Genetic evaluation under parental uncertainty

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Genetic evaluation under parental uncertainty

• Robert J. Tempelman
• Michigan State University, East Lansing, MI
• National Animal Breeding Seminar Series
• December 6, 2004.

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Key papers from our lab

• Cardoso, F.F., and R.J. Tempelman. 2003.
  Bayesian inference on genetic merit under
  uncertain paternity. Genetics, Selection,
  Evolution 35469-487.
• Cardoso, F.F., and R.J. Tempelman. 2004. Genetic
  evaluation of beef cattle accounting for
  uncertain paternity. Livestock Production
  Science 89 109-120.

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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).

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Multiple sires The situation
x
x
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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 3269.
• A aij where aij is the genetic relationship
  between animals i and j. Let parents of j be sj
  and dj.

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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 661614-1621.
• aij is the genetic relationship between animals i
  and j. Suppose dam of j be known to be dj
  whereas there are vj different candidate sires
  (s1,s2,svj) with probabilities (p1,p2,pvj) of
  being the true sire

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Pedigree file example from Henderson (1988)
0 unknown
Animal Sires Sire probabilities Dam
1 0 1 0
2 0 1 0
3 1 1 2
4 1 1 2
5 3 1 4
6 3 1 0
7 3,5 0.6, 0.4 6
8 1,5 0.3, 0.7 4
9 1,4,5 0.3, 0.6, 0.1 6
10 1 1 4
Could be determined using genetic markers
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Numerator relationship matrix
Rest provided in Henderson, 1988
Animal Sires Sire probabilities Dam
7 3,5 0.6, 0.4 6
8 1,5 0.3, 0.7 4
9 1,4,5 0.3, 0.6, 0.1 6
10 1 1 4
symmetric
Note if true sire of 7 is 3, a77 1.25
otherwise a77 1.1875
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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.

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Simple sire model
y Xb Zs e
Animal Sires Sire probabilities
1 0 1
2 0 1
3 1 1
4 1 1
5 3 1
6 3 1
7 3,5 0.6, 0.4
8 1,5 0.3, 0.7
9 1,4,5 0.3, 0.6, 0.1
10 1 1
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One possibility Substitute sire probabilities
for elements of Z.
Animal Sires Sire probabilities
1 0 1
2 0 1
3 1 1
4 1 1
5 3 1
6 3 1
7 3,5 0.6, 0.4
8 1,5 0.3, 0.7
9 1,4,5 0.3, 0.6, 0.1
10 1 1
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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.
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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.

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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)
• Assumes the number of candidate sires is
  effectively infinite within a group.
• None of the phantom parents are related.
• Potential confounding problems for small groups
  (Quaas, 1988).

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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
  84173-179.
• Sullivan, P.G. 1995. Alternatives for genetic
  evaluation with uncertain paternity. Canadian
  Journal of Animal Science 7531-36.
• Greater selection response using Hendersons ANRM
  relative to phantom grouping (simulation
  studies).
• Excluding animals with uncertain paternity
  reduces expected selection response by as much as
  37.

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Uncertain paternity - objectives

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 Hendersons
   ANRM via
4. Simulation study
5. Application to Hereford PWG and WW data.

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Uncertain paternity -hierarchical Bayes model

• 1st stage

Data - y (Performance records)
y Xb Za e e N (0,Ise2)
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Uncertain paternity -hierarchical Bayes model

• 2nd stage

Residual Variance
Non-genetic effects
Animal genetic values
b N (bo,Vb)
as N (0,Assa2)
se2 se2cn-2
(Co)variances based on relationship (A), sire
assignments (s) and genetic variance (sa2)
Prior means based on literature information
Variance based on the reliability of prior
information
Prior knowledge based on literature information
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Uncertain paternity -hierarchical Bayes model

• 3rd stage

sire assignments
genetic variance
sa2 sa2cn(a)-2
Probability for sire assignments (pj)
Prior knowledge based on literature information
Could be based on marker data.
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Uncertain paternity -hierarchical Bayes model

• 4th stage

Specifying uncertainty for probability of sire
assignments
Dirichlet prior
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 34353-369)
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Uncertain paternity -joint posterior density
Data
1st stage
Residual error
Genetic effects
Non-genetic fixed effects

• 2nd stage

Prior knowledge based on literature information
Prior means (literature information) Variance
(reliability of priors)
(Co)variances (relationship, sire assignments and
genetic variances)
3rd stage
Prior probability for sire assignments
Prior knowledge based on literature information
4th stage
Reliability of priors
Markov chain Monte Carlo (MCMC)
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Simulation Study (Cardoso and Tempelman, 2003)
Totals 80 sires, 400 dams, 2000 non-parents.
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Paternity assignment
Sires averaged 23.6 progeny, Dams averaged 5.9
progeny
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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)
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Posterior probabilities of sire assignments being
equal to true sires
Multiple-sire group size Multiple-sire group size Multiple-sire group size Multiple-sire group size Multiple-sire group size Multiple-sire group size
Animal Category 2 3 4 6 8 10
Trait 1 Trait 1
Parents 0.525 0.349 0.269 0.183 0.127 0.110
Non-parents 0.517 0.345 0.268 0.178 0.134 0.105
Trait 2 Trait 2
Parents 0.521 0.352 0.280 0.188 0.138 0.111
Non-parents 0.540 0.360 0.289 0.191 0.143 0.111
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Rank correlation of predicted genetic effects
ANRM Hendersons ANRM HIER proposed
model TRUE all sires known
Sidenote Model fit criteria was clearly in favor
of HIER over ANRM
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Uncertain paternity -application to field data

• 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)

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Posterior inference for PWG genetic parameters
under ANRM versus HIER models
Parametera Posterior median 95 Credible Set 95 Credible Set
ANRM
0.231 (0.153, 0.316) (0.153, 0.316)
73.8 (48.0, 103.6) (48.0, 103.6)
246.5 (221.5, 271.2) (221.5, 271.2)
404.5 (334.3, 494.0) (334.3, 494.0)
HIER
0.244 (0.162, 0.336) (0.162, 0.336)
78.2 (51.1, 111.2) (51.1, 111.2)
242.9 (216.5, 268.2) (216.5, 268.2)
404.5 (333.9, 493.8) (333.9, 493.8)
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Uncertain paternity -Results summary

• 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

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Uncertain paternity -assessment of accuracy (PWG)

• Standard deviation of additive genetic effects

i.e. accuracies are generally slightly overstated
with Hendersons ANRM
Sire with 9 progeny
Sire with 50 progeny
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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 Hendersons
  ANRM.
• However, accuracies of EPDs overstated using
  ANRM.
• Power of inference may improve with better
  statistical assumptions (i.e. heterogeneous
  residual variances)

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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 Weabers talk.