# Quantitative genetics

Quantitative genetics is that branch of population genetics which deals with phenotypes which vary continuously (such as height or mass), rather than with phenotypes and gene-products which are discretely identifiable (such as eye-colour, or the presence of a particular biochemical). Both employ the frequencies of different alleles of a gene in breeding populations (gamodemes), and combine them with concepts arising from simple Mendelian inheritance in order to analyze inheritance patterns across generations and descendant lines. While population genetics can focus on particular genes and their subsequent metabolic products, quantitative genetics focuses more on the outward phenotypes, and makes summaries only of the underlying genetics. This, however, can be viewed as its strength, because it facilitates an interface with the biological macrocosm, including micro-evolution and artificial selection in plant and animal breeding. Both branches share some common history; and some mathematics: for example, they use expansion of the quadratic equation to represent the fertilization of gametes to form the zygote. However, because of the continuous distribution of phenotypic values, quantitative genetics needs also to employ many other statistical methods (such as the effect, the mean and the variance) in order to link the phenotype to underlying genetics principles. Some phenotypes (attributes) may be analyzed either as discrete categories or as continuous phenotypes, depending on the definition of cut-off points, or on the metric used to quantify them. [1]:27-69 Mendel himself had to discuss this matter in his famous paper,[2] especially with respect to his peas attribute tall/dwarf, which actually was "length of stem".[3][4] Analysis of quantitative trait loci, or QTL,[5] is a more recent addition to quantitative genetics, linking it more directly to molecular genetics.

## Basic principles

### Gene effects

In diploid organisms, the average genotypic outcome of a gene (locus) may be defined by the allele "effect" together with a dominance effect, and also by how genes interact with genes at other loci (epistasis). The founder of quantitative genetics - Sir Ronald Fisher - perceived much of this when he proposed the first mathematics of this branch of genetics.[6]

Gene effects and Phenotype values.

Being a statistician, he defined the gene effects as deviations from a central value: thereby enabling the use of statistical concepts such as the mean and variance, which utilize this idea.[7] The central value he chose for the gene was the midpoint between the two opposing homozygotes at the one locus. The deviation from there to the "greater" homozygous genotype can be named "+a" ; and therefore it is "-a" from that same midpoint to the "lesser" homozygote genotype. This is the "allele" effect mentioned above. The heterozygote deviation from the same midpoint can be named "d", this being the "dominance" effect referred to above.[8] The diagram depicts the idea. However, in reality we measure phenotypes, and the figure also shows how observed phenotypes relate to the gene effects. Formal definitions of these effects recognize this phenotypic focus.[9][10] Epistasis has been approached statistically as interaction (ie "inconsistencies"),[11] but epigenetics suggests a new approach may be needed.

If 0<d<a, the dominance is regarded as partial or "incomplete"; while d=a indicates full or "classical" dominance. Previously, d>a was known as "over-dominance".[12]

Mendel's pea attribute "length of stem" provides us with a good example.[3] Mendel stated that the tall true-breeding parents ranged from 6–7 feet in stem length (183 – 213 cm), giving a median of 198 cm (= P1). The short parents ranged from 0.75 - 1.25 feet in stem length (23 – 46 cm), with a rounded median of 34 cm (= P2). Their hybrid ranged from 6–7.5 feet in length (183–229 cm), with a median of 206 cm (= F1). The mean of P1 and P2 is 116 cm, this being the phenotypic value of the homozygotes midpoint (mp). The allele affect (a) is [P1-mp] = 82 cm = -[P2-mp]. The dominance effect (d) is [F1-mp] = 90 cm.[13] This historical example illustrates clearly how phenotype values and gene effects are linked.

### Allele and Genotype frequencies

To obtain means, variances and other statistics, both quantities and their occurrences are required. The gene effects (above) provide the framework for quantities: and the frequencies of the contrasting alleles in the fertilization gamete-pool provide the information on occurrences.

Analysis of Sexual reproduction.

Commonly, the frequency of the allele causing "more" in the phenotype (including dominance) is given the symbol p, while the frequency of the contrasting allele is q. An initial assumption made when establishing the algebra was that the parental population was infinite and random mating, which was made simply to facilitate the derivation. The subsequent mathematical development also implied that the frequency distribution within the effective gamete-pool was uniform: there were no local perturbations where p and q varied. Looking at the diagrammatic analysis of sexual reproduction, this is the same as declaring that pP = pg = p; and similarly for q.[12] This mating system, dependent upon these assumptions, became known as "panmixia".

