# Heritability

In genetics, heritability is the proportion of phenotypic variation in a population that is attributable to genetic variation among individuals. Variation among individuals may be due to genetic and/or environmental factors. Heritability analyses estimate the relative contributions of differences in genetic and non-genetic factors to the total phenotypic variance in a population.

## Definition

Figure 1. Relationship of phenotypic values to additive and dominance effects using a completely dominant locus.

Consider a statistical model for describing some particular phenotype:[1]

Phenotype (P) = Genotype (G) + Environment (E).

Considering variances (Var), this becomes:

Var(P) = Var(G) + Var(E) + 2 Cov(G,E).

In planned experiments, we can often take Cov(G,E) = 0. Heritability is defined as:

${\displaystyle H^{2}={\frac {Var(G)}{Var(P)}}}$ .

The parameter H2 is the broad-sense heritability and reflects all possible genetic contributions to a population's phenotypic variance. Included are effects due to allelic variation (additive variance), dominance variation or which act epistatically (multi-genic interactions), as well as maternal and paternal effects, where individuals are directly affected of their parents' phenotype (such as with milk production in mammals).

These additional terms can be included in genetic models. For example, the simplest genetic model involves a single locus with two alleles that effect some quantitative phenotype, as shown by + in Figure 1. We can calculate the linear regression of phenotype on the number of B alleles (0, 1, or 2), which is shown as the Linear Effect line. For any genotype, BiBj, the expected phenotype can then be written as the sum of the overall mean, a linear effect, and a dominance deviation:

${\displaystyle P_{ij}=\mu +\alpha _{i}+\alpha _{j}+d_{ij}}$ = Population mean + Additive Effect (${\displaystyle a_{ij}=\alpha _{i}+\alpha _{j}}$) + Dominance Deviation (${\displaystyle d_{ij}}$).

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

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

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

There is a similar relationship for variance of dominance deviations:

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

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

Narrow-sense heritability is defined as

${\displaystyle h^{2}={\frac {Var(A)}{Var(P)}}}$

and quantifies only the portion of the phenotypic variation that is additive (allelic) by nature (note upper case H2 for broad sense, lower case h2 for narrow sense). When interested in improving livestock via artificial selection, for example, knowing the narrow-sense heritability of the trait of interest will allow predicting how much 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).

## Estimating heritability

Estimating heritability is not a simple process, however, since only P can be observed or measured directly. Measuring the genetic and environmental variance requires various sophisticated statistical methods. These methods give better estimates when using data from closely related individuals - such as brothers, sisters, parents and offspring, rather than from more distantly related ones. The standard error for heritability estimates are generally very poor unless the dataset is large.

File:Heritability-from-twin-correlations1.jpg
Figure 2. Heritability for nine psychological traits as estimated from twin studies. All sources are twins raised together (sample size shown inside bars). MZ: Monozygotic twin, DZ: Dizygotic twin

In non-human populations it is often possible to collect information in a controlled way. For example, among farm animals it is easy to arrange for a bull to produce offspring from a large number of cows. Due to ethical concerns, such a degree of experimental control is impossible when gathering human data. As a result, studies of human heritability often contrast identical twins who have been separated early in life and raised in different environments (see for example Fig. 2). Such individuals have identical genotypes and can be used to separate the effects of genotype and environment. Twin studies entail problems of its own, such as: independently raised twins shared a common prenatal environment; they may have undergone intrauterine competition; the mother may be more physically stressed (less nutrients); and twins reared apart are difficult to find, and may reflect certain types of environments.

Heritability estimates are always relative to the genetic and environmental factors in the population, and are not absolute measurements of the contribution of genetic and environmental factors to a phenotype. Heritability estimates reflect the amount of variation in genotypic effects compared to variation in environmental effects. Heritability can be made larger by diversifying the genetic background, e.g., by using only very outbred individuals (which increases the Variance(G)) and/or by minimizing environmental effects (which decreases the Variance(E)). Smaller heritability, on the other hand, can be generated by using inbred individuals (which decreases the Variance(G)) or individuals reared in very diverse environments (which increases the Variance(E)). Due to such effects, different populations of a species might have different heritabilities even for the same trait.

Because of the contextual nature of measured heritabilities, paradoxes often arise. For example, the heritability of a trait could be near 100% in one study and close to zero in another. In one study, e.g., a group of unrelated army recruits may be given identical training and nutrition and then their muscular strength may be measured. The variation in strength observed after the (identical) training will translate into a high heritability estimate. In another study, whose purpose might be to assess the efficacy of various workout regimes or nutritional programs, study subjects may be first chosen to match each other as closely as possible in prior physical characteristics before some of them are put onto Program A and others onto Program B, and this will lead to a low heritability estimate.

