The mean after random fertilization

The contribution of AA is ${\textstyle p^{2}(+)a}$, that of Aa is ${\textstyle 2pqd}$, and that of aa is ${\textstyle q^{2}(-)a}$. Gathering together the two a terms and accumulating over all, the result is: ${\textstyle a(p^{2}-q^{2})+2pqd}$. Simplification is achieved by noting that ${\textstyle (p^{2}-q^{2})=(p-q)(p+q)}$, and by recalling that ${\textstyle (p+q)=1}$, thereby reducing the right-hand term to ${\textstyle (p-q)}$.

The succinct result is therefore ${\textstyle G=a(p-q)+2pqd}$.^{[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: ${\textstyle P=G+mp}$.

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 mean after long-term self-fertilization

The contribution of AA is ${\textstyle p(+a)}$, while that of aa is ${\textstyle q(-a)}$. [See above for the frequencies.] Gathering these two a terms together leads to an immediately very simple final result:

${\textstyle G_{(f=1)}=a(p-q)}$. As before, ${\textstyle P=G+mp}$.

Often, "G_(f=1)" is abbreviated to "G₁".

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).

The mean – 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:^{[13] : 77–78}

$${\displaystyle {\begin{aligned}G_{f}&=a(q-p)+[2pqd-f(2pqd)]\\&=a(p-q)+(1-f)2pqd\\&=G_{0}-f\ 2pqd\end{aligned}}}$$

Here, G₀ is G, which was given earlier. (Often, when dealing with inbreeding, "G₀" is preferred to "G".)

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 = 0.25, then, using the third version (above) of G_f:

G_0.25 = 1.15 − 0.25 (0.48) 0.12 = 1.136   cm (rounded), with P_0.25 = 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 G_f reveals that this would lead to trivial change in the Population mean. Where dominance was notable, however, there would be considerable change.

The sample gamodemes – genetic drift

Following completion of these five binomial sampling events, the resultant actual gamodemes each contained different allele frequencies—(p_k and q_k). [These are given at white label "5" in the diagram.] This outcome is actually the genetic drift itself. Notice that two samples (k = 1 and 5) happen to have the same frequencies as the base (potential) gamodeme. Another (k = 3) happens to have the p and q "reversed". Sample (k = 2) happens to be an "extreme" case, with p_k = 0.9 and q_k = 0.1 ; while the remaining sample (k = 4) is "middle of the range" in its allele frequencies. All of these results have arisen only by "chance", through binomial sampling. Having occurred, however, they set in place all the downstream properties of the progenies.

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 q_g) and the sample size (2N_k). They are tedious to obtain,^{[13]: 382–395 [30]: 55} but are of considerable interest. [See white label "6" in the diagram.] The two samples (k = 1, 5), with the allele frequencies the same as in the potential gamodeme, had higher "chances" of occurring than the other samples. Their binomial probabilities did differ, however, because of their different sample sizes (2N_k). The "reversal" sample (k = 3) had a very low Probability of occurring, confirming perhaps what might be expected. The "extreme" allele frequency gamodeme (k = 2) was not "rare", however; and the "middle of the range" sample (k=4) was rare. These same Probabilities apply also to the progeny of these fertilizations.

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: ${\textstyle p_{\centerdot }=\sum _{k}^{s}\omega _{k}\ p_{k}}$ and ${\textstyle q_{\centerdot }=\sum _{k}^{s}\omega _{k}\ q_{k}}$. (Notice that k is replaced by • for the overall result—a common practice.)^[9] The results for the example are p_• = 0.631 and q_• = 0.369 [black label "5" in the diagram]. These values are quite different to the starting ones (p_g and q_g) [white label "1"]. The sample allele frequencies also have variance as well as an average. This has been obtained using the sum of squares (SS) method ^[31] [See to the right of black label "5" in the diagram]. [Further discussion on this variance occurs in the section below on Extensive genetic drift.]

The progeny lines – dispersion

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 [(p²_k + q²_k) = (1 − 2p_kq_k)] , or by examining the level of heterozygosis (2p_kq_k), as they are complementary.^[32] Notice that samples k= 1, 3, 5 all had the same level of heterozygosis, despite one being the "mirror image" of the others with respect to allele frequencies. The "extreme" allele-frequency case (k= 2) had the most homozygosis (least heterozygosis) of any sample. The "middle of the range" case (k= 4) had the least homozygosity (most heterozygosity): they were each equal at 0.50, in fact.

The overall summary can continue by obtaining the weighted average of the respective genotype frequencies for the progeny bulk. Thus, for AA, it is ${\textstyle p_{\centerdot }^{2}=\sum _{k}^{s}\omega _{k}\ p_{k}^{2}}$, for Aa , it is ${\textstyle 2p_{\centerdot }q_{\centerdot }=\sum _{k}^{s}\omega _{k}\ 2p_{k}q_{k}}$ and for aa, it is ${\textstyle q_{\centerdot }^{2}=\sum _{k}^{s}\omega _{k}\ q_{k}^{2}}$. 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 ${\textstyle G_{k}=a(p_{k}-q_{k})+2p_{k}q_{k}d}$ [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 ${\textstyle P_{k}=G_{k}+mp}$ 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 ${\textstyle G_{\centerdot }=\sum _{k}^{s}\omega _{k}\ G_{k}}$ and ${\textstyle P_{\centerdot }=\sum _{k}^{s}\omega _{k}\ P_{k}}$. The example result for P_• is 36.94 (black label "10" in the diagram). This later is used to quantify inbreeding depression overall, from the gamete sampling. [See the next section.] However, recall that some "non-depressed" progeny means have been identified already (k = 1, 2, 5). This is an enigma of inbreeding—while there may be "depression" overall, there are usually superior lines among the gamodeme samplings.