Panmixia rarely actually occurs in nature,[14]:152–180[15][16] as gamete distribution may be limited, for example by dispersal restrictions or by behaviour, or by chance sampling (those local perturbations mentioned above). It is well-known that there is a huge wastage of gametes in Nature, which is why the diagram depicts a potential gamete-pool separately to the actual gamete-pool. Only the latter sets the definitive frequencies for the zygotes: this is the true "gamodeme" ("gamo" refers to the gametes, and "deme" derives from Greek for "population"). But, under Fisher's assumptions, the gamodeme can be effectively extended back to the potential gamete-pool, and even back to the parental base-population (the "source" population). The random sampling arising when small "actual" gamete-pools are sampled from a large "potential" gamete-pool is known as genetic drift, and will be considered subsequently.

While random fertilization (especially panmixia) may not be widely extant, it does occur, although it may be only ephemeral because of those local perturbations. It has been shown, for example, that the F2 derived from random fertilization of F1 individuals (an allogamous F2), following hybridization, is an origin of a new potentially panmictic population.[17][18] It has also been shown that if panmictic random fertilization did occur continually, it would maintain the same allele and genotype frequencies across each successive panmictic sexual generation - this being the Hardy Weinberg equilibrium.[11]:34–39[19][20][21][22] However, as soon as genetic drift was initiated by local random sampling of gametes, the equilibrium would cease.

#### Random Fertilization

Male and female gametes within the actual fertilizing pool are considered usually to have the same frequencies for their corresponding alleles. (Exceptions have been considered.) This means that when p male gametes carrying the A allele randomly fertilize p female gametes carrying that same allele, the resulting zygote will have genotype AA, and, under random fertilization, the combination will occur with a frequency of p x p (= p2). Similarly, the zygote aa will occur with a frequency of q2. Heterozygotes (Aa) can arise in two ways: when p male (A allele) randomly fertilize q female (a allele) gametes, and vice versa. The resulting frequency for the heterozygous zygotes is thus 2pq.[11]:32 Notice that such a population is never more than half heterozygous, this maximum occurring when p=q= 0.5.

In summary then, under random fertilization, the zygote (genotype) frequencies are the quadratic expansion of the gametic (allelic) frequencies: ( p + q )2 = ( p2 + 2 p q + q2) = 1. (The "=1" states that the frequencies are in fraction form, not percentages; and that there are no omissions within the framework proposed.) Also recall that "random fertilization" and "panmixia" are not synonyms, as discussed in the previous section.

#### Mendel's Research Cross - a Contrast

Mendel's pea experiments were constructed by establishing true-breeding parents with "opposite" phenotypes for each attribute.[3] This meant that each opposite parent was homozygous for its respective allele only. In our example, "tall vs dwarf", the tall parent would be genotype TT with p = 1 (and q = 0); while the dwarf parent would be genotype tt with q = 1 (and p = 0). After controlled crossing, their hybrid will be Tt, with p = q = ½. However, the frequency of this heterozygote = 1, because this is the F1 of an artificial cross: it has not arisen through random fertilization.[23] The F2 generation was produced by natural self-pollination of the F1 (with monitoring against insect contamination), resulting in p = q = ½ being maintained. Such an F2 is said to be "autogamous". However, the genotype frequencies (0.25 TT, 0.5 Tt, 0.25 tt) have arisen through a mating system very different from random fertilization, and therefore the use of the quadratic expansion has been avoided. The numerical values obtained were the same as those for random fertilization only because this is the special case of having originally crossed homozygous opposite parents.[24] We can notice that, because of the dominance of T- [frequency (0.25 + 0.5)] over tt [frequency 0.25], the 3:1 ratio is still obtained.

A cross such as Mendel's, where true-breeding (largely homozygous) opposite parents are crossed in a controlled way to produce an F1, is a special case of hybrid structure. The F1 is often regarded as being "entirely heterozygous" for the gene under consideration. However, this is an over-simplification which does not apply generally: for example when individual parents are not homozygous, or when populations inter-hybridise to form "hybrid swarms".[23] The general properties of intra-species hybrids (F1) and F2 (both "autogamous" and "allogamous") will be considered in a later section.

