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Unformatted text preview: Linkage Disequilibrium Mapping - Natural Population Put Markers and Trait Data into box below OR Simulate Data Means sigma^2 N p11 p10 p01 p00 Back Linkage Disequilibrium Mapping - Natural Population Initial value of p11, p10, p01: Estimate Back Next Linkage Disequilibrium Mapping - Natural Population Estimate Back Next Linkage disequilibrium mapping – natural population Mixture model-based likelihood Sample 1 2 3 4 5 6 7 8 Height (cm, y) 184 185 180 182 167 169 165 166 markers m1 m2 1 1 2 2 0 1 1 2 2 0 1 2 2 1 0 0 m3 … 2 0 1 2 1 1 2 0 Linkage disequilibrium mapping – natural population Association between marker and QTL -Marker, Prob(M)=p, Prob(m)=1-p -QTL, Prob(Q)=q, Prob(q)=1-q Four haplotypes: Prob(MQ)=p11=pq+D p=(p11+p10) Prob(Mq)=p10=p(1-q)-D q=(p11+p01) Prob(mQ)=p01=(1-p)q-D D=p11p00-p10p01 Prob(mq)=p00=(1-p)(1-q)+D Estimate p, q, D AND µ 2 , µ 1 , µ 0 Linkage disequilibrium mapping – natural population Mixture model-based likelihood L(y,M|Ω )=Π i=1n[π 2|if2(yi) + π 1|if1(yi) + π 0|if0(yi)] Prior prob. Sample 1 2 3 4 5 6 7 8 Height (cm, y) 184 185 180 182 167 169 165 166 Marker genotype M MM (2) MM (2) Mm (1) Mm (1) Mm (1) Mm (1) mm (0) mm (0) QTL genotype QQ Qq π2|i π1|i π2|i π1|i π2|i π1|i π2|i π1|i π2|i π1|i π2|i π1|i π2|i π1|i π2|i π1|i qq π0|i π0|i π0|i π0|i π0|i π0|i π0|i π0|i Joint and conditional (π j|i) genotype prob. between marker and QTL QQ Qq qq Obs MM p112 2p11p10 p102 n2 Mm 2p11p01 2(p11p00+p10p01) 2p10p00 n1 mm p012 2p01p00 p002 n0 MM p112 2p11p10 p102 n2 p2 p2 p2 Mm 2p11p01 2(p11p00+p10p01) 2p10p00 n1 mm 2p(1-p)2p(1-p) 2p(1-p) p012 2p01p00 p002 n0 (1-p)2 (1-p)2 (1-p)2 Linkage disequilibrium mapping – natural population Conditional probabilities of the QTL genotypes (missing) based on marker genotypes (observed) L(y,M|Ω ) = Π i=1n [π 2|if2(yi) + π 1|if1(yi) + π 0|if0(yi)] = Π i=1n2 [π 2|2f2(yi) + π 1|2f1(yi) + π 0|2f0(yi)] Conditional on 2 (n2) × Π i=1n1 [π 2|1f2(yi) + π 1|1f1(yi) + π 0|1f0(yi)] Conditional on 1 (n1) × Π i=1n0 [π 2|0f2(yi) + π 1|0f1(yi) + π 0|0f0(yi)] Conditional on 0 (n0) Linkage disequilibrium mapping – natural population Normal distributions of phenotypic values for each QTL genotype group f2(yi) = 1/(2πσ 2)1/2exp[-(yi-µ 2)2/(2σ 2)], µ2 = µ + a f1(yi) = 1/(2πσ 2)1/2exp[-(yi-µ 1)2/(2σ 2)], µ1 = µ + d f0(yi) = 1/(2πσ 2)1/2exp[-(yi-µ 0)2/(2σ 2)], µ0 = µ - a Linkage disequilibrium mapping – natural