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Griffiths AJF, Miller JH, Suzuki DT, et al. An Introduction to Genetic Analysis. 7th edition. New York: W. H. Freeman; 2000.

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An Introduction to Genetic Analysis. 7th edition.

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Quantifying heritability

If a trait is shown to have some heritability in a population, then it is possible to quantify the degree of heritability. In Figure 25-3, we saw that the variation between phenotypes in a population arises from two sources. First, there are average differences between the genotypes; second, each genotype exhibits phenotypic variance because of environmental variation. The total phenotypic variance of the population (S2p) can then be broken into two parts: the variance between genotypic means (S2g) and the remaining variance (S2e) The former is called the genetic variance, and the latter is called the environmental variance; however, as we shall see, these names are quite misleading. Moreover, the breakdown of the phenotypic variance into the sum of environmental and genetic variance leaves out the possibility of some covariance between genotype and environment. For example, suppose it were true (we do not know) that there are genes that influence musical ability. Parents with such genes might themselves be musicians, who would create a more musical environment for their children, who would then have both the genes and the environment promoting musical performance. The result would be an increase in the phenotypic variances of musical ability and an erroneous estimate of genetic and environmental variances. If the phenotype is the sum of a genetic and an environmental effect, P = G + E, then, as explained on page 768 of the Statistical Appendix, the variance of the phenotype is the sum of the genetic variance, the environmental variance, and twice the covariance between the genotypic and environmental effects.

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If genotypes are not distributed randomly across environments, there will be some covariance between genotype and environmental values, and the covariance will be hidden in the genetic and environmental variances.

The degree of heritability can be defined as the part of the total variance that is due to genetic variance:

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H2, so defined, is called the broad heritability of the character.

It must be stressed that this measure of “genetic influence” tells us what part of the population’s variation in phenotype can be assigned to variation in genotype. It does not tell us what parts of an individual’s phenotype can be ascribed to its heredity and to its environment. This latter distinction is not a reasonable one. An individual’s phenotype is a consequence of the interaction between its genes and its sequence of environments. It clearly would be silly to say that you owe 60 inches of your height to genes and 10 inches to environment. All measures of the “importance” of genes are framed in terms of the proportion of variance ascribable to their variation. This approach is a special application of the more general technique of the analysis of variance for apportioning relative weight to contributing causes. The method was, in fact, invented originally to deal with experiments in which different environmental and genetic factors were influencing the growth of plants. (For a sophisticated but accessible treatment of the analysis of variance written for biologists, see R. Sokal and J. Rohlf, Biometry, 3d ed. W. H. Freeman and Company, 1995.)

Methods of estimating H2

Genetic variance and heritability can be estimated in several ways. Most directly, we can obtain an estimate of (S2e) by making a number of homozygous lines from the population, crossing them in pairs to reconstitute individual heterozygotes, and measuring the phenotypic variance within each heterozygous genotype. Because there is no genetic variance within a genotypic class, these variances will (when averaged) provide an estimate of (S2e). This value can then be subtracted from the value of (S2p) in the original population to give (S2g). With the use of this method, any covariance between genotype and environment in the original population will be hidden in the estimate of genetic variance and will inflate it.

Other estimates of genetic variance can be obtained by considering the genetic similarities between relatives. Using simple Mendelian principles, we can see that half the genes of full siblings will (on average) be identical. For identification purposes, we can label the alleles at a locus carried by the parents differently, so that they are, say, A1/A2 and A3/A4. Now the older sibling has a probability of 1/2 of getting A1 from its father, as does the younger sibling, so the two siblings have a chance of 1/2 × 1/2 = 1/4 of both carrying A1. On the other hand, they might both have received an A2 from their father; so, again, they have a probability of 1/4 of carrying a gene in common that they inherited from their father. Thus, the chance is 1/4 + 1/4 = 1/2 that both siblings will carry an A1 or that both siblings will carry an A2. The other half of the time, one sibling will inherit an A1 and the other will inherit an A2. So, as far as paternally inherited genes are concerned, full siblings have a 50 percent chance of carrying the same allele. But the same reasoning applies to their maternally inherited gene. Averaging over their paternally and maternally inherited genes, half the genes of full siblings are identical between them. Their genetic correlation, which is equal to the chance that they carry the same allele, is 1/2.