#### Self Fertilization - an Alternative

Having noticed that the pea is naturally self-pollinated, we cannot continue to use it as an example for illustrating random fertilization properties. Self-fertilization ("selfing") is a major alternative to random fertilization, especially within Plants. Most of the Earth's cereals are naturally self-pollinated (rice, wheat, barley, for example), as well as the pulses. Considering the millions of individuals of each of these on Earth at any time, it's obvious that self-fertilization is at least as significant as random fertilization! Self-fertilization is the most intensive form of inbreeding, which arises whenever there is restricted independence in the genetical origins of gametes. Such reduction in independence arises if parents are already related, and/or from genetic drift or other spatial restrictions on gamete dispersal. Path analysis demonstrates that these are tantamount to the same thing.[25][26] Arising from this background, the inbreeding coefficient (often symbolized as F or f) quantifies the effect of inbreeding from whatever cause. There are several formal definitions of f, and some of these will be considered in later sections. For the present, note that for a long-term self-fertilized species f = 1. Natural self-fertilized populations are not single " pure lines ", however, but mixtures of such lines. This becomes particularly obvious when considering more than one gene at a time. Therefore, allele frequencies (p and q) other than 1 or 0 are still relevant in these cases (refer back to the Mendel Cross section). The genotype frequencies take a different form, however.

In general, the genotype frequencies become [p2(1-f) + pf] for AA and [2pq(1-f)] for Aa and [q2(1-f) + qf] for aa.[11] :65 Notice that the frequency of the heterozygote declines in proportion to f. When f = 1, these three frequencies become respectively p, 0 and q !! Conversely, when f = 0, they reduce to the random-fertilization quadratic expansion shown previously.

### Population Mean

The population mean shifts the central reference point from the homozygote midpoint (mp) to the mean of a sexually reproduced population. This is important not only to relocate the focus into the natural world, but also to use a measure of central tendency utilized by Statistics/Biometrics. In particular, the square of this mean is the Correction Factor, which is used to obtain the Genotypic Variances later.[7]

Population mean across all values of p, for various d effects.

For each genotype in turn, its allele effect is multiplied by its genotype frequency; and the products are accumulated across all genotypes in the model. Some algebraic simplification usually follows to reach a succinct result.

#### Random Fertilization

The contribution of AA is p2 (+)a, that of Aa is 2pq d. and that of aa is q2 (-)a. Gathering together the two a terms and accumulating over all, the result is: a (p2 - q2) + 2pq d. Simplification is achieved by noting that (p2-q2) = (p-q)(p+q), and by recalling that (p+q) = 1, thereby reducing the left-hand term to (p-q). The succinct result is therefore G = a(p-q) + 2pqd.[12] :110 This defines the population mean as an "offset" from the homozygote midpoint (recall a and d are defined as deviations from that midpoint). The Figure depicts G across all values of p for several values of d, including one case of slight over-dominance. Notice that G is often negative, thereby emphasizing that it is itself a deviation (from mp). Finally, to obtain the actual Population Mean in "phenotypic space", the midpoint value is added to this offset: P = G + mp.

An example arises from data on ear length in maize.[27]:103 Assuming for now that one gene only is represented, a = 5.45 cm, d = 0.12 cm [virtually "0", really], mp = 12.05 cm. Further assuming that p = 0.6 and q = 0.4 in this example population, then:-

G = 5.45 (0.6 - 0.4) + (0.48)0.12 = 1.15 cm (rounded); and

P = 1.15 + 12.05 = 13.20 cm (rounded).

#### Self Fertilization

The contribution of AA is p (+)a, while that of aa is q (-)a. (See above for the frequencies.) Gathering these two a terms together leads to an immediately very simple final result:- GS = a(p-q). P is obtained as above.

Mendel's peas can provide us with the allele effects and midpoint (see previously); and a mixed self-pollinated population with p = 0.6 and q = 0.4 provides example frequencies. Thus:-

GS = 82 (0.6 - .04) = 59.6 cm (rounded); and

PS = 59.6 + 116 = 175.6 cm (rounded).

#### Generalized Fertilization

A general formula incorporates the inbreeding coefficient f, and can then accommodate any situation. The procedure is exactly the same as before, using the weighted genotype frequencies given earlier. After translation into our symbols, and further rearrangement [11] :77–78 :-

Gf = a(q-p) + [2pqd - f 2pqd] = a(q-p) + (1-f) 2pqd = G - f 2pqd !

Supposing that the maize example (earlier) had been constrained on a holme (a narrow riparian meadow), and had partial inbreeding to the extent of f = 0.25, then, using the third version (above) of Gf:-

G0.25 = 1.15 - 0.25 (0.48) 0.12 = 1.136 cm (rounded), with P = 13.194 cm (rounded).

There is hardly any effect from inbreeding in this example, which arises because there was virtually no dominance in this attribute (d → 0). Examination of all three versions of Gf reveals that this would lead to trivial change in the Population mean. Where dominance was notable, however, there would be considerable change.

