Editing Quantitative genetics (section)

==Selection==
===Basic principles===
[[File:SelctnPresur.jpg|thumb|250px|right|Genetic advance and selection pressure repeated]]

Selection operates on the attribute (phenotype), such that individuals that equal or exceed a selection threshold '''(z<sub>P</sub>)''' become effective parents for the next generation. The ''proportion'' they represent of the base population is the ''selection pressure''. The ''smaller'' the proportion, the ''stronger'' the pressure. The ''mean of the selected group'' '''(P<sub>s</sub>)''' is superior to the ''base-population mean'' '''(P<sub>0</sub>)''' by the difference called the ''selection differential (S)''. All these quantities are phenotypic. To "link" to the underlying genes, a ''heritability'' '''(h<sup>2</sup>)''' is used, fulfilling the role of a ''coefficient of determination'' in the biometrical sense. The ''expected genetical change''—still expressed in ''phenotypic units of measurement''—is called the ''genetic advance (ΔG)'', and is obtained by the product of the ''selection differential (S)'' and its ''coefficient of determination'' '''(h<sup>2</sup>)'''. The expected ''mean of the progeny'' '''(P<sub>1</sub>)''' is found by adding the ''genetic advance (ΔG)'' to the ''base mean (P<sub>0</sub>)''. The graphs to the right show how the (initial) genetic advance is greater with stronger selection pressure (smaller ''probability''). They also show how progress from successive cycles of selection (even at the same selection pressure) steadily declines, because the Phenotypic variance and the Heritability are being diminished by the selection itself. This is discussed further shortly.
 
Thus  <math> \Delta G = S h^2  </math>.<ref name="Falconer 1996"/> {{rp|1710-181}}
and <math> P_1 = P_0+\Delta G  </math>.<ref name="Falconer 1996"/> {{rp|1710-181}}

The ''narrow-sense heritability (h<sup>2</sup>)'' is usually used, thereby linking to the ''genic variance (σ<sup>2</sup><sub>A</sub>) ''. However, if appropriate, use of the ''broad-sense heritability (H<sup>2</sup>)'' would connect to the ''genotypic variance (σ<sup>2</sup><sub>G</sub>)''; and even possibly an ''allelic heritability [ h<SUP>2</SUP><sub>eu</sub> = (σ<sup>2</sup><sub>a</sub>) / (σ<sup>2</sup><sub>P</sub>) ]'' might be contemplated, connecting to ('''σ<sup>2</sup><sub>a</sub>''' ). [See section on Heritability.]

To apply these concepts ''before'' selection actually takes place, and so predict the outcome of alternatives (such as choice of ''selection threshold'', for example), these phenotypic statistics are re-considered against the properties of the Normal Distribution, especially those concerning truncation of the ''superior tail'' of the Distribution. In such consideration, the ''standardized'' selection differential (i){{pprime}} and the ''standardized'' selection threshold (z){{pprime}} are used instead of the previous "phenotypic" versions. The '''phenotypic standard deviate (σ<sub>P(0)</sub>)''' is also needed. This is described in a subsequent section.

Therefore, '''ΔG = (i σ<sub>P</sub>) h<sup>2</sup>''', where  ''(i σ<sub>P(0)</sub>)'' = ''S'' previously.<ref name="Falconer 1996"/> {{rp|1710-181}}

[[File:SelctnRptd.jpg|thumb|250px|left|Changes arising from repeated selection]]

The text above noted that successive '''ΔG''' declines because the "input" [the '''phenotypic variance ( σ<sup>2</sup><sub>P</sub> )'''] is reduced by the previous selection.<ref name="Falconer 1996"/>{{rp|1710-181}} The heritability also is reduced. The graphs to the left show these declines over ten cycles of repeated selection during which the same selection pressure is asserted. The accumulated genetic advance ('''ΣΔG''') has virtually reached its asymptote by generation 6 in this example. This reduction depends partly upon truncation properties of the Normal Distribution, and partly upon the heritability together with ''meiosis determination ( b<sup>2</sup> )''. The last two items quantify the extent to which the ''truncation'' is "offset" by new variation arising from segregation and assortment during meiosis.<ref name="Falconer 1996"/> {{rp|1710-181}}<ref name="Wright 1921 a"/> This is discussed soon, but here note the simplified result for ''undispersed random fertilization (f = 0)''.

Thus : '''σ<sup>2</sup><sub>P(1)</sub> = σ<sup>2</sup><sub>P(0)</sub> [1 − i ( i-z) {{sfrac|1|2}} h<sup>2</sup>]''',  where '''i ( i-z) = K = truncation coefficient''' and '''{{sfrac|1|2}} h<sup>2</sup> = R = reproduction coefficient'''<ref name="Falconer 1996"/>{{rp|1710-181}}<ref name="Wright 1921 a"/> This can be written also as '''σ<sup>2</sup><sub>P(1)</sub> = σ<sup>2</sup><sub>P(0)</sub> [1 − K R ]''', which facilitates more detailed analysis of selection problems.

