Mapping Quantitative Trait Loci Using Multiple Families of Line Crosses

Transcription

1 Copyrigh 998 by he Geneics Sociey of America Mapping Quaniaive Trai Loci Using Muliple Families of Line Crosses Shizhong Xu Deparmen of Boany and Plan Sciences, Universiy of California, Riverside, California 95 Manuscrip received July 6, 997 Acceped for publicaion Sepember 9, 997 ABSTRACT To avoid a loss in saisical power as a resul of homozygous individuals being seleced as parens of a mapping populaion, one can use muliple families of line crosses for quaniaive rai geneic linkage analysis. Two sraegies of combining daa are invesigaed: he fixed-model and he random-model sraegies. The fixed-model approach esimaes and ess he average effec of gene subsiuion for each paren, while he random-model approach reas each effec of gene subsiuion as a random variable and direcly esimaes and ess he variance of gene subsiuion. Exensive Mone Carlo simulaions verify ha he wo sraegies perform equally well, alhough he random model is preferable in combining daa from a large number of families. Simulaions also show ha here may be an opimal sampling sraegy (number of families vs. number of individuals per family) in which QTL mapping reaches is maximum power and minimum esimaion error. Deviaion from he opimal sraegy reduces he efficiency of he mehod. L INE crossing is a common experimenal design for mapping quaniaive rai loci (QTLs) in plans and laboraory animals. Saisical mehods are well developed for QTL mapping using line-crossing daa (Lander and Bosein 989; Haley and Kno 99; Marínez and Curnow 99; Jansen 993, 994; Zeng 994). Mehods developed by hese auhors are mainly designed o handle a single cross, e.g., a single F family. Under hese mehods, he effecs of gene subsiuion (he firs momens) are esed and esimaed. Because of his, he mehods are classified by Xu and Achley (995) as he fixed-model approach. The sampling sraegy (using a single family) and he saisical mehodology (he fixed model) consequenly resrain he inference space of he parameer esimaion o he paricular cross. This is undesirable if he wo lines iniiaing he cross are no segregaing a a QTL, for hen no maer how many offspring are sampled in he F or backcross populaion, he QTL canno be deeced. If a QTL is presen, bu is no deeced because of fixaion o he same allele in boh lines, hen a ype of ype II error has occurred. This ype II error, referred o as geneic drif error by Xu (996a), has largely been ignored in he QTL mapping lieraure. A ype II error of his kind can be reduced or even prevened by using muliple families of line crosses. A he low end, Murany (996) claims ha QTL deecion in a populaion derived from wo parens is ofen less powerful han one derived from more parens. He hen demonsraes ha if QTL heerozygoe frequency in he base populaion is high enough, a maing design wih six parens should give a good sample of variance Auhor xu@geneics.ucr.edu and allow he deecion of QTL wih reasonable power. Murany (996) inroduced he idea of muliple-family QTL mapping by using an ideal siuaion in which he genoype of he QTL is known wihou error. In acualiy, he QTL genoype canno be observed, so he saisical mehod demonsraed by Murany should be modified before i is applied o genome scanning using real daa. In his paper, I propose wo sraegies for combining daa from muliple families of line crosses: he fixedmodel and he random-model approaches. I hen conduc Mone Carlo simulaions o show ha boh he fixed- and he random-model approaches work as expeced. METHODOLOGY Linear model: Consider independen F families each derived from cross of wo inbred lines (a oal of independen inbred lines are involved). The phenoypic value of a quaniaive characer can be described by he following linear model: () where y ij is he phenoypic value of a rai under consideraion for he j-h individual in he i-h family, is he overall mean, i is an unknown family-specific effec, i and i are he respecive effec of allelic subsiuion and he dominance deviaion a he QTL of ineres, and ij is he residual error disribued as N(0, ). The variables z ij and w ij are defined as follows: and y ij = µ + β i + z ij α i + w ij δ i + ij if Q Q z ij = 0 if Q Q if Q Q () Geneics 48: ( January, 998)

