Part of a series on |

Genetics |
---|

This article may require cleanup to meet Wikipedia's quality standards. The specific problem is: nested fractions probably better written with $...$ markup. Please help improve this article if you can. (February 2024) (Learn how and when to remove this message)

**Quantitative genetics** is the study of quantitative traits, which are phenotypes that vary continuously—such as height or mass—as opposed to phenotypes and gene-products that are discretely identifiable—such as eye-colour, or the presence of a particular biochemical.

Both of these branches of genetics use the frequencies of different alleles of a gene in breeding populations (gamodemes), and combine them with concepts from simple Mendelian inheritance 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 only summaries of the underlying genetics.

Due to the continuous distribution of phenotypic values, quantitative genetics must employ many other statistical methods (such as the *effect size*, the *mean* and the *variance*) to link phenotypes (attributes) to genotypes. Some phenotypes 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 derived by adding a cut-off point to "length of stem".^{[3]}^{[4]} Analysis of quantitative trait loci, or QTLs,^{[5]}^{[6]}^{[7]} is a more recent addition to quantitative genetics, linking it more directly to molecular genetics.

In diploid organisms, the average genotypic "value" (locus value) 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.^{[8]}

Being a statistician, he defined the gene effects as deviations from a central value—enabling the use of statistical concepts such as mean and variance, which use this idea.^{[9]} 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.^{[10]} 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.^{[11]}^{[12]} Epistasis has been approached statistically as interaction (i.e., inconsistencies),^{[13]} 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".^{[14]}

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 to 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.^{[15]} This historical example illustrates clearly how phenotype values and gene effects are linked.

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*.

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

Panmixia rarely actually occurs in nature,^{[16]}^{: 152–180 }^{[17]} 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 is considered subsequently.

While panmixia may not be widely extant, the *potential* for 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.^{[18]}^{[19]} It has also been shown that if panmictic random fertilization occurred continually, it would maintain the same allele and genotype frequencies across each successive panmictic sexual generation—this being the *Hardy Weinberg* equilibrium.^{[13]}^{: 34–39 }^{[20]}^{[21]}^{[22]}^{[23]} However, as soon as genetic drift was initiated by local random sampling of gametes, the equilibrium would cease.

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

In summary then, under random fertilization, the zygote (genotype) frequencies are the quadratic expansion of the gametic (allelic) frequencies: . (The "=1" states that the frequencies are in fraction form, not percentages; and that there are no omissions within the framework proposed.)

Notice that "random fertilization" and "panmixia" are *not* synonyms.

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

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 "entirely heterozygous" for the gene under consideration. However, this is an over-simplification and does not apply generally—for example when individual parents are not homozygous, or when *populations* inter-hybridise to form *hybrid swarms*.^{[24]} The general properties of intra-species hybrids (F1) and F2 (both "autogamous" and "allogamous") are considered in a later section.

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 is 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.^{[26]}^{[27]} 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

In general, the genotype frequencies become for **AA** and for **Aa** and for **aa**.^{[13]}^{: 65 }

Notice that the frequency of the heterozygote declines in proportion to * f*. When

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* used by Statistics/Biometrics. In particular, the square of this mean is the Correction Factor, which is used to obtain the genotypic variances later.^{[9]}

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.

The contribution of **AA** is , that of **Aa** is , and that of **aa** is . Gathering together the two **a** terms and accumulating over all, the result is: . Simplification is achieved by noting that , and by recalling that , thereby reducing the right-hand term to .

The succinct result is therefore .^{[14]} ^{: 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: .

An example arises from data on ear length in maize.^{[28]}^{: 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).

The contribution of **AA** is , while that of **aa** is . [See above for the frequencies.] Gathering these two **a** terms together leads to an immediately very simple final result:

. As before, .

Often, "G_{(f=1)}" is abbreviated to "G_{1}".

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:

**G _{(f=1)}** = 82 (0.6 − .04) = 59.6 cm (rounded); and

**P _{(f=1)}** = 59.6 + 116 = 175.6 cm (rounded).

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:

Here,

Supposing that the maize example [given earlier] had been constrained on a holme (a narrow riparian meadow), and had partial inbreeding to the extent of * f *=

**G _{0.25}** = 1.15 − 0.25 (0.48) 0.12 = 1.136 cm (rounded), with

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 **G _{f}** reveals that this would lead to trivial change in the Population mean. Where dominance was notable, however, there would be considerable change.

Genetic drift was introduced when discussing the likelihood of panmixia being widely extant as a natural fertilization pattern. [See section on Allele and Genotype frequencies.] Here the sampling of gametes from the *potential* gamodeme is discussed in more detail. The sampling involves random fertilization between pairs of random gametes, each of which may contain either an **A** or an **a** allele. The sampling is therefore binomial sampling.^{[13]}^{: 382–395 }^{[14]}^{: 49–63 }^{[29]}^{: 35 }^{[30]}^{: 55 } Each sampling "packet" involves **2N** alleles, and produces **N** zygotes (a "progeny" or a "line") as a result. During the course of the reproductive period, this sampling is repeated over and over, so that the final result is a mixture of sample progenies. The result is *dispersed random fertilization* These events, and the overall end-result, are examined here with an illustrative example.

The "base" allele frequencies of the example are those of the *potential gamodeme*: the frequency of **A** is **p _{g} = 0.75**, while the frequency of

Following completion of these five binomial sampling events, the resultant actual gamodemes each contained different allele frequencies—(**p _{k}** and

Because sampling involves chance, the *probabilities* ( `∫`_{k} ) of obtaining each of these samples become of interest. These binomial probabilities depend on the starting frequencies (**p _{g}** and

Here, some *summarizing* can begin. The *overall allele frequencies* in the progenies bulk are supplied by weighted averages of the appropriate frequencies of the individual samples. That is: and . (Notice that **k** is replaced by **•** for the overall result—a common practice.)^{[9]} The results for the example are **p _{•}** = 0.631 and

The *genotype frequencies* of the five sample progenies are obtained from the usual quadratic expansion of their respective allele frequencies (*random fertilization*). The results are given at the diagram's *white label* "**7**" for the homozygotes, and at *white label* "**8**" for the heterozygotes. Re-arrangement in this manner prepares the way for monitoring inbreeding levels. This can be done either by examining the level of *total* homozygosis [(**p ^{2}_{k} + q^{2}_{k}**) = (

The *overall summary* can continue by obtaining the *weighted average* of the respective genotype frequencies for the progeny bulk. Thus, for **AA**, it is , for **Aa** , it is and for **aa**, it is . The example results are given at *black label* "**7**" for the homozygotes, and at *black label* "**8**" for the heterozygote. Note that the heterozygosity mean is *0.3588*, which the next section uses to examine inbreeding resulting from this genetic drift.