### Allele shuffling - allele Substitution

The gene-model examines the heredity pathway from the point of view of "inputs" (alleles/gametes) and "outputs" (genotypes/zygotes), with fertilization being the "process" converting one to the other. An alternative viewpoint concentrates on the "process" itself, and considers the zygote genotypes as arising from allele shuffling. In particular, it regards the results as if one allele had "substituted" for the other during the shuffle, together with a residual which deviates from this view. This formed an integral part of Fisher's method,[6] in addition to his use of frequencies and effects to generate his genetical statistics.[12]

Analysis of Allele Substitution

A discursive derivation of the Allele Substitution alternative follows.[12]:113 Suppose that the usual random fertilization of gametes in a "base" gamodeme - consisting of p gametes (A) and q gmaetes (a) - is replaced by fertilization with a "flood" of gametes all containing a single allele (A or a, but not both). The zygotic results can be interpreted in terms of the "flood" allele having "substituted for" the alternative allele in the underlying "base" gamodeme. The diagram assists in following this viewpoint: the upper part pictures an A substitution, while the lower part shows an a substitution. (The diagram's "RF allele" is the allele in the "base" gamodeme.)

Consider the upper part firstly. Because base A is present with a frequency of p, the substitute A will fertilize it with a frequency of p resulting in a zygote AA with an allele effect of a. Its contribution to the outcome will therefore be the product p a. Similarly, when the substitute fertilizes base a (resulting in Aa with a frequency of q and heterozygote effect of d), the contribution will be q d. The overall result of substitution by A will therefore be pa + qd. This is now oriented towards the population mean (see earlier section) by expressing it as a deviate from that mean : (pa + qd) - G. After some algebraic simplification, this becomes α A = q [a + (q-p)d] - the substitution effect of A.

A parallel reasoning can be applied to the lower part of the diagram, taking care with the differences in frequencies and gene effects. The result is the substitution effect of a, which is α a = -p [a + (q-p)d].

The common factor inside the brackets is known as the average allele substitution effect, and is usually given as α = a + (q-p)d.[12]:113 It can be derived also in a more direct way, but the result is the same.

In subsequent sections, these substitution effects will be used to define the gene-model genotypes as consisting of a partition predicted by these new effects (substitution expectations), and a deviation between these expectations and the previous gene-model effect. The expectations are also known as the breeding value, and the deviations are also known as the dominance deviations. Ultimately, the variance arising from the substitution expectations will become the Additive Genetic variance (σ2A) [12] (also the Genic variance [28]); while the variance arising from the deviations will become known as the Dominance variance (σ2D).

## Extended principles

### Gene effects redefined

The gene-model effects (a, d and -a) are important soon in the derivation of the deviations from substitution expectations (δ), which were first discussed in the previous Allele Substitution section. However, they need to be redefined themselves before they become useful in that exercise. They firstly need to be re-centralized around the population mean (G), and secondly they need to be re-arranged as functions of α, the average allele substitution effect.

The re-centralized effect for AA, therefore, is a′ = a - G which, after simplification, becomes a′ = 2q(a-pd). The similar effect for Aa is d′ = d - G = a(q-p) + d(1-2pq), after simplification. Finally, the re-centralized effect for aa is (-a)′ = -2p(a+qd).[12]:116–119

These re-centralized effects eventually will have the genotype substitution expectations (see next section) subtracted from them in order to define subsequently the genotype substitution deviations. These expectations each are a function of the average allele substitution effect, and the present re-centralized effects have to be re-arranged still further to accommodate this last subtraction. Recalling that α = [a +(q-p)d], rearrangement gives a = [α -(q-p)d]. After substituting this for a in a′ and simplifying, the final version becomes a′′ = 2q(α-qd). Similarly, d′ becomes d′′ = α(q-p) + 2pqd; and (-a)′ becomes (-a)′′ = -2p(α+pd).[12]:118

### Genotype substitution - Expectations and Deviations

The zygote genotypes are the target of all this preparation. The homozygous genotype AA is a union of two substitution effects of A, one coming from each sex. Its substitution expectation is therefore αAA = 2αA = 2qα (see previous sections). Similarly, the substitution expectation of Aa is αAa = αA + αa = (q-p ; and for aa, αaa = 2αa = -2pα. These substitution expectations of the genotypes are known also as "Breeding values".[12]:114–116

The substitution deviations are the differences between these expectations and the gene effects after their two-stage redefinition in the previous section. Therefore, δAA = a′′ - αAA = -2q2d after simplification. Similarly, δAa = d′′ - αAa = 2pqd after simplification. Finally, δaa = (-a)′′ - αaa = -2p2d after simplification.[12]:116–119

The genotype substitution expectations will give rise ultimately to the σ2A, and the genotype substitution deviations will give rise to the σ2D.

