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Main article: Quantitative Genetics

This is a sub-article of Quantitative Genetics. It deals with Relationship measures (based upon ancestral gene-pools or autozygosity coefficients), Pedigree analysis, In-breeding amongst relatives, Resemblances between relatives and an Overview of inbreeding. These issues apply universally - to all species: but they have also particular focus for humankind. The emphasis herein may be different to that expressed by social-workers, legal-fraternity or medical-practitioners, as it considers matters only from a general Quantitative Genetics perspective. Many of these issues are important foundations within plant- and animal- breeding methodology, and also are necessary for a proper understanding of Evolution and Nature in general. The broader view of "inbreeding" includes genetic drift and dispersion, self-fertilization, and the very structure of the basic core-material of Quantitative Genetics (eg. the population mean, genotypic variances, and selection). The lead-article deals with many of those general matters.

Relationship

From the heredity perspective, relatives are individuals (plants or animals) which inherited genes from one or more common ancestors. Therefore, their "relationship" can be quantified on the basis of (A) the extent to which they share a common ancestral germplasm (gene-pool), or (B) the probability that they each have inherited a copy of an allele from the common ancestor (an autozygous allele). Both utilize the expression $\left({\tfrac {1}{2}}\right)^{n}$. This formula arises because sexual reproduction imposes the pattern: (i) two parents contribute virtually equal shares of autosomal genes (hence the ${\tfrac {1}{2}}$), and (ii) this share is successively diluted for each generation (n) between the zygote and the “focus” ancestor level.

Ancestral germplasm

Ancestral contributions

Using the expression given above, the proportion of an ancestral genepool in a genotype is:- $\gamma _{n}=\left({\frac {1}{2}}\right)^{n}$ where n = number of sexual generations between the zygote and the focus ancestor.

For example, each parent defines a genepool contributing $\left({\tfrac {1}{2}}\right)^{1}$ to its offspring.A second example is that each great-grandparent contributes $\left({\tfrac {1}{2}}\right)^{3}$ to its great-grand-offspring.

The zygote’s total genepool (Γ) is, of course, the sum of the sexual contributions to its descent. ${\begin{aligned}\Gamma &=\sum _{n=1}^{2^{n}}{\gamma _{n}}\\&=\sum _{n=1}^{2^{n}}{\left({\frac {1}{2}}\right)^{n}}\end{aligned}}$

Relatedness through ancestors

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. 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 (ie. m = 1 or 2 only) ; and an “individual index” k. Thus: ${\begin{aligned}\mathrm {P} _{k}&=\sum _{m=1}^{1,2}{\gamma _{n}}\\&=\sum _{m=1}^{1,2}{\left({\frac {1}{2}}\right)^{n}}\end{aligned}}$ 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 respective sibling parents, and they have both of these grandparents in common. [See later pedigree.] For this case, m=2 and n=2, so for each of them ${\begin{aligned}\mathrm {P} _{k}&=\sum _{m=1}^{2}{\gamma _{2}}\\&=\sum _{m=1}^{2}{\left({\frac {1}{2}}\right)^{2}}\\&={\frac {1}{2}}\end{aligned}}$ 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) [ie. a second and first cousin relationship]. For both, m=2 (they are full cousins). ${\begin{aligned}\mathrm {P} _{1}&=\sum _{m=1}^{2}{\gamma _{3}}\\&=\sum _{m=1}^{2}{\left({\frac {1}{2}}\right)^{3}}\\&={\frac {1}{4}}\end{aligned}}$ and also ${\begin{aligned}\mathrm {P} _{2}&=\sum _{m=1}^{2}{\gamma _{2}}\\&=\sum _{m=1}^{2}{\left({\frac {1}{2}}\right)^{2}}\\&={\frac {1}{2}}\end{aligned}}$ 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 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. A summary of some levels of relationship (GRC) follow, using the formulae above.

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 Cousins {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

Autozygosity coefficients

Connection between the inbreeding and co-ancestry coefficients.

