Combined high resolution linkage and association mapping of quantitative trait loci

Transcription

1 European Journal of Human Genetcs (), 5 37 ª Nature Publsng Group ll rgts reserved 8 483/ $5 wwwnaturecom/ejg RTICLE Combned g resoluton lnkage and assocaton mappng of quanttatve trat loc Ruzong Fan*,, and Momao Xong 3 Department of Statstcs, Te Texas &M Unversty, 447 locker uldng, College Staton, Texas , US; Insttute of Medcal ometry, Informatcs and Epdemology, Unversty of onn, Sgmund Freud Strasse 5, D-535 onn, Germany; 3 Human Genetcs Center, Unversty of Texas Houston, PO ox 334, Houston, Texas 775, US In ts paper, we nvestgate varance component models of bot lnkage analyss and g resoluton lnkage dsequlbrum (LD) mappng for quanttatve trat loc (QTL) Te models are based on bot famly pedgree and populaton data We consder lkeloods wc utlze flankng marker nformaton, and carry out an analyss of model buldng and parameter estmatons Te lkeloods jontly nclude recombnaton fractons, LD coeffcents, te average allele substtuton effect and allele domnant effect as parameters Hence, te model smultaneously takes care of te lnkage, LD or assocaton and te effects of te putatve trat locus Te models clearly demonstrate tat lnkage analyss and LD mappng are complementary, not exclusve, metods for QTL mappng y power calculatons and comparsons, we sow te advantages of te proposed metod: () populaton data can provde nformaton for LD mappng, and famly pedgree data can provde nformaton for bot lnkage analyss and LD mappng; () usng famly pedgree data and a sparse marker map, one may nvestgate te pror suggestve lnkage between trat locus and markers to obtan low resoluton of te trat loc, because lnkage analyss can locate a broad canddate regon; (3) wt te pror knowledge of suggestve lnkage from lnkage analyss, bot populaton and famly pedgree data can be used smultaneously n g resoluton LD mappng based on a dense marker map, snce LD mappng can ncrease te resoluton for canddate regons; (4) models of g resoluton LD mappngs usng two flankng markers ave ger power tan tat of models of usng only one marker n te analyss; (5) excludng te domnant varance from te analyss wen t does exst would lose power; (6) by performng lnkage nterval mappngs, one may get ger power tan by usng only one marker n te analyss European Journal of Human Genetcs (), 5 37 do:38/sjejg594 Keywords: varance component models; lnkage nterval mappng; lnkage dsequlbrum mappng; QTL *Correspondence: Dr Ruzong Fan; Department of Statstcs, Te Texas &M Unversty, 447 locker uldng, College Staton, Texas , US Tel: ; Fax: ; E-mal: rfanstattamuedu Receved 4 June ; revsed 4 October ; accepted November Introducton Twenty years ago, varatons n uman DN were recognzed as genetc markers n lnkage study fter tat, te advances n molecular bology and computatonal tecnology ave led to mappng several uman nerted dsease genes Usng restrcton fragment lengt polymorpsm (RFLP) markers and polymorpc mcrosatellte loc, lnkage analyss and postonal clonng ave been used successfully n mappng te cromosome locatons of Mendelan dsease genes Te success manly depends on one premse tat te dsease genes of Mendelan trats ave a large effect on te penotypes In fact, tere s usually a one-to-one correspondence between dsease gene genotypes and te dsease penotype for Mendelan trats Moreover, te correlatons

2 6 Lnkage and assocaton mappng of QTL R Fan and M Xong between genotypes and penotype of Mendelan trats are strong Gven suffcent famly data, Mendelan trats can be mapped wt g probablty by lnkage analyss Wt te encouragement of successful mappng Mendelan trat genes, tere as been growng nterests and endeavors n te study of complex trats suc as astma and dabetes For complex dseases, te nertance patterns and penotype defntons as wt genetc etology are muc more complex Te trat/affecton status s usually a contnuous varable 3 Te mappng of complex dsease genes s muc arder Novel statstcal metods suc as bot lnkage analyss and lnkage dsequlbrum (LD) mappng or assocaton study are needed n dssectng complex trats s very dense marker maps suc as sngle nucleotde polymorpsm (SNP) are avalable, 4 bot lnkage analyss and assocaton study are utlzed smultaneously for mappng complex dsease loc 5,6 lmasy et al 7 and Fulker et al 8 proposed to use combned lnkage and assocaton analyss for quanttatve trat loc (QTL) Sam et al 9 studed te power of lnkage versus assocaton analyss of quanttatve trats by analytcally calculatng non-centralty parameters of test statstcs becass et al proposed test statstcs of assocaton studes for quanttatve trats n nuclear famles, general pedgrees, and selected samples Cardon 3 studed a sbpar regresson model of LD for quanttatve trats ll tese researces concentrated on famly data wc nclude sb-pars, and used only one marker n analyss In Fan and Xong, 4 we proposed a lnear regresson metod of g resoluton mappng of quanttatve trat loc by LD mappng analyss Te metod s based on populaton data Usng two flankng markers, te regresson models ave ger power tan tat of models usng only one marker 4 It s well-known tat famly pedgree data can be used n bot lnkage analyss and assocaton study, and populaton data can be used n assocaton study Hence, t s necessary to consder