### Data

An outbred population was simulated with 1,000 generations of 1,000 individuals, which was followed by 30 generations of 150 individuals. The data used in the analysis corresponded to the last generations of the pedigree and contained 20 sire families. Each sire was mated with 10 dams and number of offspring per dam was 15 resulting in 3,000 offspring in total. Both pedigree and phenotypes of sires and dams were not provided. Genomic kinship between sires and dams indicated that sires and dams most likely descended from one population (data not shown). Only 10 out of 15 offspring per full-sib family were phenotyped for a quantitative trait that was normally distributed. All individuals were genotyped for 9,990 SNP, which were equally distributed on 5 chromosomes (size: 1 Morgan each). Monomorphic SNP (n = 2,869) and SNP with MAF (Minor Allele Frequency) <0.01 (n = 383) were excluded from the analysis. A complete description of the simulated data can be found on the website of the 15^{th} QTL-MAS workshop [5].

### QTL analysis and breeding value estimation

The model used for QTL detection and breeding value estimation simultaneously fitted polygenic and SNP effects:

$\mathbf{y}=\mu +\mathbf{Z}\mathbf{a}+{\Sigma}_{k}{\mathbf{X}}_{\mathbf{k}}{\alpha}_{\mathbf{k}}+\mathbf{e},$

where **y** is the quantitative trait and *μ* is the mean; **Z** is the incidence matrix indicating for each observation the (polygenic) genetic effects by which it is influenced; **a** is the (polygenic) genetic effects with $\mathsf{\text{a}}~N\left(0,\mathsf{\text{A}}{\sigma}_{a}^{2}\right)$, where **A** is the numerator relationship matrix between the individuals based on pedigree and ${\sigma}_{a}^{2}$ is the (polygenic) genetic variance; ∑_{
k
}X_{
k
}α_{
k
} fitted additive SNP association effects, where **α**_{
k
} is a vector with allele substitution effects with $~N\left(0,\mathsf{\text{I}}{\sigma}_{{g}_{k}}^{2}\right)$, where **I** is an identity matrix of appropriate dimensions and ${\sigma}_{{a}_{k}}^{2}$ is the additive genetic variance of SNP and **X**_{
k
} is an incidence matrix relating allele substitution effects to observed SNP genotypes; and **e** are residuals with $\mathsf{\text{e}}~N\left(0,\mathsf{\text{I}}{\sigma}_{e}^{2}\right)\phantom{\rule{1em}{0ex}}{}_{{}_{}}^{\mathrm{}}$, where **I** is an identity matrix of appropriate dimensions and ${\sigma}_{e}^{2}$ is the residual variance. SNP were also modelled to have a dominance effect on the simulated quantitative trait as well. However, no significant dominance effect was found (data not shown).

Bayesian Variable Selection implemented in the Bayz software [

4] was used to detect QTL and predict breeding values of individuals without phenotype. The applied Bayesian Variable Selection was similar to the well-known BayesC

*π* method [

6], except prior of

*π* had a uniform(0,1) distribution [

6] while we used a slightly informative prior distribution ~

*Beta*(100,1). In Bayz [

4], shrinkage of allele effects was done by applying a mixture distribution. Many SNP effects were shrunk to nearly zero to obtain high sparsity in SNP effects and only a small part of the SNP effects were less severely shrunken, thereby identifying SNP with important associations. The prior mixture distribution was:

${\alpha}_{k}~\{\begin{array}{c}N(0,{\sigma}_{g0}^{2})withprobability{\pi}_{\text{0}}\\ N(0,{\sigma}_{g1}^{2})withprobability{\pi}_{1}=(1-{\pi}_{0})\end{array},$

where the 'null' distribution modelled the majority of SNP with (virtually) no effect using prior settings *π*_{0} = 0.98 and ${\sigma}_{g0}^{2}=0.00\mathsf{\text{1}}$. The second distribution modelled SNP with large effects where prior settings were *π*_{1} = 0.02 and ${\sigma}_{g1}^{2}=0.\mathsf{\text{1}}$. Variances of the mixture distribution and other model effects were estimated using a uniform prior. A Bernoulli distribution specified probabilities for a SNP belonging to the 'null' or second distribution and proportions for the mixture were set to have a slightly informative prior distribution ~ *Beta*(100,1).

### Applied MCMC techniques

The model estimated a 'mixture indicator' that indicated per MCMC (Markov ChainMonte Carlo) cycle for each SNP whether it was estimated to belong to the 'null' (= 0) or second distribution (= 1). After averaging in the MCMC, a value ranging from 0 to 1 indicated the posterior probability of each SNP to have a large effect (${\widehat{p}}_{i}$).

Most samplers were single site Gibbs samplers. An alternative Metropolis Hastings sampler was used to speed up mixing of estimated SNP variance components. Joint updates for 2 SNP effects and 2 'mixture indicators' were made. The Metropolis Hastings sampler updated the variance of the 'null' and second distribution thereby keeping a constant ratio (1:100) to allow for fast mixing by jointly shrinking or expanding variances together with all SNP effects. Tuning of step size from the Metropolis Hastings sampler was needed to reach an acceptance rate around 0.5.

One MCMC chain of 52,000 cycles with a burn-in period of 2,000 cycles was run, which was found sufficient to obtain accurate estimates (effective number of samples was 39.6 for polygenic genetic variance and >180 for all other model effects).

### Identification of associated SNP

Bayes Factor (BF) was used to identify associated SNP as the odds ratio between the estimated posterior and prior probabilities for a SNP:

$BF=\frac{\raisebox{1ex}{${\widehat{p}}_{i}$}\!\left/ \!\raisebox{-1ex}{$\left(1-{\widehat{p}}_{i}\right)$}\right.}{\raisebox{1ex}{${\pi}_{1}$}\!\left/ \!\raisebox{-1ex}{$(1-{\pi}_{1})$}\right.},$

where ${\widehat{p}}_{i}$ = 'mixture indicator' of a SNP and *π*_{1} = prior 1/101 related to the Beta distribution. Using guidelines from Kass and Raftery [7] to judge BF, a value above 10 was considered as 'strong' evidence. SNP with BF between 3.2 and 10 were considered to be 'putative'.

In case more SNP within a region showed significant association, the size of the region and LD (Linkage Disequilibrium) (r^{2}) among the SNP were used to call a single or multiple underlying QTL. When identified SNP showed clear LD blocks (r^{2} of most SNP ≥0.7), SNP were considered to be associated with the same QTL.