Editing Quantitative genetics (section)

===Dispersion and the genotypic variance===
In the section on genetic drift, and in other sections that discuss inbreeding, a major outcome from allele frequency sampling has been the  ''dispersion'' of progeny means. This collection of means has its own average, and also has a variance: the ''amongst-line variance''. (This is a variance of the attribute itself, not of ''allele frequencies''.) As dispersion develops further over succeeding generations, this amongst-line variance would be expected to increase. Conversely, as homozygosity rises, the within-lines variance would be expected to decrease. The question arises therefore as to whether the total variance is changing—and, if so, in what direction. To date, these issues have been presented in terms of the ''genic (σ <sup>2</sup><sub>A</sub> )'' and ''quasi-dominance (σ <sup>2</sup><sub>D</sub> )'' variances rather than the gene-model components. This will be done herein as well.

The crucial ''overview equation'' comes from Sewall Wright,<ref name="Crow & Kimura"/> {{rp|99,130}}<ref name="Wright 1951"/> and is the outline of the '''inbred genotypic variance''' based on a ''weighted average of its extremes'', the weights being quadratic with respect to the ''inbreeding coefficient'' 
<math display="inline"> f </math>. This equation is:

<math display="block"> \sigma^2_{G_{f}} = \left( 1-f \right) \sigma^2_{G_{0}} + f \ \sigma^2_{G_{1}} + f \left( 1-f \right) \left[ G_0 - G_1 \right] ^2 </math>

where <math display="inline"> f </math> is the inbreeding coefficient, <math display="inline"> \sigma^2_{G_{0}} </math> is the genotypic variance at 
''f=0'', <math display="inline"> \sigma^2_{G_{1}} </math> is the genotypic variance at ''f=1'', 
<math display="inline"> G_0 </math> is the population mean at ''f=0'', and <math display="inline"> G_1 </math> is the population mean at ''f=1''.

The <math display="inline"> \left( 1-f \right) </math> component [in the equation above] outlines the reduction of variance within progeny lines. The <math display="inline"> f </math>  component addresses the increase in variance amongst progeny lines. Lastly, the <math display="inline"> f \left( 1-f \right) </math> component is seen (in the next line) to address the ''quasi-dominance'' variance.<ref name="Crow & Kimura"/> {{rp|99 & 130}} These components can be expanded further thereby revealing additional insight. Thus:-

<math display="block"> \sigma^2_{G_{f}} = \left( 1-f \right) \left[ \sigma^2_{A_{0}} + \sigma^2_{D_{0}} \right] + f \ \left( 4pq \ a^2 \right) + f \ \left( 1-f \right) \left[ 2pq \ d \right] ^2 </math>

Firstly, ''σ<sup>2</sup><sub>G(0)</sub>'' [in the equation above] has been expanded to show its two sub-components [see section on "Genotypic variance"]. Next, the ''σ<sup>2</sup><sub>G(1)</sub>'' has been converted to ''4pqa<sup>2</sup> '', and is derived in a section following. The third component's substitution is the difference between the two "inbreeding extremes" of the population mean [see section on the "Population Mean"].<ref name="Gordon 2003"/>
[[File:VADf-p5.jpg|thumb|300px|left|Dispersion and components of the genotypic variance]]

Summarising: the '''within-line''' components are <math display="inline"> \left( 1-f \right) \sigma^2_{A_{0}} </math> and <math display="inline"> \left( 1-f \right) \sigma^2_{D_{0}} </math>; and the '''amongst-line''' components are <math display="inline"> 2f \ \sigma^2_{a_{0}} </math> and <math display="inline"> \left( f - f^2 \right) \sigma^2_{D_{0}} </math>.<ref name="Gordon 2003"/>
[[File:ALWL-10p5.jpg|thumb|Development of variance dispersion]]

Rearranging gives the following:
<math display="block"> \begin{align}
\sigma^2_{A_{inbred}} & = \left( 1-f \right) \sigma^2_{A_{0}} + 2f \sigma^2_{a_{0}} \\ & = \left( 1+f \right) \sigma^2_{A_{f}}
\end{align} </math> 
The version in the last line is discussed further in a subsequent section.