a metod to combne bot populaton data and famly pedgree data n te analyss In ts paper, we propose to perform bot lnkage analyss and g resoluton LD mappng for QTL based on combned famly and populaton data Lnkage nterval mappng s based on famly data, and LD mappng s based on bot famly pedgree and populaton data ased on varance component models, we construct lkelood to analyse famly and populaton data n Secton of Models Ten, we dscuss te parameter estmatons and regresson coeffcents Te lnkage nformaton, e, recombnaton fractons, s contaned n te varance-covarance matrx, and te assocaton nformaton, e, te LD coeffcents, s contaned n te mean parameters or te regresson coeffcents We calculate te non-centralty parameters for assocaton study and lnkage analyss, respectvely Usng te noncentralty parameters, we perform power calculatons and comparsons Te tecncal detals to calculate te regresson coeffcents, parameters, non-centralty parameters are left n te ppendxes Models Consder a quanttatve trat locus Q wc as two alleles Q and Q Suppose tat te allele frequences of Q and Q are q and q, respectvely ssume tat two markers and flank te trat locus Q n an order of Q Marker as two alleles and a wt frequences P and P a, respectvely Marker as two alleles and b wt frequences P and P b, respectvely For a nuclear famly of k cldren and two parents, let us denote ter quanttatve trats by a vector y=(y f, y m, y,, y k ) t, genotypes at marker by a vector ( f, m,,, k ) t, and genotypes at marker by a vector ( f, m,,, k ) t Here y f s te trat value of te fater, f s te genotype of te fater at marker, and f s te genotype of te fater at marker Oter notatons are defned, smlarly, for te moter wt subscrpt m and for te -t cld wt subscrpt Te log-lkelood s defned by L ¼ k logðpþ logjj ðy XmÞt ðy XmÞ Te notatons of te log-lkelood are defned as follows For te mean component Xm, we consder te followng regresson equaton suc as model () n Fan and Xong 4 y ¼ b! g x a x a z z G e ; were b s overall mean, w s a row vector of covarates suc as gender and age, g s a column vector of regresson coeffcents for te covarates w, G s polygenc effect, e s error term ssume tat G s normal Nð; s G Þ, and e s normal Nð; s e Þ Moreover, G and e are ndependent x, x,z and z are dummy varables defned by 8 8 < P a f ¼ >< Pa f ¼ x ¼ P a P f ¼ a ; z : ¼ P a P f ¼ a ; >: P f ¼ aa P f ¼ aa 8 8 < P b f ¼ >< Pb f ¼ x ¼ P b P f ¼ b; z : ¼ P b P f ¼ b: >: P f ¼ bb P f ¼ bb a, a, d, and d are regresson coeffcents of te dummy varables x,x,z and z Te model matrx X s defned by 8 9 w f x f x f z f z f w m x m x m z m z m >< >= X ¼ w x x z z ¼ >: >; w k x k x k z k z k and m=(b, g t, a, a, d, d ) t s a vector of regresson coeffcents S s a (k+)6(k+) X t f X t m X t X t k ðþ ; C European Journal of Human Genetcs

3 Lnkage and assocaton mappng of QTL R Fan and M Xong 7 varance-covarance matrx defned as r r r r r r r r r r k ¼ r r r r s ; were s ¼ k C r r r k r k s g s G s e ; s g s varance explaned by te putatve QTL Q, s G s polygenc varance, and s e s error varance Te genetc varances s g ¼ s ga s gd and s G ¼ s Ga s Gd are decomposed nto addtve and domnant components r ¼ðs ga s Ga Þ=ðs Þ s correlaton between parents and cldren, r j ¼ðp jq s ga jqs gd s Ga = s Gd =4Þ=s s correlaton between te -t cld and te j-t cld, p jq s te proporton of alleles sared dentcal by descent (ID) at QTL Q by te -t cld and te j-t cld, and jq s te probablty tat bot alleles at QTL Q sared by te -t cld and te j-t cld are ID For populaton data, an ntutve ratonale of regresson model () s gven n Fan and Xong 4 In general, one can construct a varance-covarance matrx for any type of pedgree n a smlar way as above ssume tat tere are two ndependent sub-samples of data: () populaton data: n ndependent ndvduals; () famly data: I n (I4n) ndependent famles Let us lst te log-lkelood of te n ndependent ndvduals by L,, L n, and te lkelood of te I n famles by L n+,, L I Ten te overall log-lkelood s L ¼ P I ¼ L Te unknown parameters are m ¼ ðb; g; a ; a ; ; Þ t ; s ga ; s gd ; s Ga ; s Gd ; and s e Usng te lkelood rato tests, one may test statstcal sgnfcance of te parameters of nterest Parameter estmatons and regresson coeffcents Regresson coeffcents Let m j be te effect of genotype Q Q j,,j=,, m =m Denote te populaton effect mean by m ¼ m q m q q m q and defne a Q =q m +(q 7q )m 7q m, d Q =m 7m 7m Ifm =a, m =d, and m = a as n te tradtonal quanttatve genetcs, 5 a Q = a+(q q )d s te average allele substtuton effect, and d Q =d caracterzes te domnant effect In general, one may defne a=m 7(m +m )/ and d=m 7(m +m )/ It s well known tat te addtve varance s ga ¼ q q a Q and te domnant varance s gd ¼ðq q Þ Q true random effect model descrbng te trat value s y =b+w g+g +G +e, were 8 < m for genotype Q Q g ¼ m for genotype Q Q : : m for genotype Q Q Denote LD coeffcent between trat locus Q and marker by D Q =P(Q ) q P, LD coeffcent between trat locus Q and marker by D Q =P(Q ) q P, and LD coeffcent between marker and marker by D =P() P P Let te addtve and domnant varance covarance matrces be V ¼ P ap D D P b P ; and V D ¼ P a P D D Pb P : ðþ Moreover, let us denote tree ratos D =ðp ap P b P Þ¼ R ; D Q =ðp ap q q Þ¼R Q ; and D Q =ðq q P b P Þ¼R Q s n ppendx, 4 we can sow tat