The equivalent post-dispersion panmictic – inbreeding

Included in the overall summary were the average allele frequencies in the mixture of progeny lines (p_• and q_•). These can now be used to construct a hypothetical panmictic equivalent.^{[13]: 382–395 [14]: 49–63 [29]: 35} This can be regarded as a "reference" to assess the changes wrought by the gamete sampling. The example appends such a panmictic to the right of the Diagram. The frequency of AA is therefore (p_•)² = 0.3979. This is less than that found in the dispersed bulk (0.4513 at black label "7"). Similarly, for aa, (q_•)² = 0.1303—again less than the equivalent in the progenies bulk (0.1898). Clearly, genetic drift has increased the overall level of homozygosis by the amount (0.6411 − 0.5342) = 0.1069. In a complementary approach, the heterozygosity could be used instead. The panmictic equivalent for Aa is 2 p_• q_• = 0.4658, which is higher than that in the sampled bulk (0.3588) [black label "8"]. The sampling has caused the heterozygosity to decrease by 0.1070, which differs trivially from the earlier estimate because of rounding errors.

The inbreeding coefficient (f) was introduced in the early section on Self Fertilization. Here, a formal definition of it is considered: f is the probability that two "same" alleles (that is A and A, or a and a), which fertilize together are of common ancestral origin—or (more formally) f is the probability that two homologous alleles are autozygous.^[14][27] Consider any random gamete in the potential gamodeme that has its syngamy partner restricted by binomial sampling. The probability that that second gamete is homologous autozygous to the first is 1/(2N), the reciprocal of the gamodeme size. For the five example progenies, these quantities are 0.1, 0.0833, 0.1, 0.0833 and 0.125 respectively, and their weighted average is 0.0961. This is the inbreeding coefficient of the example progenies bulk, provided it is unbiased with respect to the full binomial distribution. An example based upon s = 5 is likely to be biased, however, when compared to an appropriate entire binomial distribution based upon the sample number (s) approaching infinity (s → ∞). Another derived definition of f for the full Distribution is that f also equals the rise in homozygosity, which equals the fall in heterozygosity.^[33] For the example, these frequency changes are 0.1069 and 0.1070, respectively. This result is different to the above, indicating that bias with respect to the full underlying distribution is present in the example. For the example itself, these latter values are the better ones to use, namely f_• = 0.10695.

The population mean of the equivalent panmictic is found as [a (p_•-q_•) + 2 p_•q_• d] + mp. Using the example gene effects (white label "9" in the diagram), this mean is ${\textstyle P_{\centerdot }=}$ 37.87. The equivalent mean in the dispersed bulk is 36.94 (black label "10"), which is depressed by the amount 0.93. This is the inbreeding depression from this Genetic Drift. However, as noted previously, three progenies were not depressed (k = 1, 2, 5), and had means even greater than that of the panmictic equivalent. These are the lines a plant breeder looks for in a line selection programme.^[34]

Extensive binomial sampling – is panmixia restored?

If the number of binomial samples is large (s → ∞ ), then p_• → p_g and q_• → q_g. It might be queried whether panmixia would effectively re-appear under these circumstances. However, the sampling of allele frequencies has still occurred, with the result that σ²_{p, q} ≠ 0.^[35] In fact, as s → ∞, the ${\textstyle \sigma _{p,\ q}^{2}\to {\tfrac {p_{g}q_{g}}{2N}}}$, which is the variance of the whole binomial distribution.^{[13]: 382–395 [14]: 49–63} Furthermore, the "Wahlund equations" show that the progeny-bulk homozygote frequencies can be obtained as the sums of their respective average values (p²_• or q²_•) plus σ²_{p, q}.^{[13]: 382–395} Likewise, the bulk heterozygote frequency is (2 p_• q_•) minus twice the σ²_{p, q}. The variance arising from the binomial sampling is conspicuously present. Thus, even when s → ∞, the progeny-bulk genotype frequencies still reveal increased homozygosis, and decreased heterozygosis, there is still dispersion of progeny means, and still inbreeding and inbreeding depression. That is, panmixia is not re-attained once lost because of genetic drift (binomial sampling). However, a new potential panmixia can be initiated via an allogamous F2 following hybridization.^[36]

Continued genetic drift – increased dispersion and inbreeding

Previous discussion on genetic drift examined just one cycle (generation) of the process. When the sampling continues over successive generations, conspicuous changes occur in σ²_{p, q} and f. Furthermore, another "index" is needed to keep track of "time": t = 1 .... y where y = the number of "years" (generations) considered. The methodology often is to add the current binomial increment (Δ = "de novo") to what has occurred previously.^[13] The entire Binomial Distribution is examined here. [There is no further benefit to be had from an abbreviated example.]

Dispersion via σ²_p,q

Earlier this variance (σ ²_p,q^[35]) was seen to be:-

$${\displaystyle {\begin{aligned}\sigma _{p,q}^{2}&=p_{g}q_{g}\ /\ 2N\\&=p_{g}q_{g}\left({\frac {1}{2N}}\right)\\&=p_{g}q_{g}\ f\\&=p_{g}q_{g}\ \Delta f\ \scriptstyle {\text{when used in recursive equations}}\end{aligned}}}$$

With the extension over time, this is also the result of the first cycle, and so is ${\textstyle \sigma _{1}^{2}}$ (for brevity). At cycle 2, this variance is generated yet again—this time becoming the de novo variance (${\textstyle \Delta \sigma ^{2}}$)—and accumulates to what was present already—the "carry-over" variance. The second cycle variance (${\textstyle \sigma _{2}^{2}}$) is the weighted sum of these two components, the weights being ${\textstyle 1}$ for the de novo and ${\textstyle \left(1-{\tfrac {1}{2N}}\right)}$ = ${\textstyle \left(1-\Delta f\right)}$ for the"carry-over".

