Editing Quantitative genetics (section)

===Allele shuffling – allele substitution===
The ''gene-model'' examines the heredity pathway from the point of view of "inputs" (alleles/gametes) and "outputs" (genotypes/zygotes), with fertilization being the "process" converting one to the other. An alternative viewpoint concentrates on the "process" itself, and considers the zygote genotypes as arising from allele shuffling. In particular, it regards the results as if one allele had "substituted" for the other during the shuffle, together with a residual that deviates from this view. This formed an integral part of Fisher's method,<ref name="Fisher 1918"/> in addition to his use of frequencies and effects to generate his genetical statistics.<ref name="Falconer 1996"/> A discursive derivation of the ''allele substitution'' alternative follows.<ref name="Falconer 1996"/>{{rp|113}} [[File:Allele Substn.jpg|thumb|300px|left|Analysis of allele substitution]]

Suppose that the usual random fertilization of gametes in a "base" gamodeme—consisting of '''''p''''' gametes ('''A''') and '''''q''''' gametes ('''a''')—is replaced by fertilization with a "flood" of gametes all containing a single allele ('''A''' or '''a''', but not both). The zygotic results can be interpreted in terms of the "flood" allele having "substituted for" the alternative allele in the underlying "base" gamodeme. The diagram assists in following this viewpoint: the upper part pictures an '''A''' substitution, while the lower part shows an '''a''' substitution. (The diagram's "RF allele" is the allele in the "base" gamodeme.)

Consider the upper part firstly. Because ''base'' '''A''' is present with a frequency of '''''p''''', the ''substitute'' '''A''' fertilizes it with a frequency of '''''p''''' resulting in a zygote '''AA''' with an allele effect of '''''a'''''. Its contribution to the outcome, therefore, is the product <math display="inline"> \left( p \ a \right) </math>. Similarly, when the ''substitute'' fertilizes ''base'' '''a''' (resulting in '''Aa''' with a frequency of '''''q''''' and heterozygote effect of '''''d'''''), the contribution is <math display="inline"> \left( q \ d \right) </math>. The overall result of substitution by '''A''' is, therefore, <math display="inline"> \left( p \ a + q \ d \right) </math>. This is now oriented towards the population mean [see earlier section] by expressing it as a deviate from that mean : <math display="inline"> \left( p \ a + q \ d \right) - G </math>

After some algebraic simplification, this becomes <math display="block"> \beta _A = q \ \left[a + \left( q - p \right) d \right] </math>  
- the ''substitution effect'' of '''A'''.

A parallel reasoning can be applied to the lower part of the diagram, taking care with the differences in frequencies and gene effects. The result is the  ''substitution effect'' of '''a''', which is <math display="block"> \beta _a = - \ p \left[ a + \left( q -p  \right) d \right] </math>
The common factor inside the brackets is the ''average allele substitution effect'',<ref name="Falconer 1996"/>{{rp|113}} and is <math display="block"> \beta = a + \left( q - p \right) d </math>
It can also be derived in a more direct way, but the result is the same.<ref>It is common to use "α" rather than "β" for this quantity (e.g. in the references already cited). The latter is used herein in order to minimize any confusion with "a", which frequently occurs also within these same equations.</ref>

In subsequent sections, these substitution effects help define the gene-model genotypes as consisting of a partition predicted by these new effects ('''substitution ''expectations'''''), and a residual ('''substitution deviations''') between these expectations and the previous gene-model effects. The ''expectations'' are also called the '''breeding values''' and the deviations are also called '''dominance deviations'''.

Ultimately, the variance arising from the ''substitution expectations'' becomes the so-called ''Additive genetic variance (σ<sup>2</sup><sub>A</sub>)''<ref name="Falconer 1996"/> (also the ''Genic variance'' <ref name="M&J 1971"/>)— while that arising from the ''substitution deviations'' becomes the so-called ''Dominance variance (σ<sup>2</sup><sub>D</sub>)''. It is noticeable that neither of these terms reflects the true meanings of these variances. The ''' "genic variance" ''' is less dubious than the '' additive genetic variance'', and more in line with Fisher's own name for this partition.<ref name= "Fisher 1918" /><ref name= "Fisher 1999" />{{rp|33}} A less-misleading name for the ''dominance deviations variance'' is the ''' "quasi-dominance variance" ''' [see following sections for further discussion]. These  latter terms are preferred herein.