2 58 S. Xu if Q Q w ij = if Q Q or Q Q (3) marix of y condiional on he marker informaion are E(y I M ) X Z W and Var(y I M ) Var (e) R, where R is a diagonal marix wih he elemen corresponding o y ij being where Q Q, Q Q and Q Q represen genoypes of he wo parenal lines and he F hybrid, respecively, for he i-h family a he candidae QTL. The maximum number of alleles a each locus (QTL or markers) is wo in each F family, bu his number can be arbirary in he whole populaion where he inbred lines are sampled. Since he genoype of a QTL is no observable, z ij and w ij are missing. Le p (kl)j be he condiional probabiliy ha he individual is of genoype Q k Q l given informaion of marker genoypes. This condiional probabiliy is derived based on genoypes of he neares flanking markers (Haley and Kno 99). Le E(z ij I M ) and E(w ij I M ) be he condiional expecaions of z ij and w ij given marker informaion (I M ). The linear model can be approximaed by subsiuing z ij and w ij by heir condiional expecaions (Haley and Kno 99; Marínez and Curnow 99), y ij = µ + β i + E( z ij I M )α i + E( w ij I M )δ i + e ij (4) where E(z ij I M ) ( ) p ()j (0) p ()j ( ) p ()j p ()j p ()j and E(w ij I M ) ( ) p ()j ( ) [p ()j p ()j ]. The residual variance is Var( e ij ) = Var( z ij I M )α i + Var( w ij I M )δ i + Cov( z ij w ij I M )α i δ i + (5) where Var(z ij I M ) i is he variance of he QTL effec ha is no explained because of uncerainy in z ij, Var(w ij I M ) i is he variance of he QTL effec ha is no explained because of uncerainy in w ij, and Cov(z ij w ij I M ) i i is he covariance because of uncerainy in boh z ij and w ij. All hree addiional componens in he residual variance will vanish if he genoype of he QTL is acually observed, i.e., Var(z ij I M ) Var(w ij I M ) Cov(z ij w ij I M ) 0. Oherwise, Var(z ij I M ) p ()j [ p ()j ] p ()j [ p ()j ] p ()j p ()j, Var(w ij I M ) 4p ()j [ p ()j ] and Cov(z ij w ij I M ) p ()j [ p ()j ] p ()j [ p ()j ] p ()j p () j p ()j p ()j. Fixed model sraegy: The firs sraegy of combining several differen line crosses is o esimae and es i i for i,...,. The null hypohesis is H 0 : i i 0 i. This approach is analogous o he nesed design for muliple-family analysis (Weller e al. 990; Kno e al. 996); i.e. i reas QTL effecs as nesed wihin families. Because he firs momens are esimaed, he mehod is called he fixed-model sraegy. Le n i be he number of individuals in he i-h family and N = n i be he overall sample size and define y as an N vecor of he daa. The model can be expressed in marix noaion by y = Xβ+ Zα+ Wδ + e (6) where X is an N ( ) known design marix, [,..., ] T are non-qtl effecs, Z is an N design marix filled by E(z ij I M ) a he appropriae posiions, [... ] T is a vecor of gene subsiuion effecs, W is an N design marix filled by E(w ij I M ) a he appropriae posiions, [... ] T is a vecor of dominance deviaions and e is an N vecor of residuals. Under a fixed model, he expecaion and variance (7) where i i /, i i / and i i i i /. Under he fixed model, parameers are esimaed via an ieraively reweighed leas-squares algorihm described below. Given an iniial guess of i, i and i i, marix R is considered as known. Under he preense of known R, he soluions of,, and are obained via he following equaions: and βˆ αˆ δˆ = R ij = Var( z ij I M )λ αi + Var( w ij I M )λ δi + Cov( z ij w ij I M )λ αiδi +, X T R X X T R Z X T R W = Z T R X Z T R Z Z T R W W T R X W T R Z W T R W C β C βα C βδ C αβ C α C αδ C δβ C δα C δ X T R y Z T R y W T R y X T R y Z T R y W T R y σˆ = --- ( y Xβˆ Zαˆ Wδˆ ) T R ( y Xβˆ Zαˆ Wδˆ ). (9) N Noe ha hese soluions are maximum likelihood esimaions (MLEs) under Model 6. If N in he denominaor of Equaion 9 had been replaced by N 3, as done in regression analysis, he soluions would no longer be MLEs. Wheher N or N 3 is used o esimae does no affec he es saisic. Because R depends on he unknown parameers, i mus be updaed by he esimaes of, and, and he esimaion is hen repeaed unil convergence. The leas-squares mehod of Kno e al. (996) simply ignores he correcion for he residual variance, i.e., assuming R I. When densed markers are used or he QTL effec is small or boh, his assumpion will have lile effec on he resuls (Xu 995). Compared wih he EM algorihm under a mixure model, his algorihm is exremely fas only wo o hree cycles of ieraion are required. However, hree addiional parameers are added o he model for each addiional family, so he number of parameers o esimae grows quickly as he number of families increases. Under he null hypohesis, H 0 : 0, he maximum likelihood is L 0 ( σˆ ) N R Exp ( y Xβˆ ) T R = ( y Xβˆ ). σˆ (8) Under he alernaive hypohesis, H : a leas one of i or i is no zero i, he maximum likelihood is L = ( σˆ ) N R Exp ( y Xβˆ Zαˆ Wδˆ ) T R ( y Xβˆ Zαˆ Wδˆ ). σˆ

