1. PBG 650 Advanced Plant Breeding Module 9: Best Linear Unbiased Prediction – Purelines – Single-crosses

2. 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

3. 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)

4. 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)

5. Regression in matrix notation Y = X  + ε b = (X’X) -1 X’Y Linear model Parameter estimates SourcedfSSMS Regressionpb’X’YMS R Residualn-pY’Y - b’X’YMS E TotalnY’Y

6. 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

7. BLUP for purelines – barley example Bernardo, pg 269 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)

8. Linear model for barley example Y ij =  + t i + u j + e ij t i = effect of i th set of environments u j = effect of j th cultivar In matrix notation: Y = X  + Zu + e

9. Weighted regression Y = X  + ε b = (X’X) -1 X’Y Where ε ij ~N (0, σ 2 ) When ε ij ~N (0, Rσ 2 ) Then b = (X’R -1 X) -1 X’R -1 Y For the barley example

10. Covariance structure of random effects MorexRobustExcelStander Morex11/27/1611/32 Robust127/3243/64 Excel191/128 Stander1 r = 2  XY Remember 217/811/16 1227/1643/32 7/827/16291/64 11/1643/3291/642  XY

11. Mixed Model Equations X’R -1 XX’R -1 ZX’R -1 Y Z’R -1 XZ’R -1 Z + A -1 (σ ε 2 /σ A 2 )Z’R -1 Y Rσ2Rσ2 = each matrix is composed of submatrices the algebra is the same Calculations in Excel

12. Results from BLUP 11 Set 14.82 22 Set 25.41 u1u1 Morex-0.33 u2u2 Robust-0.17 u3u3 Excel0.18 u4u4 Stander0.36 Original data BLUP estimates For fixed effects b 1 =  + t 1 b 2 =  + t 2

13. Interpretation from BLUP 11 Set 14.82 22 Set 25.41 u1u1 Morex-0.33 u2u2 Robust-0.17 u3u3 Excel0.18 u4u4 Stander0.36 BLUP estimates For a set of recombinant inbred lines from an F 2 cross of Excel x Stander Predicted mean breeding value = ½(0.18+0.36) = 0.27

14. Shrinkage estimators In the simplest case (all data balanced, the only fixed effect is the overall mean, inbreds unrelated) If h 2 is high, BLUP values are close to the phenotypic values If h 2 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

15. 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 -1 XX’R -1 ZX’R -1 Y Z’R -1 XZ’R -1 Z + A -1 (σ ε 2 /σ A 2 )Z’R -1 Y Rσ2Rσ2 = invert the matrix C 11 C 12 C 21 C 22 coefficient matrix each element of the matrix is a matrix

16. Sampling error of BLUP fixed effects random effects

17. Estimation of Variance Components (would really need a larger data set) 1. Use your best guess for an initial value of σ ε 2 /σ A 2 2. Solve for  and û 3. Use current solutions to solve for σ ε 2 and then for σ A 2 4. Calculate a new σ ε 2 /σ A 2 5. Repeat the process until estimates converge ˆ

18. BLUP for single-crosses G B73,Mo17 = GCA B73 + GCA Mo17 + SCA B73,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  + Ug 1 + Wg 2 + Ss + e Example in Bernardo, pg 277 from Hallauer et al., 1996

19. Performance of maize single crosses Iowa Stiff Stalk x Lancaster Sure Crop

20. Covariance of single crosses SC-X is jxkSC-Y is j’xk’ B73, B84, H123MO17, N197 assuming no epistasis

21. Covariance of single crosses SC-X is jxkSC-Y is j’xk’ SC-1=B73xMO17SC-2=H123xMO17 SC-3=B84xN197

22. Solutions X