### Estimation of breeding values

Simulated data of the 14^{th} QTL-MAS workshop was analyzed with univariate and bivariate applications of four different models to predict breeding values for juvenile animals without phenotypes. A linear model was assumed for both the quantitative and binary trait. Using a linear model for binary traits is expected to give breeding values that are highly related to those obtained from a threshold model, when trait incidence is moderate [e.g. 4], which is the case here with a value of 0.30. The first two models used ASREML to estimate variance components:

*y*_{
ij
} = *µ*_{
j
} + *animal*_{
ij
} + *e*_{
ij
}

where

*y*_{
ij
} is the phenotypic record of animal

*i*,

*µ*_{
j
} is the overall mean for trait

*j*,

*animal*_{
ij
} is the random polygenic effect of animal

*i* for trait

*j*, and

*e*_{
ij
} is a random residual for animal

*i*. Model A used a numerator relationship matrix for polygenic effects, while model G used a SNP based genomic relationship matrix. For G, matrix

**G** was calculated as [

5]:

,

where **Z** contained marker genotypes for all animals across loci, being -1 and 1 for either homozygote and 0 for the heterozygote genotype, corrected for allele frequency per locus in the current population.

The third and fourth model were based on Gibbs sampling and included SNP effects, next to the pedigree based relationship matrix:

where *SNP*_{
ijkl
} is a random effect for allele *l* on trait *j* at locus *k* of animal *i*. The difference between those two models is that 1 (BayesA) or 2 (BayesC) distributions for SNP effects are considered, respectively.

SNP effects, denoted as *SNP*_{
ijkl
}, were estimated in BayesA and BayesC as *q*_{
ijkl
}×*v*_{
.k
}[6], where *q*_{
ijkl
} is the effect size of allele *l* at locus *k* and *v*_{
.jk
} is the direction vector for locus *k* that scales the effect at locus *k* for trait *j*. In the original implementation [6], variance of the direction vector *v*_{
jk
}, denoted as **V**, is sampled for each trait *j* separately, without considering covariances between traits across loci. Here, both in BayesA and BayesC, in **V** covariances between traits across loci are considered.

### QTL mapping

BayesC, also known as Bayesian stochastic search variable selection (BSSVS) [7], involved sampling presence of a QTL at each SNP position from a Bernoulli distribution with probability equal to
, where P(**v**_{
j
} | **0**, **V**) is the probability of sampling **v**_{
j
} from N(**0**, **V**), and Pr_{j} is the prior probability of presence of a QTL at SNP position *j*. Pr_{j} was calculated per locus as 50 divided by the total number of SNPs, reflecting that 50 QTL were expected. Posterior QTL probabilities were calculated as proportions of cycles after burn-in that a locus was placed in the distribution with large effects and therefore was sampled from N(**0**, **V**). For more details on prior distributions and fully conditional distributions, see Meuwissen and Goddard [6].

To obtain significance thresholds for posterior QTL probabilities for the bivariate BayesC model, genotypes were permuted 2,000 times against phenotypes and pedigree.