Heritability

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_i,B_j), where B_i and B_j 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_i and B_j):

${\displaystyle {\begin{aligned}P_{ij}&=\mu +\alpha \,(B_{i}+B_{j})+\delta \,(B_{i}B_{j})\\&={\text{Population mean}}+{\text{Additive Effect }}(a_{ij}=\alpha (B_{i}+B_{j}))+{\text{Dominance Deviation }}(d_{ij}=\delta (B_{i}B_{j})).\\\end{aligned}}}$

The additive genetic variance at this locus is the weighted average of the squares of the additive effects:

${\displaystyle \mathrm {Var} (A)=f(bb)a_{bb}^{2}+f(Bb)a_{Bb}^{2}+f(BB)a_{BB}^{2},}$

where ${\displaystyle f(bb)a_{bb}+f(Bb)a_{Bb}+f(BB)a_{BB}=0.}$

There is a similar relationship for the variance of dominance deviations:

${\displaystyle \mathrm {Var} (D)=f(bb)d_{bb}^{2}+f(Bb)d_{Bb}^{2}+f(BB)d_{BB}^{2},}$

where ${\displaystyle f(bb)d_{bb}+f(Bb)d_{Bb}+f(BB)d_{BB}=0.}$

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

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.[16] There is also some empirical evidence that the additivity assumption is frequently violated in behavior genetic studies of adolescent intelligence and academic achievement.[17]

The first school of estimation uses regression and correlation to estimate heritability.

In the comparison of relatives, we find that in general,

${\displaystyle h^{2}={\frac {b}{r}}={\frac {t}{r}}}$

where r can be thought of as the coefficient of relatedness, b is the coefficient of regression and t is the coefficient of correlation.

Figure 2. Francis Galton's (1889) data showing the relationship between offspring height (928 individuals) as a function of mean parent height (205 sets of parents).

Heritability may be estimated by comparing parent and offspring traits (as in Fig. 2). The slope of the line (0.57) approximates the heritability of the trait when offspring values are regressed against the average trait in the parents. If only one parent's value is used then heritability is twice the slope. (Note that this is the source of the term "regression," since the offspring values always tend to regress to the mean value for the population, i.e., the slope is always less than one). This regression effect also underlies the DeFries–Fulker method for analyzing twins selected for one member being affected.[19]

A basic approach to heritability can be taken using full-Sib designs: comparing similarity between siblings who share both a biological mother and a father.[20] When there is only additive gene action, this sibling phenotypic correlation is an index of familiarity – the sum of half the additive genetic variance plus full effect of the common environment. It thus places an upper limit on additive heritability of twice the full-Sib phenotypic correlation. Half-Sib designs compare phenotypic traits of siblings that share one parent with other sibling groups.

Figure 3. Twin concordances for seven psychological traits (sample size shown inside bars), with DZ being fraternal and MZ being identical twins.

Heritability for traits in humans is most frequently estimated by comparing resemblances between twins. "The advantage of twin studies, is that the total variance can be split up into genetic, shared or common environmental, and unique environmental components, enabling an accurate estimation of heritability".[21] Fraternal or dizygotic (DZ) twins on average share half their genes (assuming there is no assortative mating for the trait), and so identical or monozygotic (MZ) twins on average are twice as genetically similar as DZ twins. A crude estimate of heritability, then, is approximately twice the difference in correlation between MZ and DZ twins, i.e. Falconer's formula H²=2(r(MZ)-r(DZ)).

The effect of shared environment, c², contributes to similarity between siblings due to the commonality of the environment they are raised in. Shared environment is approximated by the DZ correlation minus half heritability, which is the degree to which DZ twins share the same genes, c²=DZ-1/2h². Unique environmental variance, e², reflects the degree to which identical twins raised together are dissimilar, e²=1-r(MZ).

The second set of methods of estimation of heritability involves ANOVA and estimation of variance components.

We use the basic discussion of Kempthorne.[14] Considering only the most basic of genetic models, we can look at the quantitative contribution of a single locus with genotype G_i as

${\displaystyle y_{i}=\mu +g_{i}+e}$

where ${\displaystyle g_{i}}$ is the effect of genotype G_i and ${\displaystyle e}$ is the environmental effect.

Consider an experiment with a group of sires and their progeny from random dams. Since the progeny get half of their genes from the father and half from their (random) mother, the progeny equation is

${\displaystyle z_{i}=\mu +{\frac {1}{2}}g_{i}+e}$

Consider the experiment above. We have two groups of progeny we can compare. The first is comparing the various progeny for an individual sire (called within sire group). The variance will include terms for genetic variance (since they did not all get the same genotype) and environmental variance. This is thought of as an error term.

The second group of progeny are comparisons of means of half sibs with each other (called among sire group). In addition to the error term as in the within sire groups, we have an addition term due to the differences among different means of half sibs. The intraclass correlation is

${\displaystyle \mathrm {corr} (z,z')=\mathrm {corr} (\mu +{\frac {1}{2}}g+e,\mu +{\frac {1}{2}}g+e')={\frac {1}{4}}V_{g}}$ ,

since environmental effects are independent of each other.