3 Mapping QTL Using Combined Daa 59 The es saisic is aken as (0) In QTL mapping wih muliple families, effor is direced from a single family ino muliple families. As a consequence, one is no longer ineresed in he and of any paricular family, bu raher in and from all families. However, ˆ and ˆ are firs momen esimaions and heir magniudes are only meaningful when repored relaive o he size of he residual variance. As a resul, hey mus be convered ino variances before being considered for publicaion. In radiional QTL analysis for a single F family, he variance explained by a QTL is repored wih he F family as he reference populaion. The QTL analysis wih muliple families, however, is ineresed in he variance of QTL allelic effecs among differen F families. This variance differs from he wihin family variance by one generaion. The addiive QTL variance among he sampled families is expressed by where Similarly, he dominance variance is where () () To esimae from ˆ, assume ha he esimaion is unbiased so ha E( ˆ ) and denoe Var( ˆ ) by Vˆ. Rewrie Equaion in marix noaion by T A and define he observed variance of he esimaed s among families as ˆ ˆ T A ˆ, where A -- ii and A = = ij ( ). I is known (Seber 977) ha E( ˆ T A ˆ ) Tr(A Vˆ ) T A, where Tr() represens he race marix operaion (he sum of all diagonal elemens). Therefore, an unbiased esimaor of T A is (3) The esimaion is only asympoically unbiased because E( ˆ ) is rue only asympoically. The variance covariance marix of he esimaed parameers are obained by C Vˆ ˆ, where C is a submarix in he coefficien marix given in Equaion 8. An asympoically unbiased esimae of is analogously derived: where C Vˆ ˆ. Λ = [ ln( L 0 ) ln( L )]. σ α = α ( i α ) α = -- α i. σ δ = δ ( i δ ) σˆ α σˆδ δ = -- δ i. = αˆ TA αˆ = δˆtaδˆ Tr( AV αˆ ). Tr( AV ) δˆ (4) Two special siuaions may be noed. When only a single family is analyzed (he radiional QTL mapping sraegy), A is only a scalar of value, which leads o and. Alernaively, when here are an infinie number of families, A approaches -- I, leading o σ = α -- α and σ i δ = -- δ. i Random-model sraegy: The second sraegy of QTL mapping is o direcly es and esimae he variances of he QTL effecs, and because of his i is called he random model approach. Consider each i and i as randomly sampled from a large hypoheical populaion wih a means of zero and variances of and, respecively. Under he null hypohesis ha here is no QTL segregaing, 0. The model says he same as (6), alhough now wih differen expecaion and variance marices. Under he random model, E(y I M ) X and Var(y I M ) V ZZ T WW T R where R is a diagonal marix wih he elemen corresponding o y ij equal o R ij = Var( z ij I M )λ α + Var( w ij I M )λ δ + (5) where / and /. Derivaion of (5) is based on he assumpion ha and are uncorrelaed. If he indicaor variables, Z and W, were observed, hen he variance of y would be Var(y ZW) V ZZ T WW T I. When Z and W are replaced by heir condiional expecaions given marker informaion, his variance marix becomes Var(y I M ) E[Var(y \ ZW )] E(ZZ T ) E(WW T ) I [E(ZZ T ) E(Z)E(Z T ) E(Z)E(Z T )] [E(WW T ) E(W )E(W T ) E(W )E(W T )] I [Var(Z ) E(Z )E(Z T )] [Var(W ) E(W )E(W T )] I E(Z )E(Z T ) E(W )E(W T ) [Var(Z ) Var(W ) I ] E(Z )E(Z T ) E(W )E(W T ) [Var(Z ) Var(W ) I] E(Z )E(Z T ) E(W)E(W T ) R where R Var(Z Var(W ) I. Recall ha E(Z) is in fac E(Z I M ) and has been denoed by Z for noaional convenience. Noe ha he definiions of and are differen from hose of he fixed model. I should be noiced ha he familyspecific effecs, s, have been reaed as fixed effecs, alhough hey can be considered as random effecs wih a mean of zero and a common variance. Given he expecaion and he variance of he model and under he preense of a normal disribuion of y, we have he following likelihood funcion: L( θ y) V Exp -- T = ( y Xβ) V ( y Xβ). (6) The MLE of [ T ] T is solved using any convenien numerical algorihm. The log likelihood raio is used as he es saisic. The random-model sraegy involves invering and deerminaing V, an N N block diagonal marix, which can be ime consuming for large blocks (each block is of n n dimension). A simple algorihm developed in random maing