Here, '''i''' and '''z''' have already been defined, '''{{sfrac|1|2}}''' is the ''meiosis determination ('''b<sup>2</sup>''') for '''f=0''''', and the remaining symbol is the heritability. These are discussed further in following sections. Also notice that, more generally, '''R = b<sup>2</sup> h<sup>2</sup>'''. If the general '''meiosis determination ( b<sup>2</sup> )''' is used, the results of prior inbreeding can be incorporated into the selection. The phenotypic variance equation then becomes:

'''σ<sup>2</sup><sub>P(1)</sub> = σ<sup>2</sup><sub>P(0)</sub> [1 − i ( i-z) b<sup>2</sup> h<sup>2</sup>]'''.

The ''Phenotypic variance'' truncated by the ''selected group'' (''' σ<sup>2</sup><sub>P(S)</sub>''' ) is simply '''σ<sup>2</sup><sub>P(0)</sub> [1 − K]''', and its contained ''genic variance'' is '''(h<sup>2</sup><sub>0</sub> σ<sup>2</sup><sub>P(S)</sub>''' ). Assuming that selection has not altered the ''environmental'' variance, the ''genic variance'' for the progeny can be approximated by ''' σ<sup>2</sup><sub>A(1)</sub> = ( σ<sup>2</sup><sub>P(1)</sub> − σ<sup>2</sup><sub>E</sub>) ''' . From this, '''h<sup>2</sup><sub>1</sub> = ( σ<sup>2</sup><sub>A(1)</sub> / σ<sup>2</sup><sub>P(1)</sub> )'''. Similar estimates could be made for '''σ<sup>2</sup><sub>G(1)</sub>'''  and '''H<sup>2</sup><sub>1</sub>''', or for '''σ<sup>2</sup><sub>a(1)</sub>''' and '''h<sup>2</sup><sub>eu(1)</sub>''' if required.

====Alternative ΔG====
The following rearrangement is useful for considering selection on multiple attributes (characters). It starts by expanding the heritability into its variance components. '''ΔG = i σ<sub>P</sub> ( σ<sup>2</sup><sub>A</sub> / σ<sup>2</sup><sub>P</sub> ) '''. The ''σ<sub>P</sub>'' and ''σ<sup>2</sup><sub>P</sub>'' partially cancel, leaving a solo ''σ<sub>P</sub>''. Next, the ''σ<sup>2</sup><sub>A</sub>'' inside the heritability can be expanded as (''σ<sub>A</sub> × σ<sub>A</sub>''), which leads to :
[[File:SelctnDifrntl.jpg|thumb|300px|right|Selection differential and the normal distribution]]
'''ΔG = i σ<sub>A</sub> ( σ<sub>A</sub> / σ<sub>P</sub> )''' = '''i σ<sub>A</sub> h '''.

Corresponding re-arrangements could be made using the alternative heritabilities, giving '''ΔG = i σ<sub>G</sub> H''' or '''ΔG = i σ<sub>a</sub> h<sub>eu</sub>'''.

===== Polygenic Adaptation Models in Population Genetics =====

This traditional view of adaptation in quantitative genetics provides a model for how the selected phenotype changes over time, as a function of the selection differential and heritability.  However it does not provide insight into (nor does it depend upon) any of the genetic details - in particular, the number of loci involved, their allele frequencies and effect sizes, and the frequency changes driven by selection.  This, in contrast, is the focus of work on [[polygenic adaptation]]<ref>{{Cite journal|last1=Pritchard|first1=Jonathan K.|last2=Pickrell|first2=Joseph K.|last3=Coop|first3=Graham|date=2010-02-23|title=The genetics of human adaptation: hard sweeps, soft sweeps, and polygenic adaptation|journal=Current Biology|volume=20|issue=4|pages=R208–215|doi=10.1016/j.cub.2009.11.055|issn=1879-0445|pmc=2994553|pmid=20178769|bibcode=2010CBio...20.R208P }}</ref> within the field of [[population genetics]]. Recent studies have shown that traits such as height have evolved in humans during the past few thousands of years as a result of small allele frequency shifts at thousands of variants that affect height.<ref>{{Cite journal|last1=Turchin|first1=Michael C.|last2=Chiang|first2=Charleston W. K.|last3=Palmer|first3=Cameron D.|last4=Sankararaman|first4=Sriram|last5=Reich|first5=David|last6=Genetic Investigation of ANthropometric Traits (GIANT) Consortium|last7=Hirschhorn|first7=Joel N.|date=September 2012|title=Evidence of widespread selection on standing variation in Europe at height-associated SNPs|journal=Nature Genetics|volume=44|issue=9|pages=1015–1019|doi=10.1038/ng.2368|issn=1546-1718|pmc=3480734|pmid=22902787}}</ref><ref>{{Cite journal|last1=Berg|first1=Jeremy J.|last2=Coop|first2=Graham|date=August 2014|title=A population genetic signal of polygenic adaptation|journal=PLOS Genetics|volume=10|issue=8|pages=e1004412|doi=10.1371/journal.pgen.1004412|issn=1553-7404|pmc=4125079|pmid=25102153 |doi-access=free }}</ref><ref>{{Cite journal|last1=Field|first1=Yair|last2=Boyle|first2=Evan A.|last3=Telis|first3=Natalie|last4=Gao|first4=Ziyue|last5=Gaulton|first5=Kyle J.|last6=Golan|first6=David|last7=Yengo|first7=Loic|last8=Rocheleau|first8=Ghislain|last9=Froguel|first9=Philippe|date=2016-11-11|title=Detection of human adaptation during the past 2000 years|journal=Science|language=en|volume=354|issue=6313|pages=760–764|doi=10.1126/science.aag0776|issn=0036-8075|pmid=27738015|pmc=5182071|bibcode=2016Sci...354..760F}}</ref>