population Differentiating L with respect to each unknown parameter, setting derivatives equal zero and solving the log-likelihood equations L(y,M|Ω ) = Π i=1n[π 2|if2(yi) + π 1|if1(yi) + π 0|if0(yi)] log L(y,M|Ω ) = Σ i=1n log[π 2|if2(yi) + π 1|if1(yi) + π 0|if0(yi)] Define Π 2|i = π 2|if1(yi)/[π 2|if2(yi) + π 1|if1(yi) + π 0|if0(yi)] Π 1|i = π 1|if1(yi)/[π 2|if2(yi) + π 1|if1(yi) + π 0|if0(yi)] Π 0|i = π 0|if1(yi)/[π 2|if2(yi) + π 1|if1(yi) + π 0|if0(yi)] µ 2 = ∑ i=1n(Π 2|iyi)/ ∑ i=1nΠ 1|i µ 1 = ∑ i=1n(Π 1|iyi)/ ∑ i=1nΠ 1|i µ 0 = ∑ i=1n(Π 0|iyi)/ ∑ i=1nΠ 0|i σ 2 = 1/n∑ i=1n[Π 1|i(yi-µ1)2+Π 0|i(yi-µ0)2] (1) (2) (3) (4) (5) (6) (7) Incomplete (observed) data Posterior prob QQ Qq qq Obs MM Π2|2i Π1|2i Π0|2i n2 Mm Π2|1i Π1|1i Π0|1i n1 Π2|0i Π1|0i Π0|0i n0 mm p11=1/2n{Σi=1n2[2Π2|2i+Π1|2i]+ Σi=1n1[Π2|1i+φΠ1|1i], (8) p10=1/2n{Σi=1n2[2Π0|2i+Π1|2i]+ Σi=1n1[Π0|1i+(1-φ)Π1|1i], (9) p01=1/2n{Σi=1n0[2Π2|0i+Π1|0i]+ Σi=1n1[Π2|1i+(1-φ)Π1|1i], p00=1/2n{Σi=1n2[2Π0|0i+Π1|0i]+ Σi=1n1[Π0|1i+φΠ1|1i] (10) (11) EM algorithm (1) Give initiate values Ω (0) =(µ 2,µ 1,µ 0,σ 2,p11,p10,p01,p00)(0) (2) Calculate Π2|i(1), Π1|i(1) and Π0|i(1) using Eqs. 1-3, (3) Calculate Ω (1) using Π2|i(1), Π1|i(1) and Π0|i(1) based on Eqs. 4-11, (4) Repeat (2) and (3) until convergence. PROGRAM: Given a initial µ 2, µ 1, µ 0, σ 2, mu(1), mu(2), mu(3), s2 p11, p10, p01, p00 Do While (Abs(mu(1) - omu(1)) + Abs(p00 - p00old) > 0.00001) kkk = kkk + 1 ‘cumulate the number of iteration p00old = p00 ‘keep old value of p00 prob(1, 1) = p11 ^ 2 / p ^ 2 ‘prior conditional probability π 2|2 prob(1, 2) = 2 * p11 * p10 / p ^ 2 ‘ π 2|1 prob(1, 3) = p10 ^ 2 / p ^ 2 ‘ π 2|0 prob(2, 1) = 2 * p11 * p01 / (2 * p * q) ‘ π 1|2 prob(2, 2) = 2 * (p11 * p00 + p10 * p01) / (2 * p * q) ‘ π 1|1 prob(2, 3) = 2 * p10 * p00 / (2 * p * q) ‘ π 1|0 prob(3, 1) = p01 ^ 2 / q ^ 2 ‘ π 0|2 prob(3, 2) = 2 * p01 * p00 / q ^ 2 ‘ π 0|1 prob(3, 3) = p00 ^ 2 / q ^ 2 ‘ π 0|0 For j = 1 To 3 omu(j) = mu(j) : cmu(j) = 0 : cpi(j) = 0 : bpi(j) = 0 For i = 1 To 3 nnn(i, j) = 0 ’3 by 3 matrix to store Π2|2, Π2|1, Π2|0, …. Π0|0 Next Next j cs2 = 0 ll = 0 For i = 1 To N sss = 0 For j = 1 To 3 ’ f2(yi), f1(yi), f0(yi) f(j) = 1 / Sqr(2 * 3.1415926 * s2) * Exp(-(y(i) - mu(j)) ^ 2 / 2 / s2) sss = sss + prob(datas(i, mrk), j) * f(j) ‘[π 2|if2(yi) + π 1|if1(yi) + π 0|if0(yi)] Next j ll = ll + Log(sss) ’calculate log-likelihood For j = 1 To 3 bpi(j) = prob(datas(i, mrk), j) * f(j) / sss ’FORMULA (1-3) cmu(j) = cmu(j) + bpi(j) * datas(i, nmrk) ’ numerator of FORMULA (4-6) cpi(j) = cpi(j) + bpi(j) ’ denominator of FORMULA (46) cs2 = cs2 + bpi(j) * (y(i) - mu(j)) ^ 2 ’FORMULA (7) nnn(datas(i, mrk), j) = nnn(datas(i, mrk), j) + bpi(j) ’FORMULA (811) Next j ‘ Update µ 2, µ 1, µ 0 formula (4-6) For j = 1 To 3 mu(j) = cmu(j) / cpi(j) Next j ‘Update σ 2 formula 7 s2 = cs2 / N ‘Update p11, p10, p01, p00 FORMULA (8-11) phi = p11 * p00 / (p11 * p00 + p10 * p01) p11 = (2 * nnn(1, 1) + nnn(1, 2) + nnn(2, 1) + phi * nnn(2, 2)) / 2 / N p10 = (2 * nnn(1, 3) + nnn(1, 2) + nnn(2, 3) + (1 - phi) * nnn(2, 2)) / 2 / N p01 = (2 * nnn(3, 1) + nnn(2, 1) + nnn(3, 2) + (1 - phi) * nnn(2, 2)) / 2 / N p00 = (2 * nnn(3, 3) + nnn(2, 3) + nnn(3, 2) + phi * nnn(2, 2)) / 2 / N p = p11 + p10 q=1-p Loop LR = 2 * (ll - ll0) Linkage Disequilibrium Mapping - Natural Population Binary Trait Put Markers and Trait Data into box below OR Simulate Data f2, f1, f0 N p11 p10 p01 p00 Back Linkage Disequilibrium Mapping - Natural Population Binary Trait Initial value of p11, p10, p01: Initial value of f2, f1, f0: Estimate Back Next Linkage Disequilibrium Mapping - Natural Population Binary Trait Initial value of f2, f1, f0: Estimate Back Next Linkage Disequilibrium Mapping - Natural Population Binary Trait L(Ω |y)=Πj=02Πi=0nj log [ϖ2|ijPr{yij=1|Gij=2,Ω }yij Pr{yij=0|Gij=2,Ω }(1-yij) +ϖ1|ijPr{yij=1|Gij=1,Ω }yij Pr{yij=0|Gij=1,Ω }(1-yij) +ϖ0|ijPr{yij=1|Gij=0,Ω }yij Pr{yij=0|Gij=0,Ω }(1-yij)] =Πj=02Πi=0nj log[ϖ2|ijf2yij(1-f2)(1-yij)+ϖ1|ijf1yij(1-f1)(1-yij)+ϖ0|ijf0yij(1-f0)(1-yij)] Ω = (p11, p10, p01, p00, f2, f1, f0) (6 parameters) For j = 1 To 3 omu(j) = mu(j) : cmu(j) = 0 : cpi(j) = 0 : bpi(j) = 0 For i = 1 To 3 nnn(i, j) = 0 ’3 by 3 matrix to store Π2|2, Π2|1, Π2|0, …. Π0|0 Next Next j cs2 = 0 ll = 0 For i = 1 To N sss = 0 For j = 1 To 3 ’ f2(yi), f1(yi), f0(yi) f(j) = 1 / Sqr(2 * 3.1415926 * s2) * Exp(-(y(i) - mu(j)) ^ 2 / 2 / s2) f(j)=mu(j) ^ datas(i, nmrk) * (1 - mu(j)) ^ (1 - datas(i, nmrk)) sss = sss + prob(datas(i, mrk), j) * f(j) Next j ll = ll + Log(sss) ’calculate log-likelihood For j = 1 To 3 bpi(j) = prob(datas(i, mrk), j) * f(j) / sss ’FORMULA (1-3) cmu(j) = cmu(j) + bpi(j) * datas(i, nmrk) ’ numerator of FORMULA (4-6) cpi(j) = cpi(j) + bpi(j) ’ denominator of FORMULA (46) cs2 = cs2 + bpi(j) * (y(i) - mu(j)) ^ 2 ’FORMULA (7) nnn(datas(i, mrk), j) = nnn(datas(i, mrk), j) + bpi(j) ’FORMULA (811) ...
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