A path of coefficient of relationship is defined as

$$\rho_{AO} = \left( \frac{1}{2}\right)^n \sqrt { \frac{1+f_A}{1+f_O}}$$

This SE post discusses this definition

From this, the coefficient of relatedness between two individuals $B$ and $C$ is defined as

$$R_{BC} = \sum \rho_{AB} \rho_{AC}$$

, where $A$ is (I think) the last common ancestor of $B$ and $C$. The sum is over all possible path coefficients.

Source: Wikipedia

I am a little bit confused, I don't really understand this formula. One reason is the following. Let's assume that $B$ has 2 paths and $C$ has 3 paths to go to $A$. If we denote $\rho_{AB,i}$, the path $i$ (or the $i^{th}$ if you prefer) between $A$ and $B$,

$$R_{BC} = \rho_{AB,i} \rho_{AC,i} + \rho_{AB,j} \rho_{AC,j} + \rho_{AB,k} \rho_{AC,k}$$

But the path $\rho_{AB,k}$ does not exist because there are only 2 paths between $A$ and $B$ (which are denoted $i$ and $j$). Also, there is no reason to count the path in a specific order and not another one. If we change the order we count the paths between $C$ and $A$, we can get:

$$R_{BC} = \rho_{AB,i} \rho_{AC,j} + \rho_{AB,j} \rho_{AC,i} + \rho_{AB,k} \rho_{AC,k}$$

, which is obviously different from the previous calculation of $R_{BC}$

Can you help me making sense of these formulas?

  • 1
    $\begingroup$ genetic-genealogy.co.uk/Toc115570135.html. The notation is confusing and the Wiki site is not helpful (imo). This site is better (many examples) and mentions Wright's original paper. $\endgroup$ – daniel May 3 '14 at 13:12
  • 1
    $\begingroup$ It would be more like: $\sum_i (\rho_{AB}\rho_{AC})_i.$ $\endgroup$ – daniel May 3 '14 at 13:59
  • 1
    $\begingroup$ Wright's 1922 paper. aipl.arsusda.gov/publish/other/wright1922.pdf $\endgroup$ – daniel May 3 '14 at 14:20
  • $\begingroup$ Your comments and links helped a lot! Thanks. [genetic-genealogy](genetic-genealogy.co.uk/Toc115570135.html) offers a perfect "for dummies introduction" I don't quite know how to answer my messy question though. You can give a try if you want and I'll check it. Otherwise I might just delete it. I still don't understand this and that though. $\endgroup$ – Remi.b May 3 '14 at 15:03

The expression $f = 2h-1,$ coefficient of inbreeding (COI) is a measure of homozygosity. Since in a randomly breeding herd (of cattle) we expect 50 per cent heterozygous and 50 per cent homozygous, the minimum for $2h-1 $ is 0. If the cattle's genome is purely homozygous (aa, AA) we have $2h-1 = 1.$ (h is total homozygosity--see Wright's paper, linked below.)

As we can see from this site, a basic calculation of relationship coefficient may not deal with f at all. For two descendants B, C, the quantities $f_B,f_C$ are an attempt to factor in existing relative homozygosity that the coefficient of relation (COR) would otherwise not capture. Likewise $f_A$ measures preexisting homozygosity in the common ancestor of B and C. If animal A is the product of siblings then the homozygosity could already be quite high and this would maybe have measurable effects on the vitality of a cross between B and C.

Now $f_A$ and $f_B$ vary between 0 and 1 so $s = (\frac{1+f_A}{1+f_B})^{1/2}$ varies between $(1/2)^{1/2}$ and $(2)^{1/2}.$ Note that the important thing here is relative homozygosity of an animal and its ancestor:

a. If the ancestor A and descendent B have the same known preexisting burden of heterozygosity we can disregard s (and f) entirely. Their COR will depend only on path distance.

b. The homozygosity of the ancestor A tends to increase COR while that of B tends to decrease COR.

The expression $r_{BC }= \sum p_{AB }\cdot p_{AC}$ is somewhat confusing. On page 334 of one of Wright's earlier papers we can get a clear idea of the use of the coefficient.

He defines $p_{BA} = (1/2)^n (\frac{1+f_A}{1+f_B})^{1/2}.$ If the path from C to A is $p_{CA} = (1/2)^m (\frac{1+f_A}{1+f_C})^{1/2}$ then the total path BAC would be

$$ p_{AB,AC} = (1/2)^{n+m}(\frac{1+f_A}{1+f_B})^{1/2}(\frac{1+f_A}{1+f_C})^{1/2} $$

$$= (1/2)^{n+m}\frac{1+f_A}{[(1+f_B)(1+f_C)]^{1/2}} $$

So the expression for the sum of such paths would be

$$r_{BC} = \sum_i p_{AB,AC} = \sum_i (1/2)^{k_i}\frac{1+f_{A_i}}{[(1+f_B)(1+f_C)]^{1/2}} $$

in which $A_i$ is the ith ancestor through whom B and C are related. The total path length for a given term is $(n_i+m_i)$ which I abbreviated $k_i.$

So a less confusing expression for $r_{BC}$ might be:

$$r_{BC }= \sum_i p_{A_iB}p_{A_iC}. $$

Note that the sum is over all common ancestors back to some arbitrary generation.

A good example of this is Figure 11 at link 1 above. It's really not ``dumbed down'' at all, it just leaves out the factor s, which for a carefully maintained herd may be close to 1 anyway.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.