In an experiment with ${\displaystyle n}$ sires and ${\displaystyle r}$ progeny per sire, we can calculate the following ANOVA, using ${\displaystyle V_{g}}$ as the genetic variance and ${\displaystyle V_{e}}$ as the environmental variance:

Table 1: ANOVA for Sire experiment

Source	d.f.	Mean Square	Expected Mean Square
Between sire groups	${\displaystyle n-1}$	${\displaystyle S}$	${\displaystyle {\frac {3}{4}}V_{g}+V_{e}+r({{\frac {1}{4}}V_{g}})}$
Within sire groups	${\displaystyle n(r-1)}$	${\displaystyle W}$	${\displaystyle {\frac {3}{4}}V_{g}+V_{e}}$

The ${\displaystyle {\frac {1}{4}}V_{g}}$ term is the intraclass correlation between half sibs. We can easily calculate ${\displaystyle H^{2}={\frac {V_{g}}{V_{g}+V_{e}}}={\frac {4(S-W)}{S+(r-1)W}}}$. The expected mean square is calculated from the relationship of the individuals (progeny within a sire are all half-sibs, for example), and an understanding of intraclass correlations.

The use of ANOVA to calculate heritability often fails to account for the presence of gene–-environment interactions, because ANOVA has a much lower statistical power for testing for interaction effects than for direct effects.[22]

For a model with additive and dominance terms, but not others, the equation for a single locus is

${\displaystyle y_{ij}=\mu +\alpha _{i}+\alpha _{j}+d_{ij}+e,}$

where

${\displaystyle \alpha _{i}}$ is the additive effect of the i^th allele, ${\displaystyle \alpha _{j}}$ is the additive effect of the j^th allele, ${\displaystyle d_{ij}}$ is the dominance deviation for the ij^th genotype, and ${\displaystyle e}$ is the environment.

Experiments can be run with a similar setup to the one given in Table 1. Using different relationship groups, we can evaluate different intraclass correlations. Using ${\displaystyle V_{a}}$ as the additive genetic variance and ${\displaystyle V_{d}}$ as the dominance deviation variance, intraclass correlations become linear functions of these parameters. In general,

Intraclass correlation${\displaystyle =rV_{a}+\theta V_{d},}$

where ${\displaystyle r}$ and ${\displaystyle \theta }$ are found as

${\displaystyle r=}$P[ alleles drawn at random from the relationship pair are identical by descent], and

${\displaystyle \theta =}$P[ genotypes drawn at random from the relationship pair are identical by descent].

Some common relationships and their coefficients are given in Table 2.

Table 2: Coefficients for calculating variance components

Relationship	${\displaystyle r}$	${\displaystyle \theta }$
Identical Twins	${\displaystyle 1}$	${\displaystyle 1}$
Parent-Offspring	${\displaystyle {\frac {1}{2}}}$	${\displaystyle 0}$
Half Siblings	${\displaystyle {\frac {1}{4}}}$	${\displaystyle 0}$
Full Siblings	${\displaystyle {\frac {1}{2}}}$	${\displaystyle {\frac {1}{4}}}$
First Cousins	${\displaystyle {\frac {1}{8}}}$	${\displaystyle 0}$
Double First Cousins	${\displaystyle {\frac {1}{4}}}$	${\displaystyle {\frac {1}{16}}}$

A wide variety of approaches using linear mixed models have been reported in literature. Via these methods, phenotypic variance is partitioned into genetic, environmental and experimental design variances to estimate heritability. Environmental variance can be explicitly modeled by studying individuals across a broad range of environments, although inference of genetic variance from phenotypic and environmental variance may lead to underestimation of heritability due to the challenge of capturing the full range of environmental influence affecting a trait. Other methods for calculating heritability use data from genome-wide association studies to estimate the influence on a trait by genetic factors, which is reflected by the rate and influence of putatively associated genetic loci (usually single-nucleotide polymorphisms) on the trait. This can lead to underestimation of heritability, however. This discrepancy is referred to as "missing heritability" and reflects the challenge of accurately modeling both genetic and environmental variance in heritability models.[23]

When a large, complex pedigree or another aforementioned type of data is available, heritability and other quantitative genetic parameters can be estimated by restricted maximum likelihood (REML) or Bayesian methods. The raw data will usually have three or more data points for each individual: a code for the sire, a code for the dam and one or several trait values. Different trait values may be for different traits or for different time points of measurement.

The currently popular methodology relies on high degrees of certainty over the identities of the sire and dam; it is not common to treat the sire identity probabilistically. This is not usually a problem, since the methodology is rarely applied to wild populations (although it has been used for several wild ungulate and bird populations), and sires are invariably known with a very high degree of certainty in breeding programmes. There are also algorithms that account for uncertain paternity.

The pedigrees can be viewed using programs such as Pedigree Viewer , and analyzed with programs such as ASReml, VCE , WOMBAT , MCMCglmm within the R environment or the BLUPF90 family of programs .

Pedigree models are helpful for untangling confounds such as reverse causality, maternal effects such as the prenatal environment, and confounding of genetic dominance, shared environment, and maternal gene effects.[24][9]

When genome-wide genotype data and phenotypes from large population samples are available, one can estimate the relationships between individuals based on their genotypes and use a linear mixed model to estimate the variance explained by the genetic markers. This gives a genomic heritability estimate based on the variance captured by common genetic variants.[4] There are multiple methods that make different adjustments for allele frequency and linkage disequilibrium. Particularly, the method called High-Definition Likelihood (HDL) can estimate genomic heritability using only GWAS summary statistics,[5] making it easier to incorporate large sample size available in various GWAS meta-analysis.