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Quantitative genetics
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===Population mean=== The population mean shifts the central reference point from the homozygote midpoint ('''mp''') to the mean of a sexually reproduced population. This is important not only to relocate the focus into the natural world, but also to use a measure of ''central tendency'' used by Statistics/Biometrics. In particular, the square of this mean is the Correction Factor, which is used to obtain the genotypic variances later.<ref name="S & T"/> [[File:G mean.jpg|thumb|300px|right|Population mean across all values of p, for various d effects.]] For each genotype in turn, its allele effect is multiplied by its genotype frequency; and the products are accumulated across all genotypes in the model. Some algebraic simplification usually follows to reach a succinct result. ==== The mean after random fertilization==== The contribution of '''AA''' is <math display="inline">p^2 (+)a</math>, that of '''Aa''' is <math display="inline">2pq d</math>, and that of '''aa''' is <math display="inline">q^2 (-)a</math>. Gathering together the two '''a''' terms and accumulating over all, the result is: <math display="inline"> a(p^2-q^2) + 2pq d</math>. Simplification is achieved by noting that <math display="inline"> (p^2-q^2) = (p-q)(p+q)</math>, and by recalling that <math display="inline"> (p+q) = 1</math>, thereby reducing the right-hand term to <math display="inline">(p-q)</math>. The succinct result is therefore <math display="inline"> G = a(p-q) + 2pqd</math>.<ref name="Falconer 1996"/> {{rp|110}} This defines the population mean as an "offset" from the homozygote midpoint (recall '''a''' and '''d''' are defined as ''deviations'' from that midpoint). The Figure depicts '''G''' across all values of '''p''' for several values of '''d''', including one case of slight over-dominance. Notice that '''G''' is often negative, thereby emphasizing that it is itself a ''deviation'' (from '''mp'''). Finally, to obtain the ''actual'' Population Mean in "phenotypic space", the midpoint value is added to this offset: <math display="inline"> P = G + mp</math>. An example arises from data on ear length in maize.<ref name="Sinnott Dunn & Dobzhansky">{{cite book|last1=Sinnott|first1=Edmund W.|last2=Dunn|first2=L. C.|last3=Dobzhansky|first3=Theodosius|title=Principles of genetics.|url=https://archive.org/details/principlesofgene00sinn|url-access=registration|date=1958|publisher=McGraw-Hill|location=New York}}</ref>{{rp|103}} Assuming for now that one gene only is represented, '''a''' = 5.45 cm, '''d''' = 0.12 cm [virtually "0", really], '''mp''' = 12.05 cm. Further assuming that '''p''' = 0.6 and '''q''' = 0.4 in this example population, then: '''G''' = 5.45 (0.6 β 0.4) + (0.48)0.12 = '''1.15 cm''' (rounded); and '''P''' = 1.15 + 12.05 = '''13.20 cm''' (rounded). ==== The mean after long-term self-fertilization==== The contribution of '''AA''' is <math display="inline"> p (+a)</math>, while that of '''aa''' is <math display="inline"> q (-a)</math>. [See above for the frequencies.] Gathering these two '''a''' terms together leads to an immediately very simple final result: <math display="inline"> G_{(f=1)} = a(p-q)</math>. As before, <math display="inline"> P = G + mp</math>. Often, "G<sub>(f=1)</sub>" is abbreviated to "G<sub>1</sub>". Mendel's peas can provide us with the allele effects and midpoint (see previously); and a mixed self-pollinated population with '''p''' = 0.6 and '''q''' = 0.4 provides example frequencies. Thus: '''G<sub>(f=1)</sub>''' = 82 (0.6 β .04) = 59.6 cm (rounded); and '''P<sub>(f=1)</sub>''' = 59.6 + 116 = 175.6 cm (rounded). ==== The mean β generalized fertilization==== A general formula incorporates the inbreeding coefficient '''''f''''', and can then accommodate any situation. The procedure is exactly the same as before, using the weighted genotype frequencies given earlier. After translation into our symbols, and further rearrangement:<ref name="Crow & Kimura"/> {{rp|77β78}} <math display="block"> \begin{align} G_{f} & = a (q-p) + [2pqd-f(2pqd)] \\ & = a(p-q) + (1-f) 2pqd \\ & = G_{0} - f\ 2pqd \end{align} </math> Here, '''G<sub>0</sub>''' is '''G''', which was given earlier. (Often, when dealing with inbreeding, "G<sub>0</sub>" is preferred to "G".) Supposing that the maize example [given earlier] had been constrained on a holme (a narrow riparian meadow), and had partial inbreeding to the extent of '''''f '''''= '''0.25''', then, using the third version (above) of '''G<sub>f</sub>''': '''G<sub>''0.25''</sub>''' = 1.15 β 0.25 (0.48) 0.12 = 1.136 cm (rounded), with '''P<sub>0.25</sub>''' = 13.194 cm (rounded). There is hardly any effect from inbreeding in this example, which arises because there was virtually no dominance in this attribute ('''d''' β 0). Examination of all three versions of '''G<sub>''f''</sub>''' reveals that this would lead to trivial change in the Population mean. Where dominance was notable, however, there would be considerable change.
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