Editing Quantitative genetics (section)

==== One-parent and offspring (PO)====
This can be derived as the ''sum of cross-products''  between parent gene-effects and ''one-half'' of the progeny expectations using the allele-substitution approach. The ''one-half'' of the progeny expectation accounts for the fact that ''only one of the two parents'' is being considered. The appropriate parental gene-effects are therefore the second-stage redefined gene effects used to define the genotypic variances earlier, that is: '''a{{pprime}} = 2q(a − qd)''' and '''d{{pprime}} = (q-p)a + 2pqd''' and also '''(-a){{pprime}} = -2p(a + pd)''' [see section "Gene effects redefined"]. Similarly, the appropriate progeny effects, '''''for allele-substitution expectations''''' are one-half of the earlier ''breeding values'', the latter being: '''a<sub>AA</sub> = 2qa''', and '''a<sub>Aa</sub> = (q-p)a''' and also '''a<sub>aa</sub> = -2pa''' [see section on "Genotype substitution – Expectations and Deviations"].

Because all of these effects are defined already as deviates from the genotypic mean, the cross-product sum using {'''genotype-frequency * parental gene-effect * half-breeding-value'''} immediately provides the ''allele-substitution-expectation covariance'' between any one parent and its offspring. After careful gathering of terms and simplification, this becomes '''cov(PO)<sub>A</sub> = pqa<sup>2</sup> = {{sfrac|1|2}} s<sup>2</sup><sub>A</sub> '''.<ref name="Crow & Kimura"/> {{rp|132–141}}<ref name="Falconer 1996"/> {{rp|134–147}}

Unfortunately, the ''allele-substitution-deviations'' are usually overlooked, but they have not "ceased to exist" nonetheless! Recall that these deviations are: '''d<sub>AA</sub> = -2q<sup>2</sup> d''', and '''d<sub>Aa</sub> = 2pq d''' and also '''d<sub>aa</sub> = -2p<sup>2</sup> d''' [see section on "Genotype substitution – Expectations and Deviations"].  Consequently, the cross-product sum using {'''genotype-frequency * parental gene-effect * half-substitution-deviations'''} also immediately provides the allele-substitution-deviations covariance between any one parent and its offspring. Once more, after careful gathering of terms and simplification, this becomes '''cov(PO)<sub>D</sub> = 2p<sup>2</sup>q<sup>2</sup>d<sup>2</sup> = {{sfrac|1|2}} s<sup>2</sup><sub>D</sub> '''.

It follows therefore that: '''cov(PO) = cov(PO)<sub>A</sub> + cov(PO)<sub>D</sub> = {{sfrac|1|2}} s<sup>2</sup><sub>A</sub> + {{sfrac|1|2}} s<sup>2</sup><sub>D</sub> ''', when dominance is ''not'' overlooked !