Thus,

${\displaystyle \sigma _{2}^{2}=\left(1\right)\ \Delta \sigma ^{2}+\left(1-\Delta f\right)\sigma _{1}^{2}}$

(1)

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

${\displaystyle \sigma _{t}^{2}=p_{g}q_{g}\left[1-\left(1-\Delta f\right)^{t}\right]}$

(2)

Because it was this variation in allele frequencies that caused the "spreading apart" of the progenies' means (dispersion), the change in σ²_t over the generations indicates the change in the level of the dispersion.

Dispersion via f

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 de novo f ( Δ f ) [recall this is 1/(2N) ] and carry-over f. Therefore, ${\textstyle f_{2}=\left(1\right)\Delta f+\left(1-\Delta f\right)f_{1}}$ , which is similar to Equation (1) in the previous sub-section.

Inbreeding resulting from genetic drift in random fertilization.

In general, after rearrangement,^[13]

$${\displaystyle {\begin{aligned}f_{t}&=\Delta f+\left(1-\Delta f\right)f_{t-1}\\&=\Delta f\left(1-f_{t-1}\right)+f_{t-1}\end{aligned}}}$$

The graphs to the left show levels of inbreeding over twenty generations arising from genetic drift for various actual gamodeme sizes (2N).

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

(A) After some simplification,^[13] ${\textstyle \left(f_{t}-f_{t-1}\right)=\Delta f\left(1-f_{t-1}\right)=\delta f_{t}}$. The left-hand side is the difference between the current and previous levels of inbreeding: the change in inbreeding (δf_t). Notice, that this change in inbreeding (δf_t) is equal to the de novo inbreeding (Δf) only for the first cycle—when f_t-1 is zero.

(B) An item of note is the (1-f_t-1), which is an "index of non-inbreeding". It is known as the panmictic index.^[13][14] ${\textstyle P_{t-1}=\left(1-f_{t-1}\right)}$.

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

$${\displaystyle {\begin{aligned}\Delta f&={\frac {\delta f_{t}}{P_{t-1}}}\\&=1-{\frac {P_{t}}{P_{t-1}}}\end{aligned}}}$$

.

(D) A key link emerges between σ ²_p,q and f. Firstly...^[13]

$${\displaystyle {\begin{aligned}f_{t}&=1-\left(1-1\Delta f\right)^{t}\left(1-f_{0}\right)\end{aligned}}}$$

Secondly, presuming that f₀ = 0, the right-hand side of this equation reduces to the section within the brackets of Equation (2) at the end of the last sub-section. That is, if initially there is no inbreeding, ${\textstyle \sigma _{t}^{2}=p_{g}q_{g}f_{t}}$ ! Furthermore, if this then is rearranged, ${\textstyle f_{t}={\tfrac {\sigma _{t}^{2}}{p_{g}q_{g}}}}$. That is, when initial inbreeding is zero, the two principal viewpoints of binomial gamete sampling (genetic drift) are directly inter-convertible.

Selfing within random fertilization

Random fertilization compared to cross-fertilization

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 ${\displaystyle \left(\bigotimes \right)}$, with the remainder (N-1)/N being cross fertilization ${\displaystyle \left({\mathsf {X}}\right)}$. Following path analysis and simplification, the new view random fertilization inbreeding was found to be: ${\textstyle f_{t}=\Delta f\left(1+f_{t-1}\right)+{\tfrac {N-1}{N}}f_{t-1}}$.^[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) ${\textstyle f_{t}=\Delta f\left[1+f_{t-1}\left(2N-1\right)\right]}$; and (B) ${\textstyle f_{t}=\Delta f\left(1-f_{t-1}\right)+f_{t-1}}$.

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

$${\displaystyle f_{{\mathsf {X}}_{t}}=f_{t-1}+\Delta f\left(1+f_{t-2}-2f_{t-1}\right)}$$

.

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".

Homozygosity and heterozygosity

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 inbreeding coefficient (f) by definition. The proportion arising allozygously is therefore (1-f). For the A-bearing gametes, which are present with a general frequency of p, the overall frequency of those that are autozygous is therefore (f p). Similarly, for a-bearing gametes, the autozygous frequency is (f q).^[38] These two viewpoints regarding genotype frequencies must be connected to establish consistency.

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:

$${\displaystyle \left(1-f\right)\left[p_{0}+q_{0}\right]^{2}=\left(1-f\right)\left[p_{0}^{2}+q_{0}^{2}\right]+\left(1-f\right)\left[2p_{0}q_{0}\right]}$$

Consider next the autozygous component. As these alleles are autozygous, they are effectively selfings, and produce either AA or aa genotypes, but no heterozygotes. They therefore produce ${\textstyle fp_{0}}$ "AA" homozygotes plus ${\textstyle fq_{0}}$ "aa" homozygotes. Adding these two components together results in:

${\textstyle \left[\left(1-f\right)p_{0}^{2}+fp_{0}\right]}$ for the AA homozygote; ${\textstyle \left[\left(1-f\right)q_{0}^{2}+fq_{0}\right]}$ for the aa homozygote; and ${\textstyle \left(1-f\right)2p_{0}q_{0}}$ for the Aa heterozygote.^{[13]: 65 [14]} This is the same equation as that presented earlier in the section on "Self fertilization – an alternative". The reason for the decline in heterozygosity is made clear here. Heterozygotes can arise only from the allozygous component, and its frequency in the sample bulk is just (1-f): hence this must also be the factor controlling the frequency of the heterozygotes.

