1. Linkage Disequilibrium Mapping - Natural Population
Put Markers and Trait Data
into box below OR

2. Linkage Disequilibrium Mapping - Natural Population
Initial value of p11, p10, p01:

3. Linkage Disequilibrium Mapping - Natural Population

4. Mixture model-based likelihood
Linkage disequilibrium mapping – natural population
Mixture model-based likelihood
Height markers
Sample (cm, y) m1 m m3 …

5. Association between marker and QTL
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 2 , 1 , 0

6. Mixture model-based likelihood
Linkage disequilibrium mapping – natural population
Mixture model-based likelihood
L(y,M|)=i=1n[2|if2(yi) + 1|if1(yi) + 0|if0(yi)]
Sam- Height Marker genotype QTL genotype
ple (cm, y) M QQ Qq qq
MM (2) 2|i 1|i 0|i
MM (2) 2|i 1|i 0|i
Mm (1) 2|i 1|i 0|i
Mm (1) 2|i 1|i 0|i
Mm (1) 2|i 1|i 0|i
Mm (1) 2|i 1|i 0|i
mm (0) 2|i 1|i 0|i
mm (0) 2|i 1|i 0|i
Prior prob.

7. Joint and conditional (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
p2 p2 p2
2p(1-p) 2p(1-p) 2p(1-p)
(1-p)2 (1-p)2 (1-p)2

8. = i=1n [2|if2(yi) + 1|if1(yi) + 0|if0(yi)]
Linkage disequilibrium mapping – natural population
Conditional probabilities of the QTL genotypes (missing) based on marker genotypes (observed)
L(y,M|)
= i=1n [2|if2(yi) + 1|if1(yi) + 0|if0(yi)]
= i=1n2 [2|2f2(yi) + 1|2f1(yi) + 0|2f0(yi)] Conditional on 2 (n2)
 i=1n1 [2|1f2(yi) + 1|1f1(yi) + 0|1f0(yi)] Conditional on 1 (n1)
 i=1n0 [2|0f2(yi) + 1|0f1(yi) + 0|0f0(yi)] Conditional on 0 (n0)

9. Normal distributions of phenotypic values for each QTL genotype group
Linkage disequilibrium mapping – natural population
Normal distributions of phenotypic values for each QTL genotype group
f2(yi) = 1/(22)1/2exp[-(yi-2)2/(22)],
2 =  + a
f1(yi) = 1/(22)1/2exp[-(yi-1)2/(22)],
1 =  + d
f0(yi) = 1/(22)1/2exp[-(yi-0)2/(22)],
0 =  - a

10. 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|) = i=1n[2|if2(yi) + 1|if1(yi) + 0|if0(yi)]
log L(y,M|) = i=1n log[2|if2(yi) + 1|if1(yi) + 0|if0(yi)]
Define
2|i = 2|if1(yi)/[2|if2(yi) + 1|if1(yi) + 0|if0(yi)] (1)
1|i = 1|if1(yi)/[2|if2(yi) + 1|if1(yi) + 0|if0(yi)] (2)
0|i = 0|if1(yi)/[2|if2(yi) + 1|if1(yi) + 0|if0(yi)] (3)
2 = i=1n(2|iyi)/ i=1n1|i (4)
1 = i=1n(1|iyi)/ i=1n1|i (5)
0 = i=1n(0|iyi)/ i=1n0|i (6)
2 = 1/ni=1n[1|i(yi-1)2+0|i(yi-0)2] (7)

11. Incomplete (observed) data Posterior prob QQ Qq qq Obs
MM 2|2i 1|2i 0|2i n2
Mm 2|1i 1|1i 0|1i n1
mm 2|0i 1|0i 0|0i n0
p11=1/2n{i=1n2[22|2i+1|2i]+ i=1n1[2|1i+1|1i], (8)
p10=1/2n{i=1n2[20|2i+1|2i]+ i=1n1[0|1i+(1-)1|1i], (9)
p01=1/2n{i=1n0[22|0i+1|0i]+ i=1n1[2|1i+(1-)1|1i], (10)
p00=1/2n{i=1n2[20|0i+1|0i]+ i=1n1[0|1i+1|1i] (11)

12. EM algorithm
(1) Give initiate values (0) =(2,1,0,2,p11,p10,p01,p00)(0)
(2) Calculate 2|i(1), 1|i(1) and 0|i(1) using Eqs. 1-3,
(3) Calculate (1) using 2|i(1), 1|i(1) and 0|i(1) based on
Eqs. 4-11,
(4) Repeat (2) and (3) until convergence.