te coeffcents of regresson equaton () are gven by a ¼ V D Q a Q ¼ a D Q ¼ VD D Q D Q! Q ¼ R Q R pffffffffffffff R Q P P a R Q R R Q p ffffffffffffff P P b R Q R R Q P P a R Q R R Q P P b pffffffffffffffffffff q q a Q R ; ð3þ C q q Q R 4 : ð4þ Parameters of varance covarances Denote te recombnaton fracton between trat locus Q and marker by y Q, te recombnaton fracton between trat locus Q and marker by y Q, and te recombnaton fracton between marker and marker by y Fulker and Cardon 6 proposed to estmate te proporton p jq of allele ID at putatve QTL Q for a sb-par and j by ^p jq ¼ Eðp jq jp j ; p j Þ¼a p b p p j b p p j were p j and p j are te ID proportons of alleles sared at te marker and marker, respectvely Te coeffcents a p, b p and b p are gven by b p ¼ ð y QÞ ð y Þ ð y Q Þ ð y Þ 4 ; b p ¼ ð y QÞ ð y Þ ð y Q Þ ð y Þ 4 ; a p ¼ b p b p : Let j, j be te probablty of sarng two alleles ID at markers and for a par of sbs, respectvely In Fan, 7 we proposed to estmate jq by equaton ^ jq ¼ a b p j b p j r j r j Under te assumpton of no nterference, te coeffcents are as follows (Fan 7 ): r ¼ ð y QÞ 4 ð y Þ 4 ð y Þ 4 ð y Þ 8 ; r ¼ ð y QÞ 4 ð y Q Þ 4 ð y Þ 4 ð y Þ 8 ; b ¼ b p r ; b ¼ b p r ; ð c a ¼ Þ ð c Þ ½c c ð c Þð c ÞŠ ; were c ¼ y Q ð y QÞ and c ¼ y Q ð y QÞ : ssumng tat te postons of marker and marker are known, y can be calculated troug Haldane s map functon Ten only one of y Q and y Q s unknown snce te European Journal of Human Genetcs

4 8 Lnkage and assocaton mappng of QTL R Fan and M Xong oter can be calculated troug Trow s formula 8 For general relatves and j, lmasy and langero 9 proposed an algortm to calculate te proporton p jq of allele ID at putatve QTL Q, and te expected probablty jq tat bot alleles at QTL Q are ID In Fan, 7 we derved formulas to calculate te covarances of trat values for a few types of relatves drectly wtout performng matrx operatons ssocaton and lnkage studes From equatons (3) and (4), we can see tat te coeffcents of LD (e, D Q and D Q ) and gene effects (e, a Q and d Q ) are contaned n te regresson coeffcents Moreover, we sow n te above paragrap tat te lnkage parameters (e, recombnaton fractons y Q, y Q and y ) are contaned n te varancecovarance matrx ssume tat markers and are n LD wt te trat locus Q, e, D Q 6¼,D Q 6¼ We may smultaneously test LD of marker and marker wt trat locus Q, te gene substtuton and domnant effects by testng a =a =d =d = From equaton (3), we may test LD of markers and wt te trat locus Q and te gene substtuton effect a Q by testng a =a = From equaton (4), we may test LD of markers and wt te trat locus Q and te domnant effect by testng d =d = To test lnkage, one may use te lkelood rato test of te log-lkelood L Under te null ypotess of no lnkage between te major trat locus Q and te markers, y Q = y Q =/ Under te alternatve ypotess of lnkage, y Q 6¼/ ory Q 6¼/ y comparng te dfference of maxmum log-lkeloods under te alternatve and null ypoteses, we may use w statstc to test te lnkage We wll derve analytcal formulas to explore te lnkage nterval mappng by te nuclear famles n a smlar way to Sam et al 9 accordng to statstcal teory of lkelood rato tests Non-centralty parameters of assocaton study ssume tat tere are no covarates Ten m=(b, a, a, d, d ) t Consder te overall log-lkelood L ¼ P I ¼ L, were L s te log-lkelood of trat values y of te -t famly or ndvdual Let S be te varance-covarance matrx of y, and X be ts model matrx Denote te total trat values y ¼ðy t ; ; y^t Þ t, te total varance covarance matrx by S=dag(S,,S I ), and te model matrx X ¼ðX t ; ; Xt I Þt Let b; ^a ; ^a ; ^ ; ^ ; ^ ; ^ be te maxmum lkelood estmators of b, a, a, d, d,s, S Te estmate of m s ^m ¼ ½X t ^ XŠ X t ^ ~y ¼½ P I ¼ Xt ^ X Š P I ¼ Xt ^ ~y Let H be a q65 test matrx of rank q Suppose tat te total number of ndvduals s N y Graybll, Capter 6, te test statstc of a ypotess Hm= s non-central F(q, N 5) defned by F ¼ ðh^mþ t ½HðX t ^ XÞ H t Š ðh^mþ Y t ½^ ^ XðX t ^ XÞ X t ^ ŠY N 5 : q Te non-centralty parameter of te test statstc F can be calculated by l ¼ðHmÞ t ½H½X t XŠ H t Š Hm If te data are composed of n ndvduals of a populaton, Fan and Xong 4 worked out te non-centralty parameters to test f tere are allele substtuton and/or domnant effects and LDs between te markers and te major gene locus In te followng, we dscuss a stuaton tat te data are composed of bot ndvdual populaton data and famly data Suppose tat tere are n ndvduals of a populaton, and n s suffcently large For eac y of te n ndvduals, S =s and X =( x x z z ), =,,, n From formulas n Fan and Xong, 4 ppendx and ppendx, we may sow tat X n X t n X ¼ X n ns X t X s dagð; V ; V D Þ; ¼ ¼ were V and V D are addtve and domnant varance-covarance matrces gven n () Secondly, suppose tat tere are m tro famles, and m s suffcently large tro famly s composed of bot parents and a sngle cld Notce tat te means of x,x,z and z are Let K f =(x f x f z f z f ) and K m =(x m x m z m z m ) We sow n ppendx tat te covarance matrx between parents and ter offsprng s E Kf t K ¼ E Km t K ¼ V = O ; ð6þ O O were K =(x x z z ) and O s zero 6 matrx For eac of te tro famles, te varance covarance S s a 363 matrx wose nverse s ð5þ r r r ¼ ð r r Þs r r : ð7þ r r Usng equatons (5), (6), and (7), we sow n ppendx X t m X ð r ¼n Þs 3 4r ð3 r r ÞV C : ð3 r ÞV D Xnm Trdly, suppose tat tere are k nuclear famles eac of tem as bot parents and two offsprng, and te correlaton of te two offsprng s r ssume tat k s suffcently large For eac famly, te varance covarance S s a 464 matrx wose nverse s ¼ s Cr Cr C C Cr Cr C C C C C C Cð r Þ r ð r Þ Cðr r Þ r ð r Þ Cðr r Þ r ð r Þ Cð r Þ r ð r Þ ; C were C ¼ r ð r Þ=½ð r Þ ðr r Þ Š In ppendx C, we sow tat te covarance matrx between two offsprng s ð8þ ð9þ European Journal of Human Genetcs

