REPORT Efficient Association Mapping of Quantitative Trait Loci with Selective Genotyping

Transcription

1 REPOR Effct Assocato Mappg of Quattatv rat Loc wth Slctv Gotypg B. E. Huag ad D. Y. L Slctv gotypg (.., gotypg oly thos dvduals wth xtrm photyps) ca gratly mprov th powr to dtct ad map quattatv trat loc gtc assocato studs. Bcaus slcto dpds o th photyp, th rsultg data caot b proprly aalyzd by stadard statstcal mthods. W provd approprat lklhoods for assssg th ffcts of gotyps ad haplotyps o quattatv trats udr slctv-gotypg dsgs. W dmostrat that th lklhood-basd mthods ar hghly ffctv dtfyg causal varats ad ar substatally mor powrful tha xstg mthods. Mappg gs assocatd wth quattatv trats s a mportat stp toward gtc dsscto of complx huma dsass. Bcaus th dsas gs ar ulkly to hav vry larg ffcts o quattatv trats, powr s a major cocr assocato studs, spcally wth th d to adjust for multpl tstg. Dspt th cotug mprovmts gotypg ffccy, t s stll hghly xpsv to gotyp a larg umbr of dvduals, partcularly gomwd assocato studs. A cost-ffctv stratgy s to prfrtally gotyp dvduals whos trat valus dvat from th populato ma. Kow as slctv gotypg, ths approach ca rsult a substatal cras powr (rlatv to radom samplg wth th sam umbr of dvduals), bcaus much of th gtc formato rsds dvduals wth xtrm photyps. 1 7 Slatk suggstd gotypg a slctd sampl of dvduals wth uusually hgh valus of th quattatv trat, togthr wth a radom sampl from th study populato. Bcaus slcto dpds o th photyp, stadard statstcal mthods that assum radom samplg ar ot applcabl. Slatk dvlopd two tsts: o comparg th alll frqucs btw th slctd sampl ad th radom sampl ad o comparg th ma trat valus amog dvduals wth dffrt gotyps th slctd sampl. h two tsts ar approxmatly dpdt, so thr P valus ca b combd to form a ovrall tst. Slatk usd smulato to show that hs tsts ar mor powrful tha th smpl t tst (wh th lattr s appld to a radom sampl wth th sam umbr of dvduals). Ch t al. 5 rcommdd rplacmt of th radom sampl wth a slctd sampl of dvduals wth uusually low trat valus ad dscrbd two samplg schms to obta th slctd sampls. hy dmostratd through a smulato study that, wth Slatk s thr tsts, thr dsgs ar mor ffct tha Slatk s orgal dsg. I a rct Scc rport o obsty, 8 o of th rplcato studs gotypd dvduals from th 90th 97th prctl of th BMI dstrbuto ad thos from th 5th 1th prctl, ad aothr rplcato study gotypd dvduals from th top ad bottom quartls. I both studs, th dvduals wth hgh ad low BMI valus wr tratd as cass ad cotrols, rspctvly, ad cas-cotrol mthods (.., tstg for alll-frqucy dffrcs btw th two slctd groups) wr usd for aalyss. Cas-cotrol mthods dsrgard th actual trat valus ad ar thus ffct. Slatk s tsts do ot mak full us of th avalabl data thr dvduals who ar homozygous for th mor alll ar dscardd, ad th trat valus th radom sampl or th low-trat-valu sampl ar ot usd at all. Rctly, ths joural, Wallac t al. 7 proposd a Hotllg s tst for ormal trats, whch thy showd through smulato has crasd powr ovr Slatk s tsts. Wallac t al. s tst, 7 whch s sstally th stadard t tst th cas of a sgl markr, gors th basd samplg atur of th slctv-gotypg dsg ad thus