The next focus of interest is the dispersion itself, which refers to the "spreading apart" of the progenies' *population means*. These are obtained as [see section on the Population mean], for each sample progeny in turn, using the example gene effects given at *white label* "**9**" in the diagram. Then, each is obtained also [at *white label* "**10**" in the diagram]. Notice that the "best" line (k = 2) had the *highest* allele frequency for the "more" allele (**A**) (it also had the highest level of homozygosity). The *worst* progeny (k = 3) had the highest frequency for the "less" allele (**a**), which accounted for its poor performance. This "poor" line was less homozygous than the "best" line; and it shared the same level of homozygosity, in fact, as the two *second-best* lines (k = 1, 5). The progeny line with both the "more" and the "less" alleles present in equal frequency (k = 4) had a mean below the *overall average* (see next paragraph), and had the lowest level of homozygosity. These results reveal the fact that the alleles most prevalent in the "gene-pool" (also called the "germplasm") determine performance, not the level of homozygosity per se. Binomial sampling alone effects this dispersion.

The *overall summary* can now be concluded by obtaining and . The example result for **P _{•}** is 36.94 (

Included in the *overall summary* were the average allele frequencies in the mixture of progeny lines (**p _{•}** and

The **inbreeding coefficient** (* f*) was introduced in the early section on Self Fertilization. Here, a formal definition of it is considered:

The *population mean* of the equivalent panmictic is found as *[a (p _{•}-q_{•}) + 2 p_{•}q_{•} d] + mp*. Using the example

If the number of binomial samples is large (**s → ∞** ), then **p _{•} → p_{g}** and

Previous discussion on genetic drift examined just one cycle (generation) of the process. When the sampling continues over successive generations, conspicuous changes occur in **σ ^{2}**

Earlier this variance (σ ^{2}_{p,q}^{[35]}) was seen to be:-

With the extension over time, this is also the result of the *first* cycle, and so is (for brevity). At cycle 2, this variance is generated yet again—this time becoming the *de novo* variance ()—and accumulates to what was present already—the "carry-over" variance. The *second* cycle variance (**) is the weighted sum of these two components, the weights being for the ***de novo* and = for the"carry-over".

Thus,

(1) |

The extension to generalize to any time *t* , after considerable simplification, becomes:^{[13]}^{: 328 }-

(2) |

Because it was this variation in allele frequencies that caused the "spreading apart" of the progenies' means (*dispersion*), the change in** σ ^{2}_{t}** over the generations indicates the change in the level of the

The method for examining the inbreeding coefficient is similar to that used for *σ ^{2}_{p,q}*. The same weights as before are used respectively for

In general, after rearrangement,^{[13]}

The graphs to the left show levels of inbreeding over twenty generations arising from genetic drift for various

Still further rearrangements of this general equation reveal some interesting relationships.

**(A)** After some simplification,^{[13]} . The left-hand side is the difference between the current and previous levels of inbreeding: the *change in inbreeding* (**δf _{t}**). Notice, that this

**(B)** An item of note is the **(1-f _{t-1})**, which is an "index of

**(C)** Further useful relationships emerge involving the *panmictic index*.^{[13]}^{[14]}

.

Secondly, presuming that

It is easy to overlook that *random fertilization* includes self-fertilization. Sewall Wright showed that a proportion **1/N** of *random fertilizations* is actually *self fertilization* , with the remainder **(N-1)/N** being *cross fertilization* . Following path analysis and simplification, the new view *random fertilization inbreeding* was found to be: .^{[27]}^{[37]} Upon further rearrangement, the earlier results from the binomial sampling were confirmed, along with some new arrangements. Two of these were potentially very useful, namely: **(A)** ; and **(B)** .

The recognition that selfing may *intrinsically be a part of* random fertilization leads to some issues about the use of the previous *random fertilization* 'inbreeding coefficient'. Clearly, then, it is inappropriate for any species incapable of *self fertilization*, which includes plants with self-incompatibility mechanisms, dioecious plants, and bisexual animals. The equation of Wright was modified later to provide a version of random fertilization that involved only *cross fertilization* with no *self fertilization*. The proportion **1/N** formerly due to *selfing* now defined the *carry-over* gene-drift inbreeding arising from the previous cycle. The new version is:^{[13]}^{: 166 }

.

The graphs to the right depict the differences between standard *random fertilization* **RF**, and random fertilization adjusted for "cross fertilization alone" **CF**. As can be seen, the issue is non-trivial for small gamodeme sample sizes.

It now is necessary to note that not only is "panmixia" *not* a synonym for "random fertilization", but also that "random fertilization" is *not* a synonym for "cross fertilization".

In the sub-section on "The sample gamodemes – Genetic drift", a series of gamete samplings was followed, an outcome of which was an increase in homozygosity at the expense of heterozygosity. From this viewpoint, the rise in homozygosity was due to the gamete samplings. Levels of homozygosity can be viewed also according to whether homozygotes arose allozygously or autozygously. Recall that autozygous alleles have the same allelic origin, the likelihood (frequency) of which * is* the

Following firstly the *auto/allo* viewpoint, consider the *allozygous* component. This occurs with the frequency of **(1-f)**, and the alleles unite according to the *random fertilization* quadratic expansion. Thus:

Consider next the

Secondly, the *sampling* viewpoint is re-examined. Previously, it was noted that the decline in heterozygotes was . This decline is distributed equally towards each homozygote; and is added to their basic *random fertilization* expectations. Therefore, the genotype frequencies are: for the *"AA"* homozygote; for the *"aa"* homozygote; and for the heterozygote.

