1. Genetic evaluation under parental uncertainty Robert J. Tempelman Michigan State University, East Lansing, MI National Animal Breeding Seminar Series December 6, 2004.

2. Key papers from our lab: Cardoso, F.F., and R.J. Tempelman. 2003. Bayesian inference on genetic merit under uncertain paternity. Genetics, Selection, Evolution 35:469-487. Cardoso, F.F., and R.J. Tempelman. 2004. Genetic evaluation of beef cattle accounting for uncertain paternity. Livestock Production Science 89: 109-120.

3. Multiple sires – The situation Cows are mated with a group of bulls under pasture conditions Common in large beef cattle populations raised on extensive pasture conditions –Accounts for up to 50% of calves in some herds under genetic evaluation in Brazil (~25-30% on average) –Multiple sires group sizes range from 2 to 12+ (Breeding cows group size range from 50 to 300+) Common in commercial U.S. herds. –Potential bottleneck for genetic evaluations beyond the seedstock level (Pollak, 2003).

4. Multiple sires – The situation x x?? Who is the sire?

5. The tabular method for computing genetic relationships Recall basis tabular method for computing the numerator relationship matrix: –Henderson, C.R. 1976. A simple method for computing the inverse of a numerator relationship matrix used in prediction of breeding values. Biometrics 32:69. A = {a ij } where a ij is the genetic relationship between animals i and j. Let parents of j be s j and d j.

6. The average numerator relationship matrix (ANRM) Henderson, C.R. 1988. Use of an average numerator relationship matrix for multiple-sire joining. Journal of Animal Science 66:1614-1621. –a ij is the genetic relationship between animals i and j. Suppose dam of j be known to be d j whereas there are v j different candidate sires (s 1,s 2,…s vj ) with probabilities (p 1,p 2,…p vj ) of being the true sire:

7. Pedigree file example from Henderson (1988) AnimalSiresSire probabilitiesDam 1010 2010 3112 4112 5314 6310 73,50.6, 0.46 81,50.3, 0.74 91,4,50.3, 0.6, 0.16 10114 0 = unknown Could be determined using genetic markers

8. Numerator relationship matrix: symmetric Rest provided in Henderson, 1988 AnimalSiresSire probabilities Dam 73,50.6, 0.46 81,50.3, 0.74 91,4,50.3, 0.6, 0.16 10114 Note if true sire of 7 is 3, a 77 = 1.25; otherwise a 77 = 1.1875

9. How about inferring upon what might be the correct sire? Empirical Bayes Strategy: –Foulley, J.L., D. Gianola, and D. Planchenault. 1987. Sire evaluation with uncertain paternity. Genetics, Selection, Evolution. 19: 83-102. Sire model implementation.

10. Simple sire model AnimalSiresSire probabilities 101 201 311 411 531 631 73,50.6, 0.4 81,50.3, 0.7 91,4,50.3, 0.6, 0.1 1011 y =X  + Zs + e

11. One possibility: Substitute sire probabilities for elements of Z. AnimalSiresSire probabilities 101 201 311 411 531 631 73,50.6, 0.4 81,50.3, 0.7 91,4,50.3, 0.6, 0.1 1011

12. Strategy of Foulley et al. (1987) : Posterior probabilities using provided sire probabilities as “prior” probabilities and y to estimate elements of Z. - computed iteratively Limitation: Can only be used for sire models.

13. Inferring upon elements of design matrix Where else is this method currently used? Segregation analysis –Estimating allelic frequencies and genotypic effects for a biallelic locus WITHOUT molecular marker information. –Prior probabilities based on HW equilibrium for base population. –Posterior probabilities based on data. –Reference: Janss, L.L.G., R. Thompson., J.A.M. Van Arendonk. 1995. Application of Gibbs sampling for inference in a mixed major gene-polygenic inheritance model in animal populations. Theoretical and Applied Genetics 91: 1137-1147.

14. Another strategy (most commonly used) Use phantom groups (Westell et al., 1988; Quaas et al., 1988). Used commonly in genetic evaluation systems having incomplete ancestral pedigrees in order to mitigate bias due to genetic trend. –Limitations (applied to multiple sires): 1.Assumes the number of candidate sires is effectively infinite within a group. 2.None of the phantom parents are related. 3.Potential confounding problems for small groups (Quaas, 1988).

15. The ineffectiveness of phantom grouping for genetic evaluations in multiple sire pastures: Perez-Enciso, M. and R.L. Fernando. 1992. Genetic evaluation with uncertain parentage: A comparison of methods. Theoretical and Applied Genetics 84:173-179. Sullivan, P.G. 1995. Alternatives for genetic evaluation with uncertain paternity. Canadian Journal of Animal Science 75:31-36. –Greater selection response using Henderson’s ANRM relative to phantom grouping (simulation studies). – Excluding animals with uncertain paternity reduces expected selection response by as much as 37%.