### Genotypic variance

There are two major approaches to defining and partitioning the Genotypic variance : one is based on the gene-model effects,[28] while the other is based on the genotype substitution effects[12] They are algebraically inter-convertible with each other.[29] In this section, the basic random fertilization derivation is considered, with the effects of inbreeding and dispersion set aside. This will be dealt with later in order to arrive at a more general solution. Until this mono-genic treatment is replaced by a multi-genic one, and until epistasis is resolved in the light of the findings of epigenetics, the Genotypic variance will have only the components considered here.

#### Gene-model approach - Mather Jinks Hayman

Components of Genotypic variance using the gene-model effects.

It is convenient to follow the Biometrical approach : which is based on correcting the unadjusted sum of squares (USS) by subtracting the correction factor (CF). Because all of our effects have been examined through frequencies, the USS can be obtained as the sum of the products of each genotype's frequency and the square of its gene-effect. The CF in this particular case is the mean squared. The result is the sum of squares (SS), which, again because of the use of frequencies, is also immediately the variance.[7]

The USS = p2a2 + 2pqd2 + q2(-a)2 , and the CF = G2 . The SS = USS - CF = σ2G .

After partial simplification,

σ2G = 2pqa2 + (q-p)4pqad + 2pqd2 + (2pq)2 d2 = σ2a + (weighted_covariance)ad + σ2d + σ2D = ½D + ½F´ + ½H1 + ¼H2 in Mather's terminology.[28]:212 [30]

Here, σ2a represents the homozygote or allelic variance, and σ2d represents the heterozygote or gene-model dominance variance. The random-fertilization dominance variance (σ2D) is present also. These components are plotted across all values of p in the Figure accompanying. Notice that the (weighted_covariance)ad[31] (hereafter abbreviated to covad) is negative for 0.5<p.

Further gathering of terms leads to ½D + ½F´ + ½H3 + ¼H2, where ½H3 = (q-p)2 ½H1 = (q-p)22pqd2. It will be useful later in Diallel analysis, which is an experimental design for estimating these genetical statistics.[32]

If, following the last-given rearrangements, the first three terms are amalgamated together, rearranged further and simplified, the result is the variance of the Fisherian substitution expectation. That is: σ2A = σ2a + covad + σ2d, a revealing insight indeed. Notice particularly that σ2A is not σ2a.[33] Notice also that σ2D = 2pq σ2d. From the Figure, this can be visualized as accumulating σ2a, σ2d and cov to obtain σ2A, while leaving the σ2D still separated. It is clear also in the Figure that σ2D < σ2d, as expected from the equations.

Components of Genotypic variance using the allele-substitution effects.

The overall result is σ2G = 2pq [a+(q-p)d]2 + (2pq)2 d2 = σ2A + σ2D .

However, its derivation via the substitution effects themselves will be given also, in the next section.

#### Allele-substitution approach - Fisher

Reference to the several earlier sections on allele substitution reveals that the two ultimate effects are genotype substitution expectations and genotype substitution deviations. Notice that these are each defined already as deviations from the random fertilization population mean (G). For each genotype in turn, the product of the frequency and the square of the relevant effect is obtained, and these are accumulated to obtain directly a SS and σ2. Details follow.

σ2A = p2 α2AA + 2pq α2Aa + q2 α2aa which simplifies to σ2A = 2pqα2.

σ2D = p2 δ2AA + 2pq δ2Aa + q2 δ2aa which simplifies to σ2D = (2pq)2 d2.

Once again, σ2G = σ2A + σ2D .

Note that this allele-substitution approach defined the components separately, and then totaled them to obtain the final Genotypic variance. Conversely, the gene-model approach derived the whole situation (components and total) as one exercise. Bonuses arising from this were (a) the revelations about the real structure of σ2A, and (b) the relative sizes of σ2d and σ2D (see previous sub-section). It is also apparent that a "Mather" analysis is more informative, and that a "Fisher" analysis can always be constructed from it. The opposite conversion is not possible, however, because information about covad would be missing.