5 Lnkage and assocaton mappng of QTL R Fan and M Xong 9 Eðx x z z Þ t ðx x z z Þ¼ V = O : ðþ O V D =4 Usng equatons (5), (6), (9) and (), we sow n ppendx D tat k nmk X ¼nm X t X dagðd ; d V ; d 44 V D Þ=s ; ðþ were te constants are gven by d ¼ ½ 4Cr 4C C=r Š; d ¼ 4Cðr Þ Cð r r Þ=½r ð r ÞŠ; d 44 ¼ ð Cr ÞC½4ð r Þ ðr r ÞŠ=½r ð r ÞŠ: Combne te n ndvduals, m tro famles, and k famles wt two offsprng Defne a ¼ n mð r Þ ð3 4r Þkd ; a ¼ n mð r Þ ð3 r r Þ kd ; a 3 ¼ n mð r Þ ð3 r Þkd 44 Ten equatons (5), (8) and () lead to nmk X ¼ X t X dagða ; a V ; a 3 V D Þ=s : ðþ To test f tere are addtve and domnant effects, we may test te ypotess H,ad : a =a =d =d = Ten te test matrx H s defned by H ¼ C : Let us denote te correspondng F-test statstc by F,ad, and te non-centralty parameter by l,ad Ten we ave from (3), (4), and () tat l ;ad s a ða a ÞV a a a 3 ð ÞV D ¼ s a a Q ½P bp D Q D QD D Q P a P D Q Š= ðp a P P b P D Þa 3 Q ½P b P D4 Q D Q D D Q P a P D4 Q Š=ðP a P P b P D4 Þ ¼ s a s ga ½R Q R QR R Q R Q Šð R Þ a 3 s gd ½R4 Q R Q R R Q R4 Q Š=ð R4 Þ : ssume tat te two markers and are n lnkage equlbrum, ten D = Moreover, assume tat te trat locus Q s n LD wt marker but not wt marker, ten D Q = and D Q 6¼ Ten l ;ad ½=s Š a s ga R Q a 3s gd R4 Q, wc only nvolves marker and can be wrtten as l,ad Correspondngly, we denote te F-test statstc by F,ad Smlarly, l ;a ½a =s Šs ga R Q s te non-centralty parameter of a test statstc F,a To test te oter ypoteses, we may get te non-centralty parameters n a smlar way by takng approprate test matrces H To test f tere s domnant effect, we may test te ypotess H,d : d =d = Te non-centralty parameter s l ;d a3 s s gd ½R4 Q R Q R R Q R4 Q Š=ð R4 Þ To test f tere s an addtve or substtuton effect, we may test te ypotess H ;a : a ¼ a ¼ Te non-centralty parameter s l ;a a s s ga ½R Q R QR R Q R Q Š=ð R Þ Te correspondng F-test statstc s denoted by F,a Non-centralty parameters of lnkage studes Consder a nuclear famly wt k cldren and bot parents Under te null ypotess of no lnkage between te trat locus and markers, te correlaton of eac sb-par s r N ¼ s ga s s gd 4s s Ga s s Gd 4s : Te expected log-lkelood s EðL Null Þ¼ ðkþ logðps Þ log ð r Þðk Þðr N r Þ ð r N Þ k : Under te alternatve ypotess of lnkage between te trat locus and marker, te correlaton between a sb-par s C ¼ Covðy ; y jp ¼ =Þ=s ¼ðs ga s gd ÞPðp Q ¼jp ¼ =Þ= s s ga Pðp Q ¼=jp ¼=Þ=s ½s Ga = s Gd =4Š=s ; ¼; ; From Haseman and Elston, Table IV, we ave C ¼ ðs ga s gd Þc s ga c ð c Þs Ga = s Gd =4 =s C ¼ ðs ga s gd Þc ð c Þs ga ½ c ð c ÞŠ= s Ga = s Gd =4 =s C ¼ ðs ga s gd Þð c Þ s ga c ð c Þs Ga =s Gd =4 =s : ð3þ Te expected log-lkelood under te alternatve ypotess of lnkage s EðL random; Þ¼ ðkþ½logðps ÞŠ X X Pðp ÞPðp k ;k Þ: p p k ;k r r r r r r r r C p C pk log det r r C p C pk ; C r r C pk C pk were P(p j =)=P(p j =)=/4 and P(p j =/)=/ From Stuart and Ord, te non-centralty parameter for lnkage of te nuclear famly s equal to l lnkage, =E(L random, ) E(L Null ) If k=, t can be sown tat l lnkage; ¼ log½ 4r r N 4r r NŠ P ¼ Pðp ¼ =Þlog½ 4r C 4r C Š European Journal of Human Genetcs

6 3 Lnkage and assocaton mappng of QTL R Fan and M Xong Under te alternatve ypotess of lnkage between te trat locus and markers and, te correlaton between a sb-par s gven by for, j=,, C j ¼ Covðy ; y jp ¼ =; p ¼ j=þ=s ¼ ðs ga s gd ÞPðp Q ¼ jp ¼ =; p ¼ j=þ s ga Pðp Q ¼ =jp ¼ =; p ¼ j=þ s Ga = s Gd =s =4 : ð4þ To calculate C j, we need to calculate te jont dstrbuton of p, p Q and p of a sb-par under te alternatve ypotess of lnkage ssume tat tere s no nterference for dsjont regons of te cromosome Ten we ave Pðp ¼ ; p Q ¼ Q ; p ¼ Þ ¼ Pðp ¼ ; p Q ¼ Q ÞPðp ¼ jp ¼ ; p Q ¼ Q Þ ¼ Pðp ¼ jp Q ¼ Q ÞPðp Q ¼ Q ÞPðp ¼ jp Q ¼ Q Þ: ð5þ Haldane s map functon y =[ exp( l )]/ Smlarly, we may calculate te recombnaton fractons y Q and y Q by te map dstances l Q and l Q ssume tat marker and marker are n lnkage equlbrum, e, D =, te genetc dstances l =5 cm, l Q =l Q =5 cm, and te ertablty =5 Suppose we ave a sample wt n= ndvduals, m=3 tro famles, and k= nuclear famles wt two offsprng ssume tat te ID proportons sared by te two offsprng n te k= famles at bot markers and are p = p =5, and te probablty of sarng two alleles ID at markers and are = =5 Fgure sows te power of te test statstcs F,ad, F,a, F,ad, and F,a aganst te dsequlbrum coeffcent D Q wen D Q =5 for a mode of domnant nertance wt a=d=, and a mode of recessve nertance wt a=, d= 5, respectvely Several features are nterestng n te two graps of Fgure Frst, te power of F,ad and F,a are ger tan tat of F,ad and F,a Hence, te regresson mappng wc uses two markers and as ts advantage From Haseman and Elston, Table IV, we may construct te jont dstrbuton of p Q, p and p