may ot b optmal. Furthrmor, o of th xstg mthods dals wth haplotyp-basd tstg or stmato of gtc ffcts. I ths rport, w show how to proprly ad ffctly map QLs wth slctv gotypg. W drv approprat lklhoods that mak full us of th avalabl data ad that proprly rflct trat-dpdt samplg. h corrspodg frc procdurs ar vald ad ffct. Our mthods ca b usd to prform both gotyp-basd ad haplotyp-basd assocato aalyss. hr advatags ovr th xstg mthods ar dmostratd through xtsv smulato studs. W cosdr two vry gral slctv-gotypg dsgs. Udr dsg 1, th quattatv trat s masurd From th Dpartmt of Bostatstcs, Uvrsty of orth Carola, Chapl Hll Rcvd Octobr 17, 006; accptd for publcato Jauary 9, 007; lctrocally publshd Jauary 30, 007. Addrss for corrspodc ad rprts: Dr. Dayu L, Dpartmt of Bostatstcs, Uvrsty of orth Carola, McGavra-Grbrg Hall, CB #740, Chapl Hll, C E-mal: l@bos.uc.du Am. J. Hum. Gt. 007;80: by h Amrca Socty of Huma Gtcs. All rghts rsrvd /007/ $15.00 DOI: / h Amrca Joural of Huma Gtcs Volum 80 March

2 o a radom sampl of dvduals from th study populato, ad a subst of dvduals s slctd for gotypg; th slcto probablts dpd o th trat valus. Udr dsg, a radom sampl of dvduals whos trat valus fall to crta rgos s slctd for gotypg, ad th trat valus ar rtad for oly thos dvduals. hus, th ma dffrc btw th two dsgs s that th trat valus o thos dvduals who ar ot slctd for gotypg ar rtad udr dsg 1 but ot udr dsg. Udr dsg, t s ot cssary to spcfy or to ascrta th dvduals outsd th slcto rgos. Lt Y b th trat valu of th th dvdual ad G b th corrspodg multlocus gotyp dotg th umbr of mor allls at ach SP st. h assocato btw G ad Y s charactrzd by th codtoal dsty fucto P(YFG ;v) dxd by a st of paramtrs v. I th spcal cas of a sgl locus wth th addtv mod of hrtac, P(YFG ;v) may tak th famlar form of th lar rgrsso modl Y p a bg, (1) whr s zro-ma ormal wth varac j. I ths cas, v p (a,b,j ). Udr th domat (or rcssv) mod of hrtac, G quato (1) s rplacd by th dcator of whthr th th dvdual has at last o mor alll (or, for th rcssv modl, two mor allls). If thr ar multpl loc, th bg quato (1) s rplacd by a approprat lar combato of dvdual gotyp scors ad (possbly) thr cross-products. W dot th probablty fucto of th gotyp by P(G; g), whr g rprsts th (multlocus) gotyp frqucs. Udr dsg 1, th data cosst of(y,g )( p 1,,) ad Y ( p 1,,). (Wthout loss of gralty, th data ar arragd so that th frst rcords prta to th dvduals who ar slctd for gotypg ad th rmag ( ) rcords to th uslctd dvduals.) h corrspodg lklhood for v ad g ca b wrtt as p1 p1 G P(Y FG ; v)p(g ;g) P(Y FG; v)p(g; g), () whr th summato ovr G s tak ovr all possbl gotyps; a drvato s gv appdx A. Udr dsg, th data cosst oly of (Y,G )( p 1,,), whch ar a radom sampl from all th dvduals whos trat valus blog to a partcular st C. W ca us th lklhood for v ad g, P(YFG ;v)p(g ;g) P(Y,GFY C) p, (3) p1 p1 P(Y CFG; v)p(g; g) or th lklhood for v, p1 G P(YFG ;v) P(YFG,Y C) p. (4) p1 P(Y CFG ;v) If oly th dvduals whos trat valus ar lss tha th lowr thrshold cl or largr tha th uppr thrshold cu ar slctd for gotypg, th, udr quato (1), cu a bg cl a bg P(Y CFG ;v) p 1 F( F, j j ) ( ) whr F s