Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Quantitative genetics
(section)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
==Relationship== [[File:Inbreeding & Coancestry.jpg|thumb|200px|right|Connection between the inbreeding and co-ancestry coefficients.]] From the heredity perspective, relations are individuals that inherited genes from one or more common ancestors. Therefore, their "relationship" can be ''quantified'' on the basis of the probability that they each have inherited a copy of an allele from the common ancestor. In earlier sections, the ''Inbreeding coefficient'' has been defined as, "the probability that two ''same'' alleles ( '''A''' and '''A''', or '''a''' and '''a''' ) have a common origin"—or, more formally, "The probability that two homologous alleles are autozygous." Previously, the emphasis was on an individual's likelihood of having two such alleles, and the coefficient was framed accordingly. It is obvious, however, that this probability of autozygosity for an individual must also be the probability that each of its ''two parents'' had this autozygous allele. In this re-focused form, the probability is called the ''co-ancestry coefficient'' for the two individuals ''i'' and ''j'' ( '''''f'' <sub>ij</sub>''' ). In this form, it can be used to quantify the relationship between two individuals, and may also be known as the ''coefficient of kinship'' or the ''consanguinity coefficient''.<ref name="Crow & Kimura"/>{{rp|132–143}} <ref name="Falconer 1996"/>{{rp|82–92}} ===Pedigree analysis=== [[File:Pedigree Analysis.jpg|thumb|150px|left|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 inbreeding and co-ancestry. Such pedigrees actually are informal depictions of ''path diagrams'' as used in ''path analysis'', which was invented by Sewall Wright when he formulated his studies on inbreeding.<ref name="Li 1977"/>{{rp|266–298}} Using the adjacent diagram, the probability that individuals "B" and "C" have received autozygous alleles from ancestor "A" is ''1/2'' (one out of the two diploid alleles). This is the "de novo" inbreeding ('''Δf<sub>Ped</sub>''') at this step. However, the other allele may have had "carry-over" autozygosity from previous generations, so the probability of this occurring is (''de novo complement'' multiplied by the ''inbreeding of ancestor A'' ), that is ''' (1 − Δf<sub>Ped</sub> ) f<sub>A</sub> ''' = ''' (1/2) f<sub>A</sub> '''. 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 ''' (1/2) + (1/2)f<sub>A</sub> ''' = ''' (1/2) (1+f <sub>A</sub> ) '''. 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'' (''' f<sub>AA</sub> ''').<ref name="Crow & Kimura"/>{{rp|132–143}}<ref name="Falconer 1996"/>{{rp|82–92}} 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)<sup>3</sup> '''. The power that (1/2) is raised to can be viewed as "the number of intermediates in the path between '''A''' and '''X''' ", '''n<sub>B</sub> = 3 '''. Similarly, for the "C path", ''' n<sub>C</sub> = 2 ''', and the "transfer probability" is ''' (1/2)<sup>2</sup> '''. The combined probability of autozygous transfer from '''A''' to '''X''' is therefore '''[ f<sub>AA</sub> (1/2)<sup>(n<sub>B</sub>)</sup> (1/2)<sup>(n<sub>C</sub>)</sup> ] '''. Recalling that '' f<sub>AA</sub> = (1/2) (1+f <sub>A</sub> ) '', ''' f<sub>X</sub> = f<sub>PQ</sub> ''' = ''' (1/2)<sup>(n<sub>B</sub> + n<sub>C</sub> + 1)</sup> (1 + f<sub>A</sub> ) '''. In this example, assuming that f<sub>A</sub> = 0, ''' f<sub>X</sub> = 0.0156 ''' (rounded) = '''f<sub>PQ</sub> ''', one measure of the "relatedness" between '''P''' and '''Q'''. In this section, powers of ('''1/2''') were used to represent the "probability of autozygosity". Later, this same method will be used to represent the proportions of ancestral gene-pools which are inherited down a pedigree [the section on "Relatedness between relatives"]. [[File:Pedigree CrossMult.jpg|thumb|200px|right|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 (statistics)|path analysis]].