Heritability
One of the most important factors in the formulation of effective breeding plans for improving the genetic quality of crops and livestock is a knowledge of the relative contribution made by genes to the variability of a trait under consideration. The variability of phenotypic values for a quantitative trait (as expressed in the phenotypic variance) can, at least in theory, be partitioned into genetic and nongenetic (environmental) components.
Heritability (symbolized H2 or H in some texts) is the proportion of the total phenotypic variance due to all types of gene effects; H represents the corresponding ratio of standard deviations and is not used in this book.
The heritability of a given trait may be any number from 0 to 1.
EXAMPLE 8.3 If all of the phenotypic variability of a trait is genetic in nature (as is true for most classical Mendelian traits, such as blood types), then environmental effects are absent and heritability equals 1; i.e., if σ2G = σ2P, then H2 = 1.
EXAMPLE 8.4 If all of the phenotypic variability of a trait is environmental in nature (as is true for any trait within a genetically homozygous line), then heritability of the trait is zero; i.e., if σ2E = σ2P, then σ2G = 0 and H2 = 0/σ2P = 0.
EXAMPLE 8.5 If half of the phenotypic variability is due to gene effects, then heritability is 50%, i.e., if σ2G = (1/2)σ2P, then 2σ2G = σ2P and H2 = 1/2 = 50%. Note: We cannot say that because heritability of a quantitative trait (such as seed production in a given plant population) is 0.5 that 50% of any one individual's seed productivity is due to gene effects, as explained in the following scenario. One plant may have a superior genotype for seed production, but was grown in a poor environment (e.g., shade, disease, insect damage, etc.) and thus produced fewer seeds than the average number for the population. Another plant in that same population, having only an average genotype, might have grown in an enriched environment (e.g., accidentally fertilized by animal dung) and was thus able to produce a seed number near its genetic potential that actually exceeded the number of seeds produced by the aforementioned plant. Likewise, it is illogical to say that a population with a high heritability for a given trait is genetically superior to another population (in the same or different species) with a lower heritability for that same trait.
EXAMPLE 8.6 If the environmental component of variance is three times as large as the genetic component, heritability is 25%, i.e., if σ2E = 3σ2G, then
Broad-Sense and Narrow-Sense Heritability
The parameter of heritability involves all types of gene action and thus forms a broad estimate of heritability, or referred to as broad sense heritability, (H2) or the degree of genetic determination. In the case of complete dominance, when a gamete bearing the active dominant allele A2 unites with a gamete bearing the null allele A1, the resulting phenotype might be two units. When two A2 gametes unite, the phenotypic result would still be two units. On the other hand, if genes lacking dominance (additive genes) are involved, then the A2 gamete will add one unit to the phenotype of the resulting zygote regardless of the allelic contribution of the gamete with which it unites, Thus, only the additive genetic component of variance (i.e., the variance of breeding values) has the quality of predictability necessary in the formulation of breeding plans. Nonallelic (epistatic) interactions may exist, but they usually make a minor contribution to the total genetic variance, their effects are difficult to access, and they will be ignored in the simple models of this book. If they do occur, all nonadditive genetic variance will be included in the dominance variance. The concept of additive variance does not necessarily involve the assumption of additive gene action. Additive variance can be produced from genes with additive gene action and/or any degree of dominance and/or epistasis. Heritability expressed as the ratio of the additive genetic variance to the phenotypic variance is termed narrow sense heritability:
Unless otherwise specified in the problems of this book, heritability in the narrow sense is to be employed. It must be emphasized that the heritability of a trait applies only to a given population living in a particular environment. A genetically different population (perhaps a different variety, breed, race, or subspecies of the same species) living in an identical environment is likely to have a different heritability for the same trait. Likewise, the same population is likely to exhibit different heritabilities for the same trait when measured in different environments because a given genotype does not always respond to different environments in the same way. There is no one genotype that is adaptively superior in all possible environments. That is why natural selection tends to create genetically different populations within a species, each population being potentially adapted to local conditions rather than generally adapted to all environments in which the species is found.
Several methods can be used to estimate heritabilities of quantitative traits.
Variance Components
Consider the simple, single-locus model (below) with alleles b1 and b2.