th cumulatv dstrbuto fucto of th stadard ormal dstrbuto. W rfr to xprsso () as th full lklhood ad to quatos (3) ad (4) as th codtoal lklhoods. hs lklhoods proprly rflct th slctv-gotypg dsgs ad us all th avalabl data. ot that xprsso () s th sam as th lklhood for a prospctv study of sz whch gotyp data ar mssg o dvduals. Udr dsg 1, o may dsrgard th trat valus of thos dvduals who ar ot slctd for gotypg ad us th codtoal lklhoods, provdd that th gotypd dvduals ar a radom sampl from st C. h maxmum-lklhood stmators ca b obtad by th stadard wto-raphso algorthm. As show appdx A, th maxmzatos of quatos (3) ad (4) yld th sam stmator of v. By th lklhood thory, th maxmum-lklhood stmators ar approxmatly ubasd, ormally dstrbutd, ad statstcally ffct. Assocato tstg ca b prformd by usg th famlar lklhood-rato, scor, or Wald statstcs. h abov dscrpto prtas to th aalyss of gotyp-photyp assocato. It s also dsrabl to assss haplotyp-photyp assocato Lt H dot th dplotyp of th th dvdual. h ffcts of haplotyps o th trat ar charactrzd by th codtoal dsty fucto P(YFH ;v) dxd by a st of paramtrs v. Ifwar trstd assssg th ffct of a partcular haplotyp h, th P(Y FH ;v) may tak th form Y p a bz(h ), (5) whr Z(H ) s th umbr of occurrcs of h Hudr th addtv mod of hrtac, th dcator of whthr H cotas at last o h udr th domat mod of hrtac, ad th dcator of whthr H cotas two cops of h udr th rcssv mod of hrtac. O 568 h Amrca Joural of Huma Gtcs Volum 80 March 007

3 abl 1. Bas, SE, Avrag SEE, Covrag Probablty of 95% CI (CP), ad Powr at th.05 omal Sgfcac Lvl at a Caddat Locus Udr Addtv (A) ad Domat (D) Modls wth MAFs of.05 ad Rcssv (R) Modl wth MAF of. Full Lklhood Codtoal Lklhood Prospctv Lklhood Modl, b, ad c L c U Bas SE SEE CP Powr Bas SE SEE CP Powr Bas SE SEE CP Powr A: 0: : : D: 0: : : R: 0: : : OE. Each try s basd o 10,000 smulatd data sts. CC p cas-cotrol aalyss. CC Powr may also df P(YFH ;v) such a way that multpl haplotyps ar compard wth a rfrc a sgl modl. 9 Bcaus haplotyps ar ot drctly obsrvd, t s cssary to mpos som rstrctos, such as Hardy-Wbrg qulbrum (HWE), o th dplotyp dstrbuto. For k p 1,,K, lt h k dot th kth possbl haplotyp th populato ad lt p k dot th populato fr- qucy of. Udr HWE, h k P[H p (h,h )] p pp( p 1,,K). k l k l W dot th dplotyp probablty fucto by P(H ;g), whr g p (p 1,,p K). Ifrc o haplotyp ffcts must proprly accout for phas ambguty. ot that whr S(G ) P(Y,G ) p P(Y FH; v)p(h; g), HS(G ) s th st of dplotyps compatbl wth g- h Amrca Joural of Huma Gtcs Volum 80 March

4 otyp G. 9 hus, th full lklhood ad codtoal lklhood aalogous to xprssos () ad (3) ar p1 HS(G ) p1 H P(Y FH; v)p(h; g) P(Y FH; v)p(h; g) (6) ad P(YFH; v)p(h; g) HS(G ) p1 P(Y CFH; v)p(h; g) H, (7) whr th scod summato xprsso (6) ad th summato th domator of xprsso (7) ar tak ovr all possbl dplotyps. h maxmzatos of xprssos (6) ad (7) ca b prformd by th xpctatomaxmzato (EM) algorthm or th wto-raphso algorthm; s appdx A. h maxmum-lklhood stmators ar approxmatly ubasd, ormally dstrbutd, ad statstcally ffct. ot that b prtas to gtc ffct quato (1) ad to haplotyp ffct quato (5). If w ar cocrd wth o SP at