Secondly, the sampling viewpoint is re-examined. Previously, it was noted that the decline in heterozygotes was ${\textstyle f\left(2p_{0}q_{0}\right)}$. This decline is distributed equally towards each homozygote; and is added to their basic random fertilization expectations. Therefore, the genotype frequencies are: ${\textstyle \left(p_{0}^{2}+fp_{0}q_{0}\right)}$ for the "AA" homozygote; ${\textstyle \left(q_{0}^{2}+fp_{0}q_{0}\right)}$ for the "aa" homozygote; and ${\textstyle 2p_{0}q_{0}-f\left(2p_{0}q_{0}\right)}$ 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:- ${\textstyle \left[\left(1-f\right)p_{0}^{2}+fp_{0}\right]}$. 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: ${\textstyle p_{0}^{2}-f\left(p_{0}^{2}-p_{0}\right)}$. Next, for the parenthesized " p²₀ ", a (1-q) is substituted for a p, the result becoming ${\textstyle p_{0}^{2}-f\left[p_{0}\left(1-q_{0}\right)-p_{0}\right]}$. Following that substitution, it is a straightforward matter of multiplying-out, simplifying and watching signs. The end result is ${\textstyle p_{0}^{2}+fp_{0}q_{0}}$, which is exactly the result for AA in the sampling paragraph. The two viewpoints are therefore consistent for the AA homozygote. In a like manner, the consistency of the aa viewpoints can also be shown. The two viewpoints are consistent for all classes of genotypes.

"Islands" random fertilization

The breeding population consists of s small dispersed random fertilization gamodemes of sample size ${\textstyle 2N_{k}}$ ( k = 1 ... s ) with " overlaps " of proportion ${\textstyle m_{k}}$ in which non-dispersive random fertilization occurs. The dispersive proportion is thus ${\textstyle \left(1-m_{k}\right)}$. 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 ${\textstyle 2N_{k}}$ effective for ${\textstyle \left(1-m_{k}\right)}$.

"Islands" random fertilization

For brevity, the argument is followed further with the subscripts omitted. Recall that ${\textstyle {\tfrac {1}{2N}}}$ is ${\textstyle \Delta f}$ in general. [Here, and following, the 2N refers to the previously defined sample size, not to any "islands adjusted" version.]

After simplification,^[37]

$${\displaystyle ^{\mathsf {islands}}\Delta f={\frac {\left(1-m\right)^{2}}{2N-m^{2}\left(2N-1\right)}}}$$

Notice that when m = 0 this reduces to the previous Δ f. The reciprocal of this furnishes an estimate of the " ${\textstyle 2N_{k}}$ effective for ${\textstyle \left(1-m_{k}\right)}$ ", mentioned above.

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

$${\displaystyle {^{\mathsf {islands}}f_{t}}=\ {^{\mathsf {islands}}\Delta f_{t}}+\left(1-\ {^{\mathsf {islands}}\Delta f_{t}}\right)\ {^{\mathsf {islands}}f_{t-1}}}$$

where t is the index over generations, as before.

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

$${\displaystyle m_{t}=1-\left[{\frac {2N\ {^{\mathsf {islands}}\Delta f_{t}}}{\left(2N-1\right)\ {^{\mathsf {islands}}\Delta f_{t}+1}}}\right]^{\tfrac {1}{2}}}$$

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]

Gene effects redefined

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• = 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).^{[14]: 116–119}

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

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 from each sex. Its substitution expectation is therefore β_AA = 2β_A = 2qβ (see previous sections). Similarly, the substitution expectation of Aa is β_Aa = β_A + β_a = (q-p)β ; and for aa, β_aa = 2β_a = -2pβ. These substitution expectations of the genotypes are also called breeding values.^{[14]: 114–116}

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 = -2q²d after simplification. Similarly, d_Aa = d•• - β_Aa = 2pqd after simplification. Finally, d_aa = (-a)•• - β_aa = -2p²d after simplification.^{[14]: 116–119} Notice that all of these substitution deviations ultimately are functions of the gene-effect d—which accounts for the use of ["d" plus subscript] as their symbols. However, it is a serious non sequitur in logic to regard them as accounting for the dominance (heterozygosis) in the entire gene model : they are simply functions of "d" and not an audit of the "d" in the system. They are as derived: deviations from the substitution expectations!

The "substitution expectations" ultimately give rise to the σ²_A (the so-called "Additive" genetic variance); and the "substitution deviations" give rise to the σ²_D (the so-called "Dominance" genetic variance). Be aware, however, that the average substitution effect (β) also contains "d" [see previous sections], indicating that dominance is also embedded within the "Additive" variance [see following sections on the Genotypic Variance for their derivations]. Remember also [see previous paragraph] that the "substitution deviations" do not account for the dominance in the system (being nothing more than deviations from the substitution expectations), but which happen to consist algebraically of functions of "d". More appropriate names for these respective variances might be σ²_B (the "Breeding expectations" variance) and σ²_δ (the "Breeding deviations" variance). However, as noted previously, "Genic" (σ ²_A) and "Quasi-Dominance" (σ ²_D), respectively, will be preferred herein.

