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In genetics

Genetics

Genetics, , a discipline of biology, is the science of heredity and variation in living organisms. The fact that living things inherit traits from their parents has been used since prehistoric times to improve crop plants and animals through selective breeding...

, Heritability is the proportion of phenotypic variation

Phenotype

A phenotype is any observable characteristic or trait of an organism: such as its morphology, development, biochemical or physiological properties, or behavior. Phenotypes result from the expression of an organism's genes as well as the influence of environmental factors and possible interactions...

in a population that is attributable to genetic variation

Genotype

The genotype is the genetic constitution of a cell, an organism, or an individual usually with reference to a specific character under consideration...

among individuals. Variation among individuals may be due to genetic and/or environmental factor

Environmental factor

In epidemiology, environmental factors are those determinants of disease that are not transmitted genetically. Apart from the true monogenic genetic disorders, environmental factors may determine the development of disease in those genetically predisposed to a particular condition...

s. Heritability analyses estimate the relative contributions of differences in genetic and non-genetic factors to the total phenotypic variance in a population.

Definition

Consider a statistical model

Statistical model

A statistical model is a set of mathematical equations which describe the behavior of an object of study in terms of random variables and their associated probability distributions...

for describing some particular phenotype:

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 then defined as: .

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

Dominance relationship

In genetics, dominance describes a relationship between different forms of a gene at a particular physical location on a chromosome. Typical plants and animals have two sets of chromosomes, one set inherited from each parent, and are described as diploid. They therefore have two alleles at each...

or which act epistatically

Epistasis

Epistasis is the interaction between genes. Epistasis takes place when the effects of one gene are modified by one or several other genes, which are sometimes called modifier genes. The gene whose phenotype is expressed is said to be epistatic, while the phenotype altered or suppressed is said to...

(multi-genic interactions), as well as maternal

Maternal effect

This article concerns the legitimate scientific concept of genes that are expressed only when carried by the female parent. It is not to be confused with the generally discredited theory of maternal impression....

and paternal effects, where individuals are directly affected by 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 affect some quantitative phenotype, as shown by + in Figure 1. We can calculate the linear regression

Linear regression

In statistics, linear regression refers to any approach to modeling the relationship between one or more variables denoted y and one or more variables denoted X, such that the model depends linearly on the unknown parameters to be estimated from the data...

of phenotype on the number of B alleles (0, 1, or 2), which is shown as the Linear Effect line. For any genotype, B_iB_j, the expected phenotype can then be written as the sum of the overall mean, a linear effect, and a dominance deviation: = Population mean

Mean

In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....

+ Additive Effect + Dominance Deviation .

The additive genetic variance is the weighted average

Weighted mean

The weighted mean is similar to an arithmetic mean , where instead of each of the data points contributing equally to the final average, some data points contribute more than others...

of the squares of the additive effects:

where

There is a similar relationship for variance of dominance deviations:

where

Narrow-sense heritability is defined as

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

Artificial selection

Artificial selection describes intentional breeding for certain traits, or combination of traits. The term was utilized by Charles Darwin in contrast to natural selection, in which the differential reproduction of organisms with certain traits is attributed to improved survival or reproductive...

, 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, since only P can be observed or measured directly. Measuring the genetic

Genetic

Genetic may refer to:*Genetics, in biology, the science of genes, heredity, and the variation of organisms**Genetic, used as an adjective, refers to heredity of traits...

and environmental variance requires various sophisticated statistical methods

Statistics

Statistics is a branch of mathematics concerned with collecting and interpreting data. According to other definitions, it is a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data. Statisticians improve the quality of data with the...

. 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

Standard error

Standard error can refer to:* Standard error , the estimated standard deviation or error of a series of measurements* Standard error stream, one of the standard streams in Unix-like operating systems...

for heritability estimates are generally very poor unless the dataset is large.

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

Twin

Twins are two offspring resulting from the same pregnancy, usually born in close succession. They can be the same or different sex. Twins can either be monozygotic or dizygotic ....

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

Twin study

Twin studies are one of a family of designs in behavior genetics which aid the study of individual differences by highlighting the role of environmental and genetic causes on behavior....

entail problems of their 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

Inbreeding

Inbreeding is breeding between close relatives, whether plant or animal. If practiced repeatedly, it can lead to exposure of recessive, deleterious traits. This generally leads to a decreased fitness of a population, which is called inbreeding depression. Deleterious alleles causing inbreeding...

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.

In observational studies

Observational study

In statistics, an observational study draws inferences about the possible effect of a treatment on subjects, where the assignment of subjects into a treated group versus a control group is outside the control of the investigator...