Similarly,
<math display="block"> \begin{align}
\sigma^2_{D_{inbred}} & = \left( 1-f \right) \sigma^2_{D_{0}} + \left( f - f^2 \right) \sigma^2_{D_{0}} \\ 
 & = \left( 1 - f^2 \right) \sigma^2_{D_{0}}
\end{align} </math>
 
Graphs to the left show these three genic variances, together with the three quasi-dominance variances,  across all values of '''f''', for '''p = 0.5''' (at which the quasi-dominance variance is at a maximum). Graphs to the right show the '''Genotypic''' variance partitions (being the sums of the respective ''genic'' and ''quasi-dominance'' partitions) changing over ten generations with an example ''f = 0.10''.

Answering, firstly, the questions posed at the beginning about the '''total variances''' [the '''Σ''' in the graphs] : the ''genic variance'' rises linearly with the ''inbreeding coefficient'', maximizing at twice its starting level. The ''quasi-dominance variance'' declines at the rate of ''(1 − f<sup>2</sup> )'' until it finishes at zero. At low levels of ''f'', the decline is very gradual, but it accelerates with higher levels of ''f''.

Secondly, notice the other trends. It is probably intuitive that the '''within line''' variances decline to zero with continued inbreeding, and this is seen to be the case (both at the same linear rate ''(1-f)'' ). The '''amongst line''' variances both increase with inbreeding up to ''f = 0.5'', the ''genic variance'' at the rate of ''2f'', and the ''quasi-dominance variance'' at the rate of ''(f − f<sup>2</sup>)''. At ''f > 0.5'', however, the trends change. The '''amongst line''' ''genic variance'' continues its linear increase until it equals the '''total''' ''genic variance''. But, the '''amongst line''' ''quasi-dominance variance'' now declines towards ''zero'',  because ''(f − f<sup>2</sup>)'' also declines with ''f > 0.5''.<ref name="Gordon 2003"/>

==== Derivation of ''σ<sup>2</sup><sub>G(1)</sub>''====
Recall that when ''f=1'', heterozygosity is zero, within-line variance is zero, and all genotypic variance is thus ''amongst-line'' variance and deplete of dominance variance. In other words, '''σ<sup>2</sup><sub>G(1)</sub>''' is the variance amongst fully inbred line means. Recall further [from "The mean after self-fertilization" section] that such means (G<sub>1</sub>'s, in fact) are '''G = a(p-q)'''. Substituting ''(1-q)'' for the ''p'', gives '''G<sub>1</sub> = a (1 − 2q)''' = '''a − 2aq'''.<ref name="Falconer 1996"/>{{rp|265}} Therefore, the '''σ<sup>2</sup><sub>G(1)</sub>''' is the '''σ<sup>2</sup><sub>(a-2aq)</sub>''' actually. Now, in general, the ''variance of a difference (x-y)'' is ''' [ σ<sup>2</sup><sub>x</sub> + σ<sup>2</sup><sub>y</sub> − 2 cov<sub>xy</sub> ]'''.<ref name="Snedecor & Cochran">{{cite book|last1=Snedecor|first1=George W.|last2=Cochran|first2=William G.|title=Statistical methods.|date=1967|publisher=Iowa State University Press|location=Ames|isbn=0-8138-1560-6|edition=Sixth}}</ref>{{rp|100}}<ref name="Kendall & Stuart">{{cite book|last1=Kendall|first1=M. G.|last2=Stuart|first2=A.|title=The advanced theory of statistics. Volume 1.|date=1958|publisher=Charles Griffin|location=London|edition=2nd}}</ref> {{rp|232}} Therefore, '''σ<sup>2</sup><sub>G(1)</sub> = [ σ<sup>2</sup><sub>a</sub> + σ<sup>2</sup><sub>2aq</sub> − 2 cov<sub>(a, 2aq)</sub> ] '''. But '''a''' (an allele ''effect'') and '''q''' (an allele ''frequency'') are ''independent''—so this covariance is zero. Furthermore, '''a''' is a constant from one line to the next, so '''σ<sup>2</sup><sub>a</sub>''' is also zero. Further, '''2a''' is another constant (k), so the '''σ<sup>2</sup><sub>2aq</sub>''' is of the type ''σ<sup>2</sup><sub>k X</sub>''. In general, the variance ''σ<sup>2</sup><sub>k X</sub>'' is equal to '''k<sup>2</sup> σ<sup>2</sup><sub>X</sub> '''.<ref name="Kendall & Stuart"/>{{rp|232}} Putting all this together reveals that ''' σ<sup>2</sup><sub>(a-2aq)</sub> = (2a)<sup>2</sup> σ<sup>2</sup><sub>q</sub> '''. Recall [from the section on "Continued genetic drift"] that ''σ<sup>2</sup><sub>q</sub> = pq f ''. With ''f=1'' here within this present derivation, this becomes ''pq 1'' (that is '''pq'''), and this is substituted into the previous.

