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 nongenetic factors to the total phenotypic variance in a population.
Definition
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:
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): H^2 = \frac{Var(G)}{Var(P)} .
The parameter H^{2} is the broadsense 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 (multigenic 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, B_{i}B_{j}, the expected phenotype can then be written as the sum of the overall mean, a linear effect, and a dominance deviation:
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): P_{ij} = \mu + \alpha_i + \alpha_j + d_{ij} = Population mean + Additive Effect (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): a_{ij}=\alpha_i+\alpha_j ) + Dominance Deviation (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): d_{ij} ).
The additive genetic variance is the weighted average of the squares of the additive effects:
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): Var(A) = f(bb)a^2_{bb}+f(Bb)a^2_{Bb}+f(BB)a^2_{BB},
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): f(bb)a_{bb}+f(Bb)a_{Bb}+f(BB)a_{BB} = 0.
There is a similar relationship for variance of dominance deviations:
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): Var(D) = f(bb)d^2_{bb}+f(Bb)d^2_{Bb}+f(BB)d^2_{BB},
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): f(bb)d_{bb}+f(Bb)d_{Bb}+f(BB)d_{BB} = 0.
Narrowsense heritability is defined as
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): 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 H^{2} for broad sense, lower case h^{2} for narrow sense). When interested in improving livestock via artificial selection, for example, knowing the narrowsense 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 narrowsense 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.
In nonhuman 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 nonscientific 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
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, h^{2}=R/S as in Fig. 3.
Comparison of close relatives
In the comparison of relatives, we find that in general,
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): 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.
Parentoffspring 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", since the offspring values always tend to regress to the mean value for the population, i.e., the slope is always less than one).
Fullsib comparison
Fullsib 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:Sectionstub
Halfsib comparison
Halfsib designs compare phenotypic traits of siblings that share one parent with other sibling groups. Template:Sectionstub
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 between MZ and DZ twins, h^{2}=2(r(MZ)r(DZ)). The effect of shared environment, c^{2}, 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, c^{2}=DZ1/2h^{2}. Unique environmental variance, e^{2}, reflects the degree to which identical twins raised together are dissimilar, e^{2}=1r(MZ).
The classical twin study has been severely criticized and is used less and less frequently nowadays.
Large, complex pedigrees
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
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): y_i = \mu + g_i + e
where
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): g_i is the effect of genotype G_{i}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): 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
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): 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
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): n sires and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): r progeny per sire, we can calculate the following ANOVA, using Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): V_g as the genetic variance and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): V_e as the environmental variance:
Source  d.f.  Mean Square  Expected Mean Square 

Among sire groups  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): n1  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): S  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \frac{3}{4}V_g + V_e + r({\frac{1}{4}V_g}) 
Within sire groups  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): n(r1)  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): W  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \frac{3}{4}V_g + V_e 
The Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \frac{1}{4}V_g term is the intraclass correlation among half sibs. We can easily calculate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): H^2 = \frac{V_g}{V_g+V_e} = \frac{4(SW)}{S+(r1)W} . The Expected Mean Square is calculated from the relationship of the individuals (progeny within a sire are all halfsibs, 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
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): y_{ij} = \mu + \alpha_i + \alpha_j + d_{ij} + e,
where
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \alpha_i is the additive effect of the i^{th} allele, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \alpha_j is the additive effect of the j^{th} allele, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): d_{ij} is the dominance deviation for the ij^{th} genotype, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): V_a as the additive genetic variance and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): V_d as the dominance deviation variance, intraclass correlations become linear functions of these parameters. In general,
 Intraclass correlationFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): = r V_a + \theta V_d,
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): r and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \theta are found as
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): r = P[alleles drawn at random from the relationship pair are identical by descent], and
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \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.
Relationship  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): r  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \theta 

Identical Twins  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): 1  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): 1 
ParentOffspring  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \frac{1}{2}  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle 0} 
Half Siblings  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \frac{1}{4}  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle 0} 
Full Siblings  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \frac{1}{2}  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \frac{1}{4} 
First Cousins  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \frac{1}{8}  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle 0} 
Double First Cousins  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \frac{1}{4}  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \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].
External links
 Stanford Encyclopedia of Philosophy entry on Heredity and Heritability
 Quantitative Genetics Resources website, including the two volume book by Lynch and Walsh. Free access
References
Notes
Further reading
 Falconer, D. S. & Mackay TFC (1996). Introduction to Quantitative Genetics. Fourth edition. Addison Wesley Longman, Harlow, Essex, UK
 Gillespie, G. H. (1997). Population Genetics: A Concise Guide. Johns Hopkins University Press.
 Joseph, J. (2004). The Gene Illusion: Genetic Research in Psychiatry and Psychology Under the Microscope.New York: Algora. (2003 United Kingdom Edition by PCCS Books) (Chapter 5 contains a critique of the heritability concept)
 Joseph, J. (2006). The Missing Gene: Psychiatry, Heredity, and the Fruitless Search for Genes.New York: Algora.
 Kempthorne, O (1957 [1969]) An Introduction to Genetic Statistics. John Wiley. Reprinted, 1969 by Iowa State University Press.
 Lynch, M. & Walsh, B. 1997. Genetics and Analysis of Quantitative Traits. Sinauer Associates. ISBN 0878934812.
 Malécot, G. 1948. Les Mathématiques de l'Hérédité. Masson, Paris.
 Wahlsten, D. (1994) The intelligence of heritability. Canadian Psychology 35, 244258.
Topics in quantitative genetics


heritability  quantitative trait locus  candidate gene  effective population size 
Related topics: population genetics  genomics 