# Heritability

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In genetics, heritability is the proportion of phenotypic variation in a population that is due to genetic variation. Variation among individuals may be due to genetic and/or environmental factors. Heritability analyses estimate the relative importance of variation in each of these factors.

The equation for heritability is derived from the equation Phenotype (P) = Genotype (G) + Environment (E):

H2 = Variance(G) / Variance(P)

This is called the "broad-sense" heritability and reflects all possible genetic contributions to the population's phenotypic variance. Included are such things as dominance, epistasis, maternal effects, and paternal effects.

The "narrow-sense" heritability (h2), is a technical statistical parameter. Instead of including all genetic variation, it only includes additive genetic variation (note upper case H2 for broad sense, lower case h2 for narrow sense). When one is interested, e.g., in trying to improve livestock via artificial selection, knowing the narrow-sense heritability of the trait of interest will allow one to predict by 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. Calculating the actual progress from selection can give an estimate of the realized heritability, which is an estimate of the narrow-sense 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 variation requires statistical methods, usually involving a complex technique called variance component estimation. In common with many statistical methods better estimates can be obtained with large volumes of data, but the method works better if data are available on closely related individuals - such as brothers, sisters, parents and children rather than more distantly related individuals.

In some populations it is possible to collect information in a controlled way. For example among farm animals it is easy to arrange for (say) a bull sire to produce offspring from a large number of cows. This allows a degree of experimental control. Due to ethical concerns, this type of control is impossible among human populations. As a result, studies of heritability in humans has often involved searches for identical twins that have been separated early in life and raised in different environments. Such individuals have identical genotypes and can be used to separate the effects of genotype and environment. Twin analysis offers problems of its own, not the least of which is that they share a common prenatal environment (see for example Fig. 1).

Heritability measures the amount of genetic and environmental variation, not a direct measurement of genetics and environments. Heritability reflects the amount of variation in genotypic effects compared to variation in environmental effects. To make the heritability larger, an experimenter might include a diverse background for the organism (increasing the Variance(G)) or restricting the environmental variation (decreasing the Variance(E)). To make the heritability smaller, the experimenter might use inbred strains (low Variance(G)) or include a wide diversity of environments (high Variance(E)). Even for the same trait, differing populations might be expected to have different heritabilities due to such effects.

There can be some apparent paradoxes in the application of this strict definition. For example, the heritability of the same phenotype could be near 100% in one study, and close to zero in another: In one study, a group of army recruits (unrelated to each other) are all given identical training and nutrition, and their muscular strength is later measured, then those differences that are observed after the (identical) training will give high heritability; In another study, whose purpose might be to assess the efficacy of various workout regimes, or various nutritional programs, a group of subjects is chosen who match each other as closely as possible in prior physical characteristics, and then some of these are put onto Program A, and others onto Program B, giving a low heritability.

In the case of scholastic ability, how well you do in the final school exams depends on both what and how well you were taught, how hard you have studied, how ‘naturally’ smart you are, and of course, a fair bit on luck. The actual heritability estimated depends on the subjects used (reflecting genetic variation) and the 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 genetic variation for many human traits, and corresponding lower environmental variation within the confines of the test.

In human genetics, much use is made of twin studies in the analysis of heritability – monozygous (‘identical’) twins are clones of each other, and have effectively identical genotypes, and similarities between sets of monozygous twins can be compared with those between dizygous (‘fraternal’, or non-identical) twins, who have only a ½ coefficient of relatedness to each other.

Heritability is often misunderstood when presented in the non-scientific media. Heritability only describes how much variation in the phenotype is attributable to variation in genotypes compared to the variation in environments, and not how much the genotype and environment as a whole actually effects phenotype.

## Estimation methods

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

The first was orignally developed by Sir Ronald 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.

The second 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.

Various methods have been developed to estimate the heritability of a trait.

### Selection experiments

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) 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. 2.

### Comparison of close relatives

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

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

Heritability may be estimated by comparing parent and offspring traits (as In Fig 3). 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 parents 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,ie the slope is always less than one).

#### Twin studies

Heritability for traits in humans is most frequently estimated by comparing resemblances between twins (Fig. 1 & 4). 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).

### Large, complex pedigrees

When a large, complex pedigree is available for estimating heritability, the most efficient use of the data is in a restricted maximum likelihood 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 artificial breeding programmes.

The pedigrees can be viewed using programs such as Pedigree Viewer [1], and analysed with programs such as ASReml.