The midparent value m = (1/2)(b1b1 + b2b2). If the heterozygote does not have a phenotypic value equal to m, some degree of dominance (d) exists. If no dominance exists, then the alleles are completely additive. However, quantitative traits are governed by many loci and it might be possible that genotype b1b2 is dominant in a positive direction whereas genotype c1c2 is dominant in a negative direction, so that they cancel each other, giving the illusion of additivity. Dominance of all types can be estimated from the variances of F2 and backcross generations. All of the phenotypic variance within pure lines b1b1 and b2b2, as well as in their genetically uniform F1 (b1b2), is environmental. Hence, the phenotypic variances of each pure parental line (VP1 and VP2) as well as that of the F1 (VF1) serve to estimate the environmental variance (VE). The F2 segregates (1/4)b1b1 : (1/2)b1b2 : (1/4)b2b2. If each genotype departs from the midparent value as shown in the above model, then the average phenotypic value of F2 should be (1/4)(–a) + 1/2 (+d) + 1/4 (+a) = (1/2)d. The contribution that each genotype makes to the total is its squared deviation from the mean (m) multiplied by its frequency 
. Therefore, the total F2 variance (all genetic in this model) is the mean of squared deviations from the mean:
If we let a2 = A; d2 = D, and E = environmental component, then the total F2 phenotypic variance (VF2) = (1/2)A + (1/4)D+E, representing the additive genetic variance (VA) + the dominance genetic variance (VD) + the environmental variance (VE), respectively. Likewise it can be shown that VB1 (the variance of backcross progeny F1 × P1) or VB2 (the variance of backcross progeny F1 × P1), = (1/4)A + (1/4)D+E, and VB1 + VB2 = (1/2)A + (1/2)D+2E. The degree of dominance is expressed as
Heritability can be easily calculated from these variance components. The same is true of variance components derived from studies of identical (monozygotic) vs. nonidentical (fraternal, dizygotic) twins. If twins reared together tend to be treated more alike than unrelated individuals, the heritabilities will be overestimated. This problem, and the fact that the environmental variance of fraternal twins tends to be greater than for identical twins, can be largely circumvented by studying twins that have been reared apart.
Genetic Similarity of Relatives
If offspring phenotypes were always exactly intermediate between the parental values regardless of the environment, then such traits would have a narrow heritability of 1.0. On the other hand, if parental phenotypes (or phenotypes of other close relatives) could not be used to predict (with any degree of accuracy) the phenotypes of offspring (or other relatives), then such traits must have very low (or zero) heritabilities.
Regression Analysis
The regression coefficient (b) is an expression of how much (on the average) one variable (Y) may be expected to change per unit change in some other variable (X).
EXAMPLE 8.7 If for every egg laid by a group of hens (X) the average production by their respective female progeny (Y) is 0.2, then the regression line of Y on X would have a slope (b) of 0.2 (symbol Δ = increment of change).
The regression line of Y on X has the formula
where a is the "Y intercept" (the point where the regression line intersects the Y axis), and 
and 
are the respective mean values. The regression line also goes through the point (, ) and establishing these two points allows the regression line to be drawn. Any X value can then be used to predict the corresponding Y value. Let 
= estimate of Y from X; then
Since daughters receive only a sample half of their genes from each parent, the daughter-dam regression estimates only one-half of the narrow heritability of a trait (e.g., egg production in chickens). If the variances in the two populations are equal (Sx = Sy), then
Similarly, the regression of offspring on the average of their parents (midparent) is also an estimate of heritability
Full sibs (having the same parents) are expected to share 50% of their genes in common; half-sibs share 25% of their genes. Therefore,
If the variances of the two populations are unequal, the data can be converted to standardized variables (as discussed later in this chapter) and the resulting regression coefficients equated to heritabilities as described above.
Correlation Analysis
The statistical correlation coefficient (r) measures how closely two sets of data are associated, is dimensionless, and has the limits ±1. If all of the data points fall on the regression line, there is complete correlation. The regression coefficient (b) and the correlation coefficient (r) always have the same sign. The correlation coefficient (r) of Y on X is defined as the linear change of Y, in standard deviations, for each increase of one standard deviation in X. The covariance (cov) of X and Y can be calculated from the following formula:
The covariance becomes the numerator in the formula for the correlation coefficient.
Notice that the numerators in the formulas for r and b are equivalent. Regression and correlation coefficients are related by
so that if the variances of X and Y are identical, then b = r. If the data are first converted to standardized variables, then the sample has a mean of 0 and a standard deviation of ±1. Using standardized variables, regression and correlation coefficients become identical. Heritabilities can be estimated from r just as they can from b.
