1	Efficient Estimation of Breeding Values from Dense Genomic Data
2	Genomic Calculations Genotypes soon available from BFGL: 50,000 SNPs / animal 3,000 animals, many more possible Need efficient computing algorithms Traditional PTAs available from AIPL: PTAs combine phenotypes and pedigree SNP effects evaluated in second step using deregressed PTAs weighted by reliability
3	Genomic Computer Programs Simulate SNPs and QTLs Compare SNP numbers, size of QTLs Calculate genomic EBVs Use selection index, G instead of A Use iteration on data for SNP effects Form haplotypes from genotypes? Not tested yet, SNP regression used
4	Simulation Program Save memory by processing each chromosome separately 3,000 Holstein bulls to genotype 17,000 ancestors in pedigree file 1 billion (20,000 x 50,000 SNPs) genotypes simulated per replicate Only 150 million (3,000 x 50,000) genotypes stored for evaluation
5	Linear Estimates using Markers Selection index equations for EBV u^ = Cov(u,y) Var(y)^-1 (y – Xb) u^ = Z Z’ [Z Z’ + R]^-1 (y – Xb) R has diagonals = (1 / Reliability) - 1 BLUP equations for marker effects, sum to get EBV u^ = Z [Z’R^-1Z + I k]^-1 Z’R^-1(y – Xb) k = var(u) / var(m)
6	Non-linear vs Linear Models
7	Marker Effect Prior Distribution Nonlinear Model
8	Iteration on Data Simple trick to reduce time from quadratic to linear with # SNPs Sum coefficients x solutions once Sum – diagonal = 3 off-diagonals Janss and de Jong, 1999 conference Rediscovered by Legarra and Misztal Elements of Z are –p and (1 – p), where p is frequency of 2^nd allele
9	Computer Memory Inversion including G matrix Animals x markers to hold genotypes Animals² to hold elements of G <1 Gbyte for 50,000 SNPs, 3000 bulls Iteration on genotype data Markers + animals <.1 Gbyte for 50,000 SNPs, 3000 bulls Little memory required for either
10	Computing Times Inversion including G matrix Animals² x markers to form G matrix Animals³ to invert selection index 10 hours for 3000 bulls, 50,000 SNPs Iteration on genotype data Markers x animals x iterations 16 hours for 1000 iterations .997 correlation with inversion
11	Convergence with iteration on data Jacobi iteration Use previous round coefficients x solutions Adaptive under-relaxation Increase relax if convergence improving Decrease relax (each round) if diverging Solution convergence reasonable SD of change < .0001 after 350 rounds SD of change < .000001 after 1700 rounds
12	Potential Results Simulation of 50,000 SNPs, 100 QTLs
13	Reliability from Genotyping Daughter equivalents DE_Total = DE_PA + DE_Prog + DE_YD + DE_G DE_G is additional DE from genotype REL = DE_total / (DE_Total + k) Gains in reliability DE_G could be about 15 for Net Merit More for traits with low heritability Less for traits with high heritability
14	Conclusions Predictions from 50,000 SNPs using: Selection index equations, or Iteration on genotype data Predictions correlated by up to .9999 Linear and nonlinear costs OK Convergence within 200 to 2500 rounds Nonlinear regression improved reliabilities Real data predictions available soon