In the case of scholastic ability, how well one does in the final school exams depends on both what and how well one was taught, how hard one has studied, how ‘naturally’ smart one is and, of course, on a fair bit on luck. The actual heritability estimate will depend on the subjects used (reflecting genetic variation) and the testing conditions (reflecting environmental variation).

Much the same goes for intelligence tests. The conclusions from studies involving intelligence tests often conclude that intelligence has high heritability. This is probably due to inherent problems with human twin studies, as well as reflecting a high level of genetic variation for many human traits, and corresponding lower environmental variation within the confines of the test.

Heritability is often misunderstood when presented in the non-scientific media. Heritability only quantifies how much of the total phenotypic variation in a population is attributable to variation among individual genotypes compared to the variation in their environment. Heritability does not quantify the extent to which genes and environment actually determine a phenotype, let alone the extent to which changes in genes and environment could change phenotypic values.

## Estimation methods

There are essentially two schools of thought regarding estimation of heritability.

One school of thought was developed by Sewall Wright at The University of Chicago, and further popularized by C. C. Li (University of Chicago) and J. L. Lush (Iowa State University). It is based on the analysis of correlations and, by extension, regression. Path Analysis was developed by Sewall Wright as a way of estimating heritability.

The second was originally developed by R. A. Fisher and expanded at The University of Edinburgh, Iowa State University, and North Carolina State University, as well as other schools. It is based on the analysis of variance of breeding studies, using the intraclass correlation of relatives. Various methods of estimating components of variance (and, hence, heritability) from ANOVA are used in these analyses.

## Regression/correlation methods of estimation

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

### Selection experiments

File:Resp-to-sel.jpg
Figure 3. Strength of selection (S) and response to selection (R) in an artificial selection experiment, h2=R/S.

Calculating the strength of selection, S (the difference in mean trait between the population as a whole and the selected parents of the next generation, also called the selection differential [2]) and response to selection R (the difference in offspring and whole parental generation mean trait) in an artificial selection experiment will allow calculation of realized heritability as the response to selection relative to the strength of selection, h2=R/S as in Fig. 3.

### Comparison of close relatives

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 the coefficient of correlation.

#### Parent-offspring regression

Figure 4. Sir Francis Galton's (1889) data showing the relationship between offsping 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. 4). 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).

#### Full-sib comparison

Full-sib designs compare phenotypic traits of siblings that share a mother and a father with other sibling groups. The estimate of the sibling phenotypic correlation is an index on familiality which is equal to half the additive genetic variance plus the common environment variance when there is only additive gene action. Template:Section-stub

#### Half-sib comparison

Half-sib designs compare phenotypic traits of siblings that share one parent with other sibling groups. Template:Section-stub

#### Twin studies

File:Twin-concordances.jpg
Figure 5. Twin concordances for seven psychological traits (sample size shown inside bars).

Heritability for traits in humans is most frequently estimated by comparing resemblances between twins (Fig. 2 & 5). Identical twins (MZ twins) are twice as genetically similar as fraternal twins (DZ twins) and so heritability is approximately twice the difference in correlation between MZ and DZ twins, 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 degee 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 classical twin study has been severely criticized and is used less and less frequently nowadays.

## Analysis of variance methods of estimation

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

### Basic model

We use the basic discussion of Kempthorne (1957 [1969]). Considering only the most basic of genetic models, we can look at the quantitative contribution of a single locus with genotype Gi as

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

where

${\displaystyle g_{i}}$ is the effect of genotype Gi

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

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

#### Intraclass correlations

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

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

since environmental effects are independent of each other.

#### The ANOVA

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
Among 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 among 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.

### Model with additive and dominance terms

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 ith allele, ${\displaystyle \alpha _{j}}$ is the additive effect of the jth allele, ${\displaystyle d_{ij}}$ is the dominance deviation for the ijth 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: Coeffients 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 ${\displaystyle {\frac {1}{4}}}$ ${\displaystyle {\frac {1}{16}}}$

### Larger models

When a large, complex pedigree is available for estimating heritability, the most efficient use of the data is in a restricted maximum likelihood (REML) model. The raw data will usually have three or more datapoints 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 timepoints 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 [1], and analysed with programs such as ASReml, VCE [2] or WOMBAT [3].