The final result is: '''σ<sup>2</sup><sub>G(1)</sub> = σ<sup>2</sup><sub>(a-2aq)</sub> = 4a<sup>2</sup> pq = 2(2pq a<sup>2</sup>) = 2 σ<sup>2</sup><sub>a</sub> '''.

It follows immediately that '''''f'' σ<sup>2</sup><sub>G(1)</sub> = ''f'' 2 σ<sup>2</sup><sub>a</sub> '''. [This last ''f'' comes from the ''initial Sewall Wright equation'' : it is '''''not''''' the ''f '' just set to "1" in the derivation concluded two lines above.]

==== Total dispersed genic variance – σ<sup>2</sup><sub>A(f)</sub> and β<sub>f</sub> ====
Previous sections found that the '''within line''' ''genic variance'' is based upon the ''substitution-derived'' genic variance '''( σ<sup>2</sup><sub>A</sub> )'''—but the ''amongst line'' ''genic variance'' is based upon the ''gene model'' allelic variance '''( σ<sup>2</sup><sub>a</sub> )'''. These two cannot simply be added to get ''total genic variance''. One approach in avoiding this problem was to re-visit the derivation of the ''average allele substitution effect'', and to construct a version, '''( β<sub> ''f'' </sub> )''', that incorporates the effects of the dispersion. Crow and Kimura achieved this<ref name ="Crow & Kimura"/> {{rp|130–131}} using the re-centered allele effects ('''a•, d•, (-a)• ''') discussed previously ["Gene effects re-defined"]. However, this was found subsequently to under-estimate slightly the ''total Genic variance'', and a new variance-based derivation led to a refined version.<ref name="Gordon 2003"/>
  
The ''refined'' version is: ''' β<sub> ''f'' </sub> = { a<sup>2</sup> + [(1−''f'' ) / (1 + ''f'' )] 2(q − p ) ad + [(1-''f'' ) / (1 + ''f'' )] (q − p )<sup>2</sup> d<sup>2</sup> } <sup>(1/2)</sup>'''

Consequently, '''σ<sup>2</sup><sub>A(f)</sub> = (1 + ''f'' ) 2pq β<sub>f</sub> <sup>2</sup> ''' does now agree with '''[ (1-f) σ<sup>2</sup><sub>A(0)</sub> +  2f σ<sup>2</sup><sub>a(0)</sub> ]''' exactly.

==== Total and partitioned dispersed quasi-dominance variances====
The ''total genic variance'' is of intrinsic interest in its own right. But, prior to the refinements by Gordon,<ref name="Gordon 2003"/> it had had another important use as well. There had been no extant estimators for the "dispersed" quasi-dominance. This had been estimated as the difference between Sewall Wright's ''inbred genotypic variance'' <ref name="Wright 1951"/> and the total "dispersed" genic variance [see the previous sub-section]. An anomaly appeared, however, because the ''total quasi-dominance variance'' appeared to increase early in inbreeding despite the decline in heterozygosity.<ref name="Falconer 1996"/> {{rp|128}} {{rp|266}}

The refinements in the previous sub-section corrected this anomaly.<ref name="Gordon 2003"/> At the same time, a direct solution for the ''total quasi-dominance variance'' was obtained, thus avoiding the need for the "subtraction" method of previous times. Furthermore, direct solutions for the ''amongst-line'' and ''within-line'' partitions of the ''quasi-dominance variance'' were obtained also, for the first time. [These have been presented in the section "Dispersion and the genotypic variance".]