Editing Quantitative genetics (section)

===Genotypic variance===
There are two major approaches to defining and partitioning ''genotypic variance''. One is based on the ''gene-model effects'',<ref name="M&J 1971">{{cite book|last1=Mather|first1=Kenneth|last2=Jinks|first2=John L.|title=Biometrical Genetics |date=1971|publisher=Chapman & Hall|location=London|isbn=0-412-10220-X|edition=2}}</ref> while the other is based on the ''genotype substitution effects''<ref name="Falconer 1996">{{cite book |last1=Falconer |first1=D. S. |last2=Mackay |first2=Trudy F. C. |title=Introduction to quantitative genetics |date=1996 |publisher=Longman |location=Harlow |isbn=978-0582-24302-6 |edition=Fourth |url=https://archive.org/details/introductiontoqu00falc}}
*{{cite journal |author=William G Hill |author2=Trudy F C Mackay |title=D. S. Falconer and Introduction to Quantitative Genetics |journal=Genetics |volume=167 |issue=4 |date=August 2004 |pages=1529–1536 |doi=10.1093/genetics/167.4.1529 |pmid=15342495 |pmc=1471025 |doi-access=free}}</ref> They are algebraically inter-convertible with each other.<ref name="Gordon 2003">{{cite journal| last=Gordon| first = I.L.| title = Refinements to the partitioning of the inbred genotypic variance| year = 2003| journal = Heredity| volume = 91| pages = 85–89| doi = 10.1038/sj.hdy.6800284| pmid = 12815457| issue=1| doi-access = free}}</ref> In this section, the basic '''''random fertilization''''' derivation is considered, with the effects of inbreeding and dispersion set aside. This is dealt with later to arrive at a more general solution. Until this ''mono-genic'' treatment is replaced by a ''multi-genic'' one, and until ''epistasis'' is resolved in the light of the findings of ''epigenetics'', the Genotypic variance has only the components considered here.

==== Gene-model approach – Mather Jinks Hayman====
[[File:Model Var 2.jpg|thumb|300px|right|Components of genotypic variance using the gene-model effects.]]
It is convenient to follow the biometrical approach, which is based on correcting the ''unadjusted sum of squares (USS)'' by subtracting the ''correction factor (CF)''. Because all effects have been examined through frequencies, the USS can be obtained as the sum of the products of each genotype's frequency' and the square of its ''gene-effect''. The CF in this case is the mean squared. The result is the SS, which, again because of the use of frequencies, is also immediately the ''variance''.<ref name="S & T"/>

The <math display="inline">\mathsf{USS} = p^2a^2 + 2pqd^2 + q^2(-a)^2</math>,  and the <math display="inline">\mathsf{CF} = \mathsf{G}^2</math>.  The <math display="inline">\mathsf{SS} = \mathsf{USS} - \mathsf{CF}</math>

After partial simplification,
<math display="block">
\begin{align}
\sigma^2_G & = 2pq a^2 + (q-p) 4pq ad + 2pq d^2 + (2pq)^2 d^2 \\
 & = \sigma^2_a + (\text{weighted-covariance})_{ad} + \sigma^2_d + \sigma^2_D \\
 & = \tfrac{1}{2}\mathsf{D} + \tfrac{1}{2}\mathsf{F}^\prime + \tfrac{1}{2}\mathsf{H}_1 + \tfrac{1}{4}\mathsf{H}_2
\end{align}</math> 
The last line is in Mather's terminology.<ref name="M&J 1971"/>{{rp|212}}<ref>In Mather's terminology, the fraction in front of the letter is ''a part of the label'' for the component.</ref><ref>In each line of these equations, the components are presented in the same order. Therefore, vertical comparison by component gives the definition of each in various forms. The Mather components have been translated thereby into Fisherian symbols: thus facilitating their comparison. The translation has been derived formally as well. See Gordon 2003.</ref>

Here, '''σ<sup>2</sup><sub>a</sub>''' is the ''homozygote'' or '''allelic''' variance, and '''σ<sup>2</sup><sub>d</sub>''' is the ''heterozygote'' or '''dominance''' variance. The ''substitution deviations'' variance ('''σ<sup>2</sup><sub>D</sub>''') is also present. The ''(weighted_covariance)<sub>ad</sub>''<ref name = "cov">Covariance is the co-variability between two sets of data. Similarly to the variance, it is based on a ''sum of cross-products (SCP)'' instead of a SS. From this, it is clear therefore that the variance is but a special form of the covariance.</ref> is abbreviated hereafter to " '''cov<sub>ad</sub>''' ".

These components are plotted across all values of '''p''' in the accompanying figure. Notice that ''cov<sub>ad</sub>'' is '''negative''' for ''p > 0.5''.