a tm, howvr, th modls quatos (1) ad (5) ar th sam. I that cas, lklhoods of xprssos (6) ad (7) dffr from xprssos () ad (3) that th formr mpos HWE ad allow mssg gotyp valus, whras th lattr do ot mpos HWE ad xclud subjcts wth mssg gotyp valus. hus, th formr yld mor ffct aalyss, provdd that HWE s a rasoabl assumpto. W coductd xtsv smulato studs to assss th prformac of th proposd mthods. W cosdrd both dsgs 1 ad. Spcfcally, w gratd a radom sampl of p 5,000 dvduals from th jot dstrbuto of th trat valu ad gotyp, ad w dtfd th subst of all th dvduals whos trat valus ar!c L or 1c U. W th slctd a radom sampl of p 500 d- vduals from that subst. By sttg th gotyps of th uslctd dvduals to mssg, w obtad th data Fgur 1. Emprcal powr for dtctg causal haplotyp 11 at th omal sgfcac lvl of.05 udr -SP modls wth MAFs of.3 ad.4 ad wth (c L,c U) p (,1) as a fucto of th LD. h sold ad dottd rd curvs corrspod to th codtoal ad prospctv lklhoods, rspctvly, udr th domat modl wth b p., whras th sold ad dottd blu curvs corrspod to th codtoal ad prospctv lklhoods, rspctvly, udr th rcssv modl wth b p.3. udr dsg 1; by dltg th uslctd dvduals altogthr, w obtad th data udr dsg. W valuatd both th full-lklhood ad codtoal-lklhood mthods. hs valuatos provdd formato about th rlatv ffccy of usg full lklhood vrsus codtoal lklhood udr dsg 1 or, quvaltly, th rlatv ffccy of dsg 1 vrsus dsg. For comparso, w also valuatd th stadard mth- abl. yp I Error ad Powr at a Markr Locus Lkd to a QL c L c U Addtv Modl Domat Modl Rcssv Modl b Full Cod Pros CC b Full Cod Pros CC b Full Cod Pros CC OE. h MAFs of th QL ad markr locus ar.05 ad.06 udr th addtv ad domat modls ad ar. ad.5 udr th rcssv modl. h stadardzd LD coffct (D ) btw th two loc s.9. Each try s basd o 10,000 smulatd data sts. Full p full lklhood; Cod p codtoal lklhood; Pros p prospctv lklhood; CC p cas-cotrol aalyss. 570 h Amrca Joural of Huma Gtcs Volum 80 March 007

5 Fgur. Emprcal typ I rror for tstg ull haplotyp 10 at th omal sgfcac lvl of.05 udr -SP addtv modls wth MAFs of.3 ad.4 ad D of.75 as a fucto of th ffct sz of causal haplotyp 11. h sold ad dottd rd curvs corrspod to th codtoal ad prospctv lklhoods, rspctvly, udr (c L,c U) p (,1), whras th sold ad dottd blu curvs corrspod to th codtoal ad prospctv lklhoods, rspctvly, udr (c L,c U) p (1,1). A sold black rfrc l s draw at th omal sgfcac lvl of.05. ods, whch ar basd o th prospctv lklhoods. For gotyp-basd aalyss, th prospctv lklhood 7 s smply p1 P(YFG ;v) ; for haplotyp-basd aalyss, th pro spctv lklhood s th frst trm xprsso (6). 10 I addto, w valuatd th cas-cotrol tsts, whch rgard th uppr ad lowr trat valus as cass ad cotrols, rspctvly. I our frst study, w gratd th trat valus from quato (1) wth a p 0, j p 1, ad b p 0, 0.1, 0., 0.3, 0.4, ad 0.5. W st (c L,c U ) to (0.5,0.5), (1.0,1.0), (1.5,0.5) or (.0,1.0). Udr th codto that b p 0, th thrsholds of.0, 1.5, 1.0, 0.5, 0.5, ad 1.0 corrspod approxmatly to th d, 7th, 16th, 31st, 69th, ad 84th prctls of th trat dstrbuto, rspctvly. W cosdrd thr mods of hrtac addtv, domat, ad rcssv ad varous valus of th moralll frqucy (MAF). h gotyps