FIC

The inbreeding coefficient $F_{IC}$ is a measure of deviation from the Hardy Weinberg Equilibrium in terms of the excess of heterozygotes observed. A value of 0 implies no deviation, a negative value implies an excess of heterozygotes and a positive value implies an excess of homozygotes. $F_{IC}$ ranges from -1 to 1.

The following equation gives the estimate of F where the observed genotypes are available. $g_{i,j,k}$ is the genotype composed of alleles $i$ and $j$ for the $k$th individual. $P(G_{i,j}|\text{p})$ is the estimated genotype allele frequency for genotype $G_{i,j}$ under HWE assumption. $I[i\ne j]$ is an identity function for heterozygote genotypes.

$\begin{array}{rl}{F}_{IC}& =1-\frac{O[Het]}{E[Het|\text{p}]}\\ & =1-\frac{\sum _{i,j,k}{g}_{i,j,k}I[i\ne j]}{\sum _{i,j}P(G_{i,j}|\text{p})I[i\ne j]}\\ \end{array}$

The following equation gives the estimate of F where genotype likelihoods are available. $P(R_k|G_{i,j})$ is the genotype likelihood.

$\begin{array}{rl}{F}_{IC}& =1-\frac{O[Het]}{E[Het|\text{p}]}\\ & =1-\frac{\sum _{i,j,k}P(G_{i,j}|R_k,\text{p})I[i\ne j]}{\sum _{i,j}P(G_{i,j}|\text{p})I[i\ne j]}\\ & =1-\frac{\sum _{i,j,k}\frac{P(R_k|G_{i,j})P(G_{i,j}|\text{p})}{\sum _{i^{\prime },j^{\prime }}P(R_k|G_{i^{\prime },j^{\prime }})P(G_{i^{\prime },j^{\prime }}|\text{p})}I[i\ne j]}{\sum _{i,j}P(G_{i,j}|\text{p})I[i\ne j]}\\ \end{array}$

Derivation

Adrian with much help from Hyun.

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