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PBG 650 Advanced Plant Breeding
Module 9: Best Linear Unbiased Prediction– Purelines– Single-crosses
Best Linear Unbiased Prediction (BLUP)
• Allows comparison of material from different populations evaluated in different environments
• Makes use of all performance data available for each genotype, and accounts for the fact that some genotypes have been more extensively tested than others
• Makes use of information about relatives in pedigree breeding systems
• Provides estimates of genetic variances from existing data in a breeding program without the use of mating designs
Bernardo, Chapt. 11
BLUP History
• Initially developed by C.R. Henderson in the 1940’s
• Most extensively used in animal breeding
• Used in crop improvement since the 1990’s, particularly in forestry
• BLUP is a general term that refers to two procedures
– true BLUP – the ‘P’ refers to prediction in random effects models (where there is a covariance structure)
– BLUE – the ‘E’ refers to estimation in fixed effect models (no covariance structure)
B-L-U
• “Best” means having minimum variance
• “Linear” means that the predictions or estimates are linear functions of the observations
• Unbiased
– expected value of estimates = their true value
– predictions have an expected value of zero (because genetic effects have a mean of zero)
Regression in matrix notation
Y = X + ε
b = (X’X)-1X’Y
Linear model
Parameter estimates
Source df SS MS
Regression p b’X’Y MSR
Residual n-p Y’Y - b’X’Y MSE
Total n Y’Y
BLUP Mixed Model in Matrix Notation
• Fixed effects are constants– overall mean– environmental effects (mean across trials)
• Random effects have a covariance structure– breeding values– dominance deviations– testcross effects– general and specific combining ability effects
Y = X + Zu + e
Design matrices
Random effectsFixed effects
Classification for the purposes of BLUP
BLUP for purelines – barley example
Bernardo, pg 269
Cultivar Grain Yield t/haSet 1 18 Morex (1) 4.45Set 1 18 Robust (2) 4.61Set 1 18 Stander (4) 5.27Set 2 9 Robust (2) 5.00Set 2 9 Excel (3) 5.82Set 2 9 Stander (4) 5.79
Environments
Parameters to be estimated• means for two sets of environments – fixed
effects– we are interested in knowing effects of these particular
sets of environments
• breeding values of four cultivars – random effects– from the same breeding population– there is a covariance structure (cultivars are related)
Linear model for barley example
Yij = + ti + uj + eij
ti = effect of ith set of environmentsuj = effect of jth cultivar
In matrix notation: Y = X + Zu + e
4.45 1 0 1 0 0 0 e11
4.61 1 0 0 1 0 0 u1 e12
5.27 = 1 0 b1 + 0 0 0 1 u2 + e14
5.00 0 1 b2 0 1 0 0 u3 e22
5.82 0 1 0 0 1 0 u4 e23
5.79 0 1 0 0 0 1 e24
Weighted regression
Y = X + ε
b = (X’X)-1X’Y
Where εij ~N (0, σ2)
When εij ~N (0, Rσ2)
Then b = (X’R-1X)-1X’R-1Y
18 0 0 0 0 0
0 18 0 0 0 0R-1= 0 0 18 0 0 0
0 0 0 9 0 0
0 0 0 0 9 0
0 0 0 0 0 9
For the barley example
Covariance structure of random effects
Morex Robust Excel Stander
Morex 1 1/2 7/16 11/32
Robust 1 27/32 43/64
Excel 1 91/128
Stander 1
2D
2ArCovXY r = 2XYRemember
2 1 7/8 11/16
1 2 27/16 43/32
7/8 27/16 2 91/64
11/16 43/32 91/64 2
2
A 2
A
2
u A
XY
Mixed Model Equations
X’R-1X X’R-1Z X’R-1Y
Z’R-1X Z’R-1Z + A-1(σε2/σA
