Editing Quantitative genetics (section)

==== The equivalent post-dispersion panmictic – inbreeding====
Included in the ''overall summary'' were the average allele frequencies in the mixture of progeny lines ('''p<sub>•</sub>''' and '''q<sub>•</sub>'''). These can now be used to construct a hypothetical panmictic equivalent.<ref name="Crow & Kimura"/>{{rp|382–395}}<ref name="Falconer 1996"/>{{rp|49–63}}<ref name="Fisher 1999"/>{{rp|35}} This can be regarded as a "reference" to assess the changes wrought by the gamete sampling. The example appends such a panmictic to the right of the Diagram. The frequency of '''AA''' is therefore '''(p<sub>•</sub>)<sup>2</sup>''' = 0.3979. This is less than that found in the dispersed bulk (0.4513 at ''black label'' "'''7'''"). Similarly, for '''aa''', '''(q<sub>•</sub>)<sup>2</sup>''' = 0.1303—again less than the equivalent in the progenies bulk (0.1898). Clearly, ''genetic drift'' has increased the overall level of homozygosis by the amount (0.6411 − 0.5342) = 0.1069. In a complementary approach, the heterozygosity could be used instead. The panmictic equivalent for '''Aa''' is '''2 p<sub>•</sub> q<sub>•</sub>''' = 0.4658, which is ''higher'' than that in the sampled bulk (0.3588) [''black label'' "'''8'''"]. The sampling has caused the heterozygosity to decrease by 0.1070, which differs trivially from the earlier estimate because of rounding errors.

The '''inbreeding coefficient''' ('''''f''''') was introduced in the early section on Self Fertilization. Here, a formal definition of it is considered: '''''f''''' is the probability that two "same" alleles (that is '''A''' and '''A''', or '''a''' and '''a'''), which fertilize together are of common ancestral origin—or (more formally) '''''f''''' is the probability that two homologous alleles are autozygous.<ref name="Falconer 1996"/><ref name="Wright 1921 a"/> Consider any random gamete in the ''potential'' gamodeme that has its syngamy partner restricted by binomial sampling. The probability that that second gamete is homologous autozygous to the first is '''1/(2N)''', the reciprocal of the gamodeme size. For the five example progenies, these quantities are 0.1, 0.0833, 0.1, 0.0833 and 0.125 respectively, and their weighted average is '''0.0961'''. This is the ''inbreeding coefficient'' of the example progenies bulk, provided it is ''unbiased'' with respect to the full binomial distribution. An example based upon ''s = 5'' is likely to be biased, however, when compared to an appropriate entire binomial distribution based upon the sample number (''s'') approaching infinity (''s → ∞''). Another derived definition of '''''f''''' for the full Distribution is that '''''f''''' also equals the rise in homozygosity, which equals the fall in heterozygosity.<ref>See the earlier citations.</ref> For the example, these frequency changes are ''0.1069'' and ''0.1070'', respectively. This result is different to the above, indicating that bias with respect to the full underlying distribution is present in the example. For the example ''itself'', these latter values are the better ones to use, namely '''''f<sub>•</sub>''''' = '''0.10695'''.

The ''population mean'' of the equivalent panmictic is found as ''[a (p<sub>•</sub>-q<sub>•</sub>) + 2 p<sub>•</sub>q<sub>•</sub> d] + mp''. Using the example ''gene effects'' (''white label'' "'''9'''" in the diagram), this mean is <math display="inline"> P_{\centerdot} = </math> 37.87. The equivalent mean in the dispersed bulk is 36.94 (''black label'' "'''10'''"), which is depressed by the amount ''0.93''. This is the ''inbreeding depression'' from this Genetic Drift. However, as noted previously, three progenies were ''not'' depressed (k = 1, 2, 5), and had means even greater than that of the panmictic equivalent. These are the lines a plant breeder looks for in a line selection programme.<ref name="Allard 1960">{{cite book|last1=Allard|first1=R. W.|title=Principles of plant breeding|url=https://archive.org/details/principlesofplan0000alla|url-access=registration|date=1960|publisher=John Wiley & Sons|location=New York}}</ref>