Editing Quantitative genetics (section)

===Heritability and repeatability===
The [[heritability]] of a trait is the proportion of the total (phenotypic) variance (σ<sup>2</sup> <sub>P</sub>) that is attributable to genetic variance, whether it be the full genotypic variance, or some component of it. It quantifies the degree to which phenotypic variability is due to genetics: but the precise meaning depends upon which genetical variance partition is used in the numerator of the proportion.<ref>In biometry, it is a variance-ratio in which a part is expressed as a fraction of the whole: that is, a [[coefficient of determination]]. Such coefficients are used particularly in [[regression analysis]]. A standardized version of regression analysis is [[path analysis (statistics)|path analysis]]. Here, standardizing means that the data were first divided by their own experimental standard errors to unify the scales for all attributes. This genetical usage is another major appearance of coefficients of determination.</ref> Research estimates of heritability  have standard errors, just as have all estimated statistics.<ref name="Gordon Byth Balaam 1972">{{cite journal|last1=Gordon|first1=I. L.|last2=Byth|first2=D. E.|last3=Balaam|first3=L. N.|title=Variance of heritability ratios estimated from phenotypic variance components.|journal=Biometrics|date=1972|volume=28|issue=2|pages=401–415|doi=10.2307/2556156|jstor=2556156|pmid=5037862}}</ref>

Where the numerator variance is the whole Genotypic variance (''' σ<sup>2</sup><sub>G</sub> '''), the heritability is known as the "broadsense" heritability (''H<sup>2</sup>''). It quantifies the degree to which variability in an attribute is determined by genetics as a whole. <math display="block"> \begin{align} H^2 & = \frac {\sigma ^2_G} {\sigma ^2_P} \\ & = \frac {\sigma ^2_A + \sigma ^2_D}{\sigma ^2_P} \\ & = \frac {\left[ \sigma ^2_a + \sigma ^2_d + cov_{ad} \right] + \sigma ^2_D}{\sigma ^2_P} \end{align} </math> [See section on the Genotypic variance.]

If only genic variance ('''σ<sup>2</sup><sub>A</sub>''') is used in the numerator, the heritability may be called "narrow sense" (h<sup>2</sup>). It quantifies the extent to which phenotypic variance is determined by Fisher's ''substitution expectations'' variance. <math display="block"> \begin{align} h^2 & = \frac {\sigma ^2_A}{\sigma ^2_P} \\ & = \frac {\sigma ^2_a + \sigma ^2_d + cov_{ad}}{\sigma ^2_P} \end{align} </math>Fisher proposed that this narrow-sense heritability might be appropriate in considering the results of natural selection, focusing as it does on change-ability, ''that is'' upon "adaptation".<ref name="Fisher 1999">{{cite book|last1=Fisher|first1=R. A.|title=The genetical theory of natural selection.|date=1999|edition=variorum|publisher=Oxford University Press|location=Oxford|isbn=0-19-850440-3}}</ref> He proposed it with regard to quantifying Darwinian evolution.

Recalling that the '''allelic''' variance (''σ <sup>2</sup><sub>a</sub>'') and the '''dominance''' variance (''σ <sup>2</sup><sub>d</sub>'') are eu-genetic components of the gene-model [see section on the Genotypic variance], and that ''σ <sup>2</sup><sub>D</sub>'' (the ''substitution deviations'' or '' "quasi-dominance" '' variance) and ''cov<sub>ad</sub>'' are due to changing from the homozygote midpoint ('''mp''') to the population mean ('''G'''), it can be seen that the real meanings of these heritabilities are obscure. The heritabilities <math display="inline"> H^2_{eu} = \tfrac {\sigma ^2_a + \sigma ^2_d}{\sigma ^2_P} </math>  and  <math display="inline"> h^2_{eu} = \tfrac {\sigma ^2_a}{\sigma ^2_P} </math> have unambiguous meaning.

Narrow-sense heritability has been used also for predicting generally the results of '''artificial selection'''. In the latter case, however, the broadsense heritability may be more appropriate, as the whole attribute is being altered: not just adaptive capacity. Generally, advance from selection is more rapid the higher the heritability. [See section on "Selection".] In animals, heritability of reproductive traits is typically low, while heritability of disease resistance and production are moderately low to moderate, and heritability of body conformation is high.

Repeatability (r<sup>2</sup>) is the proportion of phenotypic variance attributable to differences in repeated measures of the same subject, arising from later records. It is used particularly for long-lived species. This value can only be determined for traits that manifest multiple times in the organism's lifetime, such as adult body mass, metabolic rate or litter size. Individual birth mass, for example, would not have a repeatability value: but it would have a heritability value. Generally, but not always, repeatability indicates the upper level of the heritability.<ref name="Dohm 2002">{{cite journal|last1=Dohm|first1=M. R.| title=Repeatability estimates do not always set an upper limit to heritibility.|journal=Functional Ecology| date=2002|volume=16|issue=2|pages=273–280|doi=10.1046/j.1365-2435.2002.00621.x|doi-access=free|bibcode=2002FuEco..16..273M }}</ref>

r<sup>2</sup> = (s<sup>2</sup><sub>G</sub> + s<sup>2</sup><sub>PE</sub>)/s<sup>2</sup><sub>P</sub>

where s<sup>2</sup><sub>PE</sub> = phenotype-environment interaction = repeatability.

The above concept of repeatability is, however, problematic for traits that necessarily change greatly between measurements. For example, body mass increases greatly in many organisms between birth and adult-hood. Nonetheless, within a given age range (or life-cycle stage), repeated measures could be done, and repeatability would be meaningful within that stage.