The "Inbreeding" coefficient (Φ) 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." 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 contained this autozygous allele. [See diagram at right.] In this re-focused form, the probability is called the co-ancestry coefficient for the two individuals i and j ( Φ _{i j} ). In this form, it can be used to quantify relationship between two individuals, and may also be known as the coefficient of kinship or the consanguinity coefficient.

Pedigree analysis

Introduction to pedigrees

Illustrative pedigree.

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 autozygosity 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.

Using the diagram to the left, the probability that individuals "B" and "C" have received autozygous alleles from ancestor "A" is ${\textstyle {\tfrac {1}{2}}}$ (one out of the two diploid alleles). This is the "de novo" autozygosity ${\textstyle \Delta \Phi _{Ped}}$ at this step. However, the other allele may have had "carry-over" autozygosity from previous generations, so the probability of this occurring is ${\textstyle \left(1-\Delta \Phi _{Ped}\right)}$ (the "non de novo" weight) multiplied by ${\textstyle \Phi _{A}}$ (the autozygosity of ancestor A ). Therefore, the total probability of autozygosity in B and C, following the bi-furcation of the pedigree, is the sum of these two components, namely

$${\begin{aligned}\Phi _{BC}&={\frac {1}{2}}+{\frac {1}{2}}\Phi _{A}\\&={\frac {1}{2}}\left(1+\Phi _{A}\right)\end{aligned}}$$

This can be viewed as the probability that two random gametes from ancestor A carry autozygous alleles, and in that context is called the coefficient of parentage (Φ_AA ). It appears often in the following paragraphs.

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 A and X ", n_B = 3 . Similarly, for the "C path", n_C = 2 , and the "transfer probability" is (1/2)² . The combined probability of autozygous transfer from A to X is therefore ${\textstyle \Phi _{AA}\ {\frac {1}{2}}^{n_{B}}\ {\frac {1}{2}}^{n_{C}}}$. Recalling that Φ_AA = (1/2) (1+Φ_A),

$$\Phi _{X}=\Phi _{PQ}=\left({\frac {1}{2}}\right)^{n_{B}+n_{C}+1}\left(1+\Phi _{A}\right)$$

In this example, assuming that Φ_A = 0, Φ_X = 0.0156 (rounded) = Φ_PQ , one measure of the "relatedness" between P and Q.

Cross-multiplication rules

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. 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 diagram to the right, Cross-multiplier 1 is that Φ_PQ = average of ( Φ_AC , Φ_AD , Φ_BC , Φ_BD ) = (1/4) [Φ_AC + Φ_AD + Φ_BC + Φ_BD ] = Φ_Y .

In a similar fashion, cross-multiplier 2 states that Φ_PC = (1/2) [ Φ_AC + Φ_BC ]

while cross-multiplier 3 states that Φ_PD = (1/2) [Φ_AD + Φ_BD ].

Returning to the first multiplier, it can now be seen also to be Φ_PQ = (1/2) [Φ_PC + Φ_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 .

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 diagram to the right, 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) ] .

The graphs at left include this half-sib (HS) inbreeding over twenty successive generations.

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)

Self fertilization inbreeding

A pedigree diagram for selfing is on the right. It is so straightforward it doesn't 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) ] .

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. 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.

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). 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.

Applicatiions (Backcrossing)

This routine is commonly used in Plant- and Animal- 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.

Resemblances between relatives

These, in like manner to the Genotypic variances, can be derived through either the gene-model ("Mather") approach or the allele-substitution ("Fisher") approach. See Quantitative genetics for discussion on that. Here, each method is demonstrated for alternate cases. The results have been given in terms of Fisher's Substitution components (σ²_A for Expectations - the Genic variance; and σ²_D for Deviations - the quasi-Dominance variance), as is commonly done.

Parent-offspring covariance

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).

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(β - qd) and d•• = (q-p)β + 2pqd and also (-a)•• = -2p(β + 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: β_AA = 2qβ, and β_Aa = (q-p)β and also β_aa = -2pβ [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 ${\textstyle cov\left(PO\right)_{A}=pq\beta ^{2}={\tfrac {1}{2}}\sigma _{A}^{2}}$

Unfortunately, the allele-substitution-deviations are usually overlooked in standard texts, but they have not "ceased to exist" nonetheless! Recall that [see section on "Genotype substitution - Expectations and Deviations] these deviations are: d_AA = -2q² d, and d_Aa = 2pq d and also d_aa = -2p² d. 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 ${\textstyle cov\left(PO\right)_{D}=2p^{2}q^{2}d^{2}={\tfrac {1}{2}}\sigma _{D}^{2}}$.

