### Additive-dominance Model

Randomly select two heterozygous lines as parents P

_{1} and P

_{2} to produce a full-sib family, in which a QTL will form four genotypes if the two lines have completely different allele systems. Let

*μ*
_{
uv
}be the value of a QTL genotype inheriting allele

*u* (

*u =* 1,2) from parent P

_{1} and allele

*v* (

*v* = 3, 4) from parent P

_{2}. Based on quantitative genetic theory, this genotypic value can be partitioned into the additive and dominant effects as follows:

where *μ* is the overall mean, *α*
_{
u
}and *β*
_{
v
}are the allelic (additive) effects of allele *u* and *v*, respectively, and *γ*
_{
uv
}is the interaction (dominant) effect at the QTL. Considering all possible alleles and allele combinations between the two parent, there are a total of four additive effects (*α*
_{1} and *α*
_{2} from parent P_{1} and *β*
_{3} and *β*
_{4} from parent P_{2} and four dominant effects (*γ*
_{13}, *γ*
_{14}, *γ*
_{23} and *γ*
_{34}). But these additive and dominant effects are not independent and, therefore, are not estimable. After parameterization, there are two independent additive effects, *α* = *α*
_{1} = -*α*
_{2} and *β*
_{3} = *β*
_{3} = -*β*
_{4}, and one dominant effect, *γ* = *γ*
_{13} = -*γ*
_{14} = -*γ*
_{23} = *γ*
_{24}, to be estimated.

Let

**u** = (

*μ*
_{
uv
})

_{4 × 1} and

**a** = (

*μ, α, β, γ*)

^{T}, which can be connected by a design matrix

**D**. We have

The expression of a can be obtained from the expression of

**u** by

### Additive-dominance-epistatic Model

If there are two segregating QTL in the full-sib family, the epistatic effects due to their nonallelic interactions should be considered. The theory for epistasis in an inbred family [

16] can be readily extended to specify different epistatic components for outbred crosses. Consider two epistatic multiallelic QTL, each of which has four different genotypes, 13, 14, 23, and 24, in the outbred progeny. Let

be the genotypic value for QTL genotype

*u*
_{1}
*v*
_{1}/

*u*
_{2}
*v*
_{2} for

*u*
_{1},

*u*
_{2} = 1,2 and

*v*
_{1},

*v*
_{2} = 3,4 and

be the corresponding mean vector. The two-QTL genotypic value is partitioned into different components as follows:

where

(1) *μ* is the overall mean;

(2) *α*
_{1} is the additive effect due to the substitution from allele 1 to 2 at the first QTL;

(3) *β*
_{1} is the additive effect due to the substitution from allele 3 to 4 at the first QTL;

(4) *γ*
_{1} is the dominant effect due to the interaction between alleles from different parents;

(5) *α*
_{2} is the additive effect due to the substitution from allele 1 to 2 at the second QTL;

(6) *β*
_{2} is the additive effect due to the substitution from allele 3 to 4 at the second QTL;

(7) *γ*
_{2} is the dominant effect due to the interaction between alleles from different parents;

(8) *I*
_{
αα
}is the additive × additive epistatic effect due to the interaction between the substitutions from allele 1 to 2 at the first and second QTLs;

(9) *I*
_{
αβ
}is the additive × additive epistatic effect due to the interaction between the substitutions from allele 1 to 2 at the first QTL and from allele 3 to 4 at the second QTL;

(10) *I*
_{
βα
}is the additive × additive epistatic effect due to the interaction between the sub-stitutions from allele 3 to 4 at the first QTL and from allele 1 to 2 at the second QTL;

(11) *I*
_{
αβ
}is the additive × additive epistatic effect due to the interaction between the sub-stitutions from allele 3 to 4 at the first and second QTLs;

(12) *J*
_{
αγ
}is the additive × dominant epistatic effect due to the interaction between the substitutions from allele 1 to 2 at the first QTL and the dominant effect at the second QTL;

(13) *J*
_{
βγ
}is the additive × dominant epistatic effect due to the interaction between the substitutions from allele 3 to 4 at the first QTL and the dominant effect at the second QTL;

(14) *K*
_{
γα
}is the dominant × additive epistatic effect due to the interaction between the dominant effect at the first QTL and the substitutions from allele 1 to 2 at the second QTL;

(15) *K*
_{
γβ
}is the dominant × additive epistatic effect due to the interaction between the dominant effect at the first QTL and the substitutions from allele 3 to 4 at the second QTL;

(16) *L*
_{
γγ
}is the dominant × dominant epistatic effect due to the interaction between the dominant effects at the first and second QTLs.

Genetic effect parameters for two interacting QTL are arrayed in

**a** = (

*μ, α*
_{1},

*β*
_{1},

*γ*
_{1},

*α*
_{2},

*β*
_{2},

*γ*
_{2},

*I*
_{
αα
},

*I*
_{
αβ
},

*I*
_{
βα
},

*I*
_{
ββ
},

*J*
_{
αγ
},

*J*
_{
βγ
},

*K*
_{
γα
},

*K*
_{
γβ
},

*L*
_{
γγ
})

^{T}. We relate the genotypic value vector and genetic effect vector by

Thus, the genetic effect vector can be expressed, in terms of the genotypic value vector, as

If we have alleles 1 = 3 and 2 = 4 for an outbred family, Equations 1 and 3 will be reduced to traditional biallelic additive-dominant and biallelic additive-dominant-epistatic genetic models, respectively [20].