Most of these components are affected by the change of central focus from ''homozygote mid-point'' ('''mp''') to ''population mean'' ('''G'''), the latter being the basis of the ''Correction Factor''. The ''cov<sub>ad</sub>'' and ''substitution deviation'' variances are simply artifacts of this shift. The ''allelic'' and ''dominance'' variances are genuine genetical partitions of the original gene-model, and are the only eu-genetical components. Even then, the algebraic formula for the ''allelic'' variance is effected by the presence of ''G'': it is only the ''dominance'' variance (i.e. σ<sup>2</sup><sub>d</sub> ) which is unaffected by the shift from ''mp'' to ''G''.<ref name="Gordon 2003"/> These insights are commonly not appreciated.

Further gathering of terms [in Mather format] leads to <math display="inline">\tfrac{1}{2}\mathsf{D} + \tfrac{1}{2}\mathsf{F}^\prime + \tfrac{1}{2}\mathsf{H}_3 + \tfrac{1}{4}\mathsf{H}_2</math>, where <math display="inline">\tfrac{1}{2}\mathsf{H}_3 = (q-p)^2 \tfrac{1}{2}\mathsf{H}_1 = (q-p)^2 2pq d^2</math>. It is useful later in Diallel analysis, which is an experimental design for estimating these genetical statistics.<ref name="Hayman 1960">{{cite journal|last1=Hayman|first1=B. I.|title=The theory and analysis of the diallel cross. III.|journal=Genetics|date=1960|volume=45|issue=2|pages=155–172|doi=10.1093/genetics/45.2.155|pmid=17247915|pmc=1210041}}</ref>

If, following the last-given rearrangements, the first three terms are amalgamated together, rearranged further and simplified, the result is the variance of the Fisherian ''substitution expectation''.

That is:  <math>\sigma^2_A = \sigma^2_a + \mathsf{cov}_{ad} + \sigma^2_d</math>

Notice particularly that '''σ<sup>2</sup><sub>A</sub>''' is not  '''σ<sup>2</sup><sub>a</sub>'''. The first is the  ''substitution expectations'' variance, while the second is the ''allelic'' variance.<ref>It has been observed that when '''p''' = '''q''', or when '''d''' = '''0''', '''β''' [= a+(q-p)d] "reduces" to '''a'''. In such circumstances, '''σ<sup>2</sup><sub>A</sub>''' = '''σ<sup>2</sup><sub>a</sub>'''—but only '''''numerically'''''. They still have not ''become'' the one and the same identity. This would be a similar ''non sequitur'' to that noted earlier for the "substitution deviations" being regarded as the "dominance" for the gene-model.</ref>  Notice also that '''σ<sup>2</sup><sub>D</sub>''' (the ''substitution-deviations'' variance) is ''not'' '''σ<sup>2</sup><sub>d</sub>''' (the ''dominance'' variance), and recall that it is an artifact arising from the use of ''G'' for the Correction Factor. [See the "blue paragraph" above.] It now will be referred to as the "quasi-dominance" variance.

Also note that  '''σ<sup>2</sup><sub>D</sub>''' < '''σ<sup>2</sup><sub>d</sub>''' ("2pq" being always a fraction); and note that (1) '''σ<sup>2</sup><sub>D</sub>''' = '''2pq σ<sup>2</sup><sub>d</sub>''', and that (2) '''σ<sup>2</sup><sub>d</sub>''' = '''σ<sup>2</sup><sub>D</sub> / (2pq)'''. That is: it is confirmed that σ<sup>2</sup><sub>D</sub> does not quantify the dominance variance in the model. It is σ<sup>2</sup><sub>d</sub> which does that. However, the dominance variance (σ<sup>2</sup><sub>d</sub>) can be estimated readily from the σ<sup>2</sup><sub>D</sub> if ''2pq'' is available.

From the Figure, these results can be visualized as accumulating '''σ<sup>2</sup><sub>a</sub>''', '''σ<sup>2</sup><sub>d</sub>''' and '''cov<sub>ad</sub>'''  to obtain '''σ<sup>2</sup><sub>A</sub>''', while leaving the '''σ<sup>2</sup><sub>D</sub>''' still separated. It is clear also in the Figure that '''σ<sup>2</sup><sub>D</sub>''' < '''σ<sup>2</sup><sub>d</sub>''', as expected from the equations.

The overall result (in Fisher's format) is
<math display="block">
\begin{align}
\sigma^2_G & = 2pq \left[ a+(q-p)d \right]^2 + \left( 2pq \right)^2 d^2 \\ & =  \sigma^2_A + \sigma^2_D \\ & = \left[ \left( \sigma^2_a + \mathsf{cov}_{ad} + \sigma^2_d \right) \right] + \left[ 2pq \ \sigma^2_d \right]
\end{align} </math> 
The Fisherian components have just been derived, but their derivation via the ''substitution effects'' themselves is given also, in the next 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.