### Phenotypic data

The quantitative trait was measured on 4665 individuals with mean and variance estimated to be 1.36 and 4.42, respectively (Table 1). The generation number and sex of each individual were provided as non-genetic variables that might be included in the analyses. Individuals in generations 4–6 did not have phenotypes available and these individuals were excluded from the linkage analyses. Preliminary analyses revealed that across all generations jointly there was no sex effect on the phenotype, however, in the oldest generation (0) the phenotypic means of males and females differed, i.e., 2.18 versus 0.89 (Table 1). The phenotypic means for generations 0 and 1 were relatively low (1.01) and high (1.47), respectively.

### Marker data

The haplotype data on the 165 individuals of generation 0 were analysed by HapBlock software [2] to identify putative haplotype blocks. Neither this combined analysis of males and females jointly nor the analyses of males (n = 15) and females (n = 150) separately revealed clear Linkage Disequilibrium structures to exist across the genome and therefore a pragmatic thinning of markers was applied. Two subsets from the total of 6000 SNP markers were selected by picking every 10^{th} or 50^{th} SNP along the genome, resulting in 600 or 120 loci, respectively.

### Statistical model for linkage analysis

The QTL was assumed to be bi-allelic, allowing three genotypes to be distinguished, i.e., QQ, Qq, and qq, having genotypic values equal to + *α*, *δ* and -*α*, respectively. The variables *α* and *δ* represent the additive and dominance effects of a single gene. The allele frequency of the positive allele Q is denoted by *f*_{
α
}, and may take any value between 0 and 1 with equal prior probability.

The linear model in our Bayesian analysis is similar to Bink et al. [

3] and may be given as follows,

$y~N\left(X\beta +W{\alpha}_{qtl},{\sigma}_{e}^{2}\right)$

(1)

where *β* is a vector containing an overall mean (*μ*) and all non-genetic variables affecting the trait of interest, i.e., sex and generation. The vectors *α*_{
qtl
}represent the additive and dominant genetic contributions of a QTL. The incidence matrices **X**, **W** connect the phenotypes to non-genetic and QTL variables, respectively. The entry values of matrix **W** depend on the genotype assigned to each individual. For the genotypes {QQ, Qq, qq} these values equal {+1, 0, -1} and {0,1,0}, for additive and dominant effects, respectively. Note that the sign of QTL effects are relative to the QTL genotypes and therefore a QTL cannot be assigned to contribute positively or negatively to the trait. The number of columns in **W** depends on the number of QTL in the model. Treating the number of QTL as a random variable in a Bayesian framework was facilitated by the use of the Reversible Jump sampler [4, 5]. The positions of putative QTL are specified in centiMorgan (cM) [6] and denoted by *λ*_{
QTL
}.

The prior distributions on model parameters were taken similar to those by Bink et al. [3], here we only report results for the prior assumption that the expected number of QTL, i.e., the mean of the Poisson distribution, equals five. The influence of the prior mean appeared to be minimal when model selection was based on Bayes Factors for competing models with different numbers of QTL (results not shown).

### Joint posterior distribution

Let

**P** and

**M** denote the pedigree and marker data, respectively, and

$\theta =\left(\beta ,{\alpha}_{QTL},{\sigma}_{e}^{2}\right)$, then the joint posterior distribution of all unknowns can be written as (omitting matrix

**X**),

$\begin{array}{c}p\left(\theta ,{f}_{\alpha},{N}_{QTL},{\lambda}_{QTL},W|y,M,P\right)\\ \propto p\left(y|\theta ,W\right)p\left(W|{f}_{\partial},{N}_{QTL},{\lambda}_{QTL},M,P\right)p\left(\theta ,{f}_{\alpha},{N}_{QTL},{\lambda}_{QTL}\right)\end{array},$

(2)

where the first term on the right hand side is the conditional distribution of the phenotypic data given all unknowns from (1). The second term is the probability distribution of QTL genotypic states (genotypes) conditional on the number and locations of QTL, the QTL allele frequencies, and the pedigree and marker data. The final term in equation (2) is the joint prior distribution of the model variables.