4 50 S. Xu designs (S. Xu, unpublished daa) can be adoped here. The algorihm provides he following marix equivalencies: and V = σ [ H HZ( B )Z T Hλ α ] V ( σ ) N R = B U (7) (8) where U W T R W I, H R R WU W T R and B Z T HZ I. Marices R and U are diagonal while marix B can be expanded as B = Z T HZλ α + I = Z T ( R R WU W T R λ δ )Zλ α + I = Z T R Zλ Z T R WU W T R Zλ α δ λ α + I. Since each of Z T R Z, Z T R W and U is diagonal, B mus also be diagonal. Solving for he inverses and deerminans of diagonal marices is rivial. NUMERICAL COMPARISON Design of simulaions: In his secion, he wo saisical mehods are verified and compared numerically via Mone Carlo simulaions. The crieria of verificaion are sandard errors of he parameer esimaion and he saisical powers. Facors considered include () marker heerozygosiy; () relaive posiion of QTL; (3) mode of QTL inheriance; (4) QTL variances; (5) disribuion of he QTL allelic effec and (6) sampling sraegy (family number vs. family size). Only a single chromosome segmen of lengh 00 cm covered by evenly spaced codominan markers is simulaed. The oal number of individuals [N family number () family size (n)] is se a 500 in all simulaions. Under each condiion, he simulaion is repeaed for 00 imes. The sandard deviaion of an esimaed parameer among he 00 replicaes provides a measure of he sandard error of parameer esimaion. The saisical power is deermined by couning he number of runs (over he 00 replicaes) ha have es saisics greaer han an empirical hreshold. The empirical hreshold value under each condiion is obained by choosing he 95h percenile of he highes es saisic over 000 addiional runs under he null model (no QTL is segregaing). Marker heerozygosiy in he populaion in which he inbred lines are sampled is simulaed a hree levels: () wo alleles, () four alleles and (3) eigh alleles. All alleles are equally frequen so ha he marker heerozygosiies represened by he hree siuaions are one half, hree quarers and seven eighhs, respecively. A single QTL is locaed a one of he hree possible posiions (measured from he lef end of he chromosome): 0 cm (overlapping wih he firs marker), 5 cm (beween markers 3 and 4) and 50 cm (in he middle of he chromosome). The esimaed QTL locaion akes he poin of he chromosome segmen ha has he highes es saisic value. The mode of QTL inheriance is deermined by he raio of o : addiive mode ( : :0); mixed mode ( : :); dominance mode ( : 0:). The variance explained by he QTL is q, which is simulaed a hree levels: () q 0., corresponding o h q /( q q ) 0.0; () q 0.5, corresponding o h q 0.0; and (3) q 0.43, corresponding o h q In all simulaions, he parameric value of is se a. Three disribuions of he allelic effec of he QTL are con- sidered. The firs is uniform disribuion wih 0 equally frequen alleles. Each allele is assigned a value beween 0 and 9. The F hybrid of each family is generaed by randomly sampling wo from he 0 alleles wih replacemen. F individuals are hen generaed by selfing he F hybrid. The addiive value of an F individual is he sum of effecs of he wo alleles. The dominance effec akes he produc of he wo parenal alleles. These geneic values (addiive and dominance) are finally rescaled so ha hey have a mean of zero and he assigned variances. The second is normal disribuion wih infinie number of alleles. An F hybrid is made of wo random alleles, each being assigned a value sampled from N(0,) disribuion. The dominance effec beween any wo sampled alleles akes he produc of he wo allelic effecs. When F individuals are generaed, heir geneic values a he QTL are rescaled so ha hey have he appropriae assigned variances. The hird disribuion is 0 alleles, each having a value beween 0 and 9. The frequency of an allele, however, scales exponenially wih is assigned effec. Le p j be he frequency of he j-h allele for j 0,..., 9, hen p j c j 9 = c k k = 0 where c 0.5. Again, he geneic values of an F individual are rescaled. Noe ha he firs disribuion is a special case of he hird disribuion wih c. The las bu mos imporan facor considered in he simulaions is he sampling sraegy: family number vs. family size (N n 500). Eigh levels are considered: () n 500; () n 3 67; (3) n 6 83; (4) n 0 50; (5) n 5 33; (6) n 0 5; (7) n 50 0; and (8) n Insead of performing simulaions under all possible cases, I simulaed a siuaion in which he cenral level is chosen for each facor considered. This paricular siuaion is hen referred o as he sandard, which is described as follows: () four equally frequen alleles for each marker locus; () he QTL locaed a 5 cm; (3) mixed mode of QTL inheriance, i.e., 0.5; (4) he oal QTL variance of q , corresponding o h 0.0; (5) TABLE Empirical hreshold values for significance es a 0.05, where is he ype I error rae Mehod Fixed model q Random model Marker alleles Two Four Eigh Sampling sraegy 500 a b a Number of families number of individuals per family. b Simulaions are no conduced in hese wo cases.

