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