by relaton (5), and te results are presented n Table 3 of Fan 7 ased on te results, we can calculate C j,,j=,,, wc are gven n ppendx D of Fan 7 Te expected log-lkelood under te alternatve ypotess of lnkage s E(L random, )= (k+)[log(ps )+] S p S p S pk,k S pk7,k P(p )P(p ) P(p k,k ) P(p k,k ) r r r r r r r r C p;p C pk ;p k log det r r C p;p C pk ; ;p k C r r C pk ;p k C pk ;p k were P(p j =)=P(p j =)=/4 and P(p j =/)=/ suc as tose for marker From Stuart and Ord, te non-centralty parameter for lnkage of te nuclear famly s equal to l lnkage, =E(L random, ) E(L Null ) If k=, t can be sown tat l lnkage; ¼ log½ 4r r N 4r r NŠ P ;j¼ Pðp ¼ =ÞPðp ¼ j=þlog½ 4r C j 4r C jš Power calculaton and comparson Let us denote ertablty by wc s defned by ¼ s ga =s In te power calculatons, we take te addtve polygenc varance s Ga ¼ :, polygenc domnant varance s Gd ¼ :5, te equal allele frequences P =q =P =5 at te two markers and, and te QTL Q Moreover, suppose tat m =a, m =m =d and m = a Suppose tat te map dstance l between marker and marker s known Under te assumpton of no nterference, we may calculate te recombnaton fracton by Fgure Power of test statstcs F,ad, F,a, F,ad, and F,a aganst dsequlbrum coeffcent D Q at sgnfcant level, wen q =P =P =5, D =, D Q =5, =5, n=, m=3, k=, p =p = = =5, l =5 cm, l Q =l Q =5 cm, Ga ¼ :; Gd ¼ :5 for a mode of domnant nertance a=d=, and a mode of recessve nertance a=, d= 5, respectvely European Journal of Human Genetcs

7 Lnkage and assocaton mappng of QTL R Fan and M Xong 3 over te one marker mappng wc only uses one marker or Second, te statstc F,ad as ger power tan tat of F,a, and te statstc F,ad as ger power tan tat of F,a Tus, excludng te domnant varance from te analyss wen t does exst would lose power Trd, as expected, wen D Q = te power to detect LD usng only marker s mnmal More nterestngly, wen D Q =5 te power s stll ger usng te flankng two markers tan usng only marker Fgure sows te power of te test statstcs F,ad, F,a, F,ad, and F,a aganst te ertablty wen D = and D Q =D Q =5 for a mode of domnant nertance wt a=d=, and a mode of recessve nertance wt a=, d= 5, respectvely Te oter parameters are te same as tose of Fgure mong te features observed n Fgure, te power s reasonably g wen te ertablty s bgger tan 5 To compare te power of populaton based and famly based metods, Fgure 3 sows te power of te test statstcs F,ad and F,a for a mode of domnant nertance wt a=d=, and a mode of recessve nertance wt a=, d= 5, respectvely For Fgure 3, populaton data contan n=5 ndvduals, but no famly data (m=k=) For domnant nertance of Fgure 3, te data contan m=84 tro famles (n=k=) For recessve nertance of Fgure 3, te data contan k=63 nuclear famles eac as two offsprng (n=m=) Notce tat m=84 or k=63 famly data contan 5 ndvduals, and tus te number of ndvduals s te same as tat of te populaton data We can see tat populaton based metod s more powerful tan te famly based metod for te same number of ndvduals In a populaton, te LD can exst due to mutatons at te trat locus In te absence of tgt lnkage between te trat locus and a marker, te recombnaton between te marker locus and te trat locus can rapdly dsspate te dsequlbrum from generaton to generaton Denote te frequency of aplotype Q at te generaton wen te mutatons occur by P(Q)() Ten LD coeffcent s D Q ()= P(Q)() q P for te generaton wen te mutatons occur For te followng generatons, te dsequlbrum Fgure Power of test statstcs F,ad, F,a, F,ad, and F,a aganst ertablty at sgnfcant level, wen q =P =P = 5, D =, D Q =D Q =5, n=, m=3, k=, p =p = = =5, l =5 cm, l Q =l Q =5 cm, Ga ¼ :; Gd ¼ :5 for a mode of domnant nertance a=d=, and a mode of recessve nertance a=, d= 5, respectvely Fgure 3 Power of test statstcs F,ad and F,a aganst ertablty at sgnfcant level for a mode of domnant nertance a=d=, and a mode of recessve nertance a=, d= 5, respectvely For populaton data n=5, m=k=; for domnant famly data n=k=, m=84; for recessve famly data n=m=, k=63 Oter parameters are te same as tose of Fgure European Journal of Human Genetcs

8 3 Lnkage and assocaton mappng of QTL R Fan and M Xong coeffcent s reduced by a factor 7y Q n eac generaton 3 Suppose tat te mutaton s already T generaton old Ten te LD coeffcent s D Q (T)=D Q ()( y Q ) T Smlarly, te oter LD coeffcents are D (T)=D ()( y ) T and D Q (T)=D Q ()( y Q ) T ssume tat te map dstance between marker and marker s l =5 cm, and te oter parameters are gven by D ()=,D Q ()=D Q ()=5, =5, l =5 cm, n=, m=3, k=, T=3, p = p =5, = =5 Fgure 4 sows te power of te test statstcs F,ad, F a, F ad, and F,a aganst te recombnaton fracton y Q for a mode of domnant nertance wt a=d=, and a mode of recessve nertance wt a=, d= 5, respectvely We can see tat te power curves of F,ad and F,a are very g, altoug te power curves of F ad and F,a decrease very rapdly as te recombnaton fracton y Q ncreases Hence, g resoluton LD mappngs ave advantage to do fne gene mappngs, and approprate for te dense marker maps suc as sngle nucleotde polymorpsms on uman genome To nvestgate te effect of te age of te mutaton on te power, Fgure 5 sows te power curves aganst te poston of markers In te Fgure, te QTL locates at 5 cm wc s flanked by two markers and One marker s one te rgt-and sde of te QTL, and te oter s on te