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 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 ${\textstyle {\mathsf {USS}}=p^{2}a^{2}+2pqd^{2}+q^{2}(-a)^{2}}$, and the ${\textstyle {\mathsf {CF}}={\mathsf {G}}^{2}}$. The ${\textstyle {\mathsf {SS}}={\mathsf {USS}}-{\mathsf {CF}}}$

After partial simplification,

$${\displaystyle {\begin{aligned}\sigma _{G}^{2}&=2pqa^{2}+(q-p)4pqad+2pqd^{2}+(2pq)^{2}d^{2}\\&=\sigma _{a}^{2}+({\text{weighted-covariance}})_{ad}+\sigma _{d}^{2}+\sigma _{D}^{2}\\&={\tfrac {1}{2}}{\mathsf {D}}+{\tfrac {1}{2}}{\mathsf {F}}^{\prime }+{\tfrac {1}{2}}{\mathsf {H}}_{1}+{\tfrac {1}{4}}{\mathsf {H}}_{2}\end{aligned}}}$$

The last line is in Mather's terminology.^{[40]: 212 [41][42]}

Here, σ²_a is the homozygote or allelic variance, and σ²_d is the heterozygote or dominance variance. The substitution deviations variance (σ²_D) is also present. The (weighted_covariance)_ad^[43] is abbreviated hereafter to " cov_ad ".

These components are plotted across all values of p in the accompanying figure. Notice that cov_ad is negative for p > 0.5.

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 substitution deviation variances are simply artifacts of this shift. The allelic and dominance variances are genuine genetical partitions of the original gene-model, and are the only eu-genetical components. Even then, the algebraic formula for the allelic variance is effected by the presence of G: it is only the dominance variance (i.e. σ²_d ) which is unaffected by the shift from mp to G.^[36] These insights are commonly not appreciated.

Further gathering of terms [in Mather format] leads to ${\textstyle {\tfrac {1}{2}}{\mathsf {D}}+{\tfrac {1}{2}}{\mathsf {F}}^{\prime }+{\tfrac {1}{2}}{\mathsf {H}}_{3}+{\tfrac {1}{4}}{\mathsf {H}}_{2}}$, where ${\textstyle {\tfrac {1}{2}}{\mathsf {H}}_{3}=(q-p)^{2}{\tfrac {1}{2}}{\mathsf {H}}_{1}=(q-p)^{2}2pqd^{2}}$. 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: ${\displaystyle \sigma _{A}^{2}=\sigma _{a}^{2}+{\mathsf {cov}}_{ad}+\sigma _{d}^{2}}$

Notice particularly that σ²_A is not σ²_a. The first is the substitution expectations variance, while the second is the allelic variance.^[45] Notice also that σ²_D (the substitution-deviations variance) is not σ²_d (the dominance variance), and recall that it is an artifact arising from the use of G for the Correction Factor. [See the "blue paragraph" above.] It now will be referred to as the "quasi-dominance" variance.

Also note that σ²_D < σ²_d ("2pq" being always a fraction); and note that (1) σ²_D = 2pq σ²_d, and that (2) σ²_d = σ²_D / (2pq). That is: it is confirmed that σ²_D does not quantify the dominance variance in the model. It is σ²_d which does that. However, the dominance variance (σ²_d) can be estimated readily from the σ²_D if 2pq is available.

From the Figure, these results can be visualized as accumulating σ²_a, σ²_d and cov_ad to obtain σ²_A, while leaving the σ²_D still separated. It is clear also in the Figure that σ²_D < σ²_d, as expected from the equations.

The overall result (in Fisher's format) is

$${\displaystyle {\begin{aligned}\sigma _{G}^{2}&=2pq\left[a+(q-p)d\right]^{2}+\left(2pq\right)^{2}d^{2}\\&=\sigma _{A}^{2}+\sigma _{D}^{2}\\&=\left[\left(\sigma _{a}^{2}+{\mathsf {cov}}_{ad}+\sigma _{d}^{2}\right)\right]+\left[2pq\ \sigma _{d}^{2}\right]\end{aligned}}}$$

The Fisherian components have just been derived, but their derivation via the substitution effects themselves is given also, in the next section.

Allele-substitution approach – Fisher

Components of genotypic variance using the allele-substitution effects.

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 σ².^[46] Details follow.

σ²_A = p² β_AA² + 2pq β_Aa² + q² β_aa², which simplifies to σ²_A = 2pqβ²—the Genic variance.

σ²_D = p² d_AA² + 2pq d_Aa² + q d_aa², which simplifies to σ²_D = (2pq)² d²—the quasi-Dominance variance.

Upon accumulating these results, σ²_G = σ²_A + σ²_D . These components are visualized in the graphs to the right. The average allele substitution effect is graphed also, but the symbol is "α" (as is common in the citations) rather than "β" (as is used herein).

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 σ²_A, and (b) the real meanings and relative sizes of σ²_d and σ²_D (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 cov_ad would be missing.