G and E may be correlated, giving rise to gene environment correlation. Depending on the methods used to estimate heritability, correlations between genetic factors and shared or non-shared environments may or may not be included in the total heritability estimate.

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

Physical strength

Physical strength is the ability of a person to exert force on physical objects using muscles. Increasing physical strength is the goal of strength training.-Overview:...

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.

Heritability estimates are often misinterpreted. Heritability refers to the proportion of variation between individuals in a population that is influenced by genetic factors. Heritability describes the population, not individuals within that population. For example, It is incorrect to say that since the heritability of a personality trait is about .6, that means that 60% of your personality is inherited from your parents and 40% comes from the environment.

The heritability estimate changes according to the genetic and environmental variability present in the population. In studies of genetically identical inbred animals, all traits have zero heritability. Heritability estimates can be much higher in outbred (genetically variable) populations under very homogeneous environments.

A highly genetically loaded trait (such as eye color

Eye color

Eye color is a polygenic phenotypic character and is determined by the amount and type of pigments in the eye's iris.Humans and other animals have many phenotypic variations in eye color, as blue, brown, green and others. These variations constitute phenotypic traits.The genetics of eye color are...

) still assumes environmental input within normal limits (a certain range of temperature, oxygen in the atmosphere, etc.). A more useful distinction than "nature vs. nurture" is "obligate vs. facultative" -- under typical environmental ranges, what traits are more "obligate" (e.g., the nose -- everyone has a nose) or more "facultative" (sensitive to environmental variations, such as specific language learned during infancy). Another useful distinction is between traits that are likely to be adaptations (such as the nose) vs. those that are byproducts of adaptations (such the white color of bones), or are due to random variation (non-adaptive variation in, say, nose shape or size).

Estimation methods

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

One school of thought

School (discipline)

A school of thought is a collection or group of people who share common characteristics of opinion or outlook of a philosophy, discipline, belief, social movement, cultural movement, or art movement. There have been several schools of economic thought throughout history.Schools are often...

was developed by Sewall Wright

Sewall Wright

Sewall Green Wright was an American geneticist known for his influential work on evolutionary theory and also for his work on path analysis. With R. A. Fisher and J.B.S. Haldane, he was a founder of theoretical population genetics. He is the discoverer of the inbreeding coefficient and of...

at The University of Chicago

University of Chicago

The University of Chicago is a private, coeducational research university in Chicago, Illinois, USA. It was founded by oil magnate and benefactor John D...

, and further popularized by C. C. Li (University of Chicago

University of Chicago

The University of Chicago is a private, coeducational research university in Chicago, Illinois, USA. It was founded by oil magnate and benefactor John D...

) and J. L. Lush (Iowa State University

Iowa State University

The Iowa State University of Science and Technology, more commonly known as Iowa State University , is a public land-grant and space-grant research university located in Ames, Iowa, United States. Iowa State has produced a number of astronauts, scientists, Nobel laureates, Pulitzer Prize winners,...

). It is based on the analysis of correlations and, by extension, regression. Path Analysis was developed by Sewall Wright

Sewall Wright

as a way of estimating heritability.

The second was originally developed by R. A. Fisher

Ronald Fisher

Sir Ronald Aylmer Fisher, FRS was an English statistician, evolutionary biologist, eugenicist and geneticist. He was described by Anders Hald as "a genius who almost single-handedly created the foundations for modern statistical science," and Richard Dawkins described him as "the greatest of...

and expanded at The University of Edinburgh

University of Edinburgh

The University of Edinburgh founded in 1582, is an internationally renowned centre for teaching and research in Edinburgh, Scotland, UK. It is the sixth university to be established in the British Isles, making it one of the ancient universities of the United Kingdom.The university is amongst the...

, Iowa State University

Iowa State University

, and North Carolina State University

North Carolina State University

North Carolina State University at Raleigh is a public, coeducational, extensive research university located in Raleigh, North Carolina, United States...

, as well as other schools. It is based on the analysis of variance

Analysis of variance

In statistics, analysis of variance is a collection of statistical models, and their associated procedures, in which the observed variance is partitioned into components due to different explanatory variables...

of breeding studies, using the intraclass correlation of relatives. Various methods of estimating components of variance (and, hence, heritability) from ANOVA

Analysis of variance

are used in these analyses.

Regression/correlation methods of estimation

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

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, also called the selection differential ) 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,

where r can be thought of as the coefficient of relatedness

Coefficient of relationship

In population genetics, Sewall Wright's coefficient of relationship or coefficient of relatedness or relatedness or r is defined as 2 times the coefficient of kinship...

, 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. 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

Regression analysis

In statistics, regression analysis includes any techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables...