Thirdly, the *consistency* between the two previous viewpoints needs establishing. It is apparent at once [from the corresponding equations above] that the heterozygote frequency is the same in both viewpoints. However, such a straightforward result is not immediately apparent for the homozygotes. Begin by considering the **AA** homozygote's final equation in the *auto/allo* paragraph above:- . Expand the brackets, and follow by re-gathering [within the resultant] the two new terms with the common-factor *f* in them. The result is: . Next, for the parenthesized " *p ^{2}_{0}* ", a

In previous sections, dispersive random fertilization (*genetic drift*) has been considered comprehensively, and self-fertilization and hybridizing have been examined to varying degrees. The diagram to the left depicts the first two of these, along with another "spatially based" pattern: *islands*. This is a pattern of *random fertilization* featuring *dispersed gamodemes*, with the addition of "overlaps" in which *non-dispersive* random fertilization occurs. With the *islands* pattern, individual gamodeme sizes (**2N**) are observable, and overlaps (**m**) are minimal. This is one of Sewall Wright's array of possibilities.^{[37]} In addition to "spatially" based patterns of fertilization, there are others based on either "phenotypic" or "relationship" criteria. The *phenotypic* bases include *assortative* fertilization (between similar phenotypes) and *disassortative* fertilization (between opposite phenotypes). The *relationship* patterns include *sib crossing*, *cousin crossing* and *backcrossing*—and are considered in a separate section. *Self fertilization* may be considered both from a spatial or relationship point of view.

The breeding population consists of **s** small **dispersed random fertilization** gamodemes of sample size ( **k** = 1 ... *s* ) with " *overlaps* " of proportion in which **non-dispersive random fertilization ** occurs. The * dispersive proportion * is thus . The bulk population consists of *weighted averages* of sample sizes, allele and genotype frequencies and progeny means, as was done for genetic drift in an earlier section. However, each *gamete sample size* is reduced to allow for the *overlaps*, thus finding a effective for .

For brevity, the argument is followed further with the subscripts omitted. Recall that is in general. [Here, and following, the *2N* refers to the *previously defined* sample size, not to any "islands adjusted" version.]

After simplification,^{[37]}

Notice that when

This Δf is also substituted into the previous *inbreeding coefficient* to obtain ^{[37]}

where

The effective *overlap proportion* can be obtained also,^{[37]} as

The graphs to the right show the *inbreeding* for a gamodeme size of *2N = 50* for *ordinary dispersed random fertilization * **(RF)** *(m=0)*, and for *four overlap levels ( m = 0.0625, 0.125, 0.25, 0.5 )* of **islands** *random fertilization*. There has indeed been reduction in the inbreeding resulting from the *non-dispersed random fertilization* in the overlaps. It is particularly notable as **m → 0.50**. Sewall Wright suggested that this value should be the limit for the use of this approach.^{[37]}

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 that deviates from this view. This formed an integral part of Fisher's method,^{[8]} in addition to his use of frequencies and effects to generate his genetical statistics.^{[14]} A discursive derivation of the *allele substitution* alternative follows.^{[14]}^{: 113 }

Suppose that the usual random fertilization of gametes in a "base" gamodeme—consisting of * p* gametes (

Consider the upper part firstly. Because *base* **A** is present with a frequency of * p*, the

After some algebraic simplification, this becomes

- the

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

The common factor inside the brackets is the

It can also be derived in a more direct way, but the result is the same.

In subsequent sections, these substitution effects help define the gene-model genotypes as consisting of a partition predicted by these new effects (**substitution expectations**), and a residual (

Ultimately, the variance arising from the *substitution expectations* becomes the so-called *Additive genetic variance (σ ^{2}_{A})*

The gene-model effects (**a**, **d** and **-a**) are important soon in the derivation of the *deviations from substitution*, 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 **β**, the *average allele substitution effect*.

Consider firstly the re-centralization. The re-centralized effect for **AA** is **a• = a - G** which, after simplification, becomes **a• = 2 q(a-pd)**. The similar effect for

Secondly, consider the re-arrangement of these re-centralized effects as functions of **β**. Recalling from the "Allele Substitution" section that β = [a +(q-p)d], rearrangement gives **a = [β -(q-p)d]**. After substituting this for **a** in **a•** and simplifying, the final version becomes **a•• = 2q(β-qd)**. Similarly, **d•** becomes **d•• = β(q-p) + 2pqd**; and **(-a)•** becomes **(-a)•• = -2p(β+pd)**.^{[14]}^{: 118 }

The zygote genotypes are the target of all this preparation. The homozygous genotype **AA** is a union of two *substitution effects of A*, one from each sex. Its *substitution expectation* is therefore **β _{AA} = 2β_{A} = 2qβ** (see previous sections). Similarly, the

*Substitution deviations* are the differences between these *expectations* and the *gene effects* after their two-stage redefinition in the previous section. Therefore, **d _{AA} = a•• - β_{AA} = -2q^{2}d** after simplification. Similarly,

The "substitution expectations" ultimately give rise to the **σ ^{2}_{A}** (the so-called "Additive" genetic variance); and the "substitution deviations" give rise to the

There are two major approaches to defining and partitioning *genotypic variance*. One is based on the *gene-model effects*,^{[40]} while the other is based on the *genotype substitution effects*^{[14]} They are algebraically inter-convertible with each other.^{[36]} In this section, the basic * random fertilization* derivation is considered, with the effects of inbreeding and dispersion set aside. This is dealt with later to arrive at a more general solution. Until this

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 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 case is the mean squared. The result is the SS, which, again because of the use of frequencies, is also immediately the *variance*.^{[9]}

The , and the . The

After partial simplification,

The last line is in Mather's terminology.