5 Mapping QTL Using Combined Daa 5 TABLE Esimaes of QTL parameers and empirical powers ( 0.05) under differen levels of marker polymorphism Mehod Marker alleles cm A Fixed model Two 7.84 (0.3) 0.30 (0.099) 0.07 (0.0) 0.98 (0.06) 89 Four 4.99 (8.65) 0.30 (0.074) 0.4 (0.44) 0.93 (0.057) 94 Eigh 5.76 (9.08) 0.3 (0.078) 0.43 (0.47) 0.9 (0.064) 93 Random model Two 7.08 (.76) 0.8 (0.09) 0.04 (0.088) (0.066) 89 Four 5.6 (8.46) 0.3 (0.067) 0.44 (0.54) (0.063) 94 Eigh 5.64 (8.88) 0.0 (0.075) 0.50 (0.44) (0.068) 94 Sandard errors of he esimaes, given in parenheses, are calculaed by he sandard deviaions among 00 replicaed simulaions. cm A, esimaed QTL posiion in cm;, esimaed addiive variance of he QTL, he rue being 0.5;, esimaed dominance variance of he QTL, he rue being 0.5;, esimaed residual variance, he rue being.0. he QTL allelic effec normally disribued; and (6) 0 families, each having 50 individuals, i.e., n When he influence of differen levels of a facor on he performances of he wo saisical mehods are examined, all oher facors are se o he sandard levels. Resuls of simulaions: The empirical hreshold values a a ype I error rae of 0.05 are given in Table. The number of alleles per marker locus does no seem o have an influence on he hreshold values. As he number of families increases, he hreshold value increases under he fixed-model sraegy considerably more han i does under he random-model sraegy. This is expeced because increasing he number of families increases he number of parameers esed under he fixed model while he number of parameers esed does no change under he random-model sraegy. When each marker has wo equally frequen alleles in he populaion in which he parenal lines are sampled, he wo models have similar esimaion errors and saisical powers (Table ). The esimaion of he QTL posiion, however, is biased and wih large error in boh mehods. The saisical powers are also low, wih wo marker alleles relaive o more marker alleles. The fixed-model sraegy generally provides a biased esimae for he residual variance, as shown in his and subsequen ables. The proporion of he phenoypic variance explained by he QTL (h q ) does no have an impac on he comparison of he wo mehods (Table 3). Boh mehods produce biased es- imaes of he QTL posiion and low saisical powers when h q is low. Table 4 shows ha when he QTL is locaed a one end of he chromosome segmen, esimaion of he QTL posiion is biased oward he cener and also wih large error in boh mehods. There is lile change in he power o deec a QTL as he rue QTL posiion varies. Mixed mode of QTL inheriance (addiive and dominance) seems o have a higher saisical power han eiher of he addiive or he dominance mode of inheriance. The esimaion of he QTL posiion is biased and wih large error under he dominance mode of inheriance. Again, he wo mehods do no show any major difference (see Table 5). Disribuion of he QTL allelic effec does no affec he comparison of he wo mehods (Table 6). I does, however, have an effec on he saisical power and he esimaion errors of QTL parameers. The uniform disribuion produces resuls similar o (in fac, slighly beer han) he normal disribuion. The exponenial disribuion decreases he saisical power and increases errors of parameer esimaion. Finally, he sampling sraegy has a major impac on he performance (Table 7). Firs, here seems o be an opimal sampling sraegy (0 50) ha leads o he highes saisical power and smalles esimaion errors of QTL parameers. Second, he sampling sraegy of a single family causes a severe loss in power and huge biases and errors of QTL parameer esimaion. Third, he residual variance is underesimaed as TABLE 3 Esimaes of QTL parameers and empirical powers ( 0.05) under differen levels of heriabiliy of he QTL Mehod h q cm A Fixed model (5.3) (0.06) (0.049) 0.95 (0.057) (4.65) 0.30 (0.074) 0.4 (0.44) 0.93 (0.057) (5.) 0.0 (0.8) 0.99 (0.4) 0.95 (0.060) 98 Random model (5.9) (0.057) (0.057) 0.97 (0.06) (4.46) 0.3 (0.067) 0.44 (0.54) (0.063) (5.3) 0.3 (0.3) 0.67 (0.64) (0.067) 00 Sandard errors of he esimaes, given in parenheses, are calculaed by he sandard deviaions among 00 replicaed simulaions. h, proporion of oal phenoypic variance explained by he QTL. q

