Modifying the Schwarz Bayesian Information Criterion to locate multiple interacting Quantitative Trait Loci 1. M.Bogdan, J.K.Ghosh and R.W.Doerge, Genetics 2004 167: 989-999. 2. M.Bogdan and R.W.Doerge “Mapping multiple interacting QTL by multidimensional genome searches’’
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X ia - genotype of i-th individual at locus a X ia = 1/2 - individual is heterozygous at locus a
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Modifying the Schwarz Bayesian Information Criterion to locate multiple interacting
Quantitative Trait Loci
1. M.Bogdan, J.K.Ghosh and R.W.Doerge,Genetics 2004 167: 989-999.
2. M.Bogdan and R.W.Doerge “Mapping multiple interacting QTL by multidimensional genome searches’’
Xia- genotype of i-th individual at locus a
Xia = 1/2 - individual is heterozygous at locus a
Xia = -1/2 - individual is homozygous at locus a
dab=10 cM - ρ (Xia, Xib) = 0.81
Data for QTL mapping
Y1,...,Yn - vector of trait values for n backcross individuals
X=[Xij], 1 ≤ i ≤ n, 1 ≤ j ≤ m - genotypes of m markers
Standard methods of QTL mapping One QTL model
2(1) Q , (0, )
Q (-1/2,1/2) - QTL genotypei i i i
i
Y N
1. Search over markers - fit model (1) at each marker and choose markers for which the likelihood exceeds a preestablished threshold value as candidate
QTL locations.
Interval mapping Lander and Botstein (1989)
• Consider a fixed position between markers
- state of flanking markers
1 1 1 1 1 1 1 1, , , , , , ,
2 2 2 2 2 2 2 2
1(Q | ) easy to compute
2
i
i
i i i
I
I
p P I
2
2 2
1
Q , (0, )
1 1( | ) ( , ) (1 ) ( , )
2 2
( | ) ( | )
i i i i
i i i i
n
i ii
Y N
f Y I p N p N
L Y I f Y I
1. Estimate μ, β, and σ by EM algorithm and compute the corresponding likelihood.
2. Repeat this procedure for a new possible QTL location.
3. Plot the resulting likelihoods as the function of assumed QTL position.
• Problems with interval mapping
a) Not able to distingush closely linked QTL
b) Not able to detect epistatic QTL (involved only in interactions)
• Solution
Estimate the location of several QTL at once using multiple regression model (Kao et al. 1999)
p r
i j ij jl ij ilj 1 1 j<l m
Y μ β γ εiQ Q Q
Problem : estimation of the number of additive and interaction terms
iεXXγXβμY jjj iuik
p
1j
r
1jjihji
Xij - genotype of j-th marker
average number of markers - (200,400)
Bayesian Information Criterion
• Choose the model which maximizes
log L -1/2 k log n
L – likelihood of the data for a given model
k – number of parameters in the model
n – sample size
Broman (1997) and Broman and Speed (2002) – BIC overestimates QTL number