + E( x E( y P + ( p P P ( p + P Var( x ( P Var( y ( P [ P P + Pa ( Pa ] [ p ( p + D ] + P [ PP + Pb ( Pb ] [ p ( p + D ] + P ( P ( P P P where D P p and D P p are Hardy-Wenberg dsequlbrum coeffcents [Wer, 996] and P,..., P are frequences of genotypes,,. Then Corr ( x, y ( p ( p + D ( p ( p + D The ndcator varables x, y are mnus the sums of those gven by Wer [979], ( x + x and y + y. These varables keep track of the number of copes of and ( on the two gametes. Wer showed that the correlaton of the sums s the same as the expresson for Corr( x, y.

7 7 When > 0, the pars { x, y } and { x, y } ncrease the resultng value of the cross-product, x y, whle the pars { x, y 0},{ x 0, y } do not add to the value, and the pars { x, y },{ x, y } decrease the value. Ths suggests a way to derve the bounds for by fndng the permutaton of x relatve to y that wll mze the absolute value of, or equvalently, C( x, y. These bounds are for fxed one-locus counts. The permutaton should mze the number of x, y } pars wth the same sgn of x and y and dstrbute the remanng x wth y n { such a way that wll not decrease the value of the cross-product. The second term n C( x, y,.e. the product x y, s not affected by the way the vectors are permuted. When < 0, the pars x, y } are arranged n a way that mzes the number of pars wth the dfferent sgn for x and y. { Consder the calculaton for the case > 0 n more detal. The mum possble number of pars of the same sgn s d mn( n, n + mn( n, n ( whch s the sum of the largest possble number of pars { x, y } and { x, y }. The remanng number of pars s ( n d, so potentally ths s the amount by whch the value of the cross-product gven by ( can be reduced. However, these pars can be arranged so that as many values of x 0 or y 0 (heterozygotes as

8 8 possble are matched n pars wth the second of the two values beng dfferent from zero. Then the overall reducton of the ( from the value n ( s s n d [( n d ( n a + nb,0] mn( n d, n + n a b Therefore, the mum value for the cross-product s ( d s and the bound for > 0 s For the case d s n n d s a n d s n ( ( < 0 ( n n ( n n ( p p ( p p ( p ( p b, the mnmum s obtaned the same way, but the value d s calculated as d mn( n, n + mn( n, n to match x, y } pars wth the opposte sgns. Ths gves the mum absolute value: ( d + s n d s + n ( p ( p ( p ( p { Puttng together, the possble values of genotype counts as are bounded by functons of sngle-locus d s + n n where d mn( n d s n n where d mn( n ( n n ( n n, n, ( n n ( n n, n + mn( n + mn( n, n, n, < 0 > 0 (3

9 9 where s n d mn( n d, n a + n b. The second part of the bounds s a functon of sample allele frequences, n ( n n ( n n ( p ( p. The populaton value as the functon of one-locus populaton frequences s d s + where d mn( P d s where d mn( P ( p ( p, P ( p ( p, P + mn( P + mn( P, P, P,, < 0 > 0 (4 and s d mn( d, P a + P. b The sample standardzed value wth bounds as gven n (3 s. δ Comparson of Standardzed Coeffcents The correspondence between the values of (mum lkelhood estmator based on the HWE assumpton and the standardzed composte LD n large samples should be noted. The problem wth the drect comparson of and D s that D bounds are for the fxed frequences of alleles, whereas bounds are for the one-locus frequences of genotypes. In ths sense, δ s more comparable to r whch ncorporates varances that nclude one-locus devatons from HWE. Numercally, δ and r coeffcents are closely correlated, although δ D δ s more stretched because t can stll reach extreme (- to values even f allele frequences at two loc are unequal. Ths s not necessarly a vrtue δ

10 0 of the standardzed measure ( δ. The coeffcent r has well-defned statstcal and populaton-genetc propertes. It s useful n contexts where an allele at one locus s to be regarded as a proxy for an allele at another locus, for example durng the selecton of markers for assocaton studes [e.g. Meng et al., 003]. Ths requres dependency (LD as well as the closeness of sngle-locus frequences whch s then reflected by large absolute values of. r To make a far comparson wth D, equaton (3 s modfed so that the bounds are calculated as the mum possble values gven frequences of alleles rather than genotypes. In ths case the standardzed composte LD can be compared to drectly. D To obtan these bounds, the new values of ( d s (n n (3 should be computed. Ths s done by replacng the sngle-locus counts n d wth ther mum values gven the counts of alleles and notcng that n ths case s n d, so that ( d s (n (d n (n. Then the bounds are d n ( ( + p p n, n nb na n where d mn(, + mn(, d n ( ( p p n, n n na nb where d mn(, + mn(, < 0 > 0 fter expressng counts va sample frequences, ths further smplfes to mn mn [ p p, ( p ( ], [( p p p, p ( p ], < 0 > 0 (5

11 Therefore, these bounds are twce the bounds for D. Defne the standardzed coeffcent as, when the bounds are computed for the fxed allele frequences, usng (5. When the populaton s n HWE, the two estmators are related as. The reason for the value to be twce as D small s that the composte dsequlbrum s a sum of the usual LD, measured by D plus the covarance between alleles at two dfferent chromosomes, D. Ths second term s zero f the populaton s n HWE, however the mum value of stll needs to account for D. One can only clam that reached a certan proporton of ts mum value, wthout attrbutng that proporton to ether the gametc or the nongametc part. The allele frequences n and D are estmated n the same way, therefore the effcency of these two coeffcents n estmaton of the populaton value of D largely determned by the performance of the correspondng LD estmators. Schad (004 argued for superorty of the composte LD by examnng propertes of the test H : 0 D 0. Therefore t s expected that performs well compared to D. When the populaton s not n HWE, s stll a vald estmator for ( D + where the second term s the normalzed D. Not unexpectedly, D s not an approprate estmator. Fgure s an llustraton of ths. To create each data pont on the graphs, ten possble d-locus populaton genotype frequences were drawn from the Drchlet D s