===Background===
====Standardized selection – the normal distribution====

The entire ''base population'' is outlined by the normal curve<ref name="Balaam 1972"/>{{rp|78–89}} to the right. Along the '''Z axis''' is every value of the attribute from least to greatest, and the height from this axis to the curve itself is the frequency of the value at the axis below. The equation for finding these frequencies for the "normal" curve (the curve of "common experience") is given in the ellipse. Notice it includes the mean ('''μ''') and the variance ('''σ<sup>2</sup>'''). Moving infinitesimally along the z-axis, the frequencies of neighbouring values can be "stacked" beside the previous, thereby accumulating an area that represents the '''probability''' of obtaining all values within the stack. [That's '''integration''' from calculus.] Selection focuses on such a probability area, being the shaded-in one from the ''selection threshold (z)'' to the end of the superior tail of the curve. This is the ''selection pressure''. The selected group (the effective parents of the next generation) include all phenotype values from '''z''' to the "end" of the tail.<ref>Theoretically, the tail is '''''infinite''''', but in practice there is a ''quasi-end''.</ref> The mean of the '''''selected group''''' is '''μ<sub>s</sub>''', and the difference between it and the base mean ('''μ''') represents the '''selection differential (S)'''.  By taking partial integrations over curve-sections of interest, and some rearranging of the algebra, it can be shown that the "selection differential" is '''S = [ y (σ / Prob.)] ''', where '''y''' is the ''frequency'' of the value at the "selection threshold" '''z''' (the ''ordinate'' of ''z'').<ref name="Crow & Kimura"/>{{rp|226–230}} Rearranging this relationship gives '''S / σ = y / Prob.''', the left-hand side of which is, in fact, ''selection differential divided by standard deviation''—that is the ''standardized selection differential (i)''. The right-side of the relationship provides an "estimator" for '''i'''—the ordinate of the ''selection threshold'' divided by the ''selection pressure''. Tables of the Normal Distribution<ref name="Snedecor & Cochran"/> {{rp|547–548}} can be used, but tabulations of '''i''' itself are available also.<ref name="Becker 1967">{{cite book|last1=Becker|first1=Walter A.|title=Manual of procedures in quantitative genetics.|date=1967|publisher=Washington State University|location=Pullman|edition=Second}}</ref>{{rp|123–124}} The latter reference also gives values of '''i''' adjusted for small populations (400 and less),<ref name="Becker 1967"/>{{rp|111–122}} where "quasi-infinity" cannot be assumed (but ''was'' presumed in the "Normal Distribution" outline above). The ''standardized selection differential ('''i''')'' is known also as the '''''intensity of selection'''''.<ref name="Falconer 1996"/>{{rp|174; 186}}

Finally, a cross-link with the differing terminology in the previous sub-section may be useful: '''μ''' (here) = "P<sub>0</sub>" (there), '''μ<sub>S</sub>''' = "P<sub>S</sub>" and '''σ<sup>2</sup>''' = "σ<sup>2</sup><sub>P</sub>".

====Meiosis determination – reproductive path analysis====
[[File:ReproDetmntn.jpg|thumb|200px|right|Reproductive coefficients of determination and inbreeding]]
[[File:ReproPaths.jpg|thumb|300px|left|Path analysis of sexual reproduction.]]