wr gratd udr HWE, ad th aalyss wr prformd both wth ad wthout ths assumpto. h rsults wthout th HWE assumpto ar summarzd tabl 1. h rsults wth HWE ar smlar ad thus omttd. Both th full ad codtoal lklhoods provd (vrtually) ubasd stmators of gtc ffcts ad corrct typ I rror. h SE stmators (SEEs) accuratly rflct th tru varatos, ad th CIs hav propr covrags. h codtoal lklhood has arly th sam powr as th full lklhood. As xpctd, th powr s substatally hghr udr th addtv ad domat modls tha udr th rcssv modl (gv th sam MAF ad th sam ffct sz). h powr crass as slcto bcoms mor xtrm. Also, th powr tds to b hghr wh c L ad c U ar of th sam dstac from th populato ma (as opposd to uqual dstacs), whch mpls that th optmal sampl-sz rato btw th uppr ad lowr ds should b 1:1 (as th cas of th cas-cotrol dsg). I practc, th populato ma may b ukow, or t may b asr to rcrut subjcts wth hgh trat valus tha thos wth low trat valus, or vc vrsa. hus, t may ot b fasbl to st cl ad cu th sam dstac from th pop- ulato ma. I th prsc of a causal varat, both th stmator of th gtc ffct ad th SEE basd o th prospctv lklhood ar basd upward, ad th covrags of th CIs may b substatally blow or abov th dsrd lvls. h prospctv lklhood appars to prsrv th typ I rror. h powr of th prospctv lklhood tds to b lowr tha that of th full ad codtoal lklhoods, spcally wh (c L,c U) p (,1) ad udr th rcssv mod of hrtac. Wh (c L,c U) p (,1), th full ad codtoal lklhoods hav powr of 75% to dtct ffct sz of 0.3 udr th addtv ad domat modls wth MAF p 0.05, ad thy hav powr of 80% to dtct ffct sz of 0.5 udr th rcssv modl wth MAF p 0.. By cotrast, th prospctv lklhood has!70% powr thos two cass. ot surprsgly, th cascotrol tsts, whch dsrgard th actual trat valus, ar substatally lss powrful tha th proposd mthods. I th scod study, w gratd data th sam way as th frst study, but w prformd th aalyss at a markr locus that s lkag dsqulbrum (LD) wth th pottal causal SP. h rsults ar show tabl. h basc coclusos ar th sam as th frst study. As xpctd, th powr s dcrasd wh tstg s prformd at a markr locus rathr tha at th caddat locus. h thrd study was cocrd wth haplotyp ffcts. W cosdrd two SPs wth varyg dgrs of LD. h 11 haplotyp that s, th haplotyp cosstg of th mor alll at ach st had a pottal ffct o th trat valu. W gratd th trat valus from quato (5) wth a p 0, j p 1, ad b p 0, 0.1, 0., 0.3, 0.4, ad 0.5. W cosdrd thr mods of hrtac: addtv, domat, ad rcssv. HWE was assumd both th data grato ad th aalyss. W prformd two typs of aalyss: th frst aalyss compard th 11 haplotyp wth th othr thr haplotyps, ad th scod aalyss compard haplotyps 11, 10, ad 01 wth haplotyp 00. Som of th tstg rsults ar dsplayd fgurs 1 ad. h full ad codtoal lklhoods provd (vrtually) ubasd stmators of haplotyp ffcts. h SEEs ar vry accurat, ad th CIs hav corrct covrags. h two mthods hav propr cotrol of th typ I rror ad vry h Amrca Joural of Huma Gtcs Volum 80 March

