Genetics ## Simple calculation of the inbreeding coefficient

Even though there are computer programs, which will calculate the inbreeding coefficient automatically, based on pedigrees entered into a database, it is still handy to be able to calculate the inbreeding in less complicated pedigrees manually. Here is a short description on how to calculate the inbreeding coefficient from a pedigree by using the path coefficient method.

## The Path Coefficient Method

Start drawing an "arrow pedigree", where every individual only appears once. The arrows should point from parents to offspring and always point downwards, or diagonally downwards.

Example 1 Example 1: Ordinary pedigree and arrow pedigree.

In the arrow pedigree we search all paths that go from one parent to the other without passing any individual more than once. For each path we count the number of individuals involved. The contribution from each path to the inbreeding coefficient is 1/2 raised to the number of individuals involved in the path.

From the arrow pedigree in example 1 we then get two paths from one parent to the other:

CAD -> (1/2)³ = 1 / (2 x 2 x 2) = 1/8
CBD -> (1/2)³ = 1 / (2 x 2 x 2) = 1/8

The inbreeding = 1/8 + 1/8 = 1/4 = 0.25 = 25%

If the common ancestor, i.e. where the path turns and goes downwards again, is inbred we have to take that into account. (For the two paths in the example above the common ancestors, A and B respectively, are underlined.) Calculate the inbreeding coefficient for the inbred common ancestor by using the path coefficient method. Add 1 to this inbreeding coefficient and multiply with the contribution given by the path in question.

So if the number of individuals along a path is n, and the inbreeding coefficient for the common ancestor is F, the total contribution to the inbreeding coefficient will be (1/2)^n x (1+F).

The sum of the contributions of all the paths will then be the inbreeding coefficient.

Example 2 Example 2: Ordinary pedigree and arrow pedigree.

The existing paths are BDC, BDFEC, BEFDC, BDEC, BEDC, BEC.

The common ancestor D is inbred and the arrow pedigree for him looks like this: Arrow pedigree for D

The inbreeding in D is then (1/2)² = 1/4.

The contribution to the inbreeding coefficient from each path will then be:

PathnF(1/2)^n x (1+F)total
BDC31/4(1/2)³ x (1+1/4) = 1/8 x 5/4 = 5/32= 0.15625
BDFEC50(1/2)^5 x (1+0) = 1/32= 0.03125
BEFDC50(1/2)^5 x (1+0) = 1/32= 0.03125
BDEC40(1/2)^4 x (1+0) = 1/16= 0.0625
BEDC40(1/2)^4 x (1+0) = 1/16= 0.0625
BEC30(1/2)³ x (1+0) = 1/8= 0.125

The inbreeding = the sum of all contributions = 0.46875 = 46.875%

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