6 5 S. Xu TABLE 4 Esimaes of QTL parameers and empirical powers ( 0.05) under hree differen locaions of he QTL Mehod cm T cm A Fixed model (5.8) 0.5 (0.080) 0.37 (0.5) 0.97 (0.063) (4.65) 0.30 (0.074) 0.4 (0.44) 0.93 (0.057) (5.57) 0.38 (0.080) 0.8 (0.04) (0.066) 93 Random model (9.83) 0.4 (0.074) 0.5 (0.096) (0.067) (4.46) 0.3 (0.067) 0.44 (0.54) (0.063) (5.5) 0.33 (0.079) 0.8 (0.04) (0.07) 95 Sandard errors of he esimaes, given in parenheses, are calculaed by he sandard deviaions among 00 replicaed simulaions. cm T, rue posiion of he QTL measured in cenimorgans from he lef end of he chromosome. he number of families increases. This is especially so for he fixed model. Overall, he wo sraegies of QTL mapping perform equally well, excep ha he fixed-model approach is difficul o implemen for large number of families. DISCUSSION Unless i is known ha he parens are heerozygous a mos QTLs for a rai of ineres, i is generally recommended o use a leas a few independen families for QTL analysis. Using more han a single family for QTL mapping may reduce a ype II error caused by homogeneous parens being sampled. In radiional QTL mapping using a single-line cross, lile aenion has been paid o he ype II error of his kind. This is because he wo parenal lines involved are no randomly seleced from a pool of available srains; insead, hey are seleced o be a he opposie exremes for he rai of ineres. As a consequence, i is almos guaraneed ha mos QTLs are heerozygous in he F parens, and hus a ype II error of his kind is likely avoided. A nonrandom selecion of parenal lines can increase he saisical power for deecing QTLs responsible for he rai used as he selecion crierion, bu i may no be helpful in deecing QTLs responsible for oher rais. In addiion, one mus be careful abou he saisical inference space of he parameer esimaion: because of he nonrandom selecion, esimaion of he QTL effec is biased and can only be inferred upon he wo parenal lines, no he pool of available srains where he wo lines were seleced. Alhough he wo sraegies of consensus QTL mapping appear o perform equally well, he fixed-model approach is generally less preferable for he following reasons. Wih muliple-family QTL mapping, one is no longer ineresed in he effec of gene subsiuion in any paricular family, bu raher is ineresed in he variance of he subsiuion effec among differen families. In oher words, he average effec of gene subsiuion is considered o be a random variable wih variance. Raher han esimaing and esing, he fixed-model approach esimaes and ess each observaion of he random variable. I is concepually incorrec o esimae and es values of a random variable. Furher- Mehod TABLE 5 Esimaes of QTL parameers and empirical powers ( 0.05) under differen modes of inheriance of he QTL Mode of inheriance cm A Fixed model A 5.7 (8.0) 0.4 (0.35) (0.07) (0.064) 9 A D 4.99 (4.65) 0.30 (0.074) 0.4 (0.44) 0.93 (0.057) 97 D 7.0 (.49) (0.08) 0.30 (0.08) 0.9 (0.060) 9 Random model A 5.3 (5.96) 0.50 (0.35) (0.00) (0.066) 9 A D 5.6 (4.46) 0.3 (0.067) 0.44 (0.54) (0.063) 96 D 7.49 (3.35) 0.0 (0.08) 0.50 (0.33) (0.063) 93 Sandard errors of he esimaes, given in parenheses, are calculaed by he sandard deviaions among 00 replicaed simulaions. Abbreviaions: A, addiive; D, dominance; A D, boh.

