In this section we present the traditional single locus VC model that includes dominance effects of the QTL and where all base QTL allele effects are assumed to be uncorrelated [13, 14]. Thereafter, we present our FIA model which was previously developed for additive QTL effects [3] and show how dominance can be included.

### Traditional VC model including dominance QTL effects

The VC model including QTL effects with dominance is given by:

where

*y* is the vector of individual phenotypes (length

*n*),

*b* is a vector of fixed effects and

*X* is the corresponding design matrix,

*v* is a vector of additive random individual QTL effects (length

*n*) in position

*τ*,

*d* is a vector of random individual QTL effects for dominance (length

*n*), and

*e* is a vector of residual effects (length

*n*). The variance-covariance matrix of

*y*, assuming independent allelic effects in the base generation, is (e.g. [

15]):

where Π is the genotype IBD-matrix (size

*n* ×

*n*) calculated in position

*τ*,

is the corresponding genotype QTL variance for additive effects, Δ is the dominance IBD-matrix (size

*n* ×

*n*) calculated in position

*τ*,

is the QTL variance for dominance effects,

*I* is the identity matrix of size

*n* ×

*n*, and

is the residual variance. An element in row

*i* and column

*j* of Δ can be calculated directly from the gametic IBD-matrix (e.g. [

16]) as:

where the values *g*
_{
ij
}(*k*, *l*) are the gametic IBDs between individual *i* and *j* for the maternal/paternal alleles *k* and *l*.

### Including dominance in the VC QTL model

Rönnegård and Carlborg [

17] described the VC model in eq.

1 in terms of independent base generation effects, where:

Here

*v** is a vector of base generation allele effects and

*d** is a vector of dominance effects for all pairwise base allele combinations. These dominance effects are assumed to be randomly sampled from an infinite population of dominance effects with a variance of

. Furthermore the random dominance effects for homozygotes and heterozygotes are assumed to be sampled from the same distribution. The incidence matrices

*Z* and

*W* relate individuals with their corresponding additive and dominance effects. We thereby have a variance-covariance matrix for the random effects given by:

Moreover, with this notation we have the relationships (see [

17])

Hence, for a single QTL model there is no covariance between additive and dominance effects. The estimates of
and
may be strongly correlated, however, since the IBD-values in Π and Δ are correlated [9].

### FIA model with additive effects

FIA extends the traditional VC model to include within-line correlations of the QTL allele effects. The FIA model without dominance effects is given by [

3]:

where the variance-covariance matrix of

*y* is:

Here, Π_{
I
}is the genotypic IBD-matrix assuming independent QTL allele effects in the base generation and Π_{
J
}is the IBD-matrix that assumes fixation of QTL alleles within founder lines. Hence, the analysis using FIA requires an IBD estimation program that allows for different base generation structures. We used the same IBD-matrix estimation program as in [3], which is based on the deterministic algorithm published by [16].

### FIA model with additive and dominance effects

Dominance is included in FIA by using the same linear model as in (1) but the variance-covariance matrix is not the same as in (2):

where the variance-covariance matrix of

*y* is:

Here, Δ_{
I
}is the dominance IBD-matrix assuming independent QTL allele effects in the base generation and Δ_{
J
}is the dominance IBD-matrix that assumes fixation of QTL alleles within founder lines. The above formula for the variance-covariance matrix *V* was derived following the derivation of eq. (4) in Rönnegård et al. [3].

We let the variance components be independent of each other. This assumption gives the variance-covariance matrix of *y* as a linear function of the variance components. This is a simplification since
is the same within-line correlation as
and the variance-covariance matrix of *y* is not strictly a linear function of the variance components.

### Calculating the score for the FIA model

FIA utilizes the score statistic [

18–

20]

where *D* is the gradient and *F* is the information matrix calculated under the null hypothesis of no QTL effects, i.e.
.

The elements of the gradient

*D* of the log-likelihood function

*L* are given by [

21]:

where

and

. The partial derivatives of

*V* are:

, and

. Furthermore,

*P* is the projection matrix given by:

The elements of the information matrix

*F* are given by [

21]:

### Calculation of genome-wide significance thresholds

The significance thresholds for the genome scan were calculated by means of permutation testing (as in [3]). Residuals were calculated from a null model assuming no QTL effect. These residuals were then permuted giving a new vector *ĕ*. Replicates of the phenotypic data were simulated with
where
is the vector of fixed effects estimated from the null model *y* = *Xb* + *e*. For each replicate, the score statistic was calculated at every tested position (5 cM apart) along the genome using 12. The empirical distribution of the maximum score value from each replicate was used to obtain significance thresholds. 2000 replicates were simulated.

### Simulation setup

In the power analyses, level of fixation within founder lines and degree of dominance were varied to evaluate the differences between FIA and HK-regression. The methods were compared by their power to detect a QTL at a given position at a 5% significance level.

The structure for the base generation was designed to mimic the pedigree of a Red Jungle Fowl – White Leghorn *F*
_{2} Cross [11] with one Jungle Fowl male mated to three Leghorn females, and 800 *F*
_{2} individuals. Four different cases (Table 1) were studied by varying the fixation level within lines for a biallelic QTL. The QTL was simulated at a position having a fully-informative marker so that the QTL alleles could be traced through the pedigree unambiguously.

The phenotype of an *F*
_{2} individual *i* was simulated with *y*
_{
i
}= *A*
_{1i
}+ *A*
_{2i
}+ *D*
_{
i
}+ *e*
_{
i
}where *A*
_{1i
}is the QTL allele effect on the paternally inherited chromosome and *A*
_{2i
}is the QTL allele effect on the maternally inherited chromosome, *D*
_{
i
}is the dominance effect and *e*
_{
i
}is an iid normally distributed residual effect with a variance equal to 98. A biallelic QTL was simulated where the additive effects for the two alternative alleles were 0 and *a*, and the dominance effects for heterozygotes was *d*. The values of *a* and *d* were varied from 0 to 2.

6000 replicates were calculated for each of the four cases in Table 1 and for varying degrees of dominance.

### Analysis of experimental data: Red Jungle Fowl × White Leghorn *F*
_{2} Cross

In a Red Jungle Fowl × White Leghorn F2 cross, we performed a full genome scan using FIA with additive and dominance effects. In this pedigree, one Red Jungle Fowl male was mated to three White Leghorn females producing 756 *F*
_{2} offspring with measured genotypes and phenotypes. We used an updated marker map to those reported in [11] including 439 markers (Leif Andersson, personal communication) covering chromosomes 1 to 28. We analyzed body weight at 200 days of age. In our previous study using FIA with only additive effects we found six QTL at a 5% genomwide significance. These QTL were located at: 102 cM on chromosome 1, 488 cM on chromosome 1, 32 cM on chromosome 5, 30 cM on chromosome 6, 21 cM on chromosome 27 and 35 cM on chromosome 28. The data are described in detail in [11].