Here, **σ ^{2}_{a}** is the

These components are plotted across all values of **p** in the accompanying figure. Notice that *cov _{ad}* is

Most of these components are affected by the change of central focus from *homozygote mid-point* (**mp**) to *population mean* (**G**), the latter being the basis of the *Correction Factor*. The *cov _{ad}* and

Further gathering of terms [in Mather format] leads to , where . It is useful later in Diallel analysis, which is an experimental design for estimating these genetical statistics.^{[44]}

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:

Notice particularly that **σ ^{2}_{A}** is not

Also note that **σ ^{2}_{D}** <

From the Figure, these results can be visualized as accumulating **σ ^{2}_{a}**,

The overall result (in Fisher's format) is

The Fisherian components have just been derived, but their derivation via the

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 already defined as deviations from the *random fertilization* population mean (**G**). For each genotype in turn therefore, the product of the frequency and the square of the relevant effect is obtained, and these are accumulated to obtain directly a **SS** and **σ ^{2}**.

**σ ^{2}_{A}** =

**σ ^{2}_{D}** =

Upon accumulating these results, **σ ^{2}_{G}** =

Once again, however, refer to the earlier discussions about the true meanings and identities of these components. Fisher himself did not use these modern terms for his components. The *substitution expectations* variance he named the ** "genetic" ** variance; and the *substitution deviations* variance he regarded simply as the unnamed **residual** between the "genotypic" variance (his name for it) and his "genetic" variance.^{[8]}^{[29]}^{: 33 }^{[47]}^{[48]} [The terminology and derivation used in this article are completely in accord with Fisher's own.] Mather's term for the *expectations* variance—**"genic"**^{[40]}—is obviously derived from Fisher's term, and avoids using "genetic" (which has become too generalized in usage to be of value in the present context). The origin is obscure of the modern misleading terms "additive" and "dominance" variances.

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 **σ ^{2}_{A}**, and (b) the real meanings and relative sizes of

In the section on genetic drift, and in other sections that discuss inbreeding, a major outcome from allele frequency sampling has been the *dispersion* of progeny means. This collection of means has its own average, and also has a variance: the *amongst-line variance*. (This is a variance of the attribute itself, not of *allele frequencies*.) As dispersion develops further over succeeding generations, this amongst-line variance would be expected to increase. Conversely, as homozygosity rises, the within-lines variance would be expected to decrease. The question arises therefore as to whether the total variance is changing—and, if so, in what direction. To date, these issues have been presented in terms of the *genic (σ ^{2}_{A} )* and

The crucial *overview equation* comes from Sewall Wright,^{[13]} ^{: 99, 130 }^{[37]} and is the outline of the **inbred genotypic variance** based on a *weighted average of its extremes*, the weights being quadratic with respect to the *inbreeding coefficient*
. This equation is:

where is the inbreeding coefficient, is the genotypic variance at
*f=0*, is the genotypic variance at *f=1*,
is the population mean at *f=0*, and is the population mean at *f=1*.

The component [in the equation above] outlines the reduction of variance within progeny lines. The component addresses the increase in variance amongst progeny lines. Lastly, the component is seen (in the next line) to address the *quasi-dominance* variance.^{[13]} ^{: 99 & 130 } These components can be expanded further thereby revealing additional insight. Thus:-

Firstly, *σ ^{2}_{G(0)}* [in the equation above] has been expanded to show its two sub-components [see section on "Genotypic variance"]. Next, the

Summarising: the **within-line** components are and ; and the **amongst-line** components are and .^{[36]}

Rearranging gives the following:

The version in the last line is discussed further in a subsequent section.

Similarly,

Graphs to the left show these three genic variances, together with the three quasi-dominance variances, across all values of **f**, for **p = 0.5** (at which the quasi-dominance variance is at a maximum). Graphs to the right show the **Genotypic** variance partitions (being the sums of the respective *genic* and *quasi-dominance* partitions) changing over ten generations with an example *f = 0.10*.

Answering, firstly, the questions posed at the beginning about the **total variances** [the **Σ** in the graphs] : the *genic variance* rises linearly with the *inbreeding coefficient*, maximizing at twice its starting level. The *quasi-dominance variance* declines at the rate of *(1 − f ^{2} )* until it finishes at zero. At low levels of

Secondly, notice the other trends. It is probably intuitive that the **within line** variances decline to zero with continued inbreeding, and this is seen to be the case (both at the same linear rate *(1-f)* ). The **amongst line** variances both increase with inbreeding up to *f = 0.5*, the *genic variance* at the rate of *2f*, and the *quasi-dominance variance* at the rate of *(f − f ^{2})*. At

Recall that when *f=1*, heterozygosity is zero, within-line variance is zero, and all genotypic variance is thus *amongst-line* variance and deplete of dominance variance. In other words, **σ ^{2}_{G(1)}** is the variance amongst fully inbred line means. Recall further [from "The mean after self-fertilization" section] that such means (G

The final result is: **σ ^{2}_{G(1)} = σ^{2}_{(a-2aq)} = 4a^{2} pq = 2(2pq a^{2}) = 2 σ^{2}_{a} **.

It follows immediately that ** f σ^{2}_{G(1)} = f 2 σ^{2}_{a} **. [This last

Previous sections found that the **within line** *genic variance* is based upon the *substitution-derived* genic variance **( σ ^{2}_{A} )**—but the

The *refined* version is: ** β _{ f } = { a^{2} + [(1−f ) / (1 + f )] 2(q − p ) ad + [(1-f ) / (1 + f )] (q − p )^{2} d^{2} } ^{(1/2)}**

Consequently, **σ ^{2}_{A(f)} = (1 + f ) 2pq β_{f} ^{2} ** does now agree with

The *total genic variance* is of intrinsic interest in its own right. But, prior to the refinements by Gordon,^{[36]} it had had another important use as well. There had been no extant estimators for the "dispersed" quasi-dominance. This had been estimated as the difference between Sewall Wright's *inbred genotypic variance* ^{[37]} and the total "dispersed" genic variance [see the previous sub-section]. An anomaly appeared, however, because the *total quasi-dominance variance* appeared to increase early in inbreeding despite the decline in heterozygosity.^{[14]} ^{: 128 } ^{: 266 }

