BLUP for Purelines

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PBG 650 Advanced Plant Breeding
Module 9 Best Linear Unbiased Prediction
Purelines Single-crosses
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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
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BLUP History

• Initially developed by C.R. Henderson in the
  1940s
• Most extensively used in animal breeding
• Used in crop improvement since the 1990s,
  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)

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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)

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Regression in matrix notation
Y X? e
Linear model
b (XX)-1XY
Parameter estimates
Source df SS MS
Regression p bXY MSR
Residual n-p YY - bXY MSE
Total n YY
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BLUP Mixed Model in Matrix Notation
Design matrices
Y X? Zu e
Random effects
Fixed effects

• 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

Classification for the purposes of BLUP
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BLUP for purelines barley example

• 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)

Bernardo, pg 269
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Linear model for barley example
Yij ? ti uj eij
ti effect of ith set of environments uj
effect of jth cultivar
Y X? Zu e
In matrix notation
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Weighted regression
Y X? e
Where eij N (0, s2)
b (XX)-1XY
For the barley example
When eij N (0, Rs2) Then b (XR-1X)-1XR-1Y
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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
?XY
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
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Mixed Model Equations
-1
XR-1X XR-1Z XR-1Y
ZR-1X ZR-1Z A-1(se2/sA2) ZR-1Y

Rs2

• each matrix is composed of submatrices
• the algebra is the same

Calculations in Excel
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Results from BLUP
Original data
?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 fixed effects b1 ? t1 b2 ? t2
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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(0.180.36)
0.27
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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

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Sampling error of BLUP
-1
XR-1X XR-1Z XR-1Y
ZR-1X ZR-1Z A-1(se2/sA2) ZR-1Y


Rs2
invert the matrix

C11 C12
C21 C22

each element of the matrix is a matrix
coefficient matrix

• Diagonal elements of the inverse of the
  coefficient matrix can be used to estimate
  sampling error of fixed and random effects

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Sampling error of BLUP

fixed effects
random effects
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Estimation of Variance Components

• (would really need a larger data set)

1. Use your best guess for an initial value of
   se2/sA2
2. Solve for ? and û
3. Use current solutions to solve for se2 and then
   for sA2
4. Calculate a new se2/sA2
5. Repeat the process until estimates converge

ˆ
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BLUP for single-crosses

• Performance of a single cross
• BLUP Model
• Sets of environments are fixed effects
• GCA and SCA are considered to be random effects

GB73,Mo17 GCAB73 GCAMo17 SCAB73,Mo17
Y X? Ug1 Wg2 Ss e
Example in Bernardo, pg 277 from Hallauer et al.,
1996
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Performance of maize single crosses
Iowa Stiff Stalk x Lancaster Sure Crop
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Covariance of single crosses

• SC-X is jxk SC-Y is jxk

B73, B84, H123
MO17, N197
assuming no epistasis
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Covariance of single crosses

• SC-X is jxk SC-Y is jxk

SC-1B73xMO17
SC-2H123xMO17
SC-3B84xN197
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Solutions
-1
X