A Note on the Dependence of Heritability on Variances of Genetic and Environmental Components

The mathematics of population genetics is highly developed and has led to many important insights. Nevertheless, it is possible that some simple algebraic expressions, not widely known, can enable researchers to view existing findings from a somewhat different perspective. These simple derivations perhaps also could have benefit in education in explaining the meaning of heritability and its role in genetics and elsewhere. The derivations are presented herewith.

The concept of heritability certainly has been widely misunderstood and has been the subject of controversies, especially in the context of the nature nurture controversy in psychology and various social sciences (see, e.g., Bouchard and McGue 2003; Gould 1996; Herrnstein and Murray 1994; Kempthorne 1978; Lewontin 1974; Schönemann 1997). There is general agreement in population genetics that it applies to populations but not to individuals (see, e.g., Beurton, Falk, and Rheinberger 2000; Falconer and Mackay 1996; Gillespie 1997; Houle 1992; Morange 2001; Plomin, DeFries, McClearn, and McGuffin 2001; Visscher, Hill, and Wray 2008). Because heritability is defined as a ratio of variances, it is meaningless to assign its value to a particular individual and to consider a trait of the individual to be proportionally determined by that ratio. Furthermore, many authors have emphasized that its value changes as the variance of its genetic component, the environmental component, or both changes.

A convenient way to express this relationship mathematically is that the assignment of heritability values to populations with different variances is a relation, but not a function, because no unique assignment is possible without specifying and holding constant the environmental variability associated with the population. Despite agreement about the dependence on variance changes, it is difficult to determine exactly how much a heritability value changes as component variances change. The present derivation results in general equations that allow quantitative statements to be made about the degree of change in heritability values to be expected as component variances change in a specified way.

Let P = G + E, where P is a measure of the phenotype and G and E are uncorrelated measures associated with the genetic and environmental components. Heritability is defined as the ratio of the variances of G and P. Therefore,

We adopt the usual convention of employing H² for broad-sense heritability. Defining θ = σ_G/σ_E, the ratio of the standard deviations of genetic and environmental components, the definition of heritability also can be put in the form H² = θ/(θ + θ⁻¹).

Consider also the correlation between the genetic measures (G) and the combined measures constituting the phenotype (P). We obtain

because G and E are uncorrelated, and

Therefore, heritability can be interpreted as the squared correlation between genotype and phenotype, which in fact was given by Wright (1921). Again this equality makes it clear that heritability can be viewed as a correlation coefficient referring to an entire population and cannot meaningfully be assigned to an individual.

In fact, it is often emphasized in the literature that the heritability of a trait depends on a particular population in a particular environment, so that its numerical value changes as one or both component variances change. As an extreme example, in a population in which every individual has exactly the same value of a genetic component, heritability is zero, even if the environmental contribution is relatively small. Modifications of lesser degree can be expected in various contexts of practical interest.

We want to find explicit equations for this relationship, so that specific numerical estimates of changes in heritability to be expected can be found when stipulated changes in variances of G and E occur. First, assume that G remains fixed, so that a change in variability of the phenotype is entirely accounted for by a change in the variability of E.

Let H₁² = 1 − σ_E1²/(σ_G² + σ_E1²) and H₂² = 1 − σ_E2²/(σ_G² + σ_E2²). Solving these equations for σ_G² and setting the resulting expressions equal, gives

Solving for H₂² then gives the result.

or, defining λ = σ_E1²/σ_E2², H...