12 dstrbuton wth all ten parameters equal to ¼ pror to obtanng each multnomal sample of 500 ndvduals. The Drchlet(¼,,¼ samplng creates a nearly unform (- to dstrbuton of and across populatons from whch each multnomal sample s D D obtaned. Fgure s a smlar plot wth all populaton dsequlbrum due to the gametc part, D. Such populatons are created by samplng four gamete frequences from the Drchlet(,,, dstrbuton and parng them at random. The resultng dstrbuton of D s close to unform on (- to. The estmator D was obtaned by numercally solvng the lkelhood formed under the assumpton of HWE. Such soluton s feasble n the case of two loc and s preferred to the common alternatve usng an EM algorthm [Wer, Cockerham, 979]. Under HWE (Fgure the coeffcent D s estmatng half of the gametc LD term,. oth fgures mply that s performng well as an estmator of the populaton value, ( D +, whle abs( s takng many possble values between D D abs( D + D and when the populaton s not n equlbrum. Performance of further mproves wth the ncreased sample sze (data not shown. Note that Fgure samplng of d-locus genotypes results n devatons from equlbrum at the level of two loc, whch ncludes non-zero, sngle-locus HWE devatons, as well as the hgher order dsequlbra. Wer [996] gves defntons of all correspondng coeffcents. Thus, estmator appears to perform well even when two-locus frequences devate from the equlbrum condtons. D

13 3 Dscusson s wth the usual normalzed LD coeffcent, D, cauton should be taken when values of or are compared for two pars of loc, or when values for the same par of δ loc are compared between dfferent populatons. Smlar evolutonary forces wll not guarantee that normalzed coeffcents should attan smlar values gven two dfferent sets of allele frequences [Lewontn, 988]. Indeed, t s the values at the bounds that are completely determned by the sngle-locus parameters, and values n the mddle reman ndetermnate. Nevertheless, the standardzed coeffcent can be useful n the sense of ts defnton as the proportonal measure of strength of assocaton between alleles at two loc. It s worthwhle to note that the nter-gametc coeffcent D can be non-zero f samplng s condtonal on the phenotype, such as the case-control samplng. If alleles, are jontly predctve for the case category, the value D can be non-zero among cases, even f the populaton value s zero. If the prevalence of the case category s w, the allelc prevalence for the allele can be defned as w j P j w j, where j ndexes genotypes and P j, w j are the genotype frequency and ts susceptblty. Suppose the case probablty for the ndvduals carryng both alleles and s w,. Then, among the cases, D wp w, p pww whch s non-zero f w, ww. Smlarly, w D n cases can be away from the populaton value. The composte coeffcent as well

14 4 as ts standardzed value can be examned n cases and compared to dsequlbrum n controls or the populaton value. On the other hand, when the gametc phase n unknown, the lkelhood (wth HWE assumpton used for the calculaton of D and D, e.g. by the means of EM algorthm, s generally not sutable for evaluaton of dsequlbrum n cases. Ths s because the equlbrum proportons n cases (ncludng sngle-locus HWE are lkely to be dstorted [Nelsen et al., 999]. Schad [004] showed that n such stuatons the ncorrect HWE assumpton can lead to grossly based results of the test H : 0 D 0, wth ether extremely conservatve or lberal type-i error rates. On the other hand, tests based on the composte coeffcent have optmal power and mantan the correct sze.

15 5 References Clark G, Nelsen R, Sgnorovtch, J, Matse TC, Glanowsk S, Hel J, Wnn-Deen ES, Holden L, La E Lnkage dsequlbrum and nference of ancestral recombnaton n 538 sngle-nucleotde polymorphsm clusters across the human genome. m J Hum Genet 73: Hedrck PH Gametc dsequlbrum measures: proceed wth cauton. Genetcs 7: Lewontn RC On measures of gametc dsequlbrum. Genetcs 0: Meng Z, Zaykn DV, Xu CF, Wagner M, Ehm MG. 003 Selecton of genetc markers for assocaton analyses, usng lnkage dsequlbrum and haplotypes m J Hum Genet 73: Nelsen DM, Ehm MG, Wer S Detectng marker-dsease assocaton by testng for Hardy-Wenberg dsequlbrum at a marker locus. m J Human Genet. 63: Schad DJ 004. Lnkage dsequlbrum testng when lnkage phase s unknown. Genetcs 66:

16 6 Wer S Genetc Data nalyss II, Snauer ssocates, Inc. Wer S Inferences about lnkage dsequlbrum. ometrcs 35: Wer S, Cockerham CC Estmaton of lnkage dsequlbrum n randomly matng populatons. Heredty 4:05.

17 7 cknowledgements Two revewers provded useful comments that mproved qualty of ths manuscrpt.

18 8 Fgure legends Fgure. Left fgure: plot of the populaton value of ( D + D vs. the sample value of. Rght fgure: plot of the populaton value of ( D + vs. the sample value of D. D Fgure. Left fgure: plot of the populaton value of vs. the sample value of. Rght fgure: plot of the populaton value of vs. the sample value of D. D D

19 Fgure. 9

20 Fgure. 0