6 smlar powr. ot surprsgly, th powr crass as LD bcoms hghr ad as slcto bcoms mor xtrm. h prospctv lklhood ylds basd stmato of haplotyp ffcts ad approprat CIs. As show fgur 1, th prospctv lklhood s lss powrful tha th full ad codtoal lklhoods, spcally udr a rcssv mod of hrtac. Furthrmor, th prospctv lklhood ylds flatd typ I rror for tstg ull haplotyps. h flato of th typ I rror bcoms mor svr as th ffct of th causal haplotyp crass, as llustratd fgur. Aga, th cas-cotrol mthods 9 ar much lss powrful tha th proposd mthods (data ot show). h two dsgs cosdrd ths rport ar qut gral ad flxbl. Sc th smulato studs dcatd that codtoal lklhoods ar arly as ffct as full lklhoods, o may smply adopt dsg ad rta th trat valus for th gotypd dvduals oly. h chocs of th slcto thrsholds do ot rqur prcs kowldg of th trat dstrbuto, although th ffccy of th dsg wll dpd o whch prctls th thrsholds corrspod to. h lklhoods prstd hr ca b asly modfd to clud a radom sampl, as th orgal Slatk dsg, or to allow svral slcto rgos wth dffrt samplg probablts. Although w hav focusd o ormally dstrbutd trats, our mthods ca b appld to ay trat dstrbutos. W focusd o th aalyss of a sgl markr or a small st of markrs. Assocato studs typcally volv may markrs, so a larg umbr of tsts s prformd. Adjustmts for multpl tstg ca b mad by prmutato or Mot Carlo mthods. 11 W ca corporat vromtal covarats to th modls ad lklhoods of ths rport. I th prsc of covarats, th lklhoods gv formulas (), (3), (6), ad (7) wll volv th covarat dstrbuto. h corrspodg umrcal algorthms ar mor complcatd ad wll b prstd lswhr. Ackowldgmts hs rsarch was supportd by th atoal Isttuts of Halth. W ar gratful to Dr. Dogl Zg for hlpful dscussos. Appdx A Drvato of Exprsso () h data for dsg 1 ca b wrtt as (Y,R,RG)( p 1,,), whr R dcats, by th valus 1 vrsus 0, whthr th th dvdual s slctd for gotypg. h lklhood fucto p1 P(Y,R,RG) ca b xprssd as or R, whch s proportoal to R or R 1R p1 P(Y,R )P(RGFY,R ) p1 P(Y )P(RFY )P(GFY ) p1 P(Y )P(GFY ) p1 P(Y,G ) P(Y ), bcaus th slcto probablts P(R FY ) ar costats. hs justfs xprsso (). Equvalc of Equatos (3) ad (4) Estmatg v It suffcs to show that th profl lklhood for v that s, th maxmum of xprsso (3) ovr g for fxd v s quvalt to quato (4). By dfg ggp P(G p g; g), gp p1 I(Gp g), ad P g(v) p P(Y CFG p g; v), w ca wrt th logarthm of xprsso (3) as p1 logp(yfg ;v) ggloggg loggggp g(v). It th follows from smpl algbrac mapulatos that th profl log-lklhood for v s p1 logp(yfg ;v) gglogp g(v) gglog ( g/), whch s xactly th logarthm of quato (4), up to th costat glog ( g/). g EM Algorthm for Maxmzg Exprsso (6) W prst a EM algorthm for th maxmzato of xprsso (6) by tratg th data log-lklhood s p1 I[H p (h,h )]{logp[y F(h,h ); v] logp[(h,h ); g]}, k l k l k l H as mssg data. h complt- whr I(7) s th dcator fucto. Df pkl p P[H p (h k,h l)fy,g ], whr G s ukow for p 1,,. h I[(h k,h l) S(G )]P[YF(h k,h l); v]p[(h k,h l); g] pkl p, I[(h k,h l) S(G )]P[YF(h k,h l); v]p[(h k,h l); g)] 57 h Amrca Joural of Huma Gtcs Volum 80 March 007