left-and sde wt equal dstance to te QTL Te power decreases quckly wen te age of te mutaton ncreases For a mutaton wc s 3 generatons old, one sould expect very low power f te markers locate 5 cm away from te QTL To explore te lnkage nterval mappng, we take a sample of k=5 nuclear famles eac as two offsprng Multplyng l lnkage, and l lnkage, by k, we may calculate te non-centralty parameters for te lnkage mappng usng marker and te lnkage nterval mappng usng markers and Moreover, assume tat te genetc dstances are l =3 cm, and l Q =l Q =5 cm, e, te QTL Q s rgt n te mddle between markers and Fgure 6 gves power curves of lnkage nterval mappng by markers and, and lnkage mappng by marker aganst ertablty for a mode of domnant nertance wt Fgure 4 Power curves of te test statstcs F,ad, F,a, F,ad, and F,a aganst te recombnaton fracton y Q at sgnfcant level, wen q =P =P =5, D ()=, D Q ()=D Q ()= 5, =5, l =5 cm, T=3, n=, m=3, k=, p =p =5, = =5, Ga ¼ :; Gd ¼ :5 for a mode of domnant nertance a=d=, and a mode of recessve nertance a=, d= 5, respectvely Fgure 5 Power curves of te test statstcs F,ad aganst te poston of markers at sgnfcant level for a mode of domnant nertance a=d=, and a mode of recessve nertance a=, d= 5, respectvely Te QTL locates at 5 cm wc s flanked by two markers and Here te mutaton age T=, 3, 4, 6, and te oter parameters are te same as tose n Fgure 4 European Journal of Human Genetcs

9 Lnkage and assocaton mappng of QTL R Fan and M Xong 33 Fgure 6 Power curves of te lnkage nterval mappng by markers and, and lnkage mappng by marker aganst te ertablty, wen q =P =P =5, l =3 cm, l Q =l Q = 5 cm, k=5, Ga ¼ :; Gd ¼ :5, at 5 sgnfcant level for a mode of domnant nertance a=d=, and a mode of recessve nertance a=, d= 5, respectvely a=d=, and a mode of recessve nertance wt a=, d= 5, respectvely It s clear tat te power of nterval lnkage mappng usng bot markers and s ger tan tat of lnkage mappng usng only one marker Dscusson In ts paper, we nvestgate varance component models of bot g resoluton LD mappng and lnkage analyss for QTL Te models are based on famly pedgree and populaton data We consder lkeloods wc utlzes flankng marker nformaton Te lkeloods jontly nclude recombnaton fractons, LD coeffcents, te average allele substtuton effect and allele domnant effect as parameters Te lnkage parameters are contaned n te varance-covarance matrx Te parameters of LD and gene effects are contaned n te regresson coeffcents 8,9,, Te model smultaneously takes care of te lnkage, LD and te effects of te putatve trat locus Q, and ence clearly demonstrates tat lnkage analyss and LD mappng are complmentary, not exclusve, metods for QTL mappng Te famly data wc ave at least two offsprng contan nformaton for bot lnkage and assocaton, and populaton data and tro famly data wc ave two parents and only one offsprng contan nformaton for assocaton y combnng te famly and populaton data n te analyss, one may expect to get better results tan tat by analysng tem separately Lnkage analyss can localze genetc trat loc n broad cromosome regons of a few cm (5 cm), and s less senstve to populaton admxture tan LD mappng In practce, one may carry out lnkage analyss as a frst step to obtan pror suggestve lnkage based on a sparse marker map y performng lnkage nterval mappngs, one may get ger power tan tat of usng only one marker Wt pror lnkage n and, LD mappng can be used to get g resoluton of te genetc trat loc based on a dense marker map We ave sown tat models of g resoluton LD mappngs usng two flankng markers ave ger power tan tat of models of usng only one marker Hence, g resoluton LD mappngs ave te advantages to do fne gene mappngs, and approprate for te dense marker maps suc as SNPs on uman genome Performng bot LD mappng and lnkage analyss as potental to avod false postves due to populaton story or envronmental effects In te meantme, t takes te advantage of g resoluton of LD mappng Te power of assocaton study depends on te exstence of LD between trat locus and markers In te absence of LD, te power of LD mappngs s very low To ncrease te probablty of detectng LD, one may need to carry out sutable desgn for a genetc study 4 It s well known tat te level of LD s eavly affected by populaton stratfcaton On te one and, te famly based metods are less lkely nfluenced by populaton stratfcaton tan tose of populaton data based metods On te oter and, a famly based assocaton study s less powerful tan tat of populaton based study for te same number of ndvduals Combnng te famly and populaton data, one may expect more nformaton, and take te advantage of populaton data and famly data More nvestgaton s needed to explore te populaton stratfcaton effect on g resoluton LD mappng of QTL, and to develop robust metods to dentfy assocaton between multple markers and QTL n te presence of populaton stratfcaton To our knowledge, tere s not muc researc on statstcal analyss about g resoluton LD mappng of QTL Usng only one b-allelc marker, te statstcal analyss of LD mappng as been studed by a few colleagues 8 3 Relatvely, multpont lnkage mappng as been studed more ntensvely 6,9,5 It s our ope tat te current researc may sed more lgt on te g resoluton assocaton study, and stmulate more nterests to utlze te advantage of LD mappng n fne resoluton of genetc studes In te Secton of power calculaton and comparson, we manly explore a set of scenaros of LD mappng For several sets of parameters, we compare te power of four test statstcs European Journal of Human Genetcs

