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Composite Interval Mapping Program

F2

Backcross

- Genetic Design

- Data and Options

Means sigma^2

QTL positionsin each group

N

Haldane

KosambiMap Function

Parameters Here for Simulation Study Only

QTL Searching Step cM

H2

Calculate H2 or Sigma2 Back

Cumulative Marker Distance (cM)

Composite Interval Mapping Program

- Data

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Put Markers and Trait Data into box below OR Simulate Data

Composite Interval Mapping Program

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Analyze

Regression ModelMixture Model

Composite Interval Mapping Program

For CIM, Controlled Background Markers by Within cM Or Markers

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LR

IM CIM1 CIM2 CIM3 CIM4 CIM5

Composite Interval Mapping Program

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- Permutation Test

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Composite Interval Mapping Program

Backcross Population – Two Point

Freq Qq qq

Mm 1/2 (1-r)/2 r/2

mm 1/2 r/2 (1-r)/2

Backcross Population – Three Point

Freq Qq qq

Mm Nn (1-r)/2 (1-r1)(1-r2) r1*r2

Mm nn r/2 (1-r1)r2 r1 (1-r2)

mm Nn r/2 r1 (1-r2) (1-r1)r2

mm nn (1-r)/2 r1*r2 (1-r)/2

M Q N

F2 Population – Two Point

Freq QQ Qq qq

MM 1/4 (1-r)2/4 (1-r)r/2 r2/4

Mm 1/2 (1-r)r/2 ½-(1-r)r (1-r)r/2

mm 1/4 r2/4 (1-r)r/2 (1-r)2/4

F2 Population – Three PointFreq QQ Qq qq

MM NN (1-r)2/4 1/4(1-a)2(1-b)2 1/2a(1-a)b(1-b) 1/4a2b2

Nn (1-r)r/2 1/2(1-a)2b(1-b) 1/2a(2b2-2b+1)(1-a)

1/2a2b(1-b)

nn r2/4 1/4(1-a)2b2 1/2a(1-a)b(1-b) 1/4a2(1-b)2

Mm NN (1-r)r/2 1/2a(1-a)(1-b)2 1/2b(1-2a+2a2)(1-b)

1/2a(1-a)b2

Nn ½-(1-r)r a(1-a)b(1-b) 1/2(2b2-2b+1)(1-2a+2a2)

a(1-a)b(1-b)

Nn (1-r)r/2 1/2a(1-a)b2 1/2b(1-2a+2a2)(1-b)

1/2a(1-a)(1-b)2

mm NN r2/4 1/4a2(1-b)2 1/2a(1-a)b(1-b) 1/4(1-a)2b2

Nn (1-r)r/2 1/2a2b(1-b) 1/2a(2b2-2b+1)(1-a)

1/2(1-a)2b(1-b)

nn (1-r)2/4 1/4a2b2 1/2a(1-a)b(1-b) 1/4(1-a)2(1-b)2

M a Q b Nr=a+b-2ab

Composite model for interval mapping and regression analysis

yi = + a* zi + km-2bkxik + ei

Expected means:Qq: + a* + kbkxik = a* + XiB

qq: + kbkxik = XiB

Xi = (1, xi1, xi2, …, xi(m-2))1x(m-1)

B = (, b1, b2, …, bm-2)T

zi: QTL genotypexik: marker genotype

M1 x1

M1m1 1 +b1

m1m1 0

zi – conditional probability of Qq given markers of individual i and QTL positionxik – coding for ‘effect’ of k-th marker of i

Backcross: xik =1 if k-th marker of i is Mm =0 or –1 if k-th marker of i is mm

km-2 – summation over all markers except

two markers of current interval

We want estimate a* and test if abs(a*) is big enough to claim that there is a QTL at the given location in an interval. The estimate B is not very important

Likelihood based CIML(y,M|) = i=1

n[1|if1(yi) + 0|if0(yi)]log L(y,M|) = i=1

n log[1|if1(yi) + 0|if0(yi)]

f1(yi) = 1/[(2)½]exp[-½(y-1)2], 1= a*+XiBf0(yi) = 1/[(2)½]exp[-½(y-0)2], 0= XiB

Define1|i = 1|if1(yi)/[1|if1(yi) + 0|if0(yi)] (1)0|i = 0|if1(yi)/[1|if1(yi) + 0|if0(yi)] (2)

a* = i=1n1|i(yi-a*-XiB)/ i=1

n1|i (3) = 1 (Y-XB)´/c

B = (X´X)-1X´(Y-1a*) (4)

2 = 1/n (Y-XB)´(Y-XB) – a*2 c (5)

= (i=1n21|i +i=1

n30|i)/(n2+n3) (6)

Y = {yi}nx1, = {1|i}nx1, c = i=1n1|i

Hypothesis testH0: a*=0 vs H1: a*0

L0 = i=1nf(yi) B = (X´X)-1X´Y, 2=1/n(Y-XB)´(Y-XB)

L1= i=1n[1|if1(yi) + 0|if0(yi)]

LR = -2(lnL0 – lnL1)LOD = -(logL0 – logL1)

Likelihood based CIM for BC and F2L(y,M|) = i=1

n k=1g [k|ifk(yi)]

log L(y,M|) = i=1n log[k=1

g k|ifk(yi)] g=2 for BC, 3 for F2fk(yi) = 1/[(2)½]exp[-½(y-k)2], k= g

k +XiB k=1,…,g

Define

k|i = k|if1(yi)/[k=1g k|if1(yi)] (1)

B = k=1g k(X´X)-1X´(Y- g

k ) (2)

gk = i=1

nk|i(yi-XiB)/ i=1nk|i (3)

2 = 1/n i=1n k=1

g k|i(yi-XiB - gk )2 (4)

Y = {yi}nx1, k = {k|i}nx1