Editing Quantitative genetics (section)

====Half-sibs of the same common-parent (HS)====
The sum of the cross-products '''{ common-parent frequency * half-breeding-value of one half-sib * half-breeding-value of any other half-sib in that same common-parent-group }''' immediately provides one of the required covariances, because the effects used [''breeding values''—representing the allele-substitution expectations] are already defined as deviates from the genotypic mean [see section on "Allele substitution – Expectations and deviations"]. After simplification. this becomes: ''' cov(HS)<sub>A</sub> = {{sfrac|1|2}} pq a<sup>2</sup> = {{sfrac|1|4}} s<sup>2</sup><sub>A</sub> '''.<ref name="Crow & Kimura"/> {{rp|132–141}}<ref name="Falconer 1996"/> {{rp|134–147}} However, the ''substitution deviations'' also exist, defining the sum of the cross-products '''{ common-parent frequency * half-substitution-deviation of one half-sib * half-substitution-deviation of any other half-sib in that same common-parent-group }''', which ultimately leads to: ''' cov(HS)<sub>D</sub> = p<sup>2</sup> q<sup>2</sup> d<sup>2</sup> = {{sfrac|1|4}} s<sup>2</sup><sub>D</sub> '''. Adding the two components gives:

'''cov(HS) = cov(HS)<sub>A</sub> + cov(HS)<sub>D</sub> = {{sfrac|1|4}} s<sup>2</sup><sub>A</sub> + {{sfrac|1|4}} s<sup>2</sup><sub>D</sub> '''.