10 34 Lnkage and assocaton mappng of QTL R Fan and M Xong for LD mappng Moreover, we compare te power of LD mappng of usng populaton data and famly data We also nvestgate te effect of mutaton age on te power For lnkage mappng, we only nclude one fgure to make power comparson of lnkage nterval mappng usng two markers wt lnkage mappng usng only one marker 9 Ts reflects te need for more researc on g resoluton LD mappng of QTL, snce te researc on lnkage nterval/multpont mappng s more mature In ts paper, we treat LD as a fxed effect snce only two markers are consdered In general, nference about te LD structure n te populaton are desrable, and LD sould be modeled as a random effect wen multple markers/aplotypes are used n analyss, wc would need more nvestgaton We assume tat te data of all famly members are avalable For some late-onset dseases, te data for te parents or former famly members may no longer be avalable In prncple, one can use smlar metods as te ones proposed n ts paper to perform g resoluton LD mappng for sb-par data of late-onset dseases Ts s an area wc s of mportance and needs more researc Due to te lengt of ts paper, we do not pursue tese ssues n dept, and tey wll be explored n oter projects cknowledgements We tank two revewers, and Dr Gert-Jan van Ommen for ter elpful comments to mprove te paper R Fan was supported partally by a researc fellowsp from te lexander von Humboldt Foundaton, Germany, and an Internatonal Researc Travel ssstance Grant of te Texas &M Unversty M Xong was supported by NIH grant R-GM5655, and MH5958 Ms JS Jung kndly wrtes te SS mcro, wc s avalable from R Fan upon request References otsten D, Wte RL, Skolnck MH, Davs RW: Constructon of a genetc lnkage map n man usng restrcton fragment lengt polymorpsms m J Hum Genet 98; 3: Morton NE: Sequental tests for te detecton of lnkage m J Hum Genet 955; 7: Morton NE: Sgnfcance levels n complex nertance m J Hum Genet 998; 6: Te Internatonal SNP Map Workng Group: map of uman genome sequence varaton contanng 4 mllon sngle nucleotde polymorpsms Nature ; 49: Rsc N, Merkangas K: Te future of genetc studes of complex uman dseases Scence 996; 73: becass GR, Cerny SS, Cookson WOC, Cardon LR: Merln rapd analyss of dense genetc maps usng sparse gene flow tress Nature Genetcs ; 3: 97 7 lmasy L, Wllams JT, Dyer TD, langero J: Quanttatve trat locus detecton usng combned lnkage/dsequlbrum analyss Genetc Epdemology 999; 7(Suppl ): S3 S36 8 Fulker DW, Cerny SS, Sam PC, Hewtt JK: Combned lnkage and assocaton sb-par analyss for quanttatve trats m J Hum Genet 999; 64: Sam PC, Cerny SS, Purcell S, Hewtt JK: Power of lnkage versus assocaton analyss of quanttatve trats, by use of varancecomponents models, for sbsp data m J Hum Genet ; 66: becass GR, Cardon LR, Cookson WOC: general test of assocaton for quanttatve trats n nuclear famles m J Hum Genet ; 66: 79 9 becass GR, Cookson WOC, Cardon LR: Pedgree tests of lnkage dsequlbrum Eur J Hum Genet ; 8: becass GR, Cookson WOC, Cardon LR: Te power to detect lnkage dsequlbrum wt quanttatve trats n selected samples m J Hum Genet ; 68: Cardon LR: sb-par regresson model of lnkage dsequlbrum for quanttatve trats Hum Hered ; 5: Fan R, Xong M: Hg resoluton mappng of quanttatve trat loc by lnkage dsequlbrum analyss Eur J Hum Gen ; : Falconer DS, Mackay TFC: Introducton to Quanttatve Genetcs London: Longman, 996, 4t edn 6 Fulker DW, Cardon LR: sb-par approac to nterval mappng of quanttatve trat loc m J Hum Genet 994; 54: Fan R: Interval mappng of quanttatve trat loc ttp:// stattamuedu/*rfan/papertml/nterval_mappngpdf 8 Lange K: Matematcal and Statstcal Metods for Genetc nalyss New York: Sprnger-Verlag, lmasy L, langero J: Multpont quanttatve trat lnkage analyss n general pedgrees m J Hum Genet 998; 6: 98 Stuart, Ord JK: Kendall s dvanced Teory of Statstcs, Vol : Classcal Inference and Relatonsps Oxford, 99, 5t edn Graybll F: Teory and pplcaton of te Lnear Model Calforna: Pacfc Grove, 976 Haseman JK, Elston RC: Te nvestgaton of lnkage between a quanttatve trat and a marker locus eavor Genetcs 97; : Hartl DL, Clark G: Prncples of Populaton Genetcs nd edn Snauer, oenke M, Langefeld CD: Genetc assocaton mappng based on dscordant sb pars: te dscordant-alleles test m J Hum Genet 998; 6: Pratt SC, Daly M, Kruglyak: Exact multpont quanttatve-trat lnkage analyss n pedgrees by varance components m J Hum Genet ; 66: European Journal of Human Genetcs