<ref name="Li 1977"/> 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<sub>PQ</sub>''' = average of ( '''f<sub>AC</sub>''', '''f<sub>AD</sub>''', '''f<sub>BC</sub>''', '''f<sub>BD</sub>''' ) = ''' (1/4) [f<sub>AC</sub> + f<sub>AD</sub> + f<sub>BC</sub> + f<sub>BD</sub> ] ''' = '''f<sub>Y</sub> '''. In a similar fashion, ''cross-multiplier 2'' states that '''f<sub>PC</sub> = (1/2) [ f<sub>AC</sub> + f<sub>BC</sub> ]'''—while ''cross-multiplier 3'' states that '''f<sub>PD</sub> = (1/2) [ f<sub>AD</sub> + f<sub>BD</sub> ]''' . Returning to the first multiplier, it can now be seen also to be '''f<sub>PQ</sub> = (1/2) [ f<sub>PC</sub> + f<sub>PD</sub> ]''', 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)==== [[File:Inbreeding- Sibs.jpg|thumb|200px|right|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<sub>Y</sub> = f<sub>P1,P2</sub> = (1/4) [ f<sub>AA</sub> + 2 f<sub>AB</sub> + f<sub>BB</sub> ] '''. Recall that '''f<sub>AA</sub>''' and '''f<sub>BB</sub>''' were defined earlier (in Pedigree analysis) as ''coefficients of parentage'', equal to '' (1/2)[1+f<sub>A</sub> ] '' and '' (1/2)[1+f<sub>B</sub> ] '' 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<sub>(t-2)</sub> ] '''. [[File:Crossing Inbreeding.jpg|thumb|200px|left|Inbreeding from full-sib and half-sib crossing, and from selfing.]] Now, examine '''f<sub>AB</sub> '''. Recall that this also is ''f<sub>P1</sub> '' or ''f<sub>P2</sub> '', and so represents ''their '' generation - '''f<sub>(t-1)</sub> '''. Putting it all together, '''f<sub>t</sub> = (1/4) [ 2 f<sub>AA</sub> + 2 f<sub>AB</sub> ] ''' = ''' (1/4) [ 1 + f<sub>(t-2)</sub> + 2 f<sub>(t-1)</sub> ] '''. That is the ''inbreeding coefficient '' for ''Full-Sib crossing'' .<ref name="Crow & Kimura"/>{{rp|132–143}}<ref name="Falconer 1996"/>{{rp|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<sub>Y</sub> = f<sub>(P1,P2)</sub> = (1/4) [ f<sub>AA</sub> + f<sub>AC</sub> + f<sub>BA</sub> + f<sub>BC</sub> ] '''. There is just one ''coefficient of parentage'' this time, but three ''co-ancestry coefficients'' at the (t-2) level (one of them—f<sub>BC</sub>—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<sub>A</sub> ] ''', and the three ''co-ancestries'' each represent '''f<sub>(t-1)</sub> '''. Recalling that '' f<sub>A</sub> '' represents '' f<sub>(t-2)</sub> '', the final gathering and simplifying of terms gives ''' f<sub>Y</sub> = f<sub>t</sub> = (1/8) [ 1 + f<sub>(t-2)</sub> + 6 f<sub>(t-1)</sub> ] '''.<ref name="Crow & Kimura"/>{{rp|132–143}}<ref name="Falconer 1996"/>{{rp|82–92}} The graphs at left include this ''half-sib (HS) inbreeding'' over twenty successive generations. [[File:Inbreeding- Selfing.jpg|thumb|200px|right|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<sub>(P1, P2)</sub> '''. ====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<sub>Y</sub> = f<sub>(P1, P1)</sub> = f<sub>t</sub> = (1/2) [ 1 + f<sub>(t-1)</sub> ] '''.<ref name="Crow & Kimura"/>{{rp|132–143}}<ref name="Falconer 1996"/>{{rp|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==== [[File:Inbreeding- Cousins First.jpg|thumb|175px|right|Pedigree analysis first cousins]] These are derived with methods similar to those for siblings.