==== Gene effects redefined====
The gene-model effects ('''a''', '''d''' and '''-a''') are important soon in the derivation of the ''deviations from substitution'', which were first discussed in the previous ''Allele Substitution'' section. However, they need to be redefined themselves before they become useful in that exercise. They firstly need to be re-centralized around the population mean ('''G'''), and secondly they need to be re-arranged as functions of '''β''', the ''average allele substitution effect''.

Consider firstly the re-centralization. The re-centralized effect for '''AA''' is '''a• = a - G''' which, after simplification, becomes '''a• = 2''q''(a-''p''d)'''. The similar effect for '''Aa''' is '''d• = d - G = a(''q''-''p'') + d(1-2''pq'')''', after simplification. Finally, the re-centralized effect for '''aa''' is '''(-a)• = -2''p''(a+''q''d)'''.<ref name="Falconer 1996"/>{{rp|116–119}}

Secondly, consider the re-arrangement of these re-centralized effects as functions of '''β'''. Recalling from the "Allele Substitution" section that β = [a +(q-p)d], rearrangement gives '''a = [β -(q-p)d]'''. After substituting this for '''a''' in '''a•''' and simplifying, the final version becomes '''a•• = 2q(β-qd)'''. Similarly, '''d•''' becomes '''d•• = β(q-p) + 2pqd'''; and  '''(-a)•''' becomes '''(-a)•• = -2p(β+pd)'''.<ref name="Falconer 1996"/>{{rp|118}}

==== Genotype substitution – expectations and deviations====
The zygote genotypes are the target of all this preparation. The homozygous genotype '''AA''' is a union of two ''substitution effects of A'', one from each sex. Its ''substitution expectation'' is therefore '''β<sub>AA</sub> = 2β<sub>A</sub> = 2''q''β''' (see previous sections). Similarly, the ''substitution expectation'' of '''Aa''' is '''β<sub>Aa</sub> = β<sub>A</sub> + β<sub>a</sub> = (''q''-''p'')β'''; and for '''aa''', '''β<sub>aa</sub> = 2β<sub>a</sub> = -2''p''β'''. These ''substitution expectations'' of the genotypes are also called ''breeding values''.<ref name="Falconer 1996"/>{{rp|114–116}}

''Substitution deviations'' are the differences between these ''expectations'' and the ''gene effects'' after their two-stage redefinition in the previous section. Therefore, '''d<sub>AA</sub> = a•• - β<sub>AA</sub> = -2''q''<sup>2</sup>d''' after simplification. Similarly, '''d<sub>Aa</sub> = d•• - β<sub>Aa</sub> = 2''pq''d''' after simplification. Finally, '''d<sub>aa</sub> = (-a)•• - β<sub>aa</sub> = -2''p''<sup>2</sup>d''' after simplification.<ref name="Falconer 1996"/>{{rp|116–119}} Notice that all of these ''substitution deviations'' ultimately are functions of the gene-effect ''d''—which accounts for the use of ["d" plus subscript] as their symbols. However, it is a serious ''non sequitur'' in logic to regard them as accounting for the dominance (heterozygosis) in the entire gene model : they are simply ''functions'' of "d" and not an ''audit'' of the "d" in the system. They ''are'' as derived: ''deviations from the substitution expectations''!

The "substitution expectations" ultimately give rise to the '''σ<sup>2</sup><sub>A</sub>''' (the so-called "Additive" genetic variance); and the "substitution deviations" give rise to the '''σ<sup>2</sup><sub>D</sub>''' (the so-called "Dominance" genetic variance). Be aware, however, that the average substitution effect (β) also contains "d" [see previous sections], indicating that dominance is also embedded within the "Additive" variance [see following sections on the Genotypic Variance for their derivations]. Remember also [see previous paragraph] that the "substitution deviations" do not account for the dominance in the system (being nothing more than deviations from the ''substitution expectations''), but which happen to consist algebraically of functions of "d". More appropriate names for these respective variances might be '''σ<sup>2</sup><sub>B</sub>''' (the  "Breeding expectations" variance) and '''σ<sup>2</sup><sub>δ</sub>''' (the "Breeding deviations" variance). However, as noted previously, "Genic" (σ <sup>2</sup><sub>A</sub>) and "Quasi-Dominance" (σ <sup>2</sup><sub>D</sub>), respectively, will be preferred herein.