Monday, 27 February 2017

(156) Coefficient of Relationship (C.O.R.)

Basic Dimension

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Disclaimer: 

- Genetics and genealogy are serious professions with a lot of hidden rules which can change the calculations considerably. 

An important rule is the difference between independent chances (multiplication: ½x½=¼ [(2^-1)*(2^-1)=(2^-2)] and dependent chances (the addition of main effects supplied with multiplication for covariance). 




- Another strategy is to calculate the conditional probability of an allele from the mother, given the same allele by the father:
Alternative strategy first cousins from the Coefficient of Inbreeding (F):
Take, for example, the mating of first cousins who, by definition, share a set of grandparents. For any particular gene in the male, the chance that his female first cousin inherited the same gene from the same source is 1/8. [AB x (BC or BD)=.5^3] or [AB x BC x CD=.5^3]:


Then, for any gene the man passes to his child, the chance is 1/8 that the woman has the same gene and ½ that she transmits that gene to the child so, 1/8 x ½ = 1/16. Thus, a first-cousin marriage has a coefficient of inbreeding F =1/16= 6.25%.





Below two more alternatives for the Coefficient of Inbreeding:





- Also important are direct relations between parties (breeding: parents) and collateral relationships (no breeding: aunt-nephew).

I have not found all compelling relations yet. Anyway, if you are not a professional you likely will not notice. So as usual, it can be easily followed 😑.


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https://www.youtube.com/watch?v=A-OU0JcuX6U

https://en.wikipedia.org/wiki/Coefficient_of_relationship

The coefficient of relationship (R) is a measure of the degree of consanguinity (or biological relationship) between two individuals. A coefficient of inbreeding (F) can be calculated for an individual, as a measure for the amount of pedigree collapse within that individual's genealogy. The term coefficient of relationship was defined by Sewall Wright in 1922, and was derived from his definition of the coefficient of inbreeding of 1921. The measure is most commonly used in genetics and genealogy.





[The more inbreeding (F) in the family the more the pedigree shrinks, collapses into a simpler model. In the limit of identical twins, people are equal, like Adam and Eve who at least had different sex chromosomes. Their pedigree crunched almost totally. Identical twins cannot breed, but Adam and Eve could, that's the power of their myth, which proves male sexual fantasy as an insane lust to inbreeding.]





The coefficient of inbreeding (as proposed by Sewell Wright in 1922) is the probability that two alleles at a randomly chosen locus are identical by descent. Note that alleles may be identical for other reasons, but the inbreeding coefficient (F) is just looking at the mathematical probability that the alleles have come from a common ancestor.




The Coefficient of Relationship (R) looks more a descriptive statistic of consanguinity relations within families. For example a father and his child have 50% of their genes in common. The same for two siblings. So it is not directly a probability for specific individual situations:








Below we see the effect of both ancestors A and B on sibling C (green). We see the product rule of independent chances (of A and B) on an unspecified allele in sibling C:







Well, in statistics we can be flexible, it's no religion:















common ancestor is any preceding individual in the pedigree who passes genes directly to two or more persons through different independent routes or paths:


http://www.genetic-genealogy.co.uk/supp/DirectRelationships.html http://www.genetic-genealogy.co.uk/Toc115570136.html

http://www.genetic-genealogy.co.uk/Toc115570135.html


CALCULATION OF THE COEFFICIENT OF RELATIONSHIP

Common Ancestors Defined

Common ancestors are usually only shared by collateral [not interbreeding] relatives, since with direct [breeding] relationships one individual is the ancestor of the other. Exceptions are shown here.
In its literal sense a 'common ancestor' is any person who passes genes directly to two or more individuals. However, for the purposes of calculating the coefficient of relationship (R) and the coefficient of inbreeding (F), it has a more restricted definition:

In its narrow sense a common ancestor is any preceding individual in the pedigree who passes genes directly to two or more persons through different independent routes or paths. In the following diagram there are three individuals who satisfy this definition: i.e. B, H and I, who are common ancestors of J and K. (Note that parents are common ancestors in this context):





http://www.genetic-genealogy.co.uk/supp/DirectRelationships.html


DIRECT RELATIONSHIPS


In male kin bonded tribes, genes are inherited by the male line systematically. But direct relations exist only between successive generations. Also in every generation there is an influx of new genes from fresh juvenile females from other tribes, which are a counterweight against inbreeding, these are also called direct relations but only for the immediate offspring:




There are fewer kinds of direct relationship than collaterals and they are usually less complicated. This is because there are no half or first degree forms, which are a common feature of all collaterals. Furthermore, all direct relationships are intergenerational while collaterals can be members of the same or different generations. Finally with direct relationships, gene transmission is forward and unidirectional, in contrast to collaterals whose connections run in both directions of time through common ancestors.