The refinements in the previous sub-section corrected this anomaly.^{[36]} At the same time, a direct solution for the *total quasi-dominance variance* was obtained, thus avoiding the need for the "subtraction" method of previous times. Furthermore, direct solutions for the *amongst-line* and *within-line* partitions of the *quasi-dominance variance* were obtained also, for the first time. [These have been presented in the section "Dispersion and the genotypic 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 ("Sites" and "Years"); or into partitions such as "litter" or "family", and "culture" or "history". 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

i.e., σ^{2}_{P} = σ^{2}_{G} + σ^{2}_{E} + σ^{2}_{GE} + σ^{2}

* or*
σ^{2}_{P} = σ^{2}_{A} + σ^{2}_{D} + σ^{2}_{I} + σ^{2}_{E} + σ^{2}_{GE} + σ^{2}

after partitioning the genotypic variance (G) into component variances "genic" (A), "quasi-dominance" (D), and "epistatic" (I).^{[51]}

The environmental variance will appear in other sections, such as "Heritability" and "Correlated attributes".

The heritability of a trait is the proportion of the total (phenotypic) variance (σ^{2} _{P}) that is attributable to genetic variance, whether it be the full genotypic variance, or some component of it. It quantifies the degree to which phenotypic variability is due to genetics: but the precise meaning depends upon which genetical variance partition is used in the numerator of the proportion.^{[52]} Research estimates of heritability have standard errors, just as have all estimated statistics.^{[53]}

Where the numerator variance is the whole Genotypic variance (** σ ^{2}_{G} **), the heritability is known as the "broadsense" heritability (

[See section on the Genotypic variance.]

If only genic variance (**σ ^{2}_{A}**) is used in the numerator, the heritability may be called "narrow sense" (h

Fisher proposed that this narrow-sense heritability might be appropriate in considering the results of natural selection, focusing as it does on change-ability,

Recalling that the **allelic** variance (*σ ^{2}_{a}*) and the

Narrow-sense heritability has been used also for predicting generally the results of **artificial selection**. In the latter case, however, the broadsense heritability may be more appropriate, as the whole attribute is being altered: not just adaptive capacity. Generally, advance from selection is more rapid the higher the heritability. [See section on "Selection".] 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 (r^{2}) 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.^{[54]}

r^{2} = (s^{2}_{G} + s^{2}_{PE})/s^{2}_{P}

where s^{2}_{PE} = 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.

From the heredity perspective, relations are individuals that inherited genes from one or more common ancestors. Therefore, their "relationship" can be *quantified* on the basis of the probability that they each have inherited a copy of an allele from the common ancestor. In earlier sections, the *Inbreeding coefficient* has been defined as, "the probability that two *same* alleles ( **A** and **A**, or **a** and **a** ) have a common origin"—or, more formally, "The probability that two homologous alleles are autozygous." Previously, the emphasis was on an individual's likelihood of having two such alleles, and the coefficient was framed accordingly. It is obvious, however, that this probability of autozygosity for an individual must also be the probability that each of its *two parents* had this autozygous allele. In this re-focused form, the probability is called the *co-ancestry coefficient* for the two individuals *i* and *j* ( ** f _{ij}** ). In this form, it can be used to quantify the relationship between two individuals, and may also be known as the

*Pedigrees* are diagrams of familial connections between individuals and their ancestors, and possibly between other members of the group that share genetical inheritance with them. They are relationship maps. A pedigree can be analyzed, therefore, to reveal coefficients of inbreeding and co-ancestry. Such pedigrees actually are informal depictions of *path diagrams* as used in *path analysis*, which was invented by Sewall Wright when he formulated his studies on inbreeding.^{[55]}^{: 266–298 } Using the adjacent diagram, the probability that individuals "B" and "C" have received autozygous alleles from ancestor "A" is *1/2* (one out of the two diploid alleles). This is the "de novo" inbreeding (**Δf _{Ped}**) at this step. However, the other allele may have had "carry-over" autozygosity from previous generations, so the probability of this occurring is (

Following the "B" path, the probability that any autozygous allele is "passed on" to each successive parent is again **(1/2)** at each step (including the last one to the "target" **X** ). The overall probability of transfer down the "B path" is therefore ** (1/2) ^{3} **. The power that (1/2) is raised to can be viewed as "the number of intermediates in the path between

In this section, powers of (**1/2**) were used to represent the "probability of autozygosity". Later, this same method will be used to represent the proportions of ancestral gene-pools which are inherited down a pedigree [the section on "Relatedness between relatives"].

In the following sections on sib-crossing and similar topics, a number of "averaging rules" are useful. These derive from path analysis.^{[55]} The rules show that any co-ancestry coefficient can be obtained as the average of *cross-over co-ancestries* between appropriate grand-parental and parental combinations. Thus, referring to the adjacent diagram, *Cross-multiplier 1* is that **f _{PQ}** = average of (

In much of the following, the grand-parental generation is referred to as **(t-2)** , the parent generation as **(t-1)** , and the "target" generation as **t**.

The diagram to the right shows that *full sib crossing* is a direct application of *cross-Multiplier 1*, with the slight modification that *parents A and B* repeat (in lieu of *C and D*) to indicate that individuals *P1* and *P2* have both of *their* parents in common—that is they are *full siblings*. Individual **Y ** is the result of the crossing of two full siblings. Therefore, **f _{Y} = f_{P1,P2} = (1/4) [ f_{AA} + 2 f_{AB} + f_{BB} ] **. Recall that

Now, examine **f _{AB} **. Recall that this also is

Derivation of the *half sib crossing* takes a slightly different path to that for Full sibs. In the adjacent diagram, the two half-sibs at generation (t-1) have only one parent in common—parent "A" at generation (t-2). The *cross-multiplier 1* is used again, giving **f _{Y} = f_{(P1,P2)} = (1/4) [ f_{AA} + f_{AC} + f_{BA} + f_{BC} ] **. There is just one

As before, this also quantifies the *relatedness* of the two half-sibs at generation (t-1) in its alternative form of **f _{(P1, P2)} **.

