Genotype Likelihood based Inbreeding Coefficient

Introduction

Inbreeding Coefficient is an important statistic in the study of genetic variants. This page details EM algorithms to estimate inbreeding coefficients from genotype likelihoods in NGS data.

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 in individual $k$ assuming $G_{i,j}$ is the underlying true genotype for that particular locus.

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

Derivation

Adrian with much help from Hyun.

Maintained by

This page is maintained by Adrian.