EXAMPLE 8.8 The correlation coefficient of Y offspring and midparent (X) is equivalent to narrow heritability; h2 = r.
EXAMPLE 8.9 If all the variation between offspring and one parent (e.g., their sires) is genetic, then r should equal 0.5; if r = 0.2, then h2 = 2(0.2) = 0.4.
EXAMPLE 8.10 If litter mates were phenotypically correlated for a trait by r = 0:15, then h2 = 2 (0.15) = 0.3
EXAMPLE 8.11 If the correlation coefficient for half-sibs is 0.08, then h2 = 4 (0.08) = 0.32.
All unbiased estimates of heritability based on correlations between relatives depend upon the assumption that there are no environmental correlations between relatives. Experimentally, this can be fostered by randomly assigning all individuals in the study to their respective environments (field plots, pens, etc.), but this obviously is not possible for humans. Relatives such as full sibs usually share the same maternal and family environment and are likely to show a greater correlation among themselves in phenotype than should rightly be attributed to common heredity. For this reason, the phenotypic correlation between sire and offspring in non-human animals is more useful for calculating heritabilities because sires often do not stay in the same environments with their offspring while mothers or siblings are prone to do so.
Response to Selection
Let us assume we wished to increase the birth weight of beef cattle by selecting parents who themselves were relatively heavy at birth. Assume our initial population (
1) has a mean birth weight of 80 lb with a 10 lb standard deviation [Fig. 8-8(a)]. Further suppose that we will save all animals for breeding purposes that weigh over 95 lb at birth. The mean of these animals that have been selected to be parents of the next generation (p) is 100 lb.
Fig. 8-8. Selection for birth weight in beef cattle. (a) Parental generation. Shaded area represents individuals selected to produce the next generation. (b) Progeny generation (right) compared with parental gene.
The difference (p) – P1, is called the selection differential, symbolized ΔP (read "delta P"), and is sometimes referred to as "reach." Some individuals with an inferior genotype are expected to have high birth weights largely because of a favorable intrauterine environment. Others with a superior genotype may possess a low birth weight because of an unfavorable environment. In a large, normally distributed population, however, the plus and minus effects produced by good and poor environments are assumed to cancel each other so that the average phenotype (1) reflects the effects of the average genotype (
1). Random mating among the selected group produces an offspring generation [Fig. 8-8(b)], with its phenotypic mean (2) also reflecting its average genotypic mean (2). Furthermore, the mean genotype of the parents (P) will be indicated in the mean phenotype of their offspring (2) because only genes are transmitted from one generation to the next. Assuming the environmental effects remain constant from one generation to the next, we can attribute the difference 2 – 1 to the selection of genes for high birth weight in the individuals that we chose to use as parents for the next generation. This difference (2 – 1) is called genetic gain or genetic response, symbolized ΔG. If all of the variability in birth weight exhibited by a population was due solely to additive gene effects, and the environment was contributing nothing at all, then by selecting individuals on the basis of their birth weight records we would actually be selecting the genes that are responsible for high birth weight. That is, we will not be confused by the effects that a favorable environment can produce with a mediocre genotype or by the favorable interaction of a certain combination of genes that will be broken up in subsequent generations. Realized heritability is defined as the ratio of the genetic gain to the selection differential:
EXAMPLE 8.12 If we gained in the offspring all that we "reached" for in the parents, then heritability is unity; i.e., if 2 – 1 = 100–80 = 20, and ΔP = p – 1 = 100–80 = 20, then h2 = ΔG/ΔP = 20/20 = 1.
EXAMPLE 8.13 If selection of parents with high birth weights fails to increase the mean birth weight of their offspring over that of the mean in the previous generation, then heritability is zero; i.e., if 2 and 1 = 80, then ΔG = 2 – 1 = 0 and h2 = ΔG/ΔP = 0/20 = 0.
EXAMPLE 8.14 If the mean weight of the offspring is increased by half the selection differential, then heritability of birth weight is 50%; i.e., if ΔG = 1/2ΔP; ΔP = 2ΔG, h2 = 0.5 = 50%. This is approximately the heritability estimate actually found for birth weight in one population of beef cattle [Fig. 8-8(b)].
Most quantitative traits are not highly heritable. What is meant by high or low heritability is not rigidly defined, but the following values are generally accepted.
Practice problems for these concepts can be found at:
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