### Dispersion and the Genotypic variance

In many apparently large breeding groups, separate gamete restrictions, together with genetic drift (discrete random sampling), will occur during a mating cycle, each one leading to a line. These lines also will vary with respect to their mean phenotypes, and the process is called dispersion. The inbreeding coefficient quantifies the increase in homozygosity which results. The values of this coefficient for a wide variety of situations (e.g. islands, "onion-skin" aggregates, linear strips, matings of related parents) are available.[34] As well as a general rise in homozygosity, the dispersed lines vary in their allele frequencies because of gamete sampling. However, the mean of the frequencies across all lines from the one source will be the same as the original frequencies in the source population. The phenotypic mean of all of these lines is less than that of the original source, this being inbreeding depression. The genetic variances also change relative to those of panmixia. Variance-within-lines decreases, but the variance-amongst-lines and the total-variance-in-the-system both increase (Mackay et al.; Gordon 2003). The first of these facts is common knowledge, but the latter two are not. Many of these lines will be inferior in phenotype: but, some lines will be superior, and some will be about average (Chapter 13 in Falconer et al.). Selection assisted by dispersion leads to maximum genetic advance (see previous references). Plant and animal breeders utilize these properties routinely, and have devised breeding methods especially to do so (e.g. line breeding, pure-line breeding, backcrossing).[35] The role of dispersion in natural selection has not received much attention.

### Environmental variance

The environmental variance is phenotypic variability which cannot be ascribed to genetics. This sounds simple, but the experimental design needed to separate the two needs very careful planning. Even the "external" environment can be divided into spatial and temporal components, as well as partitions such as "litter" or "family" and "culture" or "history". Where does epigenetic variance get placed? Is it embedded within epistasis: or is it "internal environment"? These components are very dependent upon the actual experimental model used to do the research. Such issues are very important when doing the research itself, but in this article on quantitative genetics this overview may suffice.

It is an appropriate place, however, for a summary:

Phenotypic variance = genotypic variances + environmental variances + genotype-environment interaction + experimental "error" variance

ie σ²P = σ²G + σ²E + σ²GE + σ²

or σ²P = σ²A + σ²D + σ²I + σ²E + σ²GE + σ²

after partitioning the genotypic variance (G) into the components of "additive" (A), "dominance" (D), and "epistasic" (I) variance mentioned above.[36]

### Heritability and repeatability

The heritability of a trait is the proportion of the total (phenotypic) variance (σ²P) that is explained by the total genotypic variance (σ²G). This is known as the "broad sense" heritability (H2).[37] If only additive genetic variance (σ²A) is used in the numerator, the heritability is called "narrow sense" (h2).

The broad sense heritability indicates the proportion of the phenotypic variance due to the whole genotypical variance. In colloquial terms, it indicates the extent of "nature" while (1-H2) indicates the extent of "nurture". Narrow sense heritability indicates the proportion of the phenotypic variance attributable to the "additive" genetic variance, discussed above. It was pointed out there that this variance arises through substitution (ie phenotypic change) following fertilization. Fisher proposed that this narrow-sense heritability might be appropriate in considering the results of natural selection, focusing as it does on change-ability, and hence adaptation.[38] It has been used also for predicting generally the results of artificial selection. In the latter case, however, the broad sense heritability may be more appropriate, as the whole attribute is being altered: not just adaptive capacity. Generally, advance from selection is more rapid with higher heritability. In animals, heritability of reproductive traits is typically low, while heritability of disease resistance and production are moderately low to moderate, and heritability of body conformation is high.

Repeatability (r2) is the proportion of phenotypic variance attributable to differences in repeated measures of the same subject, arising from later records. It is used particularly for long-lived species. This value can only be determined for traits that manifest multiple times in the organism's lifetime, such as adult body mass, metabolic rate or litter size. Individual birth mass, for example, would not have a repeatability value: but it would have a heritability value. Generally, but not always, repeatability indicates the upper level of the heritability.[39]

r2 = (σ²G + σ²PE)/σ²P

where σ²PE = phenotype-environment interaction ≡ repeatability.

The above concept of repeatability is, however, problematic for traits that necessarily change greatly between measurements. For example, body mass increases greatly in many organisms between birth and adult-hood. Nonetheless, within a given age range (or life-cycle stage), repeated measures could be done, and repeatability would be meaningful within that stage.

## Relationship

### Resemblance between relatives

Central in estimating the variances for the various components is the principle of relatedness. A child has a father and a mother. Consequently, the child and father share 50% of their alleles, as do the child and the mother. However, the mother and father normally do not share alleles as a result of shared ancestors. Similarly, two full siblings share also on average 50% of the alleles with each other, while half siblings share only 25% of their alleles. This variation in relatedness can be used to estimate which proportion of the total phenotypic variance (σ²P) is explained by the above-mentioned components.