Derivation of σ²_G(1)

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, σ²_G(1) is the variance amongst fully inbred line means. Recall further [from "The mean after self-fertilization" section] that such means (G₁'s, in fact) are G = a(p-q). Substituting (1-q) for the p, gives G₁ = a (1 − 2q) = a − 2aq.^[14]: 265 Therefore, the σ²_G(1) is the σ²_(a-2aq) actually. Now, in general, the variance of a difference (x-y) is [ σ²_x + σ²_y − 2 cov_xy ].^{[49]: 100 [50] : 232} Therefore, σ²_G(1) = [ σ²_a + σ²_2aq − 2 cov_{(a, 2aq)} ] . But a (an allele effect) and q (an allele frequency) are independent—so this covariance is zero. Furthermore, a is a constant from one line to the next, so σ²_a is also zero. Further, 2a is another constant (k), so the σ²_2aq is of the type σ²_{k X}. In general, the variance σ²_{k X} is equal to k² σ²_X .^[50]: 232 Putting all this together reveals that σ²_(a-2aq) = (2a)² σ²_q . Recall [from the section on "Continued genetic drift"] that σ²_q = pq f . With f=1 here within this present derivation, this becomes pq 1 (that is pq), and this is substituted into the previous.

The final result is: σ²_G(1) = σ²_(a-2aq) = 4a² pq = 2(2pq a²) = 2 σ²_a .

It follows immediately that f σ²_G(1) = f 2 σ²_a . [This last f comes from the initial Sewall Wright equation : it is not the f just set to "1" in the derivation concluded two lines above.]

Total dispersed genic variance – σ²_A(f) and β_f

Previous sections found that the within line genic variance is based upon the substitution-derived genic variance ( σ²_A )—but the amongst line genic variance is based upon the gene model allelic variance ( σ²_a ). These two cannot simply be added to get total genic variance. One approach in avoiding this problem was to re-visit the derivation of the average allele substitution effect, and to construct a version, ( β _f ), that incorporates the effects of the dispersion. Crow and Kimura achieved this^{[13] : 130–131} using the re-centered allele effects (a•, d•, (-a)• ) discussed previously ["Gene effects re-defined"]. However, this was found subsequently to under-estimate slightly the total Genic variance, and a new variance-based derivation led to a refined version.^[36]

The refined version is: β _f = { a² + [(1−f ) / (1 + f )] 2(q − p ) ad + [(1-f ) / (1 + f )] (q − p )² d² } ^(1/2)

Consequently, σ²_A(f) = (1 + f ) 2pq β_f ² does now agree with [ (1-f) σ²_A(0) + 2f σ²_a(0) ] exactly.

Total and partitioned dispersed quasi-dominance variances

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".]

Cross-multiplication rules

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 ( f_AC , f_AD , f_BC , f_BD ) = (1/4) [f_AC + f_AD + f_BC + f_BD ] = f_Y . In a similar fashion, cross-multiplier 2 states that f_PC = (1/2) [ f_AC + f_BC ]—while cross-multiplier 3 states that f_PD = (1/2) [ f_AD + f_BD ] . Returning to the first multiplier, it can now be seen also to be f_PQ = (1/2) [ f_PC + f_PD ], which, after substituting multipliers 2 and 3, resumes its original form.

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.

Full-sib crossing (FS)

Inbreeding in sibling relationships

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 f_AA and f_BB were defined earlier (in Pedigree analysis) as coefficients of parentage, equal to (1/2)[1+f_A ] and (1/2)[1+f_B ] respectively, in the present context. Recognize that, in this guise, the grandparents A and B represent generation (t-2) . Thus, assuming that in any one generation all levels of inbreeding are the same, these two coefficients of parentage each represent (1/2) [1 + f_(t-2) ] .

Inbreeding from full-sib and half-sib crossing, and from selfing.

Now, examine f_AB . Recall that this also is f_P1 or f_P2 , and so represents their generation - f_(t-1) . Putting it all together, f_t = (1/4) [ 2 f_AA + 2 f_AB ] = (1/4) [ 1 + f_(t-2) + 2 f_(t-1) ] . That is the inbreeding coefficient for Full-Sib crossing .^{[13]: 132–143 [14]: 82–92} The graph to the left shows the rate of this inbreeding over twenty repetitive generations. The "repetition" means that the progeny after cycle t become the crossing parents that generate cycle (t+1 ), and so on successively. The graphs also show the inbreeding for random fertilization 2N=20 for comparison. Recall that this inbreeding coefficient for progeny Y is also the co-ancestry coefficient for its parents, and so is a measure of the relatedness of the two Fill siblings.

Half-sib crossing (HS)

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 coefficient of parentage this time, but three co-ancestry coefficients at the (t-2) level (one of them—f_BC—being a "dummy" and not representing an actual individual in the (t-1) generation). As before, the coefficient of parentage is (1/2)[1+f_A ] , and the three co-ancestries each represent f_(t-1) . Recalling that f_A represents f_(t-2) , the final gathering and simplifying of terms gives f_Y = f_t = (1/8) [ 1 + f_(t-2) + 6 f_(t-1) ] .^{[13]: 132–143 [14]: 82–92} The graphs at left include this half-sib (HS) inbreeding over twenty successive generations.

Self fertilization inbreeding

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

Self fertilization (SF)

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) ] .^{[13]: 132–143 [14]: 82–92} This is the fastest rate of inbreeding of all types, as can be seen in the graphs above. The selfing curve is, in fact, a graph of the coefficient of parentage.

Cousins crossings

Pedigree analysis first cousins

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 + f₁₂ + f_CD + f_C2 ]. After substitution with corresponding inbreeding coefficients, gathering of terms and simplifying, this becomes f_t = (1/4) [ 3 f_(t-1) + (1/4) [2 f_(t-2) + f_(t-3) + 1 ]] , which is a version for iteration—useful for observing the general pattern, and for computer programming. A "final" version is f_t = (1/16) [ 12 f_(t-1) + 2 f_(t-2) + f_(t-3) + 1 ] .