### Posterior computations

We used the FlexQTL™ software http://www.flexqtl.nl that performs Markov chain Monte Carlo (MCMC) simulation [7–9] to obtain draws from the joint posterior distribution. For all simulations, a Markov chain was executed for 500 K iterations and every 100^{th} iteration samples were stored for posterior inference. The chromosomes were divided into small intervals (1 cM-bins) and the number of QTL per bin per cycle was used to calculate the posterior QTL intensity [10]. This procedure was used independent from the marker density (1 or 5 cM spacing). For the posterior inference on the chromosomal positions of the QTL we use 0.90 Highest Posterior Intensity (abbreviated to HPI90) [3]. Posterior mean and 90% quantiles for QTL effects were computed for those chromosomal bins that contained sufficient intensity (samples).

The samples of QTL genotypes of the first 30 individuals of the dataset, i.e., 15 males and the first 15 females of generation 0, were stored and used to compute posterior probabilities along the genome using 5 cM bins. A color-coding was applied to indicate probability of genotype assignment,

$\{\begin{array}{lll}P\left(QQ|y\right)\hfill & >0.8(0.6)\hfill & dark(light)\phantom{\rule{1em}{0ex}}red\hfill \\ P\left(Qq|y\right)\hfill & >0.8(0.6)\hfill & dark(light)\phantom{\rule{1em}{0ex}}green\hfill \\ P\left(qq|y\right)\hfill & >0.8(0.6)\hfill & dark(light)\phantom{\rule{1em}{0ex}}blue\hfill \\ else\hfill & gray\hfill \end{array}$

(3)

The individuals' genotypes and QTL effects (additive and, if included, dominance) were multiplied to estimate the individuals' genotypic values (or breeding value) along the genome. These breeding values were subsequently weighted by the posterior evidence of a QTL being present at a specific chromosomal bin. A heat-coloring scheme was applied where the degree of redness (blueness) indicated more positive (negative) values. The additive and dominant genetic variance explained by all QTL jointly were calculated as

${\sum}_{j}^{{N}_{QTL}}2\left({f}_{\alpha}(1-{f}_{\alpha}){\left[{\alpha}_{j}+{\delta}_{j}(1-2{f}_{\alpha})\right]}^{2}\right)},$

(4)

${\sum}_{j}^{{N}_{QTL}}\left({\left[2{f}_{\alpha}(1-{f}_{\alpha}){\delta}_{j}\right]}^{2}\right)},$

(5)

where Hardy Weinberg equilibrium was assumed in the initial founder population [11] and linkage equilibrium among QTL.

### Model selection

In respect of model selection, we use Bayes factors [12] as a measure of evidence coming from the data for different QTL models. More specifically twice the natural logarithm (2ln) of a Bayes Factor was used as this was on the same scale as the familiar deviance and likelihood ratio test statistics. The Bayes factor is the ratio of the marginal likelihood under one model to the marginal likelihood under a second model and was computed from the prior and posterior odds ratios for the competing models[12]. The Bayes factors for two competing models can be interpreted as follows: 2ln(BF) = [0–2, 2–5, 5–10, >10] corresponds to [hardly any, positive, strong, decisive] evidence against 1^{st} model, respectively. QTL with positive or stronger evidence are reported in this study.

### Types of genetic models

The default in this study was the additive genetic model with a prior mean for the number of QTL equal to 5, denoted as **Q5a**. This prior mean reflects our expectation that there are likely 5 QTL affecting the quantitative trait in an additive manner. The models in which the QTL affect the trait in both additive and dominant manner are denoted **Q5ad**. As outlined above, we studied two marker densities, i.e., 1 cM and 5 cM, and we explored the power to map QTL when only part of the phenotypic data was used, i.e., only data on the first 2 generations of individuals.