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PBG 650 Advanced Plant Breeding

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

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PBG 650 Advanced Plant Breeding

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  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 estimationin 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 Linear model Y = X + ε Parameter estimates b = (X’X)-1X’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 Design matrices Y = X + Zu + e Fixed effects Random effects Classification for the purposes of BLUP

  7. 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

  8. 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:

  9. Weighted regression Y = X + ε Where εij ~N (0, σ2) b = (X’X)-1X’Y For the barley example When εij ~N (0, Rσ2) Then b = (X’R-1X)-1X’R-1Y

  10. r = 2XY Remember Covariance structure of random effects XY

  11. Mixed Model Equations • each matrix is composed of submatrices • the algebra is the same -1 = Rσ2 Calculations in Excel

  12. Results from BLUP Original data BLUP estimates For fixed effects b1 =  + t1 b2 =  + t2

  13. Interpretation from BLUP 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

  14. 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

  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 -1 = invert the matrix Rσ2 each element of the matrix is a matrix coefficient matrix

  16. Sampling error of BLUP fixed effects random effects

  17. Estimation of Variance Components (would really need a larger data set) • Use your best guess for an initial value ofσε2/σA2 • Solve for  and û • Use current solutions to solve for σε2and then forσA2 • Calculate a newσε2/σA2 • Repeat the process until estimates converge ˆ

  18. 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

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

  20. Covariance of single crosses SC-X is jxkSC-Yis j’xk’ B73, B84, H123 MO17, N197 assuming no epistasis

  21. Covariance of single crosses SC-X is jxkSC-Yis j’xk’ SC-1=B73xMO17 SC-2=H123xMO17 SC-3=B84xN197

  22. Solutions -1 X

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