<ref name="Crow & Kimura"/>{{rp|132–143}}<ref name="Falconer 1996"/>{{rp|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<sub>Y</sub> = f<sub>t</sub> = f<sub>P1,P2</sub> = (1/4) [ f<sub>1D</sub> + f<sub>12</sub> + f<sub>CD</sub> + f<sub>C2</sub> ]'''. After substitution with corresponding inbreeding coefficients, gathering of terms and simplifying, this becomes ''' f<sub>t</sub> = (1/4) <nowiki>[</nowiki> 3 f<sub>(t-1)</sub> + (1/4) <nowiki>[</nowiki>2 f<sub>(t-2)</sub> + f<sub>(t-3)</sub> + 1 <nowiki>]]</nowiki> ''', which is a version for iteration—useful for observing the general pattern, and for computer programming. A "final" version is ''' f<sub>t</sub> = (1/16) [ 12 f<sub>(t-1)</sub> + 2 f<sub>(t-2)</sub> + f<sub>(t-3)</sub> + 1 ] '''. [[File:Inbreeding- Cousins Second.jpg|thumb|175px|left|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<sub>Y</sub> = f<sub>t</sub> = f<sub>P1,P2</sub> = (1/4) [ f<sub>3F</sub> + f<sub>34</sub> + f<sub>EF</sub> + f<sub>E4</sub> ]'''. After working through the appropriate algebra, this becomes ''' f<sub>t</sub> = (1/4) <nowiki>[</nowiki> 3 f<sub>(t-1)</sub> + (1/4) <nowiki>[</nowiki>3 f<sub>(t-2)</sub> + (1/4) <nowiki>[</nowiki>2 f<sub>(t-3)</sub> + f<sub>(t-4)</sub> + 1 <nowiki>]]]</nowiki> ''', which is the iteration version. A "final" version is ''' f<sub>t</sub> = (1/64) [ 48 f<sub>(t-1)</sub> + 12 f<sub>(t-2)</sub> + 2 f<sub>(t-3)</sub> + f<sub>(t-4)</sub> + 1 ] '''. [[File:Cousin Inbreeding.jpg|thumb|250px|left|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<sub>t</sub> = (1/4)[2 f<sub>(t-1)</sub> + f<sub>(t-2)</sub> + 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<sub>(t-(1+k))</sub> + f<sub>(t-(2+k))</sub> + 1] '''. Stacked in front of this ''last term'' are one or more ''iteration increments'' in the form '''(1/4) [ 3 f<sub>(t-j)</sub> + ... ''', 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 '''k'''th ''level'' full cousins, '''f{k}<sub>t</sub> = ''Ιter''<sub>j = 1</sub><sup>k</sup> { (1/4) [ 3 f<sub>(t-j)</sub> + }<sub>j</sub> + (1/4) [ 2 f<sub>(t-(1+k))</sub> + f<sub>(t-(2+k))</sub> + 1] '''. At the commencement of iteration, all f<sub>(t-''x'')</sub> 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. [[File:Inbreeding- Csns Half.jpg|thumb|200px|left|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<sub>Y</sub> = f<sub>t</sub> = f<sub>P1,P2</sub> = (1/4) [ f<sub>3D</sub> + f<sub>34</sub> + f<sub>CD</sub> + f<sub>C4</sub> ]'''. After working through the appropriate algebra, this becomes ''' f<sub>t</sub> = (1/4) <nowiki>[</nowiki> 3 f<sub>(t-1)</sub> + (1/8) <nowiki>[</nowiki>6 f<sub>(t-2)</sub> + f<sub>(t-3)</sub> + 1 <nowiki>]]</nowiki> ''', which is the iteration version. A "final" version is ''' f<sub>t</sub> = (1/32) [ 24 f<sub>(t-1)</sub> + 6 f<sub>(t-2)</sub> + f<sub>(t-3)</sub> + 1 ] '''. The iteration algorithm is similar to that for ''full cousins'', except that the last term is '''(1/8) [ 6 f<sub>(t-(1+k))</sub> + f<sub>(t-(2+k))</sub> + 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)==== [[File:Inbreeding- Backcross.jpg|thumb|250px|left|Pedigree analysis: backcrossing]] [[File:Backcrossing 1.jpg|thumb|250px|right|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<sub>D</sub> = f<sub>t</sub> = f<sub>CR</sub> = (1/2) [ f<sub>RB</sub> + f<sub>RR</sub> ] ''', using ''cross-multiplier 2'' previously given. The ''f<sub>RB</sub>'' just defined is the one that involves generation ''(t-1)'' with ''(t-2)''. However, there is another such ''f<sub>RB</sub>'' 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<sub>(t-1)</sub>'''. The remaining '''f<sub>RR</sub>''' is the ''coefficient of parentage'' of the ''recurrent parent'', and so is ''(1/2) [1 + f<sub>R</sub> ] ''. Putting all this together : '''f<sub>t</sub> = (1/2) [ (1/2) [ 1 + f<sub>R</sub> ] + f<sub>(t-1)</sub> ] ''' = ''' (1/4) [ 1 + f<sub>R</sub> + 2 f<sub>(t-1)</sub> ] '''. 