A further fundamental difference is that inbreeding plays a more active role in the formation of double and multiple direct relationships. Most double and multiple collaterals are possible without inbreeding, but with direct relationships all plural forms result from matings between fairly close relatives.

Collateral relatives are related through common ancestors. They hold a proportion of their genes in common because they have received them from a common source. The important point is that no one receives any genes from collateral relatives.




Collateral Relationship

An important rule must be observed for collaterals:

a) The path must go back against the arrows from one collateral relative to the common ancestor, and then follow the arrows forward to the other collateral relative. For example, in the next pedigree diagram, one of the paths between G and H is:
G_1_D_2_A_3_E_4_H


In this case trace the path back from one cousin to the common ancestor and then forward to the other cousin. Remember, a qualifying common ancestor is someone who passes genes directly to both collateral relatives along different routes. With collateral relatives there is often more than one common ancestor.

Find RGH

Common Ancestors [3]
Paths
(1/2)n
A
G_1_D_2_A_3_E_4_H
(1/2)4
=
1/16
B
G_1_D_2_B_3_E_4_H
(1/2)4
=
1/16
Σ(1/2)n
=
1/8

Therefore, RGH0.125

Cousin marriages will again increase the number of paths as they did with direct
relationships. 










Combined Relationships

It is possible, if there has been any inbreeding such as a cousin marriage, for two people to be both directly and collaterally related. Below we see the calculation of the Coefficient of Inbreeding for cousin marriage. B is directly related to I (grandmother). But B is also collaterally related to I (great aunt). The second relationship results from the fact that B and I have common ancestors in C and D:




Other strategy first cousins:


Take, for example, the mating of first cousins who, by definition, share 
a set of grandparents. For any particular gene in the male, the chance
that his female first cousin inherited the same gene from the same source
is 1/8. [AB x (BC or BD)=.5^3]. Further, for any gene the man passes to his child,
the chance is 1/8 that the woman has the same gene and ½ that she transmits 
that gene to the child so 1/8 x ½ = 1/16. Thus, a first-cousin marriage has a 
coefficient of inbreeding F =1/16= 6.25%.







http://www.genetic-genealogy.co.uk/supp/DirectRelationships.html http://www.genetic-genealogy.co.uk/Toc115570136.html

http://www.genetic-genealogy.co.uk/Toc115570135.html


CALCULATION OF THE COEFFICIENT OF RELATIONSHIP

Common Ancestors Defined

Common ancestors are usually only shared by collateral relatives, since with direct relationships one individual is the ancestor of the other. Exceptions are shown here.
In its literal sense a 'common ancestor' is any person who passes genes directly to two or more individuals. However, for the purposes of calculating the coefficient of relationship (R) and the coefficient of inbreeding (F), it has a more restricted definition:


In its narrow sense a common ancestor is any preceding individual in the pedigree who passes genes directly to two or more persons through different independent routes or paths. In the following diagram there are three individuals who satisfy this definition: i.e. B, H and I, who are common ancestors of J and K. (Note that parents are common ancestors in this context).

The coefficient of relationship ("r") between two individuals J and K is obtained by a summation of coefficients calculated for every line by which they [together] are connected to their common ancestors. Each such line connects the two individuals via a common ancestor, passing through no individual which is not a common ancestor more than once. 







B passes genes directly to J and K through the following independent routes:

(B  E  H  J)  and  (B  F   I   K)                          
(B  E  H  K) and  (B   F  I   J)




Similarly H and I pass genes to J and K as follows:

(H  J) and (H  K)
(I   J) and (I  K)





It can be seen that although A, C, D, E, F and G all pass genes directly to both J and K, they do not do so by different independent routes. e.g. The paths from A to J and K are:

(A  E  H  J) and (A  E  H  K)

They do not, therefore, qualify as common ancestors when calculating R . This distinction is important in finding all the relevant common ancestors in the next section. The contributions of A, C, D, E, F and G towards the degree of relationship between J and K are accounted for by the paths using H and I as common ancestors. 


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