If we apply this reasoning to half-siblings, say, with a common father but with different mothers, we get a different result. Again, the two siblings have a 50 percent chance of inheriting an identical gene from their father, but this time they have no way of inheriting the same gene from their mothers because they have two different mothers. Averaging the maternally inherited and paternally inherited genes thus gives a probability of (1/2 + 0)/2 = 1/4 that these half-siblings will carry the same gene.

We might be tempted to use the theoretical correlation between, say, siblings to estimate H2. If the observed phenotypic correlation were, for example, 0.4 and we expect on purely genetic grounds a correlation of .05, then an estimate of heritability would be 0.4/0.5 = 0.8. But such an estimate fails to take into account the fact that siblings may also be environmentally correlated. Unless we are careful to raise the siblings in independent environments, the estimate of H2 would be too large and could even exceed 1 if the observed phenotypic correlation were greater than 0.5. To get around this problem, we use the differences between phenotypic correlations of different relatives. For example, the difference in genetic correlation between full and half-siblings is 1/2 − 1/4 = 1/4. Let’s contrast this with their phenotypic correlations. If the environmental similarity is the same for half- and full siblings—a very important condition for estimating heritability—then environmental similarities will cancel out if we take the difference in correlation between the two kinds of siblings. This difference in phenotypic correlation will then be proportional to how much of the variance is genetic. Thus:

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so an estimate of H2 is:

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where the correlation here is the phenotypic correlation.

We can use similar arguments about genetic similarities between parents and offspring and between twins to obtain two other estimates of H2:

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These formulas are derived from considering the genetic similarities between relatives. They are only approximate and depend on assumptions about the ways in which genes act. The first two formulas, for example, assume that genes at different loci add together in their effect on the character. The last formula also assumes that the alleles at each locus show no dominance (see the discussion of components of variance on pages 760–762).

All these estimates, as well as others based on correlations between relatives, depend critically on the assumption that environmental correlations between individuals are the same for all degrees of relationship. If closer relatives have more similar environments, as they do in humans, the estimates of heritability are biased. It is reasonable to assume that most environmental correlations between relatives are positive, in which case the heritabilities would be overestimated. Negative environmental correlations also can exist. For example, if the members of a litter must compete for food that is in short supply, there could be negative correlations in growth rates among siblings.

The difference in correlation between monozygotic and dizygotic twins is commonly used in human genetics to estimate H2 for cognitive or personality traits. Here the problem of degree of environmental similarity is very severe. Identical (monozygotic) twins are generally treated more similarly to each other than are fraternal (dizygotic) twins. People often give their identical twins names that are similar, dress them alike, treat them identically, and, in general, accent their similarities. As a result, heritability is overestimated.

Meaning of H2

Attention to the problems of estimating broad heritability distracts from the deeper questions about the meaning of the ratio when it can be estimated. Despite its widespread use as a measure of how “important” genes are in influencing a trait, H2 actually has a special and limited meaning.

There are two conclusions that can be drawn from a properly designed heritability study. First, if there is a nonzero heritability, then, in the population measured and in the environments in which the organisms have developed, genetic differences have influenced the variation between individuals, so genetic differences do matter to the trait. This finding is not trivial and is a first step in a more detailed investigation of the role of genes. It is important to notice that the reverse is not true. Finding no heritability for the trait is not a demonstration that genes are irrelevant; rather, it demonstrates that, in the particular population studied, there is no genetic variation at the relevant loci or that the environments in which the population developed were such that different genotypes had the same phenotype. In other populations or other environments, the trait might be heritable.


In general, the heritability of a trait is different in each population and in each set of environnents; it cannot be extrapolated from one population and set of environments to another.

Moreover, we must distinguish between genes being relevant to a trait and genetic differences being relevant to differences in the trait. The experiment of immigration to North America has proved that the ability to pronounce the sounds of North American English, rather than French, Swedish, or Russian, is not a consequence of genetic differences between our immigrant ancestors. But, without the appropriate genes, we could not speak any language at all.

Second, the value of the H2 provides a limited prediction of the effect of environmental modification under particular circumstances. If all the relevant environmental variation is eliminated and the new constant environment is the same as the mean environment in the original population, then H2 estimates how much phenotypic variation will still be present. So, if the heritability of performance on an IQ test were, say, 0.4, then, if all children had the same developmental and social environment as the “average child,” about 60 percent of the variation in IQ test performance would disappear and 40 percent would remain.

