FIC

Formulation

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.$P(G_{i,j}|{\textbf {p}})$ is the estimated genotype allele frequency for genotype $G_{i,j}$ under HWE assumption. $I[i\neq j]$ is an indicator function for heterozygote genotypes.

${\begin{aligned}F_{IC}&=1-{\frac {O[Het]}{E[Het|{\textbf {p}}]}}\\&=1-{\frac {\sum _{i,j,k}{g_{i,j,k}I[i\neq j]}}{\sum _{i,j}{P(G_{i,j}|{\textbf {p}})I[i\neq j]}}}\\\end{aligned}}$

The following equation gives the estimate of F where genotype likelihoods are available. $P(R_{k}|G_{i,j})$ is the genotype likelihood for individual $k$ given genotype $G_{i,j}$. This is basically the probability of observing the reads individual $k$ assuming $G_{i,j}$ is the underlying true genotype.

${\begin{aligned}F_{IC}&=1-{\frac {O[Het]}{E[Het|{\textbf {p}}]}}\\&=1-{\frac {\sum _{i,j,k}{P(G_{i,j}|R_{k},{\textbf {p}})I[i\neq j]}}{\sum _{i,j}{P(G_{i,j}|{\textbf {p}})I[i\neq j]}}}\\&=1-{\frac {\sum _{i,j,k}{\frac {P(R_{k}|G_{i,j})P(G_{i,j}|{\textbf {p}})}{\sum _{i',j'}{P(R_{k}|G_{i',j'})P(G_{i',j'}|{\textbf {p}})}}}I[i\neq j]}{\sum _{i,j}{P(G_{i,j}|{\textbf {p}})I[i\neq j]}}}\\\end{aligned}}$

Derivation

Adrian with much help from Hyun.

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