Editing Heritability (section)

==Definition==
Any particular phenotype can be [[statistical model|modeled]] as the sum of genetic and environmental effects:<ref name = "Kempthorne_1957">{{cite book |last1=Kempthorne |first1=Oscar | name-list-style = vanc |title=An introduction to genetic statistics |date=1957 |publisher=Iowa State Univ. Press |location=Ames, Iowa |edition=1st | oclc = 422371269 }}</ref>
:Phenotype (''P'') = Genotype (''G'') + Environment (''E'').
Likewise the phenotypic variance in the trait – Var (P) – is the sum of effects as follows:
:Var(''P'') = Var(''G'') + Var(''E'') + 2 Cov(''G'',''E'').
In a planned experiment Cov(''G'',''E'') can be controlled and held at 0. In this case, heritability, <math>H^2,</math> is defined as<ref>{{cite web |last1=Stephen Downes and Lucas Matthews |title=Heritability |url=https://plato.stanford.edu/entries/heredity/ |website=Stanford Encyclopedia of Philosophy |publisher=Stanford University |access-date=2020-02-20 |archive-date=2020-02-25 |archive-url=https://web.archive.org/web/20200225132941/https://plato.stanford.edu/entries/heredity/ |url-status=live }}</ref>
:<math>H^2 = \frac{\mathrm{Var}(G)}{\mathrm{Var}(P)}</math>

''H''<sup>2</sup> is the broad-sense heritability. This reflects all the genetic contributions to a population's phenotypic variance including additive, [[dominance relationship|dominant]], and [[epistasis|epistatic]] (multi-genic interactions), as well as [[maternal effect|maternal and paternal effects]], where individuals are directly affected by their parents' phenotype, such as with [[milk]] production in mammals.

A particularly important component of the genetic variance is the additive variance, Var(A), which is the variance due to the average effects (additive effects) of the [[allele]]s. Since each parent passes a single allele per [[locus (genetics)|locus]] to each offspring, parent-offspring resemblance depends upon the average effect of single alleles. Additive variance represents, therefore, the genetic component of variance responsible for parent-offspring resemblance. The additive genetic portion of the phenotypic variance is known as Narrow-sense heritability and is defined as
:<math>h^2 = \frac{\mathrm{Var}(A)}{\mathrm{Var}(P)}</math>

An upper case ''H''<sup>2</sup> is used to denote broad sense, and lower case ''h''<sup>2</sup> for narrow sense.

For traits which are not continuous but dichotomous such as an additional toe or certain diseases,  the contribution of the various alleles can be considered to be a sum, which past a threshold, manifests itself as the trait, giving the [[Threshold model#Liability threshold model|liability threshold model]] in which heritability can be estimated and selection modeled.

Additive variance is important for [[natural selection|selection]]. If a selective pressure such as improving livestock is exerted, the response of the trait is directly related to narrow-sense heritability. The mean of the trait will increase in the next generation as a function of how much the mean of the selected parents differs from the mean of the population from which the selected parents were chosen. The observed ''response to selection'' leads to an estimate of the narrow-sense heritability (called '''realized heritability'''). This is the principle underlying [[artificial selection]] or breeding.

===Example===
[[Image:Additive and Dominance Effects.png|250px|thumbnail|Figure 1. Relationship of phenotypic values to additive and dominance effects using a completely dominant locus.]]

The simplest genetic model involves a single locus with two alleles (b and B) affecting one quantitative phenotype.

The number of '''B''' alleles can be 0, 1, or 2. For any genotype, (''B''<sub>i</sub>,''B''<sub>j</sub>), where ''B''<sub>i</sub> and ''B''<sub>j</sub> are either 0 or 1, the expected phenotype can then be written as the sum of the overall mean, a linear effect, and a dominance deviation (one can think of the dominance term as an ''interaction'' between ''B''<sub>i</sub> and ''B''<sub>j</sub>):

<math>
\begin{align}
P_{ij} & = \mu + \alpha \, (B_i + B_j) + \delta \, (B_i B_j) \\
& = \text{Population mean} + \text{Additive Effect } (a_{ij} = \alpha (B_i + B_j)) + \text{Dominance Deviation } (d_{ij} = \delta (B_i B_j)). \\
\end{align}
</math>

The additive genetic variance at this locus is the [[Weighted mean|weighted average]] of the squares of the additive effects:
:<math>\mathrm{Var}(A) = f(bb)a^2_{bb}+f(Bb)a^2_{Bb}+f(BB)a^2_{BB},</math>

where <math>f(bb)a_{bb}+f(Bb)a_{Bb}+f(BB)a_{BB} = 0.</math>

There is a similar relationship for the variance of dominance deviations:
:<math>\mathrm{Var}(D) = f(bb)d^2_{bb}+f(Bb)d^2_{Bb}+f(BB)d^2_{BB},</math>

where <math>f(bb)d_{bb}+f(Bb)d_{Bb}+f(BB)d_{BB} = 0.</math>

The [[linear regression]] of phenotype on genotype is shown in Figure 1.

===Assumptions===
Estimates of the total heritability of human traits assume the absence of epistasis, which has been called the "assumption of additivity". Although some researchers have cited such estimates in support of the existence of "[[missing heritability]]" unaccounted for by known genetic loci, the assumption of additivity may render these estimates invalid.<ref>{{cite journal | vauthors = Zuk O, Hechter E, Sunyaev SR, Lander ES | title = The mystery of missing heritability: Genetic interactions create phantom heritability | journal = Proceedings of the National Academy of Sciences of the United States of America | volume = 109 | issue = 4 | pages = 1193–8 | date = January 2012 | pmid = 22223662 | pmc = 3268279 | doi = 10.1073/pnas.1119675109 | bibcode = 2012PNAS..109.1193Z | doi-access = free }}</ref> There is also some empirical evidence that the additivity assumption is frequently violated in behavior genetic studies of adolescent intelligence and [[academic achievement]].<ref>{{cite journal | vauthors = Daw J, Guo G, Harris KM | title = Nurture net of nature: Re-evaluating the role of shared environments in academic achievement and verbal intelligence | journal = Social Science Research | volume = 52 | pages = 422–39 | date = July 2015 | pmid = 26004471 | pmc = 4888873 | doi = 10.1016/j.ssresearch.2015.02.011 }}</ref>