The requirement that the new constant environment be at the mean of the old environmental distribution is absolutely essential to this prediction. If the environment is shifted toward one end or the other of the environmental distribution or a new environment is introduced, nothing at all can be predicted. In the example of IQ performance, the heritability gives us no information at all about how variable performance would be if children’s developmental and social environments were generally enriched. To understand why this is so, we must return to the concept of the norm of reaction.

The separation of variance into genetic and environmental components S2g and S2e does not really separate the genetic and environmental causes of variation. Consider Figure 25-9b. When the environment is poor (50), corn variety 2 is much higher yielding than variety 1, so a population made up of a mixture of the two varieties would have a lot of genetic variance for yield. But, in an environment scoring 80, there is no difference in yield between genotypes 1 and 2, so a mixed population would have no genetic variance at all for yield in that environment. Thus, genetic variance has been changed by changing the environment. On the other hand, variety 2 is less sensitive to environment than variety 1, as shown by the slopes of the lines. So a population made up mostly of genotype 2 would have a lower environmental variance than one made up mostly of genotype 1. So, environmental variance in the population is changed by changing the proportion of genotypes.


Because genotype and environment interact to produce phenotype, no partition of variation can actually separate causes of variation.

As a consequence of the argument just given, knowledge of the heritability of a trait does not permit us to predict how the distribution of that trait will change if either genotypic frequencies or environmental factors change markedly.


A high heritability does not mean that a trait is unaffected by its environment.

All that high heritability means is that, for the particular population developing in the particular distribution of environments in which the heritability was measured, average differences between genotypes are large compared with environmental variation within genotypes. If the environment is changed, there may be large differences in phenotype.

Perhaps the most well known example of the erroneous use of heritability arguments to make claims about the changeability of a trait is the case of human IQ performance and social success. In 1969, an educational psychologist, A. R. Jensen, published a long paper in the Harvard Educational Review, asking the question (in its title) “How much can we boost IQ and scholastic achievement?” Jensen’s conclusion was “not much.” As an explanation and evidence of this unchangeability, he offered a claim of high heritability for IQ performance. A great deal of criticism has been made of the evidence offered by Jensen for the high heritability of IQ scores. But, irrespective of the correct value of H2 for IQ performance, the real error of Jensen’s argument lies in his equation of high heritability with unchangeability. In fact, the heritability of IQ is irrelevant to the question raised in the title of his article.

To see why this is so, let us consider the results of adoption studies in which children are separated from their biological parents in infancy and reared by adoptive parents. Although results may vary quantitatively from study to study, there are three characteristics in common. First, adopting parents generally have higher IQ scores than those of the biological parents. Second, the adopted children have higher IQ scores than those of their biological parents. Third, the adopted children show a higher correlation of IQ scores with their biological parents than with their adoptive families. The following table is a hypothetical data set that shows all these characteristics, in idealized form, to illustrate the concepts. The scores given for parents are meant to be the average of mother and father.

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First, we can see that the children have a high correlation with their biological parents but a low correlation with their adoptive parents. In fact, in our hypothetical example, the correlation of children with biological parents is r = 1.00, but, with adoptive parents, it is r = 0. (The correlation between two sets of numbers does not mean that the two sets are identical but that, for each unit increase in one set, there is a constant proportion increase in the other set. See page 768 of the Statistical Appendix at the end of this chapter.) This perfect correlation with biological parents and zero correlation with adoptive parents means that H2 = 1, given the arguments developed on page 755. All the variation in IQ score between the children is explained by the variation between the biological parents.

Second, however, we notice that each of the IQ scores of the children is 20 points higher than the IQ scores of their respective biological parents and that the mean IQ of the children is equal to the mean IQ of the adoptive parents. Thus, adoption has raised the average IQ of the children 20 points higher than the average IQ of their biological parents; so, as a group, the children resemble their adoptive parents. So we have perfect heritability, yet high environmental plasticity.

An investigator who is seriously interested in knowing how genes might constrain or influence the course of development of any trait in any organism must study directly the norms of reaction of the various genotypes in the population over the range of projected environments. No less detailed information will do. Summary measures such as H2 are not first steps toward a more complete analysis and therefore are not valuable in themselves.


Heritability is not the opposite of phenotypic plasticity. A character may have perfect heritability in a population and still be subject to great changes resulting from environmental variation.

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By agreement with the publisher, this book is accessible by the search feature, but cannot be browsed.

Copyright © 2000, W. H. Freeman and Company.
Bookshelf ID: NBK21866


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