Pedigree analysis second cousins

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 + f₃₄ + f_EF + f_E4 ]. After working through the appropriate algebra, this becomes f_t = (1/4) [ 3 f_(t-1) + (1/4) [3 f_(t-2) + (1/4) [2 f_(t-3) + f_(t-4) + 1 ]]] , which is the iteration version. A "final" version is f_t = (1/64) [ 48 f_(t-1) + 12 f_(t-2) + 2 f_(t-3) + f_(t-4) + 1 ] .

Inbreeding from several levels of cousin crossing.

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 cousin level as k = 1 (for First cousins), = 2 (for Second cousins), = 3 (for Third cousins), etc., etc.; and = 0 (for Full Sibs, which are "zero level cousins"). The last term can be written now as: (1/4) [ 2 f_(t-(1+k)) + f_(t-(2+k)) + 1] . Stacked in front of this last term are one or more iteration increments in the form (1/4) [ 3 f_(t-j) + ... , where j is the iteration index and takes values from 1 ... k over the successive iterations as needed. Putting all this together provides a general formula for all levels of full cousin possible, including Full Sibs. For kth level full cousins, f{k}_t = Ιter_{j = 1}^k { (1/4) [ 3 f_(t-j) + }_j + (1/4) [ 2 f_(t-(1+k)) + f_(t-(2+k)) + 1] . At the commencement of iteration, all f_(t-x) are set at "0", and each has its value substituted as it is calculated through the generations. The graphs to the right show the successive inbreeding for several levels of Full Cousins.

Pedigree analysis half cousins

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 + f₃₄ + f_CD + f_C4 ]. After working through the appropriate algebra, this becomes f_t = (1/4) [ 3 f_(t-1) + (1/8) [6 f_(t-2) + f_(t-3) + 1 ]] , which is the iteration version. A "final" version is f_t = (1/32) [ 24 f_(t-1) + 6 f_(t-2) + f_(t-3) + 1 ] . The iteration algorithm is similar to that for full cousins, except that the last term is (1/8) [ 6 f_(t-(1+k)) + f_(t-(2+k)) + 1 ] . Notice that this last term is basically similar to the half sib equation, in parallel to the pattern for full cousins and full sibs. In other words, half sibs are "zero level" half cousins.

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.

Backcrossing (BC)

Pedigree analysis: backcrossing

Backcrossing: basic inbreeding levels

Following the hybridizing between A and R, the F1 (individual B) is crossed back (BC1) to an original parent (R) to produce the BC1 generation (individual C). [It is usual to use the same label for the act of making the back-cross and for the generation produced by it. The act of back-crossing is here in italics. ] Parent R is the recurrent parent. Two successive backcrosses are depicted, with individual D being the BC2 generation. These generations have been given t indices also, as indicated. As before, f_D = f_t = f_CR = (1/2) [ f_RB + f_RR ] , using cross-multiplier 2 previously given. The f_RB just defined is the one that involves generation (t-1) with (t-2). However, there is another such f_RB contained wholly within generation (t-2) as well, and it is this one that is used now: as the co-ancestry of the parents of individual C in generation (t-1). As such, it is also the inbreeding coefficient of C, and hence is f_(t-1). The remaining f_RR is the coefficient of parentage of the recurrent parent, and so is (1/2) [1 + f_R ] . Putting all this together : f_t = (1/2) [ (1/2) [ 1 + f_R ] + f_(t-1) ] = (1/4) [ 1 + f_R + 2 f_(t-1) ] . The graphs at right illustrate Backcross inbreeding over twenty backcrosses for three different levels of (fixed) inbreeding in the Recurrent 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 f_t of the backcrossing. The result is a more gradual curve rising to the asymptotes than shown in the present graphs, because the f_R is not at a fixed level from the outset.

Relationship through ancestral genepools

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:

$${\displaystyle {\begin{aligned}\mathrm {P} _{k}&=\sum _{m=1}^{1,2}{\gamma _{n}}\\&=\sum _{m=1}^{1,2}{\left({\frac {1}{2}}\right)^{n}}\end{aligned}}}$$

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

$${\displaystyle {\begin{aligned}\mathrm {P} _{k}&=\sum _{m=1}^{2}{\gamma _{2}}\\&=\sum _{m=1}^{2}{\left({\frac {1}{2}}\right)^{2}}\\&={\frac {1}{2}}\end{aligned}}}$$

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).

$${\displaystyle {\begin{aligned}\mathrm {P} _{1}&=\sum _{m=1}^{2}{\gamma _{3}}\\&=\sum _{m=1}^{2}{\left({\frac {1}{2}}\right)^{3}}\\&={\frac {1}{4}}\end{aligned}}}$$

and

$${\displaystyle {\begin{aligned}\mathrm {P} _{2}&=\sum _{m=1}^{2}{\gamma _{2}}\\&=\sum _{m=1}^{2}{\left({\frac {1}{2}}\right)^{2}}\\&={\frac {1}{2}}\end{aligned}}}$$

Notice each cousin has a different Ρ _k.

GRC – genepool relationship coefficient

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 Ρ is a fraction of a total genepool, the appropriate average for them is the geometric mean ^{[56][57]: 34–55} This average is their Genepool Relationship Coefficient—the "GRC".

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.

One-parent and offspring (PO)

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 breeding values, the latter being: a_AA = 2qa, and a_Aa = (q-p)a and also a_aa = -2pa [see section on "Genotype substitution – Expectations and Deviations"].