7 Mapping QTL Using Combined Daa 53 TABLE 6 Esimaes of QTL parameers and empirical powers ( 0.05) under differen allelic disribuions of he QTL Mehod Disribuion a cm A Fixed model Uniform 4.87 (3.66) 0.5 (0.075) 0.3 (0.084) (0.065) 98 Normal 4.99 (4.65) 0.30 (0.074) 0.4 (0.44) 0.93 (0.057) 97 Exponenial 6.86 (9.68) 0.7 (0.088) 0.54 (0.55) 0.96 (0.064) 89 Random model Uniform 4.88 (3.78) 0.9 (0.066) 0.3 (0.084) 0.99 (0.067) 99 Normal 5.6 (4.46) 0.3 (0.067) 0.44 (0.54) (0.063) 96 Exponenial 6.8 (9.93) 0.5 (0.08) 0.63 (0.55) (0.070) 88 Sandard errors of he esimaes, given in parenheses, are calculaed by he sandard deviaions among 00 replicaed simulaions. a Disribuion of he QTL allelic effec. more, he fixed-model approach involves wo seps: () esimaing he effecs and () convering he effecs ino a variance. Because of his, he fixed-model approach is compuaionally inferior o he randommodel approach when he number of families is large. The random-model approach o QTL mapping was originally developed in human geneic linkage analysis in which a large number of small families are ofen involved (Haseman and Elson 97; Goldgar 990; Schork 993; Olson and Wijsman 993; Fulker and Cardon 994; Kruglyak and Lander 995; Xu and Achley 995). Because linkage phases of markers in he parens are generally no known in small pedigrees, he random-model approach is ofen implemened hrough an idenical-by-descen (IBD) based variance componen analysis. The IBD-based mehod does no depend on informaion abou linkage phases of he parens; raher, i uilizes informaion on he number of alleles IBD shared by wo siblings. The randommodel approach proposed in his paper is closely relaed o he IBD-based mehod. Recall ha he variance covariance marix of he daa is Var(y I M ) V ZZ T WW T R, which can be reformulaed as V (ZZ T D ) (WW T D ) I I, where D diag{var(z ij I M )}, D diag{var (w ij I M )}, ZZ T D and WW T D. Marices and have been referred o as he IBD and double IBD marices, respecively, by Xu (996b). Here, one is able o pariion he IBD marix ino wo componens, ZZ T and D, because one knows he linkage phases of he markers in he parens. Decomposiion of he IBD marices allows one o apply he special algorihms of marix inversion (Equaion 7) and deerminan calculaion (Equaion 8). As a consequence, his Mehod TABLE 7 Esimaes of QTL parameers and empirical powers ( 0.05) under differen sampling sraegies (number of families number of individuals per family) Sampling sraegy cm A Fixed model (0.04) 0.4 (0.40) 0.89 (0.489). (.45) (9.85) 0.43 (0.5) 0.4 (0.39) (0.060) (0.3) 0.3 (0.09) 0. (0.6) (0.064) (4.65) 0.30 (0.074) 0.4 (0.44) 0.93 (0.057) (9.5) 0.30 (0.074) 0.5 (0.074) (0.059) (.58) 0.33 (0.069) 0.33 (0.085) 0.85 (0.058) 94 Random model (9.83) 0.99 (0.456) 0.08 (0.58) (0.084) (0.5) 0.59 (0.70) 0.66 (0.508) (0.063) (3.54) 0.33 (0.095) 0.5 (0.8) 0.97 (0.074) (4.46) 0.3 (0.067) 0.44 (0.54) (0.063) (6.48) 0.38 (0.083) 0.4 (0.3) (0.078) (.39) 0.5 (0.067) 0.34 (0.00) (0.06) (0.4) 0.4 (0.073) 0.36 (0.07) (0.08) (.00) 0.36 (0.3) 0.35 (0.074) (0.084) 6 Sandard errors of he esimaes, given in parenheses, are calculaed by he sandard deviaions among 00 replicaed simulaions.