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<sub>R</sub>''' carries-over into the '''f<sub>t</sub>''' of the backcrossing. The result is a more gradual curve rising to the asymptotes than shown in the present graphs, because the ''f<sub>R</sub>'' is not at a fixed level from the outset. === Contributions from ancestral genepools === In the section on "Pedigree analysis", <math display="inline"> \left( \tfrac {1}{2} \right)^n </math> was used to represent probabilities of autozygous allele descent over '''n''' generations down branches of the pedigree. This formula arose because of the rules imposed by sexual reproduction: '''(i)''' two parents contributing virtually equal shares of autosomal genes, and '''(ii)''' successive dilution for each generation between the zygote and the "focus" level of parentage. These same rules apply also to any other viewpoint of descent in a two-sex reproductive system. One such is the proportion of any ancestral gene-pool (also known as 'germplasm') which is contained within any zygote's genotype. Therefore, the proportion of an '''ancestral genepool''' in a genotype is: <math display="block"> \gamma_n = \left( \frac{1}{2}\right) ^n </math> where '''n''' = number of sexual generations between the zygote and the focus ancestor. For example, each parent defines a genepool contributing <math display="inline"> \left( \tfrac{1}{2} \right)^1 </math> to its offspring; while each great-grandparent contributes <math display="inline"> \left( \tfrac{ 1}{2} \right)^3 </math> to its great-grand-offspring. The zygote's total genepool ('''Γ''') is, of course, the sum of the sexual contributions to its descent. <math display="block"> \begin{align} \Gamma & = \sum_{n=1} ^{2^n} {\gamma_n} \\ & = \sum_{n=1} ^{2^n} {\left( \frac{1}{2}\right)^n} \end{align} </math> ====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: <math display="block"> \begin{align} \Rho_k & = \sum_{m=1} ^{1 , 2} {\gamma_n} \\ & = \sum_{m=1} ^{1 , 2} {\left( \frac{1}{2} \right) ^n} \end{align} </math> 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 <math display="block"> \begin{align} \Rho_k & = \sum_{m=1} ^{2} {\gamma_2} \\ & = \sum_{m=1} ^{2} { \left( \frac{1}{2} \right) ^2} \\ & = \frac{1}{2} \end{align} </math> 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). <math display="block"> \begin{align} \Rho_1 & = \sum_{m=1} ^{2} {\gamma_3} \\ & = \sum_{m=1} ^{2} {\left( \frac{1}{2}\right) ^3} \\ & = \frac{1}{4} \end{align} </math> and <math display="block"> \begin{align} \Rho_2 & = \sum_{m=1} ^{2} {\gamma_2} \\ & = \sum_{m=1} ^{2} {\left( \frac{1}{2}\right) ^2} \\ & = \frac{1}{2} \end{align} </math> Notice each cousin has a different Ρ <sub>k</sub>. ====GRC – genepool relationship coefficient==== In any pairwise relationship estimation, there is one '''Ρ<sub>k</sub>''' 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'' <ref>the square-root of their product</ref><ref name="Moroney 1956">{{cite book |last1=Moroney|first1=M.J. |title=Facts from figures|date=1956 |publisher=Penguin Books |location=Harmondsworth|edition=third}}</ref>{{rp|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.<ref name="Li 1977">{{cite book|last1=Li|first1=Ching Chun|title=Path analysis - a Primer|date=1977|publisher=Boxwood Press|location=Pacific Grove|isbn=0-910286-40-X|edition=Second printing with Corrections}}</ref>{{rp|214–298}} A summary of some levels of relationship (GRC) follow. {| class="wikitable" |- ! GRC !! Relationship examples |- | 1.00 || full Sibs |- | 0.7071 || Parent ↔ Offspring; Uncle/Aunt ↔ Nephew/Niece |- | 0.5 || full First Cousins; half Sibs; grand Parent ↔ grand Offspring |- | 0.3536 || full Cousins First ↔ Second; full First Cousins {1 remove} |- | 0.25 || full Second Cousins; half First Cousins; full First Cousins {2 removes} |- | 0.1768 || full First Cousin {3 removes}; full Second Cousins {1 remove} |- | 0.125 || full Third Cousins; half Second Cousins; full 1st Cousins {4 removes} |- |0.0884 || full First Cousins {5 removes}; half Second Cousins {1 remove} |- | 0.0625 || full Fourth Cousins; half Third Cousins |}
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)