FIC

Many data sets consist of individuals from different populations, in the cases of structured populations, this usually result in an increased number of homozygotes.

The inbreeding coefficient FIC is a measure of deviation away from the Hardy Weinberg Equilibrium. A value of 0 implies no deviation, a negative value implies an excess of heterozygotes and a positive value implies an excess of homozygotes.

The following equation gives the estimate of F where the observed genotypes are available.

${\begin{aligned}F_{IC}&=1-{\frac {O(Het)}{E(Het)}}\\&=1-{\frac {\text{No. observed HETs}}{E(Het|{\textbf {p)}}}}\\&=1-{\frac {\text{No. observed HETs}}{\sum _{i=1}^{n}{\sum _{j}{P(Het_{j}|{\textbf {p}})}}}}\\\end{aligned}}$

The following equation gives the estimate of F where genotype likelihoods are available.

${\begin{aligned}F_{IC}&=1-{\frac {O(Het)}{E(Het)}}\\&=1-{\frac {E(Het|R_{i},{\textbf {p}})}{E(Het|{\textbf {p}})}}\\&=1-{\frac {\sum _{i=1}^{n}{\sum _{j}{P(Het_{j}|R_{i},{\textbf {p}})}}}{\sum _{i=1}^{n}{\sum _{j}{P(Het_{j}|{\textbf {p}})}}}}\\&=1-{\frac {\sum _{i=1}^{n}{\sum _{j}{\frac {P(R_{i}|Het_{j},{\textbf {p}})P(Het_{j}|{\textbf {p}})}{\sum _{(k,l)}{P(R_{i}|G_{(k,l)},{\textbf {p}})P(G_{(k,l)}|{\textbf {p}})}}}}}{\sum _{i=1}^{n}{\sum _{j}{P(Het_{j}|{\textbf {p}})}}}}\\\end{aligned}}$

where:

${\begin{aligned}P(G_{(k,l)}|{\textbf {g}})&=&g_{(k,l)}\end{aligned}}$

$P(G_{(k,l)}|{\textbf {p}})={\begin{cases}p_{k}^{2},&{\text{if }}k=l\\2p_{k}p_{l},&{\text{if }}k\neq l\end{cases}}$