8 54 S. Xu implemenaion of he random-model approach is compuaionally much faser han he fixed-model approach, especially when he number of families is large. In he random-model sraegy, he family-specific effecs,, have been reaed as fixed effecs. When he number of families is large, however, i is desirable o rea as random effecs. By doing so, one only esimaes a single parameer,, insead of a large array of parameers. Assume ha are random effecs so ha he expecaion and variance marices of he daa are E(y I M ) and Var(y I M ) V XX T ZZ T WW T R, respecively. The variance of family-specific effecs,, is conribued by boh geneic and nongeneic facors. Geneic facors include polygenic effecs and heriable maernal or paernal effecs. Nongeneic facors include common environmenal effecs shared by members of he same families. Noe ha or are nuisances because hey are no QTL parameers. Therefore, hey can be removed from he model using he resriced maximum likelihood mehod (Paerson and Thompson 97). Such a reamen will significanly reduce he large bias observed in he esimae of he residual variance (see he las wo rows of Table 7). This paper demonsraes he algorihm of QTL mapping combining muliple F families as an example. Wih he random-model approach, i is easy o exend he algorihm o combine all ypes of line cross daa, e.g., backcrosses, double haploids, open pollinaed progenies. I is also no difficul o combine daa from muliple full-sib and half-sib families. The mehod provides a general ool for daa updaing; i.e., QTL linkage analysis can be consanly updaed as new daa become available. I hank Damian Gessler for helpful commens on he manuscrip. This research was suppored by he Naional Insiues of Healh gran GM553-0 and he Naional Research Iniiaive Compeiive grans program/usda LITERATURE CITED Fulker, D. W., and L. R. Cardon, 994 A sib-pair approach o inerval mapping of quaniaive rai loci. Am. J. Hum. Gene. 54: Goldgar, D. E., 990 Mulipoin analysis of human quaniaive geneic variaion. Am. J. Hum. Gene. 47: Haley, C. S., and S. A. Kno, 99 A simple regression mehod for mapping quaniaive rai loci in line crosses using flanking markers. Herediy 69: Haseman, J. K., and R. C. Elson, 97 The invesigaion of linkage beween a quaniaive rai and a marker locus. Behav. Gene. : 3 9. Jansen, R. C., 993 Inerval mapping of muliple quaniaive rai loci. Geneics 35: 05. Jansen, R. C., 994 Conrolling he ype I and ype II errors in mapping quaniaive rai loci. Geneics 38: Kno, S. A., J. M. Elsen and C. S. Haley, 996 Mehods for muliple-marker mapping of quaniaive rai loci in half-sib populaions. Theor. Appl. Gene. 93: Kruglyak, L., and E. S. Lander, 995 Complee mulipoin sibpair analysis of qualiaive and quaniaive rais. Am. J. Hum. Gene. 57: Lander, E. S., and D. Bosein, 989 Mapping Mendelian facors underlying quaniaive rais using RFLP linkage maps. Geneics : Marínez, O., and R. N. Curnow, 99 Esimaing he locaions and he sizes of he effecs of quaniaive rai loci using flanking markers. Theor. Appl. Gene. 85: Murany, H., 996 of ess for quaniaive rai loci deecion using full-sib families in differen schemes. Herediy 76: Olson, J. M., and E. M. Wijsman, 993 Linkage beween quaniaive rai and markers loci: mehods using all relaive pairs. Gene. Epidemiol. 0: Paerson, H. D., and R. Thompson, 97 Recovery of iner-block informaion when block sizes are unequal. Biomerika 58: Schork, N. J., 993 Exended mulipoin ideniify-by-descen analysis of human quaniaive rais: efficiency, power, and modeling consideraions. Am. J. Hum. Gene. 53: Seber, G. A. F., 977 Linear Regression Analysis. John Wiley & Sons, New York. Weller, J. I., Y. Kashi and M. Soller, 990 of daugher and granddaugher designs for deermining linkage beween marker loci and quaniaive rai loci in dairy cale. J. Dairy Sci. 73: Xu, S., 995 A commen on he simple regression mehod for inerval mapping. Geneics 4: Xu, S., 996a Mapping quaniaive rai loci using four-way crosses. Gene. Res. Camb. 68: Xu, S., 996b Compuaion of he full likelihood funcion for esimaing variance a a quaniaive rai locus. Geneics 44: Xu, S., and W. R. Achley, 995 A random model approach o inerval mapping of quaniaive rai loci. Geneics 4: Zeng, Z. B., 994 Precision mapping of quaniaive rai loci. Geneics 36: Communicaing edior: C.-I Wu