The '''meiosis determination (b<sup>2</sup>)''' is the ''coefficient of determination'' of meiosis, which is the cell-division whereby parents generate gametes. Following the principles of ''standardized partial regression'', of which '''path analysis''' is a pictorially oriented version, Sewall Wright analyzed the paths of gene-flow during sexual reproduction, and established the "strengths of contribution" (''coefficients of determination'') of various components to the overall result.<ref name="Wright 1921 a"/><ref name="Wright 1951"/> Path analysis includes ''partial correlations'' as well as ''partial regression coefficients'' (the latter are the ''path coefficients''). Lines with a single arrow-head are directional ''determinative paths'', and lines with double arrow-heads are ''correlation connections''. Tracing various routes according to ''path analysis rules'' emulates the algebra of standardized partial regression.<ref name="Li 1977"/>

The path diagram to the left represents this analysis of sexual reproduction. Of its interesting elements, the important one in the selection context is ''meiosis''. That's where segregation and assortment occur—the processes that partially ameliorate the truncation of the phenotypic variance that arises from selection.  The path coefficients '''b''' are the meiosis paths. Those labeled '''a''' are the fertilization paths. The correlation between gametes from the same parent ('''g''') is the ''meiotic correlation''. That between parents within the same generation is '''r<sub>A</sub>'''. That between gametes from different parents ('''f''') became known subsequently as the ''inbreeding coefficient''.<ref name="Crow & Kimura"/>{{rp|64}}  The primes ( ' ) indicate generation '''(t-1)''', and the ''un''primed indicate generation '''t'''. Here, some important results of the present analysis are given. Sewall Wright interpreted many in terms of inbreeding coefficients.<ref name="Wright 1921 a"/><ref name="Wright 1951"/>

The meiosis determination ('''b<sup>2</sup>''') is ''{{sfrac|1|2}} (1+g)'' and equals '''{{sfrac|1|2}} (1 + f<sub>(t-1)</sub>) ''', implying that '''g = f<sub>(t-1)</sub>'''.<ref>Notice that this '''b<sup>2</sup>''' is the ''coefficient of parentage ('''f<sub>AA</sub>''')'' of ''Pedigree analysis'' re-written with a "generation level" instead of an "A" inside the parentheses.</ref> With non-dispersed random fertilization, f<sub>(t-1)</sub>) = 0, giving '''b<sup>2</sup> = {{sfrac|1|2}}''', as used in the selection section above. However, being aware of its background, other fertilization patterns can be used as required. Another determination also involves inbreeding—the fertilization determination ('''a<sup>2</sup>''') equals '''1 / [ 2 ( 1 + f<sub>t</sub> ) ]''' . Also another correlation is an inbreeding indicator—'''r<sub>A</sub>''' = '''2 f<sub>t</sub> / ( 1 + f<sub>(t-1)</sub> )''', also known as the ''coefficient of relationship''. [Do not confuse this with the ''coefficient of kinship''—an alternative name for the ''co-ancestry coefficient''. See introduction to "Relationship" section.] This '''r<sub>A</sub>''' re-occurs in the sub-section on dispersion and selection.

These links with inbreeding reveal interesting facets about sexual reproduction that are not immediately apparent. The graphs to the right plot the ''meiosis'' and ''syngamy (fertilization)'' coefficients of determination against the inbreeding coefficient. There it is revealed that as inbreeding increases, meiosis becomes more important (the coefficient increases), while syngamy becomes less important. The overall role of reproduction [the product of the previous two coefficients—'''r<sup>2</sup>'''] remains the same.<ref>There is a small "wobble" arising from the fact that ''b<sup>2</sup>'' alters one generation behind ''a<sup>2</sup>''—examine their inbreeding equations.</ref> This ''increase in '''b<sup>2</sup>''''' is particularly relevant for selection because it means that the ''selection truncation of the Phenotypic variance'' is offset to a lesser extent during a sequence of selections when accompanied by inbreeding (which is frequently the case).

===Genetic drift and selection===
The previous sections treated ''dispersion'' as an "assistant" to ''selection'', and it became apparent that the two work well together. In quantitative genetics, selection is usually examined in this "biometrical" fashion, but the changes in the means (as monitored by ΔG) reflect the changes in allele and genotype frequencies beneath this surface. Referral to the section on "Genetic drift" brings to mind that it also effects changes in allele and genotype frequencies, and associated means; and that this is the companion aspect to the dispersion considered here ("the other side of the same coin"). However, these two forces of frequency change are seldom in concert, and may often act contrary to each other. One (selection) is "directional" being driven by selection pressure acting on the phenotype: the other (genetic drift) is driven by "chance" at fertilization (binomial probabilities of gamete samples). If the two tend towards the same allele frequency, their "coincidence" is the probability of obtaining that frequencies sample in the genetic drift: the likelihood of their being "in conflict", however, is the ''sum of probabilities of all the alternative frequency samples''. In extreme cases, a single syngamy sampling can undo what selection has achieved, and the probabilities of it happening are available. It is important to keep this in mind. However, genetic drift resulting in sample frequencies similar to those of the selection target does not lead to so drastic an outcome—instead slowing progress towards selection goals.