# Coefficient of Inbreeding - implementation issue

The formula for the coefficient of inbreeding is as follows... I have a family tree going back 9 generations. Say I find a common ancestor X in the 4th generation on the mothers side and in the 5th generation on the fathers side. I can work out that constituent of F... but what do I do about all the ancestors of X (they will all also be common ancestors). Do I also work out their values and add them to F?... or do I ignore them? The descriptions of the COI I have seen do not make this clear.

## 1 Answer

The $F_i$ term is the term that accounts for inbreeding of your common ancestor X. If that common ancestor is not inbred then the term is 0 and the calculation is a little easier. Using the example you have given:

 X ─┬─ Y
│
┌──┴──┐
GGGF  GGM
│     │
GGF    │
│    GM
GF     │
│     │
F ─┬─ M
│
Mick


In this case, the common ancestors are not inbred so the $F_i$ terms go away ($1+F_i = 1$). There are two paths of common ancestry (via X and Y), therefore the calculation of the two paths from the father (F) and mother (M) to X and Y is:

$F_{Mick} = \frac12^{n_{F,X}+n_{M,X}+1}(1+F_X)+\frac12^{n_{F,Y}+n_{M,Y}+1}(1+F_Y)$

$F_{Mick} = \frac12^{4+3+1}(1+0)+\frac12^{4+3+1}(1+0)$

$F_{Mick} = \frac{1}{256} + \frac{1}{256} = 0.0078 = 0.78\%$

In a more complex case, such as where ancestor X was inbred, $F_i$ cannot be ignored. In the following example, X's parents were first cousins:

  U ─┬─ V
│
┌──┴──┐
XGF   XGM
│     |
XF─┬─XM
│
X ─┬─ Y
│
┌──┴──┐
GGGF  GGM
│     │
GGF    │
│    GM
GF     │
│     │
F ─┬─ M
│
Mick


For this, we must first calculate $F_X$ (i.e. how inbred X is):

$F_X = \frac12^{n_{XF,U}+n_{XM,U}+1}(1+F_U)+\frac12^{n_{XF,V}+n_{XM,V}+1}(1+F_V)$

$F_X = \frac12^{2+2+1}(1+0)+\frac12^{2+2+1}(1+0)$

$F_X = \frac{1}{32} + \frac{1}{32} = 0.0625 = 6.25\%$

Then we can input this $F_X$ into the same calculation as done before for the simple case:

$F_{Mick} = \frac12^{n_{F,X}+n_{M,X}+1}(1+F_X)+\frac12^{n_{F,Y}+n_{M,Y}+1}(1+F_Y)$

$F_{Mick} = \frac12^{4+3+1}(1+0.0625)+\frac12^{4+3+1}(1+0)$

$F_{Mick} = 0.00415 + \frac{1}{256} = 0.0081 = 0.81\%$

A good explanation of these calculations with further examples can be found here: The Coefficient of Inbreeding (F) and its applications. The calculations can get quite complicated even with relatively simple inbreeding.

• Thank you for your explanation - but I still don't think you've answered my question. Say we are considering child C and we find a common ancestor A in the family tree of both the mother and father of C. Let us say for simplicity that A is not itself inbred. I know how to calculate all the terms to the right of the sigma in the COI equation for A, so I can add that to the summation... but what about the mother (or father) of A? That will be a common ancestor too. Do I work out everything to the right of the sigma for that too and add that to our summation? – Mick Nov 27 '15 at 13:37
• @Mick No, if A is not inbred, then you do not need to add anything for the mother or father of A (because the inbreeding coefficient for A is 0). This is what the first simple example in my answer illustrates (neither X nor Y are inbred). – Harry Vervet Nov 27 '15 at 14:20
• Ok, I see. I will assume you are correct and give you the green tick. Thank you. – Mick Nov 27 '15 at 14:51