A pedigree diagram for selfing is on the right. It is so straightforward it does not require any cross-multiplication rules. It employs just the basic juxtaposition of the *inbreeding coefficient* and its alternative the *co-ancestry coefficient*; followed by recognizing that, in this case, the latter is also a *coefficient of parentage*. Thus, ** f _{Y} = f_{(P1, P1)} = f_{t} = (1/2) [ 1 + f_{(t-1)} ] **.

These are derived with methods similar to those for siblings.^{[13]}^{: 132–143 }^{[14]}^{: 82–92 } As before, the *co-ancestry* viewpoint of the *inbreeding coefficient* provides a measure of "relatedness" between the parents **P1** and **P2** in these cousin expressions.

The pedigree for *First Cousins (FC)* is given to the right. The prime equation is **f _{Y} = f_{t} = f_{P1,P2} = (1/4) [ f_{1D} + f_{12} + f_{CD} + f_{C2} ]**. After substitution with corresponding inbreeding coefficients, gathering of terms and simplifying, this becomes

The *Second Cousins (SC)* pedigree is on the left. Parents in the pedigree not related to the *common Ancestor* are indicated by numerals instead of letters. Here, the prime equation is ** f _{Y} = f_{t} = f_{P1,P2} = (1/4) [ f_{3F} + f_{34} + f_{EF} + f_{E4} ]**. After working through the appropriate algebra, this becomes

To visualize the *pattern in full cousin* equations, start the series with the *full sib* equation re-written in iteration form: ** f _{t} = (1/4)[2 f_{(t-1)} + f_{(t-2)} + 1 ]**. Notice that this is the "essential plan" of the last term in each of the cousin iterative forms: with the small difference that the generation indices increment by "1" at each cousin "level". Now, define the

For *first half-cousins (FHC)*, the pedigree is to the left. Notice there is just one common ancestor (individual **A**). Also, as for *second cousins*, parents not related to the common ancestor are indicated by numerals. Here, the prime equation is ** f _{Y} = f_{t} = f_{P1,P2} = (1/4) [ f_{3D} + f_{34} + f_{CD} + f_{C4} ]**. After working through the appropriate algebra, this becomes

There is a tendency to regard cousin crossing with a human-oriented point of view, possibly because of a wide interest in Genealogy. The use of pedigrees to derive the inbreeding perhaps reinforces this "Family History" view. However, such kinds of inter-crossing occur also in natural populations—especially those that are sedentary, or have a "breeding area" that they re-visit from season to season. The progeny-group of a harem with a dominant male, for example, may contain elements of sib-crossing, cousin crossing, and backcrossing, as well as genetic drift, especially of the "island" type. In addition to that, the occasional "outcross" adds an element of hybridization to the mix. It is *not* panmixia.

Following the hybridizing between **A** and **R**, the **F1** (individual **B**) is crossed back (* BC1*) to an original parent (

This routine is commonly used in Animal and Plant Breeding programmes. Often after making the hybrid (especially if individuals are short-lived), the recurrent parent needs separate "line breeding" for its maintenance as a future recurrent parent in the backcrossing. This maintenance may be through selfing, or through full-sib or half-sib crossing, or through restricted randomly fertilized populations, depending on the species' reproductive possibilities. Of course, this incremental rise in **f _{R}** carries-over into the

In the section on "Pedigree analysis", was used to represent probabilities of autozygous allele descent over **n** generations down branches of the pedigree. This formula arose because of the rules imposed by sexual reproduction: **(i)** two parents contributing virtually equal shares of autosomal genes, and **(ii)** successive dilution for each generation between the zygote and the "focus" level of parentage. These same rules apply also to any other viewpoint of descent in a two-sex reproductive system. One such is the proportion of any ancestral gene-pool (also known as 'germplasm') which is contained within any zygote's genotype.

Therefore, the proportion of an **ancestral genepool** in a genotype is:

where

For example, each parent defines a genepool contributing to its offspring; while each great-grandparent contributes to its great-grand-offspring.

The zygote's total genepool (**Γ**) is, of course, the sum of the sexual contributions to its descent.

Individuals descended from a common ancestral genepool obviously are related. This is not to say they are identical in their genes (alleles), because, at each level of ancestor, segregation and assortment will have occurred in producing gametes. But they will have originated from the same pool of alleles available for these meioses and subsequent fertilizations. [This idea was encountered firstly in the sections on pedigree analysis and relationships.] The genepool contributions [see section above] of their **nearest common ancestral genepool**(an *ancestral node*) can therefore be used to define their relationship. This leads to an intuitive definition of relationship which conforms well with familiar notions of "relatedness" found in family-history; and permits comparisons of the "degree of relatedness" for complex patterns of relations arising from such genealogy.

The only modifications necessary (for each individual in turn) are in Γ and are due to the shift to "shared **common** ancestry" rather than "individual **total** ancestry". For this, define **Ρ** (in lieu of **Γ**) ; ** m = number of ancestors-in-common** at the node (i.e. m = 1 or 2 only) ; and an "individual index" **k**. Thus:

where, as before, *n = number of sexual generations* between the individual and the ancestral node.

An example is provided by two first full-cousins. Their nearest common ancestral node is their grandparents which gave rise to their two sibling parents, and they have both of these grandparents in common. [See earlier pedigree.] For this case, *m=2* and *n=2*, so for each of them

In this simple case, each cousin has numerically the same Ρ .

A second example might be between two full cousins, but one (*k=1*) has three generations back to the ancestral node (n=3), and the other (*k=2*) only two (n=2) [i.e. a second and first cousin relationship]. For both, m=2 (they are full cousins).

and

Notice each cousin has a different Ρ _{k}.

In any pairwise relationship estimation, there is one **Ρ _{k}** for each individual: it remains to average them in order to combine them into a single "Relationship coefficient". Because each

For the first example (two full first-cousins), their GRC = 0.5; for the second case (a full first and second cousin), their GRC = 0.3536.

All of these relationships (GRC) are applications of path-analysis.^{[55]}^{: 214–298 } A summary of some levels of relationship (GRC) follow.