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 = pqa² = ½ s²_A .^{[13] : 132–141 [14] : 134–147}

Unfortunately, the allele-substitution-deviations are usually overlooked, but they have not "ceased to exist" nonetheless! Recall that these deviations are: d_AA = -2q² d, and d_Aa = 2pq d and also d_aa = -2p² d [see section on "Genotype substitution – Expectations and Deviations"]. Consequently, the cross-product sum using {genotype-frequency * parental gene-effect * half-substitution-deviations} also immediately provides the allele-substitution-deviations covariance between any one parent and its offspring. Once more, after careful gathering of terms and simplification, this becomes cov(PO)_D = 2p²q²d² = ½ s²_D .

It follows therefore that: cov(PO) = cov(PO)_A + cov(PO)_D = ½ s²_A + ½ s²_D , when dominance is not overlooked !

Mid-parent and offspring (MPO)

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 correction factor (CF). After multiplying out all the various combinations, carefully gathering terms, simplifying, factoring and cancelling-out where applicable, this becomes:

cov(MPO) = pq [a + (q-p)d ]² = pq a² = ½ s²_A , with no dominance having been overlooked in this case, as it had been used-up in defining the a.^{[13] : 132–141 [14] : 134–147}

Applications (parent-offspring)

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 s²_D; and twice the cov(MPO) estimates s²_A. With appropriate experimental design and analysis,^[9][49][50] standard errors can be obtained for these genetical statistics as well. This is the basic core of an experiment known as Diallel analysis, the Mather, Jinks and Hayman version of which is discussed in another section.

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) = ½ s²_A as cov(XY), and s²_P / 2 (the variance of the mean of two parents—the mid-parent) as s²_X, it can be seen that ß_MPO = [½ s²_A] / [½ s²_P] = h² .^[60] Next, utilizing cov(PO) = [ ½ s²_A + ½ s²_D ] as cov(XY), and s²_P as s²_X, it is seen that 2 ß_PO = [ 2 (½ s²_A + ½ s²_D )] / s²_P = H² .

Analysis of epistasis has previously been attempted via an interaction variance approach of the type s²_AA , and s²_AD and also s²_DD. This has been integrated with these present covariances in an effort to provide estimators for the epistasis variances. However, the findings of epigenetics suggest that this may not be an appropriate way to define epistasis.

Half-sibs of the same common-parent (HS)

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 = ½ pq a² = ¼ s²_A .^{[13] : 132–141 [14] : 134–147} However, the substitution deviations also exist, defining the sum of the cross-products { common-parent frequency * half-substitution-deviation of one half-sib * half-substitution-deviation of any other half-sib in that same common-parent-group }, which ultimately leads to: cov(HS)_D = p² q² d² = ¼ s²_D . Adding the two components gives:

cov(HS) = cov(HS)_A + cov(HS)_D = ¼ s²_A + ¼ s²_D .

Full-sibs (FS)

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 correction factor (CF). In this case, multiplying out all combinations, carefully gathering terms, simplifying, factoring, and cancelling-out is very protracted. It eventually becomes:

cov(FS) = pq a² + p² q² d² = ½ s²_A + ¼ s²_D , with no dominance having been overlooked.^{[13] : 132–141 [14] : 134–147}

Applications (siblings)

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) / s²_{all HS together} = [¼ s²_A + ¼ s²_D ] / s²_P = ¼ H² .^[61] The correlation between full-sibs is of little utility, being r_FS = cov(FS) / s²_{all FS together} = [½ s²_A + ¼ s²_D ] / s²_P . The suggestion that it "approximates" (½ h²) is poor advice.

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)] = ¼ s²_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"].

Alternative ΔG

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 ( σ²_A / σ²_P ) . The σ_P and σ²_P partially cancel, leaving a solo σ_P. Next, the σ²_A inside the heritability can be expanded as (σ_A × σ_A), which leads to :

Selection differential and the normal distribution

ΔG = i σ_A ( σ_A / σ_P ) = i σ_A h .

Corresponding re-arrangements could be made using the alternative heritabilities, giving ΔG = i σ_G H or ΔG = i σ_a h_eu.

Standardized selection – the normal distribution

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 (σ²). Moving infinitesimally along the z-axis, the frequencies of neighbouring values can be "stacked" beside the previous, thereby accumulating an area that represents the probability of obtaining all values within the stack. [That's integration from calculus.] Selection focuses on such a probability area, being the shaded-in one from the selection threshold (z) to the end of the superior tail of the curve. This is the selection pressure. The selected group (the effective parents of the next generation) include all phenotype values from z to the "end" of the tail.^[66] The mean of the selected group is µ_s, and the difference between it and the base mean (µ) represents the selection differential (S). By taking partial integrations over curve-sections of interest, and some rearranging of the algebra, it can be shown that the "selection differential" is S = [ y (σ / Prob.)] , where y is the frequency of the value at the "selection threshold" z (the ordinate of z).^{[13]: 226–230} Rearranging this relationship gives S / σ = y / Prob., the left-hand side of which is, in fact, selection differential divided by standard deviation—that is the standardized selection differential (i). The right-side of the relationship provides an "estimator" for i—the ordinate of the selection threshold divided by the selection pressure. Tables of the Normal Distribution^{[49] : 547–548} can be used, but tabulations of i itself are available also.^{[67]: 123–124} The latter reference also gives values of i adjusted for small populations (400 and less),^{[67]: 111–122} where "quasi-infinity" cannot be assumed (but was presumed in the "Normal Distribution" outline above). The standardized selection differential (i) is known also as the intensity of selection.^{[14]: 174, 186}