11 Lnkage and assocaton mappng of QTL R Fan and M Xong 35 ppendx In ts ppendx, we sow equaton (6) ctually, we ave E½x f x Š¼P a E½x ; f ¼ ŠðP a P ÞE½x ; f ¼ aš P E½x ; f ¼ aaš ¼ P a P a P ðp a P ÞP a P P P P a ðp a P ÞP P P a P a ðp a P Þ P a P ðp a P ÞðP a P Þ P P a ðp P a =Þ ¼ P a P a P ðp a P ÞðP a P ÞP a P P ð P Pa Þ¼P ap E½x f x Š¼P a E½x ; f ¼ ŠðP a P ÞE½x ; f ¼ aš P E½x ; f ¼ aaš ¼ P a P b PðÞP P ðp b P Þ PðÞP P b PðbÞP P P PðbÞP P b ðp a P Þ P b PðÞP a P PðaÞP P ðp b P Þ PðÞP a P b PðbÞP a P PðaÞP P b PðabÞP P P PðbÞP a P b PðabÞP P b P P b PðaÞP a P ðp b P Þ PðaÞP a P b PðabÞP a P P PðabÞP a P b ¼ P a ½PðÞP P b PðbÞP P Š ðp a P Þ½PðÞP a P b PðaÞP P b PðbÞP a P PðabÞP P Š P ½PðaÞP a P b PðabÞP a P Š ¼ PðÞP a P b PðaÞP P b PðbÞP a P PðabÞP P ¼ D E½x f z Š¼P a E½z ; f ¼ ŠðP a P ÞE½z ; f ¼ aš P E½z ; f ¼ aaš ¼ P a Pa P ðp a P ÞP a P P P Pa P ap P P P a Pa ðp a P Þ Pa P P a P ðp a P Þ P P a P P a = ¼ E½x f z Š¼P a E½z ; f ¼ ŠðP a P ÞE½z ; f ¼ aš P E½z ; f ¼ aaš ¼ P a P b PðÞP P P b P PðÞP P b PðbÞP P P PðbÞP P b ðp a P Þ Pb PðÞP a P PðaÞP P P b P PðÞP a P b PðbÞP a P PðaÞP P b PðabÞP P P PðbÞP a P b PðabÞP P b P Pb PðaÞP a P P b P PðaÞP a P b PðabÞP a P P PðabÞP a P b ¼ : Smlarly, we may sow te oter terms n equaton (6) ppendx y equatons (6), (7), and large number teory, we can sow te approxmaton (8) For nstance, te approxmaton for element on te second row and te second column s m Xnm ¼m ðx f x m x Þ ðx f x m x Þ t Xnm ¼ ð ð r r Þs m Þx f r x m r x x f ¼n r x f ð r Þx m r x x m r x f r x m x x ð r ð r Þs ÞP ap 4r P a P P a P ¼ P ap ð3 r r Þ ð r Þs : For te element on te fort row and te fort column, we ave m Xnm ¼m ðz f z m z Þ z f z m z ¼ ð r Þs m Xnm ¼n ð r Þz f r z m r z z f r z f ð r Þz m r z z m r z f r z m z z P a P ð3 r Þ ð r : Þs t European Journal of Human Genetcs

12 36 Lnkage and assocaton mappng of QTL R Fan and M Xong ppendx C To prove equaton (), we frst ave Eðx x Þ¼P 4 E½x x j f ¼ ; m ¼ Š P ðp P a ÞE½x x j f ¼ ; m ¼ aš P P a E½x x j f ¼ ; m ¼ aaš ðp P a Þ E½x x j f ¼ a; m ¼ aš Pa ðp P a ÞE½x x j f ¼ a; m ¼ aaš Pa 4 E½x x j f ¼ aa; m ¼ aaš ¼ P 4 ð4p a Þ4P3 P a P a = ðp a P Þ= P P a ½P a P Š ðp P a Þ P a =4 ðp a P Þ= P =4 4Pa 3 P ðp a P Þ= P = P 4 a ð4p Þ ¼ P P a Eðx z Þ¼P 4 E½x z j f ¼ ; m ¼ Š P ðp P a ÞE½x z j f ¼ ; m ¼ aš P P a E½x z j f ¼ ; m ¼ aaš ðp P a Þ E½x z j f ¼ a; m ¼ aš Pa ðp P a ÞE½x z j f ¼ a; m ¼ aaš Pa 4 E½x z j f ¼ aa; m ¼ aaš ¼ P 4 ð P3 a Þ4P3 P a½p a = ðp a P Þ=Š ½ Pa = P ap =ŠP P a ½P a P ŠP a P ðp P a Þ P a =4 ðp a P Þ= P =4 P a =4 P ap = P =4 4P 3 a P P 4 a ðp3 Þ¼ ðp a P Þ= P = P a P = P = Eðz z Þ¼P 4 E½z z j f ¼ ; m ¼ Š P ðp P a ÞE½z z j f ¼ ; m ¼ aš P P a E½z z j f ¼ ; m ¼ aaš ðp P a Þ E½z z j f ¼ a; m ¼ aš Pa ðp P a ÞE½z z j f ¼ a; m ¼ aaš Pa 4 E½z z j f ¼ aa; m ¼ aaš ¼ P 4 ð P a Þ 4P 3 P a Pa = P ap = P P a ½P ap Š ðp P a Þ Pa =4 P ap = P =4 4Pa 3 P P a P = P = P 4 a ð P Þ ¼ P a P =4: Smlarly, we may sow tat Eðx x Þ¼P b P ; E ðx z Þ¼ ; Eðz z Þ¼Pb P =4: To sow te oter terms of (), we frst calculate te jont probabltes P(, ), n wc te frst offsprng s genotype s at marker and te second offsprng s genotype s at marker, {,a, aa}, {, b, bb}we need to consder nne possble pases {, a, aa}6{, b, bb} for eac parent t te frst glance, one needs to consder 969 possble matngs to calculate P(, ) However, many matngs do not lead to specfc genotypes (, ) of a sb par Ts elmnates many terms and reduces te amount of calculatons For nstance, a matng of ( f =, f =)6( m =, m =) only results offsprng wt genotype (,) Ten, we ave Pð ¼ ; ¼ Þ ¼ X f ; f Pð f ; f Þ X m; m Pð m ; m ÞP½ ¼ ; ¼ jð f ; f Þ; ð m ; m ÞŠ ¼ PðÞ PðÞ PðÞPðbÞ= PðÞPðaÞ= ½PðÞPðabÞ PðbÞPðaÞŠ=4 PðÞPðbÞ PðÞPðbÞ=4 PðÞPðaÞ=4 ½PðÞPðabÞ PðbÞPðaÞŠ = =4 PðÞPðaÞ PðÞPðaÞ=4 ½PðÞPðabÞ PðbÞPðaÞŠ = =4 ½PðÞPðabÞPðbÞPðaÞŠ =4 =4 ¼ðPðÞ P P Þ =4 P P PðÞ ¼ =4 P P PðÞ: ð6þ Symmetrcally, we may get te followng tree terms Pð ¼ ; ¼ bbþ ¼ =4 P P b PðbÞ; Pð ¼ aa; ¼ Þ ¼ =4 P ap PðaÞ; Pð ¼ aa; ¼ bbþ ¼ =4 P ap b PðabÞ: ð7þ Note tat P( =, =b)=p( =)7P( =, = or bb) Hence, Pð ¼ ; ¼ bþ ¼ = PðÞP P b PðbÞP P : ð8þ Smlarly, we may calculate te followng tree terms Pð ¼ aa; ¼ bþ ¼ = PðaÞP ap b PðabÞP a P Pð ¼ a; ¼ Þ ¼ = PðÞP ap PðaÞP P Pð ¼ a; ¼ bbþ ¼ = PðbÞP ap b PðabÞP P b : ð9þ Fnally, we can calculate te followng term usng equaton S S P(, )= Pð ¼ a; ¼ bþ ¼ PðÞP ap b PðbÞP a P PðaÞP P b PðabÞP P : ðþ European Journal of Human Genetcs

13 Lnkage and assocaton mappng of QTL R Fan and M Xong 37 Usng equatons (6), (7), (8), (9) and (), we may calculate E½x x Š¼P a P b P½ ¼ ; ¼ Š ðp b P ÞP½ ¼ ; ¼ bš P P½ ¼ ; ¼ bbš ðp a P Þ P b P½ ¼ a; ¼ Š ðp b P ÞP½ ¼ a; ¼ bš P P½ ¼ a; ¼ bbš P P b P½ ¼ aa; ¼ Š ðp b P ÞP½ ¼ aa; ¼ bš P P½ ¼ aa; ¼ bbš ¼ D : Smlarly, we may get E½x z Š¼; E½z z Š¼D =4 y symmetrc property, we may calculate te remanng terms n () ppendx D Let P be te matrx gven by (9) To sow te approxmaton of (), we notce tat d can be calculated by d ¼ s ð Þ P ð Þ t Te element on te second row and te second column of approxmaton () can be calculated by k nmk X ¼nm ¼ s k ðx f x m x x Þ ðx f x m x x Þ t nmk X ¼nm ð Cr Þx f Cr x m Cx Cx x f Cr x f ð Cr Þx m Cx Cx x m Cx f Cx m Cð r Þ r ð r Þ x Cðr r Þ r ð r Þ x x Cx f Cx m Cðr r Þ r ð r Þ x Cð r Þ r ð r Þ x x ð Cr s ÞP a P CP a P CP a P ðcr ÞP a P CP a P CP a P CP a P CP a P Cð r Þ r ð r Þ P ap Cðr r Þ r ð r Þ P ap CP a P CP a P Cðr r Þ r ð r Þ P ap Cð r Þ r ð r Þ P ap ¼ P a P d =s : Smlarly, te element on te fort row and te fort column of approxmaton () s k nmk X ¼nm ¼ s k ðz f z m z z Þ ðz f z m z z Þ t nmk X ¼nm ð Cr Þz f Cr z m Cz Cz z f Cr z f ð Cr Þz m Cz Cz z m Cz f Cz m Cð r Þ r ð r Þ z Cðr r Þ r ð r Þ z z Cz f Cz m Cðr r Þ r ð r Þ z Cð r Þ r ð r Þ z z s ð Cr ÞP a P Cð r Þ r ð r Þ P a P Cðr r Þ r ð r Þ P a P =4 ¼ P a P d 44=s : Te oter terms of approxmaton () can be calculated n a smlar manner European Journal of Human Genetcs