7 whr S(G ) s th st of all possbl dplotyps wh G s ukow. I th E stp of th EM algorthm, w valuat th at th currt stmats of v ad g. I th M stp, w solv th quatos p kl k p1 logp[yf(h k,h l); v] I[(h,h l) S(G )]pkl p 0 v ad k p1 logp[(h k,h l); g] I [(h,h l) S(G ) ] pkl p 0 g for v ad g, rspctvly. h lar rgrsso modl spcfs that, codtoal o Hp (h k,h l), th quattatv trat Y s ormally dstrbutd wth ma b Z(h k,h l) ad varac j, whr Z(h k,h l) s a spcfc fucto of hk ad hl ad whr b s th corrspodg st of rgrsso paramtrs. ot that Z(h k,h l) cluds th ut compot ad that b corrspods to a ad b of quato (5). If w ar trstd comparg a partcular haplotyp h wth all othrs, th Z(h k,h l) p [1,I(hkp h ) I(hlp h )] udr th addtv modl, Z(h k,h l) p [1,I(hkp h ) I(hlp h ) I(hkp hlp h )] udr th domat modl, ad Z(h,h ) p [1,I(h p h p h )] udr th rcssv modl. I ths cas, k l k l [Y b Z(h k,h l)] I [(h k,h l) S(G ) ] xp{ } pp k l j pkl p, [Y b Z(h k,h l)] I [(h k,h l) S(G ) ] xp{ pp } k l j ad th M stp has xplct solutos 1 [ kl k l k l ] [ kl k l ] p1 p1 b p p Z(h,h )Z(h,h ) Y p Z(h,h ), 1 kl[ k l ] p1 j p p Y b Z(h,h ), ad p p p. k K 1 kl p1 lp1 wto-raphso Algorthm for Maxmzg Exprsso (7) Udr th lar rgrsso modl wth thrsholds c ad c, xprsso (7) bcoms L U { } [Y b Z(h,h )] 1/ (pj ) xp k l pp k l (h,h )S(G ) j k l p1 { } [ ] [ ] k l c b Z(h,h ) c b Z(h,h ) U k l L k l 1 F j F j p p. h Amrca Joural of Huma Gtcs Volum 80 March

8 K o corporat th costrats that kp1 pkp 1 ad pk 1 0(k p 1,,K) to th calculatos, w df pk p p k/pk ad h p logp. For otatoal covc, dot j as v. Lt h p (h,,h ) ad c p (b,v,h). h th log-lklhood s k k 1 K1 1 k l k l p1 (h k,h l)s(g ) cu b Z(h k,h l) cl b Z(h k,h l) h W(h k,h l) { } (c) p logv log xp { (v) [Y b Z(h,h )] h W(h,h )} [ ] [ ] log 1 F F, v v whr ] [ I(hk p h 1) I(hl p h 1) W(h k,h l ) p _. I(h p h ) I(h p h ) k K1 l K1 Lt [ Y b Z(h k,h l) ] kl { k l } Q (c) p xp h W(h,h ), v cl b Z(h k,h l) L R kl(c) p, v cu b Z(h k,h l) U R kl(c) p, v ad { [ ] [ ]} U L hw(h k,h l) p 1 F R kl(c) F R kl(c). Also, lt a p aa, ad lt f b th stadard ormal dsty fucto. h { [ ] [ ] } hw(h k,h l) [Y b Z(h,h )] U U L L k l Q f R kl(c) R kl(c) f R kl(c) R kl(c) kl(c) v v (c) (h k,h l)s(g ) p, v v p1 Q (c) (h k,h l)s(g ) kl Y b Z(h,h ) Z(h,h ) f R (c) f R (c) Q h kl(c) U L Z(h k,h l) W(h k,h l) v [ k l ] k l { [ kl ] [ kl ]} v (c) (h k,h l)s(g ) b p1 Q kl(c) (h k,h l)s(g ) p, { [ ] [ ]} U L hw(h k,h l) Q (c)w(h,h ) 1 F R kl(c) F R kl(c) W(h k,h l) (c) kl k l (h k,h l)s(g ) h p1 Q kl(c) (h k,h l)s(g ) p, 574 h Amrca Joural of Huma Gtcs Volum 80 March 007

9 4 [Yb Z(h k,h l)] [Yb Z(h k,h l)] [Yb Z(h k,h l)] Q (c) 4 3 Q (c) kl { 4v v } kl v k l k l v v ( Q (c) { Q (c) } ) p1 kl kl (h k,h l)s(g ) (h k,h l)s(g ) (c) (h,h )S(G ) (h,h )S(G ) p ( ) f[ R (c)][[ R (c) ] 3R (c)] f[r (c)] {[ R (c) ] 3R (c)} [ U U 3 U L L 3 L kl kl kl kl kl kl 4v hw(h k,h l) U U L L { f [ R kl(c) ] R kl(c) f [ R kl(c) ] R kl(c) } v ( ) ], hw(h k,h l) 3 [ Yb Z(h k,h l) ] [ Yb Z(h k,h l) ] [ Yb Z(h k,h l) ] 3 Q (c)z(h,h ) { } Q (c) kl k l v v kl v k l k l p vb Q (c) ( p1 kl Q kl(c) (h k,h l)s(g ) (h k,h l)s(g ) [ ] { } (c) (h,h )S(G ) (h,h )S(G ) #{ [ Yb Z(h k,h l)z(h ] k,h l) Q kl(c) v (h k,h l)s(g ) Q (c) }) kl (h k,h l)s(g ) hw(h k,h l) U U L L ( ) { f(r (c)) R (c) 1 f [ R (c)][ R (c) 1]} 3/ kl kl kl kl v Z(h k,h l) ([ ] { } { }) hw(h k,h l) { f [ R kl(c) ] R kl(c) f [ R kl(c) ] R kl (c)} ( f [ R (c)] f[ R (c)]) U