The principle of relationship (R) is central to understanding the resemblances within families and can be useful when calculating inbreeding. Relationship has two definitions that can be applied: -The probable portion of genes that are the same for two individuals due to common ancestry exceeding that of the base population -Additive/numerator relationship: the relationship coefficient (Rxy¬) = twice the probability of two genes at loci in different individuals being identical by descent. Rxy values can range from 0 to 1. Relationship can be calculated in several ways; from the known relationships of the individual, from bracket pedigrees, and from pedigree path diagrams.

#### Calculating relationship from known relationships

Relationship Relationship Coefficient
Individual and itself 1.00
Individual and a monozygotic twin 1.00
Individual and parent 0.50
Full siblings 0.50
Half siblings 0.25
Individual and grandparent 0.25
Son of sire and daughter of sire 0.125
Grandson and granddaughter of sire 0.0625
• Note: if the common ancestor is inbred, multiply the relationship by (1+inbreeding coefficient)

#### Calculating relationship from pathway diagrams

RXY = Σ(.5)n(1+FCA)

n = number of segregations between X and Y through their common ancestor FCA = the inbreeding coefficient of the common ancestor

Example: calculating RAE and RBE Note: valid pathways only go through ancestors (only go against the direction of the arrow). For example, to calculate the relationship of A and B, the pathway A-D-B would be acceptable, whereas the pathway A-X-B would be not. The reason behind this is that having progeny together does not make two individuals related.

RAB: there are two possible pathways from A to E. A-D-F-E = (1/2)3 = .125 A-D-E = (1/2)2 = .25 Total: .375

RBE: there are four possible pathways from B to E. B-D-E = (1/2)2 = .25 B-D-F-E = (1/2)3 = .125 B-C-D-E = (1/2)3 = .125 B-C-D-F-E = (1/2)4 = .0625 Total: .5625

The square root of h^2 equals the correlation between additive genotype and expressed phenotype, as shown through the general procedures of Path Analysis.[citation needed]

## Correlated traits

Although some genes have only an effect on a single trait, many genes have an effect on various traits, which is termed pleiotropy. Because of this, a change in a single gene will have an effect on all those traits. This is calculated using covariances, and the phenotypic covariance (CovP) between two traits can be partitioned in the same way as the variances described above. The genetic correlation is calculated by dividing the covariance between the additive genetic effects of two traits by the square root of the product of the variances for the additive genetic effects of the two traits:

$\mbox{Genetic correlation} = \frac{\mathrm{Cov}(A_{1}, A_{2})}{\sqrt{{V_{A_1}*V_{A_2}}}}$