GRC | Relationship examples |
---|---|

1.00 | full Sibs |

0.7071 | Parent ↔ Offspring ; Uncle/Aunt ↔ Nephew/Niece |

0.5 | full First Cousins ; half Sibs ; grand Parent ↔ grand Offspring |

0.3536 | full Cousins First ↔ Second ; full First Cousins {1 remove} |

0.25 | full Second Cousins; half First Cousins; full First Cousins {2 removes} |

0.1768 | full First Cousin {3 removes}; full Second Cousins {1 remove} |

0.125 | full Third Cousins; half Second Cousins; full 1st Cousins {4 removes} |

0.0884 | full First Cousins {5 removes}; half Second Cousins {1 remove} |

0.0625 | full Fourth Cousins ; half Third Cousins |

These, in like manner to the Genotypic variances, can be derived through either the gene-model ("Mather") approach or the allele-substitution ("Fisher") approach. Here, each method is demonstrated for alternate cases.

These can be viewed either as the covariance between any offspring and *any one* of its parents (**PO**), or as the covariance between any offspring and the * "mid-parent" * value of both its parents (**MPO**).

This can be derived as the *sum of cross-products* between parent gene-effects and *one-half* of the progeny expectations using the allele-substitution approach. The *one-half* of the progeny expectation accounts for the fact that *only one of the two parents* is being considered. The appropriate parental gene-effects are therefore the second-stage redefined gene effects used to define the genotypic variances earlier, that is: **a″ = 2q(a − qd)** and **d″ = (q-p)a + 2pqd** and also **(-a)″ = -2p(a + pd)** [see section "Gene effects redefined"]. Similarly, the appropriate progeny effects, * for allele-substitution expectations* are one-half of the earlier

Because all of these effects are defined already as deviates from the genotypic mean, the cross-product sum using {**genotype-frequency * parental gene-effect * half-breeding-value**} immediately provides the *allele-substitution-expectation covariance* between any one parent and its offspring. After careful gathering of terms and simplification, this becomes **cov(PO) _{A} = pqa^{2} = 1/2 s^{2}_{A} **.

Unfortunately, the *allele-substitution-deviations* are usually overlooked, but they have not "ceased to exist" nonetheless! Recall that these deviations are: **d _{AA} = -2q^{2} d**, and

It follows therefore that: **cov(PO) = cov(PO) _{A} + cov(PO)_{D} = 1/2 s^{2}_{A} + 1/2 s^{2}_{D} **, when dominance is

Because there are many combinations of parental genotypes, there are many different mid-parents and offspring means to consider, together with the varying frequencies of obtaining each parental pairing. The gene-model approach is the most expedient in this case. Therefore, an *unadjusted sum of cross-products (USCP)*—using all products {** parent-pair-frequency * mid-parent-gene-effect * offspring-genotype-mean **}—is adjusted by subtracting the **{overall genotypic mean} ^{2} ** as

**cov(MPO) = pq [a + (q-p)d ] ^{2} = pq a^{2} = 1/2 s^{2}_{A} **, with no dominance having been overlooked in this case, as it had been used-up in defining the

The most obvious application is an experiment that contains all parents and their offspring, with or without reciprocal crosses, preferably replicated without bias, enabling estimation of all appropriate means, variances and covariances, together with their standard errors. These estimated statistics can then be used to estimate the genetic variances. Twice *the difference between the estimates of the two forms of (corrected) parent-offspring covariance* provides an estimate of **s ^{2}_{D}**; and twice the

A second application involves using *regression analysis*, which estimates from statistics the ordinate (Y-estimate), derivative (regression coefficient) and constant (Y-intercept) of calculus.^{[9]}^{[49]}^{[58]}^{[59]} The *regression coefficient* estimates the *rate of change* of the function predicting **Y** from **X**, based on minimizing the residuals between the fitted curve and the observed data (MINRES). No alternative method of estimating such a function satisfies this basic requirement of MINRES. In general, the regression coefficient is estimated as *the ratio of the covariance(XY) to the variance of the determinator (X)*. In practice, the sample size is usually the same for both X and Y, so this can be written as **SCP(XY) / SS(X)**, where all terms have been defined previously.^{[9]}^{[58]}^{[59]} In the present context, the parents are viewed as the "determinative variable" (X), and the offspring as the "determined variable" (Y), and the regression coefficient as the "functional relationship" (ß_{PO}) between the two. Taking **cov(MPO) = 1/2 s ^{2}_{A} ** as

Analysis of *epistasis* has previously been attempted via an *interaction variance* approach of the type * s ^{2}_{AA} *, and

Covariance between half-sibs (**HS**) is defined easily using allele-substitution methods; but, once again, the dominance contribution has historically been omitted. However, as with the mid-parent/offspring covariance, the covariance between full-sibs (**FS**) requires a "parent-combination" approach, thereby necessitating the use of the gene-model corrected-cross-product method; and the dominance contribution has not historically been overlooked. The superiority of the gene-model derivations is as evident here as it was for the Genotypic variances.

The sum of the cross-products **{ common-parent frequency * half-breeding-value of one half-sib * half-breeding-value of any other half-sib in that same common-parent-group }** immediately provides one of the required covariances, because the effects used [*breeding values*—representing the allele-substitution expectations] are already defined as deviates from the genotypic mean [see section on "Allele substitution – Expectations and deviations"]. After simplification. this becomes: ** cov(HS) _{A} = 1/2 pq a^{2} = 1/4 s^{2}_{A} **.

**cov(HS) = cov(HS) _{A} + cov(HS)_{D} = 1/4 s^{2}_{A} + 1/4 s^{2}_{D} **.

As explained in the introduction, a method similar to that used for mid-parent/progeny covariance is used. Therefore, an *unadjusted sum of cross-products* (USCP) using all products—{** parent-pair-frequency * the square of the offspring-genotype-mean **}—is adjusted by subtracting the **{overall genotypic mean} ^{2} ** as

**cov(FS) = pq a ^{2} + p^{2} q^{2} d^{2} = 1/2 s^{2}_{A} + 1/4 s^{2}_{D} **, with no dominance having been overlooked.

The most useful application here for genetical statistics is the *correlation between half-sibs*. Recall that the correlation coefficient (*r*) is the ratio of the covariance to the variance [see section on "Associated attributes" for example]. Therefore, ** r _{HS} = cov(HS) / s^{2}_{all HS together} ** =

Of course, the correlations between siblings are of intrinsic interest in their own right, quite apart from any utility they may have for estimating heritabilities or genotypic variances.