U L L U L hw(h k,h l) Z(h k,h l) v kl kl v, ( } ) [ { } { } [Y b Z(h,h )] 1 [Y b Z(h,h )]Z(h,h ) k l k l k l Q (c) kl { v v } Z Q kl(c) v k l k l p bb Q (c) { p1 kl Q kl(c) (h k,h l)s(g ) (h k,h l)s(g ) U U L L hw(h Z(h,h ) k,h l) k l U L h W(h k,h l) Z(h k,h l) { f[r (c)]r (c) f[r (c)]r (c)} ] f[r (c)] f[r (c)] kl kl kl kl v kl kl v (c) (h,h )S(G ) (h,h )S(G ) [ ], { }{ } [Y b Z(h,h )] [Y b Z(h,h )] (c) (h,h )S(G ) (h,h )S(G ) (h,h )S(G ) p [ ] [ ] v [ ] [ ] ( U L hw(h k,h l) 1 F R kl(c) F R kl(c) W(h k,h l) vh Q (c) Q (c) Q (c) p1 kl kl kl (h k,h l)s(g ) (h k,h l)s(g ) (h k,h l)s(g ) h W(h k,h l) U U L L U U L L { f R kl(c) R kl(c) f R kl(c) R kl(c) } W(h k,h l) { f R kl(c) R kl(c) f R kl(c) R kl(c) } { W(h k,h l) v } h { [ ] [ ]} { }, ) h Amrca Joural of Huma Gtcs Volum 80 March

10 [Yb Z(h k,h l)]z(h k,h l) Q kl(c) v W(h k,h l) p( p1 kl (h k,h l)s(g ) (c) (h k,h l)s(g ) bh Q (c) [Yb Z(h k,h l)]z(h k,h l) Q kl(c) v Q kl(c)w(h k,h l) (h k,h l)s(g ) (h k,h l)s(g ) { Q (c) }{ Q (c) } ) kl kl (h k,h l)s(g ) (h k,h l)s(g ) f R (c) f R (c) [ h U L Z(h k,h l) W(h k,h l) { [ kl ] [ kl ]} v W(h k,h l) U L Z(h,h ) h W(h,h ) U L h k l k l W(h k,h l) { f [ R kl(c) ] f[ R kl(c) ]} v { 1 F [ R kl(c) ] F[ R kl(c) ]} W(h k,h l) ( )( ) ], ad Q kl(c)w(h k,h l) Q kl(c)w(h k,h l) p hh p1 { Q (c) [ Q (c) ] } kl kl (h k,h l)s(g ) (h k,h l)s(g ) (c) (h k,h l)s(g ) (h k,h l)s(g ) { [ ] [ ]} U L hw(h k,h l) 1 F R kl(c) F R kl(c) W(h k,h l) ( U L hw(h k,h l) { 1 F [ R kl(c) ] F[ R kl(c) ]} W(h k,h l). { } ) Rfrcs 1. Lat, Kaupp P, Igatus J, Ruotsala, Daly MJ, Käärä H, Kruglyak L, Lat H, d la Chapll A, Ladr ES, t al (1997) Gtc cotrol of srum IgE lvls ad asthma: lkag ad lkag dsqulbrum studs a solatd populato. Hum Mol Gt 6: Slatk M (1999) Dsqulbrum mappg of a quattatvtrat locus a xpadg populato. Am J Hum Gt 64: va Gstl S, Houwg-Dustrmaat JJ, Adolfsso R, va Duj CM, va Brockhov C (000) Powr of slctv gotypg gtc assocato aalyss of quattatv trats. Bhav Gt 30: Xog M, Fa R, J L (00) Lkag dsqulbrum mappg of quattatv trat loc udr trucato slcto. Hum Hrd 53: Ch Z, Zhg G, Ghosh K, L Z (005) Lkag dsqulbrum mappg of quattatv-trat loc by slctv gotypg. Am J Hum Gt 77: Corsh KM, Maly, Savag R, Swaso J, Morsao D, Butlr, Grat C, Cross G, Btly L, Holls CP (005) Assocato of th dopam trasportr (DA1) 10/10-rpat gotyp wth ADHD symptoms ad rspos hbto a gral populatos sampl. Mol Psychatry 10: Wallac C, Chapma JM, Clayto DG (006) Improvd powr offrd by a scor tst for lkag dsqulbrum mappg of quattatv-trat loc by slctv gotypg. Am J Hum Gt 78: Hrbrt A, Grry P, McQu MB, Hd IM, Pfufr A, Illg, Wchma HE, Mtgr, Hutr D, Hu FB, t al (006) A commo gtc varat s assocatd wth adult ad chldhood obsty. Scc 31: L DY, Zg D, Mllka R (005) Maxmum lklhood stmato of haplotyp ffcts ad haplotyp-vromt tractos assocato studs. Gt Epdmol 9: Schad DJ, Rowlad CM, s DE, Jacobso RM, Polad GA (00) Scor tsts for assocato btw trats ad haplotyps wh lkag phas s ambguous. Am J Hum Gt 70: L DY (005) A ffct Mot Carlo approach to assssg statstcal sgfcac gomc studs. Boformatcs 1: h Amrca Joural of Huma Gtcs Volum 80 March 007