It follows therefore that:

$${\begin{aligned}cov\left(PO\right)&=cov\left(PO\right)_{A}+cov\left(PO\right)_{D}\\&={\tfrac {1}{2}}\sigma _{A}^{2}+{\tfrac {1}{2}}\sigma _{D}^{2}\end{aligned}}$$

when dominance is included and 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:

$${\begin{aligned}cov\left(MPO\right)&=pq\left[a+\left(q-p\right)d\right]^{2}\\&=pq\beta ^{2}\\&={\tfrac {1}{2}}\sigma _{A}^{2}\end{aligned}}$$

with no dominance having been overlooked in this case, as it had been used-up in defining the β (the average allele substitution effect).

Applications (parent-offspring)

(A) 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 genetical variances. Twice the difference between the estimates of the two forms of (corrected) parent-offspring covariance provides an estimate of σ²_D; and twice the cov(MPO) estimates σ²_A. With appropriate experimental design and analysis, 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.

(B) A second application involves using regression analysis. Regresion estimates from sample statistics the ordinate (Y-estimate), derivative (regression coefficient) and constant (Y-intercept) of calculus. 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. 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" (b_PO) between the two.

Taking cov(MPO) = ½ σ²_A as cov(XY), and σ²_P / 2 (the variance of the mid-parent) as σ²_X, it can be seen that b_MPO = [½ σ²_A] / [½ σ²_P] = h² (where h² is the narrow-sense heritability). Next, utilizing cov(PO) = [ ½ σ²_A + ½ σ²_D ] as cov(XY), and σ²_P as σ²_X, it is seen that 2 b_PO = [ 2 (½ σ²_A + ½ σ²_D )] / σ²_P = H² (where H² is the broad-sense heritability).

Siblings covariances

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

Half-sibs of the one 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 are already defined as deviates from the genotypic mean [see section on Allele substitution - Expectations and deviations]. After simplification. this becomes: ${\textstyle cov\left(HS\right)_{A}={\tfrac {1}{2}}pq\beta ^{2}={\tfrac {1}{4}}\sigma _{A}^{2}}$. Most texts finish the task here: however, the substitution deviations also do exist. The deviations sum-of-cross-products are defined from { 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: ${\textstyle cov\left(HS\right)_{D}=p^{2}q^{2}d^{2}={\tfrac {1}{4}}\sigma _{D}^{2}}$.

Adding the two components gives the full (and correct) result:

$${\begin{aligned}cov\left(HS\right)&=cov\left(HS\right)_{A}+cov\left(HS\right)_{D}\\&={\tfrac {1}{4}}\sigma _{A}^{2}+{\tfrac {1}{4}}\sigma _{D}^{2}\end{aligned}}$$

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:

$${\begin{aligned}cov\left(FS\right)&=pq\beta ^{2}+p^{2}q^{2}d^{2}\\&={\tfrac {1}{2}}\sigma _{A}^{2}+{\tfrac {1}{4}}\sigma _{D}^{2}\end{aligned}}$$

with no dominance having been overlooked.

Applications (siblings)

(A) 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.

(B) The most useful application here for genetical statistics is the correlation between half-sibs. Therefore, r_HS = cov(HS) / σ²_{(Σ HS components)} = [¼ σ²_A + ¼ σ²_D ] / σ²_P = ¼ H² . The correlation between full-sibs is of little utility, being r_FS = cov(FS) / σ²_{(Σ FS components)} = [½ σ²_A + ¼ σ²_D ] / σ²_P . The suggestion that it "approximates" (½ h²) is poor advice.

(C) It may be worth noting that [ cov(FS) - cov(HS)] = ¼ σ²_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" in Quantitative genetics for the rationale behind this}.