16. 1.To propose a hierarchical Bayes animal model for genetic evaluation of individuals having uncertain paternity 2.To estimate posterior probabilities of each bull in the group being the correct sire of the individual 3.To compare the proposed method with Henderson’s ANRM via 1.Simulation study 2.Application to Hereford PWG and WW data. Uncertain paternity - objectives

17. Animal genetic values – a Uncertain paternity - hierarchical Bayes model 1 st stage Data - y (Performance records) Non-genetic effects -  (Contemporary groups, age of dam, age of calf, gender) Residual terms - e (assumed to be normal) y = X  + Za + e; e ~N (0,I  e 2 )

18. Uncertain paternity - hierarchical Bayes model 2 nd stage Non-genetic effects Animal genetic values Residual Variance Prior knowledge based on literature information sire assignments (s) genetic variance (  a 2 ) (Co)variances based on relationship (A), sire assignments (s) and genetic variance (  a 2 ) Prior means based on literature information Variance based on the reliability of prior information  ~N (  o,V  )a|s ~N (0,A s  a 2 )  e 2 ~ s e 2  

19. Uncertain paternity - hierarchical Bayes model 3 rd stage sire assignments Prior knowledge based on literature information Probability for sire assignments (  j ) genetic variance  a 2 ~ s a 2   a)  Could be based on marker data.

20. Uncertain paternity - hierarchical Bayes model 4 th stage Specifying uncertainty for probability of sire assignments assignments e.g. How sure are you about the prior probabilities of 0.6 and 0.4 for Sires 3 and 5, respectively, being the correct sire? Assessment based on how much you trust the genotype based probabilities. Could also model genotyping error rates explicitly (Rosa, G.J.M, Yandell, B.S., Gianola, D. A Bayesian approach for constructing genetics maps when markers are miscoded. Genetics, Selection, Evolution 34:353-369) Dirichlet prior

21. Uncertain paternity - joint posterior density 2 nd stage Genetic effects Residual error Prior knowledge based on literature information (Co)variances (relationship, sire assignments and genetic variances) Prior means (literature information) Variance (reliability of priors) 1 st stage Data Prior knowledge based on literature information 3 rd stage Non-genetic fixed effects Markov chain Monte Carlo (MCMC) Prior probability for sire assignments Reliability of priors 4 th stage

22. Simulation Study (Cardoso and Tempelman, 2003) Generation 0 Base population Selection (20 sires & 100 dams) Breeding population Selection (15 sires & 75 dams) Selection (5 sires & 25 dams) Breeding population Offspring (500 animals) Random mating (inbreeding avoided) 1...... Offspring (360 animals) 5...... Selection (15 sires & 75 dams) Selection (5 sires & 25 dams) Breeding population Offspring (500 animals) Random mating (inbreeding avoided) 2 Totals: 80 sires, 400 dams, 2000 non-parents.

23. Paternity assignment Offspring UncertainCertain Random Assignment to Paternity Condition.3.7 Assignment to Multiple Sire Groups.2.3.2.1 2346810 Within the assigned group one of the sire is picked to be the true sire (with equal or unequal probabilities) Sire Record: Sires averaged 23.6 progeny, Dams averaged 5.9 progeny

24. Simulated traits: Ten datasets generated from each of two different types of traits: –Trait 1 (WW): –Trait 2 (PWG): Naïve prior assignments: i.e. equal prior probabilities to each candidate sire (i.e. no information based on genetic markers available)

25. Posterior probabilities of sire assignments being equal to true sires Multiple-sire group size Animal Category 2346810 Trait 1 Parents0.5250.3490.2690.1830.1270.110 Non-parents0.5170.3450.2680.1780.1340.105 Trait 2 Parents0.5210.3520.2800.1880.1380.111 Non-parents0.5400.3600.2890.1910.1430.111

26. Rank correlation of predicted genetic effects ANRM = Henderson’s ANRM HIER = proposed model TRUE = all sires known Sidenote: Model fit criteria was clearly in favor of HIER over ANRM

27.  Data set  3,402 post-weaning gain records on Hereford calves raised in southern Brazil (from 1991-1999)  4,703 animals  Paternity (57% certain; 15% uncertain & 28% unknown-base animals)  Group sizes 2, 3, 4, 5, 6, 10, 12 & 17  Methods  ANRM (average relationship)  HIER (uncertain paternity hierarchical Bayes model) Uncertain paternity - application to field data

28. Parameter a Posterior median95% Credible Set ANRM 0.231(0.153, 0.316) 73.8(48.0, 103.6) 246.5(221.5, 271.2) 404.5(334.3, 494.0) HIER 0.244(0.162, 0.336) 78.2(51.1, 111.2) 242.9(216.5, 268.2) 404.5(333.9, 493.8) Posterior inference for PWG genetic parameters under ANRM versus HIER models

29. Model choice criteria (DIC and PBF) decisively favored HIER over ANRM Very high rank correlations between genetic evaluations using ANRM versus HIER Some non-trivial differences on posterior means of additive genetic value for some animals Uncertain paternity - Results summary

30. Standard deviation of additive genetic effects Uncertain paternity - assessment of accuracy (PWG) Sire with 50 progeny Sire with 9 progeny i.e. accuracies are generally slightly overstated with Henderson’s ANRM

31. Conclusions Uncertain paternity modeling complements genetic marker information (as priors) –Reliability on prior information can be expressed (via Dirichlet). Little advantage over the use of Henderson’s ANRM. –However, accuracies of EPD’s overstated using ANRM. –Power of inference may improve with better statistical assumptions (i.e. heterogeneous residual variances)

32. Implementation issues Likely require a non-MCMC approach to providing genetic evaluations. Some hybrid with phantom grouping may be likely needed. –Candidate sires are not simply known for some animals. Bob Weaber’s talk.