13. PROGRAM:
Given a initial 2, 1, 0, 2, p11, p10, p01, p00
mu(1), mu(2), mu(3), s2
Do While (Abs(mu(1) - omu(1)) + Abs(p00 - p00old) > )
kkk = kkk ‘cumulate the number of iteration
p00old = p00 ‘keep old value of p00
prob(1, 1) = p11 ^ 2 / p ^ ‘prior conditional probability 2|2
prob(1, 2) = 2 * p11 * p10 / p ^ 2 ‘ 2|1
prob(1, 3) = p10 ^ 2 / p ^ ‘ 2|0
prob(2, 1) = 2 * p11 * p01 / (2 * p * q) ‘ 1|2
prob(2, 2) = 2 * (p11 * p00 + p10 * p01) / (2 * p * q) ‘ 1|1
prob(2, 3) = 2 * p10 * p00 / (2 * p * q) ‘ 1|0
prob(3, 1) = p01 ^ 2 / q ^ ‘ 0|2
prob(3, 2) = 2 * p01 * p00 / q ^ ‘ 0|1
prob(3, 3) = p00 ^ 2 / q ^ ‘ 0|0

14. 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 2|2, 2|1, 2|0, …. 0|0
Next
Next j
cs2 = 0
ll = 0
For i = 1 To N
sss = 0
For j = 1 To ’ f2(yi), f1(yi), f0(yi)
f(j) = 1 / Sqr(2 * * s2) * Exp(-(y(i) - mu(j)) ^ 2 / 2 / s2)
sss = sss + prob(datas(i, mrk), j) * f(j)
ll = ll + Log(sss) ’calculate log-likelihood
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 (4-6)
cs2 = cs2 + bpi(j) * (y(i) - mu(j)) ^ ’FORMULA (7)
nnn(datas(i, mrk), j) = nnn(datas(i, mrk), j) + bpi(j) ’FORMULA (8-11)
Next i
‘[2|if2(yi) + 1|if1(yi) + 0|if0(yi)]

15. ‘ Update 2, 1, 0 formula (4-6)
For j = 1 To 3
mu(j) = cmu(j) / cpi(j)
Next j
‘Update 2 formula 7
s2 = cs2 / N
‘Update p11, p10, p01, p 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)

16. Linkage Disequilibrium Mapping - Natural Population Binary Trait
Put Markers and Trait Data
into box below OR

17. Linkage Disequilibrium Mapping - Natural Population Binary Trait
Initial value of p11, p10, p01:
Initial value of f2, f1, f0:

18. Linkage Disequilibrium Mapping - Natural Population Binary Trait
Initial value of
f2, f1, f0:

19. Linkage Disequilibrium Mapping - Natural Population Binary Trait
For Marker:

20. Linkage Disequilibrium Mapping - Natural Population Binary Trait
L(|y)=j=02i=0nj
log [2|ijPr{yij=1|Gij=2,}yij Pr{yij=0|Gij=2,}(1-yij)
+1|ijPr{yij=1|Gij=1,}yij Pr{yij=0|Gij=1,}(1-yij)
+0|ijPr{yij=1|Gij=0,}yij Pr{yij=0|Gij=0,}(1-yij)]
=j=02i=0nj
log[2|ijf2yij(1-f2)(1-yij)+1|ijf1yij(1-f1)(1-yij)+0|ijf0yij(1-f0)(1-yij)]
 = (p11, p10, p01, p00, f2, f1, f0) (6 parameters)

21. 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 2|2, 2|1, 2|0, …. 0|0
Next
Next j
cs2 = 0
ll = 0
For i = 1 To N
sss = 0
For j = 1 To ’ f2(yi), f1(yi), f0(yi)
f(j) = 1 / Sqr(2 * * 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)
ll = ll + Log(sss) ’calculate log-likelihood
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 (4-6)
cs2 = cs2 + bpi(j) * (y(i) - mu(j)) ^ ’FORMULA (7)
nnn(datas(i, mrk), j) = nnn(datas(i, mrk), j) + bpi(j) ’FORMULA (8-11)
Next i