. Author manuscript; available in PMC: 2010 Nov 8.

Published in final edited form as: Mamm Genome. 2009 Aug 6;20(9-10):681–698. doi: 10.1007/s00335-009-9209-2

Table 1.

The two-locus genetic-effect design matrix (S_AB)

Genotypic values	Coefficients
	R	a_A	d_A	i_A	a_B	d_B	i_B	aa	ad	ai	da	dd	di	ia	id	ii
G₁₁₁₁	1	1	$- \frac{1}{2}$	0	1	$- \frac{1}{2}$	0	1	$- \frac{1}{2}$	0	$- \frac{1}{2}$	$\frac{1}{4}$	0	0	0	0
G₁₂₁₁	1	0	$\frac{1}{2}$	1	1	$- \frac{1}{2}$	0	0	$\frac{1}{2}$	1	0	$- \frac{1}{4}$	$- \frac{1}{2}$	0	0	0
G₂₁₁₁	1	0	$\frac{1}{2}$	−1	1	$- \frac{1}{2}$	0	0	$\frac{1}{2}$	−1	0	$- \frac{1}{4}$	$\frac{1}{2}$	0	0	0
G₂₂₁₁	1	−1	$- \frac{1}{2}$	0	1	$- \frac{1}{2}$	0	−1	$- \frac{1}{2}$	0	$\frac{1}{2}$	$\frac{1}{4}$	0	0	0	0
G₁₁₁₂	1	1	$- \frac{1}{2}$	0	0	$\frac{1}{2}$	1	0	0	0	$\frac{1}{2}$	$- \frac{1}{4}$	0	1	$- \frac{1}{2}$	0
G₁₂₁₂	1	0	$\frac{1}{2}$	1	0	$\frac{1}{2}$	1	0	0	0	0	$\frac{1}{4}$	$\frac{1}{2}$	0	$\frac{1}{2}$	1
G₂₁₁₂	1	0	$\frac{1}{2}$	−1	0	$\frac{1}{2}$	1	0	0	0	0	$\frac{1}{4}$	$- \frac{1}{2}$	0	$\frac{1}{2}$	−1
G₂₂₁₂	1	−1	$- \frac{1}{2}$	0	0	$\frac{1}{2}$	1	0	0	0	$- \frac{1}{2}$	$- \frac{1}{4}$	0	−1	$- \frac{1}{2}$	0
G₁₁₂₁	1	1	$- \frac{1}{2}$	0	0	$\frac{1}{2}$	−1	0	0	0	$\frac{1}{2}$	$- \frac{1}{4}$	0	−1	$\frac{1}{2}$	0
G₁₂₂₁	1	0	$\frac{1}{2}$	1	0	$\frac{1}{2}$	−1	0	0	0	0	$\frac{1}{4}$	$\frac{1}{2}$	0	$- \frac{1}{2}$	−1
G₂₁₂₁	1	0	$\frac{1}{2}$	−1	0	$\frac{1}{2}$	−1	0	0	0	0	$\frac{1}{4}$	$- \frac{1}{2}$	0	$- \frac{1}{2}$	1
G₂₂₂₁	1	−1	$- \frac{1}{2}$	0	0	$\frac{1}{2}$	−1	0	0	0	$- \frac{1}{2}$	$- \frac{1}{4}$	0	1	$\frac{1}{2}$	0
G₁₁₂₂	1	1	$- \frac{1}{2}$	0	−1	$- \frac{1}{2}$	0	−1	$\frac{1}{2}$	0	$- \frac{1}{2}$	$\frac{1}{4}$	0	0	0	0
G₁₂₂₂	1	0	$\frac{1}{2}$	1	−1	$- \frac{1}{2}$	0	0	$- \frac{1}{2}$	−1	0	$- \frac{1}{4}$	$- \frac{1}{2}$	0	0	0
G₂₁₂₂	1	0	$\frac{1}{2}$	−1	−1	$- \frac{1}{2}$	0	0	$- \frac{1}{2}$	1	0	$- \frac{1}{4}$	$\frac{1}{2}$	0	0	0
G₂₂₂₂	1	−1	$- \frac{1}{2}$	0	−1	$- \frac{1}{2}$	0	1	$\frac{1}{2}$	0	$\frac{1}{2}$	$- \frac{1}{4}$	0	0	0	0

Because the genetic-effect design matrix gives the relationship between the vector of genotypic values (G_AB) and the vector of genetic effects (E_AB), i.e., G_AB = S_ABE_AB, we label the columns and rows to reflect this relationship. Thus, the rows are labeled to reflect the corresponding genotypic value (G_ijkl, where ij are the paternally and maternally inherited alleles at the A locus, while kl are the paternally and maternally inherited alleles at the B locus) and the columns are labeled to reflect the 16 effects in the two-locus genetic-effect vector E_AB. The first column is reference point for the model (R). The other 15 columns correspond to the single-locus (additive, dominance, and imprinting) effects of the A locus (a_A, d_A, and i_A), the B locus (a_B, d_B, and i_B), and the interaction between these single-locus effects (e.g., id is the interaction between the imprinting effect of the A locus and the dominance effect of the B locus and therefore is referred to as imprinting-by-dominance epistasis). The order of the columns has been rearranged to conform to a more natural order (e.g., as in Kao and Zeng 2002; Zeng et al. 2005), but the natural order of the columns can be seen in Appendix 1. These index scores are used in a regression model to estimate each of the 16 parameters and are given as X(y)_ijkl, where y indicates the parameter (e.g., a_A, a_B, etc.) and ijkl indicates the particular genotypic value. For example, X(a_A)₁₁₁₁ would be the a_A index score for the G₁₁₁₁ genotypic value and would have the value 1