# Inbreeding Coefficient and Coefficient of Relationship

Wikipedia gives the following formula to calculate a "path of coefficient of relationship" between an ancestor $A$ and an offspring $O$:

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

, where $f_A$ and $f_O$ are the coefficient of inbreeding of the ancestor and the offspring respectively.

Question

Where does the term $\sqrt { \frac{1+f_A}{1+f_O}}$ comes from? Please explain why this multiplicative term is $\sqrt { \frac{1+f_A}{1+f_O}}$ and not something different such as $\frac{f_A f_O}{2}$ for example.

• In his paper Wright presents this as an alternative to other (existing) formulas. The ultimate test is whether it predicts herd health well and because of broad assumptions underlying the model it may be hard to tell how good a formula is. In general we want something close to 1 unless there is strong homozygosity somewhere. Yours won't work because it would likely be very small. – daniel May 3 '14 at 23:44
• Great Question! If you have solved this, please gives us the complete derivation. – Devashish Das Aug 1 '14 at 18:30
• No I haven't solved it yet. Thks a lot for coming back to this post. I'll read your answer in 48h as I won't have time before! – Remi.b Aug 2 '14 at 0:06