## Footnotes and references

1. ^ Anderberg, Michael R. (1973). Cluster analysis for applications. New York: Academic Press.
2. ^ Mendel, Gregor (1866). "Versuche über Pflanzen Hybriden". Verhandlungen naturforschender Verein in Brünn iv.
3. ^ a b c Mendel, Gregor; Bateson, William [translator] (1891). "Experiments in plant hybridisation". J. Roy. hort. Soc. (London) xxv: 54–78.
4. ^ The Mendel G.; Bateson W. (1891) paper, with additional comments by Bateson, is reprinted in:- Sinnott E.W.; Dunn L.C.; Dobzhansky T. (1958). "Principles of genetics"; New York, McGraw-Hill: 419-443. Footnote 3, page 422 identifies Bateson as the original translator, and provides the reference for that translation.
5. ^ A QTL is a region in the DNA genome that affects, or is associated with, quantitative phenotypic traits.
6. ^ a b Fisher, R. A. (1918). "The correlation between relatives on the supposition of Mendelian inheritance.". Trans. Roy. Soc, (Edinburgh) 52: 399–433.
7. ^ a b c Steel, R. G. D.; Torrie, J. H. (1980). Principles and procedures of statistics. (2 ed.). New York: McGraw-Hill. ISBN 0 07 060926 8.
8. ^ Other symbols are sometimes used, but these are common.
9. ^ The allele effect is the average phenotypic deviation of the homozygote from the mid-point of the two contrasting homozygote phenotypes at one locus, when observed over the infinity of all background genotypes and environments. In practice, estimates from large unbiased samples substitute for the parameter.
10. ^ The dominance effect is the average phenotypic deviation of the heterozygote from the mid-point of the two homozygotes at one locus, when observed over the infinity of all background genotypes and environments. In practice, estimates from large unbiased samples substitute for the parameter.
11. Crow, J. F.; Kimura, M. (1970). An introduction to population genetics theory. New York: Harper & Row.
12. Falconer, D. S.; Mackay, Trudy F. C. (1996). Introduction to quantitative genetics (Fourth ed.). Harlow: Longman. ISBN 978-0582-24302-6. Lay summaryGenetics (journal) (24 August 2014).
13. ^ Mendel commented on this particular tendency for F1 > P1, ie evidence of hybrid vigour in stem length. However, the difference may not be sufficient to be judged significant. (The relationship between the range and the standard deviation is known [Steel and Torrie (1980): 576], permitting an approximate significance test to be made for this present difference.)
14. ^ Richards, A. J. (1986). Plant breeding systems. Boston: George Allen & Unwin. ISBN 0 04 581020 6.
15. ^ Jane Goodall Institute. "Social structure of chimpanzees.". Chimp Central. Retrieved 20 August 2014.
16. ^ Wikipedia. "Animal mating systems.". English Wikipedia. Retrieved 21 August 2014.
17. ^ Gordon, Ian L. (2000). "Quantitative genetics of allogamous F2: an origin of randomly fertiliized populations.". Heredity 85: 43–52.
18. ^ An F2 derived by self fertilizing F1 individuals (an autogamous F2), however, is not an origin of a randomly fertilized population structure. See Gordon (2001).
19. ^ Castle, W. E. (1903). "The law of heredity of Galton and Mendel and some laws governing race improvement by selection.". Proc. Amer. Acad, Sci. 39: 233–242.
20. ^ Hardy, G. H. (1908). "Mendelian proportions in a mixed population.". Science 28: 49–50.
21. ^ Weinberg, W. (1908). "Über den Nachweis der Verebung beim Menschen.". Jahresh. Verein f. vaterl. Naturk, Württem. 64: 368–382.
22. ^ Usually in science ethics, a discovery is named after the earliest person to propose it. Castle, however, seems to have been overlooked: and later when re-found, the title "Hardy Weinberg" was so ubiquitous it seemed too late to update it. Perhaps the "Castle Hardy Weinberg" equlilbrium would be a good compromise?
23. ^ a b Gordon, Ian L. (1999). "Quantitative genetics of intraspecies hybrids.". Heredity 83: 757–764.
24. ^ Gordon, Ian L. (2001). "Quantitative genetics of autogamous F2.". Hereditas 134: 255–262.
25. ^ Wright, S. (1917). "The average correlation within subgroups of a population.". J. Wash. Acad. Sci. 7: 532–535.
26. ^ Wright, S. (1921). "Systems of mating. I. The biometric relations between parent and offspring.". Genetics 6: 111–123.
27. ^ Sinnott, Edmund W.; Dunn, L. C.; Dobzhansky, Theodosius (1958). Principles of genetics. New York: McGraw-Hill.
28. ^ a b c Mather, Kenneth; Jinks, John L. (1971). Biometrical genetics (2 ed.). London: Chapman & Hall. ISBN 0 412 10220 X.
29. ^ Gordon, I.L. (2003). "Refinements to the partitioning of the inbred genotypic variance". Heredity 91: 85–89. doi:10.1038/sj.hdy.6800284. PMID 12815457.
30. ^ These have been translated from Mather's symbols into Fisherian ones to facilitate the comparison.
31. ^ Covariance is the co-variability between two sets of data - in this case the a and the d. Similarly to the variance, it is based on a sum of cross-products (SCP) instead of a SS. From this, it is clear therefore that the variance is but a special form of the covariance!
32. ^ Hayman, B. I. (1960). "The theory and analysis of the diallel cross. III.". Genetics 45: 155–172.
33. ^ It has been observed that when p = q, or when d = 0, α [= a+(q-p)d] "reduces" to a. In such circumstances, σ2A = σ2a - but only numerically. They still have not become the one and the same identity.
34. ^ Wright, Sewall (1951). "The genetical structure of populations". Annals of Eugenics 15: 323–354. doi:10.1111/j.1469-1809.1949.tb02451.x.
35. ^ Allard, R.W. (1960). Principles of plant breeding. Wiley.
36. ^ It is common practice not to have a subscript on the experimental "error" variance.
37. ^ This type of variance-ratio is an example of a coefficient of determination. It is used particularly in regression analysis. A standardized version of regression analysis is path analysis. Standardizing here means that the data were first divided by their own experimental standard errors, in order to unify the scales for all attributes.
38. ^ Fisher, R. A. (1999). The genetical theory of natural selection. ("variorum" ed.). Oxford: Oxford University Press. ISBN 0 19 850440 3.
39. ^ Dohm, M. R. (2002). "Repeatability estimates do not always set an upper limit to heritibility.". Functional ecology 16: 273–280.