2) Z’R-1Y
Rσ2
β
u=
-1
• each matrix is composed of submatrices
• the algebra is the same
Calculations in Excel
Results from BLUP
1 Set 1 4.82
2 Set 2 5.41
u1 Morex -0.33
u2 Robust -0.17
u3 Excel 0.18
u4 Stander 0.36
Cultivar Grain Yield t/haSet 1 18 Morex 4.45Set 1 18 Robust 4.61Set 1 18 Stander 5.27Set 2 9 Robust 5.00Set 2 9 Excel 5.82Set 2 9 Stander 5.79
Environments
Original data
BLUP estimates
For fixed effectsb1 = + t1
b2 = + t2
Interpretation from BLUP
1 Set 1 4.82
2 Set 2 5.41
u1 Morex -0.33
u2 Robust -0.17
u3 Excel 0.18
u4 Stander 0.36
BLUP estimates
For a set of recombinant inbred linesfrom an F2 cross of Excel x Stander
Predicted mean breeding value = ½(0.18+0.36) = 0.27
Shrinkage estimators
• In the simplest case (all data balanced, the only fixed effect is the overall mean, inbreds unrelated)
• If h2 is high, BLUP values are close to the phenotypic values
• If h2 is low, BLUP values shrink towards the overall mean
• For unrelated inbreds or families, ranking of genotypes is the same whether one uses BLUP or phenotypic values
...)( YYhBLUP i2
i
Sampling error of BLUP
• Diagonal elements of the inverse of the coefficient matrix can be used to estimate sampling error of fixed and random effects
X’R-1X X’R-1Z X’R-1Y
Z’R-1X Z’R-1Z + A-1(σε2/σA
2) Z’R-1Y
Rσ2
β
u=
-1
invert the matrix
C11 C12
C21 C22
coefficient matrix each element of the matrix is a matrix
Sampling error of BLUP
2 2 222C
2 2 211C
C11 C12 X’R-1YC21 C22 Z’R-1Y
=βu
fixed effects
random effects
Estimation of Variance Components
(would really need a larger data set)
1. Use your best guess for an initial value of σε2/σA
2
2. Solve for and û
3. Use current solutions to solve for σε2 and then for
σA2
4. Calculate a new σε2/σA
2
5. Repeat the process until estimates converge
ˆ
BLUP for single-crosses
GB73,Mo17 = GCAB73 + GCAMo17 + SCAB73,Mo17
Performance of a single cross:
BLUP Model
• Sets of environments are fixed effects
• GCA and SCA are considered to be random effects
Y = X + Ug1 + Wg2 + Ss + e
Example in Bernardo, pg 277 from Hallauer et al., 1996
Performance of maize single crosses
Set Entry Pedigree
Grain Yield
t ha-1
1 SC-1 B73 x Mo17 7.851 SC-2 H123 x Mo17 7.361 SC-3 B84 x N197 5.612 SC-2 H123 x Mo17 7.472 SC-3 B84 x N197 5.96
7.85 1 0 1 0 0 1 0 1 0 0 e11
7.36 1 0 b1 0 0 1 gB73 1 0 gMo17 0 1 0 s1 e12
5.61 = 1 0 b2 + 0 1 0 gB84 + 0 1 gN197 + 0 0 1 s2 + e13
7.47 0 1 0 0 1 gH123 1 0 0 1 0 s3 e22
5.96 0 1 0 1 0 0 1 0 0 1 e23
Iowa Stiff Stalk x Lancaster Sure Crop
Covariance of single crosses
SC-X is jxk SC-Y is j’xk’
2 2g1 1 GCA(1)G
2SCAkkjj
22GCAkk
21GCAjjSCCov '')(')('
1 B73,B84
B73,H123
G1= B73,B84 1
B84,H123
B73,H123
B84,H123 1
1 Mo17,N197
G2= Mo17,N197 1
B73, B84, H123 MO17, N197
2 2g2 2 GCA(2)G
assuming no epistasis
Covariance of single crosses
SC-X is jxk SC-Y is j’xk’
2SCAkkjj
22GCAkk
21GCAjjSCCov '')(')('
1 B73,H123
Mo17,Mo17
B73,B84
Mo17,N197
S = B73,H123
Mo17,Mo17 1
B84,H123
Mo17,N197
B73,B84
Mo17,N197
B84,H123
Mo17,N197 1
SC-1=B73xMO17 SC-2=H123xMO17 SC-3=B84xN197
2 2s SCAS
Solutions
X'R-1X X'R-1U X'R-1W X'R-1Z X'R-1Y
U'R-1X U'R-1U + Q1 U'R-1W U'R-1Z U'R-1Y
W'R-1X W'R-1U W'R-1W + Q 2 W'R-1Z W'R-1Y
Z'R-1X Z'R-1U Z'R-1W Z'R-1Z + QS Z'R-1Y
3
2
1
197N
17Mo
123H
84B
73B
2
1
s
s
s
g
g
g
g
g
b
b
-1
X
1 2 21 1 GCA(1)
1 2 22 2 GCA(2)
1 2 2S SCA
G /
G /
S /
Q Q Q