", since the offspring values always tend to regress to the mean

Regression toward the mean

In statistics, regression toward the mean refers to the phenomenon that a variable that is extreme on its first measurement will tend to be closer to the centre of the distribution on a later measurement...

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.

Half-sib comparison

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

Twin studies

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

Correlation

In statistics, correlation indicates the strength and direction of a relationship between two random variables. The commonest use refers to a linear relationship. In general statistical usage, correlation or co-relation refers to the departure of two random variables from independence...

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 methodology of the classical twin study

Twin study

Twin studies are one of a family of designs in behavior genetics which aid the study of individual differences by highlighting the role of environmental and genetic causes on behavior....

has been criticized, but these criticisms do not take into account the methodological innovations and refinements described above.

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 G_i as
where

is the effect of genotype G_i

and 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

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

Errors and residuals in statistics

In statistics and optimization, statistical errors and residuals are two closely related and easily confused measures of "deviation of a sample from the mean": the error of a sample is the deviation of the sample from the population mean or actual function, while the residual of a sample is the...

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 ,
since environmental effects are independent of each other.

The ANOVA

In an experiment with sires and progeny per sire, we can calculate the following ANOVA, using as the genetic variance and as the environmental variance:

Table 1: ANOVA for Sire experiment
Source	d.f.	Mean Square	Expected Mean Square
Among sire groups
Within sire groups

The term is the intraclass correlation

Intraclass correlation

In statistics, the intraclass correlation is a descriptive statistic that can be used when quantitative measurements are made on units that are organized into groups. It describes how strongly units in the same group resemble each other...

among half sibs. We can easily calculate . 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
where

is the additive effect of the i^th allele, is the additive effect of the j^th allele, is the dominance deviation for the ij^th genotype, and 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 as the additive genetic variance and as the dominance deviation variance, intraclass correlations become linear function

Linear function

In mathematics, the term linear function can refer to either of two different but related concepts:* a first-degree polynomial function of one variable;* a map between two vector spaces that preserves vector addition and scalar multiplication....

s of these parameters. In general,

Intraclass correlation

where and are found as

P[ alleles drawn at random from the relationship pair are identical by descent

Identity by descent

Two or more alleles are identical by descent if they are identical copies of the same ancestral allele. This property is often used in genetic linkage to identify alleles which are potential candidates for harboring mutations causing a genetic disease....

], and

P[ genotypes drawn at random from the relationship pair are identical by descent

Identity by descent

].

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

Table 2: Coeffients for calculating variance components
Relationship
Identical Twins
Parent-Offspring
Half Siblings
Full Siblings
First Cousins
Double First Cousins

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

Restricted maximum likelihood

In statistics, restricted maximum likelihood is a method for fitting linear mixed models. In contrast to conventional maximum likelihood estimation, REML can produce unbiased estimates of variance and covariance parameters....

(REML) model. The raw data

Raw data

Raw data is a term for data collected on source which has not been subjected to processing or any other manipulation. , it is also known as primary data. It is a relative term . Raw data can be input to a computer program or used in manual analysis procedures such as gathering statistics from a...

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 http://www-personal.une.edu.au/~bkinghor/pedigree.htm, and analysed with programs such as ASReml

ASReml

ASReml is a statistical software package for fitting linear mixed models using restricted maximum likelihood, a technique commonly used in plant and animal breeding and quantitative genetics as well as other fields...

, VCE http://vce.tzv.fal.de/index.pl, WOMBAT http://agbu.une.edu.au/~kmeyer/wombat.html or BLUPF90 family's programs http://nce.ads.uga.edu/~ignacy/programs.html

Response to Selection

In selective breeding

Selective breeding

Selective breeding is the process of breeding plants and animals for particular genetic traits. Typically, strains which are selectively bred are domesticated, and the breeding is sometimes done by a professional breeder. Bred animals are known as breeds, while bred plants are known as varieties,...

of plants and animals, the expected response to selection can be estimated by the following equation:

R = h²S

In this equation, the Response to Selection (R) is defined as the realized average difference between the parent generation and the next generation. The Selection Differential (S) is defined as the average difference between the parent generation and the selected parents.

For example, imagine that a plant breeder is involved in a selective breeding project with the aim of increasing the number of kernels per ear of corn. For the sake of argument, let us assume that the average ear of corn in the parent generation has 100 kernels. Let us also assume that the selected parents produce corn with an average of 120 kernels per ear. If h² equals 0.5, then the next generation will produce corn with an average of 0.5(120-100) = 10 additional kernels per ear. Therefore, the total number of kernels per ear of corn will equal, on average, 110.