Calculating Inbreeding

The article of reference about calculating inbreeding is Wright, S. (1922). Coefficient of inbreeding and relationship. American Naturalist, 56. In this article for the coefficient for inbreeding is introduced.

Definition

The inbreeding coefficient is the probability that the two genes on a random locus are equal because they descend from the same gene in an ancestor.

The inbreeding coefficient is a probability, so it will be between 0 and 1. The inbreeding coefficient is based on common ancestry in the parents so we always start from the pedigree to calculate inbreeding. Here's the pedigree of the example we'll use in this article:

The original method for calculating the inbreeding coefficients is of course from Wright but Henderson & Cunningham have come up with a different method for calculating the same inbreeding coefficients.

The method of Wright

The original method of Wright consisted of finding all common ancestors of the parents and the paths from the parents to these common ancestors. The formula by Wright is (don't worry if you don't have a higher degree in math, I'll explain what it means right after and how it practically works):

F is the symbol commonly used for the inbreeding coefficient. F_i is the inbreeding coefficient for bird i. Bird a is a common ancestor of both parents of bird i. F_a is the inbreeding coefficient of the common ancestor a. The n₁ and n₂ are the number of generation between the common ancestor and both parents. The weird symbol in front is a summation sign. It means that you have to do this for all possible common ancestors and paths and then add the results.

So far the theory - it's far clearer if you do it by example.

Example: F_I
In this example A and B are common ancestors for the parents (G and H) of I. The paths from both parents to these common ancestors are:
For ancestor A:
 G - A (n₁ = 1)
 G - D - A (n₁ = 2)
 H - E - A (n₂ = 2)

For ancestor B:
 G - D - B (n₁ = 2)
 H - E - B (n₂ = 2)

We suppose that A and B are not inbred so (1 + FA) = 1 and (1 + FB) = 1.
The inbreeding because of common ancestor B is:
 F(I<B) = (1/2)^(2+2+1) = (1/2)⁵ = 0.03125 = 3.125%
The inbreeding because of common ancestor A is:
 F(I<A) = (1/2)^(1+2+1) + (1/2)^(2+2+1) = (1/2)⁴ + (1/2)⁵ = 0.0625 + 0.03125 = 0.09375 = 9.375%
The inbreeding coefficient of I is:
 F_I = 3.125% + 9.375% = 12.5%

Example: F_G
Calculating the inbreeding in G that is a result of father x daughter pairing is:
 A (n₁ = 0)
 D - A (n₂ = 1)
 F_G = (1/2)^(0+1+1) = (1/2)² = 0.25 = 25%

Method of Henderson & Cunningham

This method lets you calculate the inbreeding coefficient for your entire population and also gives you the kinships between individuals.

1. Order all animals from oldest to youngest:
A B D E F G H I

2. Create a square grid with a column and row for each animal:

	A	B	D	E	F	G	H	I
A
B
D
E
F
G
H
I

3. In the diagonal cells (where a column and a row of the same animal cross) write one plus the inbreeding coefficient (1 + F). For the 3 animals without pedigree information we assumed that the inbreeding coefficient is 0, so the diagonal cell becomes 1.

4. The non diagonal cells hold the additive genetic relationship a_ij. This means that the grid is symmetric because a_ij = a_ji. For the animals without pedigree we suppose that they are unrelated: a_ij = 0.

The rules for further completing the grid are easy. For diagonal cells 1 + a_ij/2 holds where i and j are the parents of the individual. For non diagonal cells holds a_ij = (a_ik + a_il)/2 if i is younger then j and k and l are the parents of i.

So for D that has parents A and B:
a_AD = (a_AA + a_AB)/2 = (1 + 0)/2 = 0.5
a_BD = (a_AB + a_BB)/2 = (0 + 1)/2 = 0.5
a_DD = 1 + a_AB/2 = 1 + 0/2 = 1

Filling these numbers into the grid:

Next we fill the column and row for E:

Next filling the column and row for G:

Next filling the column and row for H:

Next filling the column and row for I:

(C) Bert Raeymaekers