It may be worth noting that **[ cov(FS) − cov(HS)] = 1/4 s ^{2}_{A} **. Experiments consisting of FS and HS families could utilize this by using intra-class correlation to equate experiment variance components to these covariances [see section on "Coefficient of relationship as an intra-class correlation" for the rationale behind this].

The earlier comments regarding epistasis apply again here [see section on "Applications (Parent-offspring"].

Selection operates on the attribute (phenotype), such that individuals that equal or exceed a selection threshold **(z _{P})** become effective parents for the next generation. The

Thus .^{[14]} ^{: 1710–181 }
and .^{[14]} ^{: 1710–181 }

The *narrow-sense heritability (h ^{2})* is usually used, thereby linking to the

To apply these concepts *before* selection actually takes place, and so predict the outcome of alternatives (such as choice of *selection threshold*, for example), these phenotypic statistics are re-considered against the properties of the Normal Distribution, especially those concerning truncation of the *superior tail* of the Distribution. In such consideration, the *standardized* selection differential (i)″ and the *standardized* selection threshold (z)″ are used instead of the previous "phenotypic" versions. The **phenotypic standard deviate (σ _{P(0)})** is also needed. This is described in a subsequent section.

Therefore, **ΔG = (i σ _{P}) h^{2}**, where

The text above noted that successive **ΔG** declines because the "input" [the **phenotypic variance ( σ ^{2}_{P} )**] is reduced by the previous selection.

Thus : **σ ^{2}_{P(1)} = σ^{2}_{P(0)} [1 − i ( i-z) 1/2 h^{2}]**, where

Here, **i** and **z** have already been defined, **1/2** is the *meiosis determination ( b^{2}) for f=0*, and the remaining symbol is the heritability. These are discussed further in following sections. Also notice that, more generally,

**σ ^{2}_{P(1)} = σ^{2}_{P(0)} [1 − i ( i-z) b^{2} h^{2}]**.

The *Phenotypic variance* truncated by the *selected group* (** σ ^{2}_{P(S)}** ) is simply

The following rearrangement is useful for considering selection on multiple attributes (characters). It starts by expanding the heritability into its variance components. **ΔG = i σ _{P} ( σ^{2}_{A} / σ^{2}_{P} ) **. The

**ΔG = i σ _{A} ( σ_{A} / σ_{P} )** =

Corresponding re-arrangements could be made using the alternative heritabilities, giving **ΔG = i σ _{G} H** or

This traditional view of adaptation in quantitative genetics provides a model for how the selected phenotype changes over time, as a function of the selection differential and heritability. However it does not provide insight into (nor does it depend upon) any of the genetic details - in particular, the number of loci involved, their allele frequencies and effect sizes, and the frequency changes driven by selection. This, in contrast, is the focus of work on polygenic adaptation^{[62]} within the field of population genetics. Recent studies have shown that traits such as height have evolved in humans during the past few thousands of years as a result of small allele frequency shifts at thousands of variants that affect height.^{[63]}^{[64]}^{[65]}

The entire *base population* is outlined by the normal curve^{[59]}^{: 78–89 } to the right. Along the **Z axis** is every value of the attribute from least to greatest, and the height from this axis to the curve itself is the frequency of the value at the axis below. The equation for finding these frequencies for the "normal" curve (the curve of "common experience") is given in the ellipse. Notice it includes the mean (**μ**) and the variance (**σ ^{2}**). Moving infinitesimally along the z-axis, the frequencies of neighbouring values can be "stacked" beside the previous, thereby accumulating an area that represents the

Finally, a cross-link with the differing terminology in the previous sub-section may be useful: **μ** (here) = "P_{0}" (there), **μ _{S}** = "P

The **meiosis determination (b ^{2})** is the

The path diagram to the left represents this analysis of sexual reproduction. Of its interesting elements, the important one in the selection context is *meiosis*. That's where segregation and assortment occur—the processes that partially ameliorate the truncation of the phenotypic variance that arises from selection. The path coefficients **b** are the meiosis paths. Those labeled **a** are the fertilization paths. The correlation between gametes from the same parent (**g**) is the *meiotic correlation*. That between parents within the same generation is **r _{A}**. That between gametes from different parents (

The meiosis determination (**b ^{2}**) is

These links with inbreeding reveal interesting facets about sexual reproduction that are not immediately apparent. The graphs to the right plot the *meiosis* and *syngamy (fertilization)* coefficients of determination against the inbreeding coefficient. There it is revealed that as inbreeding increases, meiosis becomes more important (the coefficient increases), while syngamy becomes less important. The overall role of reproduction [the product of the previous two coefficients—**r ^{2}**] remains the same.

The previous sections treated *dispersion* as an "assistant" to *selection*, and it became apparent that the two work well together. In quantitative genetics, selection is usually examined in this "biometrical" fashion, but the changes in the means (as monitored by ΔG) reflect the changes in allele and genotype frequencies beneath this surface. Referral to the section on "Genetic drift" brings to mind that it also effects changes in allele and genotype frequencies, and associated means; and that this is the companion aspect to the dispersion considered here ("the other side of the same coin"). However, these two forces of frequency change are seldom in concert, and may often act contrary to each other. One (selection) is "directional" being driven by selection pressure acting on the phenotype: the other (genetic drift) is driven by "chance" at fertilization (binomial probabilities of gamete samples). If the two tend towards the same allele frequency, their "coincidence" is the probability of obtaining that frequencies sample in the genetic drift: the likelihood of their being "in conflict", however, is the *sum of probabilities of all the alternative frequency samples*. In extreme cases, a single syngamy sampling can undo what selection has achieved, and the probabilities of it happening are available. It is important to keep this in mind. However, genetic drift resulting in sample frequencies similar to those of the selection target does not lead to so drastic an outcome—instead slowing progress towards selection goals.