Editing Quantitative genetics (section)

==== Allele-substitution approach – Fisher====
[[File:Fisher Var 1.jpg|thumb|300px|right|Components of genotypic variance using the allele-substitution effects.]]
Reference to the several earlier sections on allele substitution reveals that the two ultimate effects are ''genotype substitution '' expectations and ''genotype substitution deviations''. Notice that these are each already defined as deviations from the ''random fertilization'' population mean ('''G'''). For each genotype in turn therefore, the product of the frequency and the square of the relevant effect is obtained, and these are accumulated to obtain directly a '''SS''' and '''σ<sup>2</sup>'''.<ref>Supporting citations have been given already in the previous sections.</ref> Details follow.

'''σ<sup>2</sup><sub>A</sub>''' = '''''p''<sup>2</sup> β<sub>AA</sub><sup>2</sup>''' + '''2''pq'' β<sub>Aa</sub><sup>2</sup>''' + '''''q''<sup>2</sup> β<sub>aa</sub><sup>2</sup>''', which simplifies to '''σ<sup>2</sup><sub>A</sub> = 2''pq''β<sup>2</sup>'''—the Genic variance.

'''σ<sup>2</sup><sub>D</sub>''' = '''''p''<sup>2</sup> d<sub>AA</sub><sup>2</sup>''' + '''2''pq'' d<sub>Aa</sub><sup>2</sup>''' + '''''q'' d<sub>aa</sub><sup>2</sup>''', which simplifies to '''σ<sup>2</sup><sub>D</sub> = (2''pq'')<sup>2</sup> d<sup>2</sup>'''—the quasi-Dominance variance.

Upon accumulating these results, '''σ<sup>2</sup><sub>G</sub>''' = '''σ<sup>2</sup><sub>A</sub> + σ<sup>2</sup><sub>D</sub> '''. These components are visualized in the graphs to the right. The ''average allele substitution'' effect is graphed also, but the symbol is "α" (as is common in the citations) rather than "β" (as is used herein).

Once again, however, refer to the earlier discussions about the true meanings and identities of these components. Fisher himself did not use these modern terms for his components. The ''substitution expectations'' variance he named the ''' "genetic" ''' variance; and the ''substitution deviations'' variance he regarded simply as the unnamed '''residual''' between the "genotypic" variance (his name for it) and his "genetic" variance.<ref name= "Fisher 1918" /><ref name= "Fisher 1999" />{{rp|33}}<ref>Fisher did note that these residuals arose through the effects of dominance: but he refrained from defining them as the "dominance variance". (See the foregoing citations.) Refer again to the earlier discussions herein.</ref><ref>While considering origins of terms: Fisher also proposed the word "variance" for this measure of variability. See Fisher (1999), p. 311 and Fisher (1918).</ref> [The terminology and derivation used in this article are completely in accord with Fisher's own.] Mather's term for the ''expectations'' variance—'''"genic"'''<ref name= "M&J 1971" />—is obviously derived from Fisher's term, and avoids using "genetic" (which has become too generalized in usage to be of value in the present context). The origin is obscure of the modern misleading terms "additive" and "dominance" variances.

Note that this allele-substitution approach defined the components separately, and then totaled them to obtain the final Genotypic variance. Conversely, the gene-model approach derived the whole situation (components and total) as one exercise. Bonuses arising from this were (a) the revelations about the real structure of '''σ<sup>2</sup><sub>A</sub>''', and (b) the real meanings and relative sizes of '''σ<sup>2</sup><sub>d</sub>''' and '''σ<sup>2</sup><sub>D</sub>''' (see previous sub-section). It is also apparent that a "Mather" analysis is more informative, and that a "Fisher" analysis can always be constructed from it. The opposite conversion is not possible, however, because information about '''cov<sub>ad</sub>''' would be missing.