HWEP

Formulation

Hardy Weinberg equilibrium is expected in a panmictic population. The following formulation is a likelihood ratio test statistic that incorporates genotype uncertainty via genotype likelihoods. $P(R_k|\text{p})$ is the probability of observing the reads for individual $k$ assuming that a locus observes HWE. $P(R_k|\text{g})$ is the probability of observing the reads for individual $k$ assuming that a locus does not observe HWE. $G_{i,j}$ denotes the genotype composed of alleles $i$ and $j$ . $k$ indexes the individuals from $1$ to $N$ . $P(R_k|G_{i,j})$ is the genotype likelihood. $P(G_{i,j}|\text{p})$ and $P(G_{i,j}|\text{g})$ are the genotype frequencies estimated with and without HWE assumption respectively.

$\begin{array}{rl}L(R|g)& =\frac{\prod _{k}P(R_k|\text{p})}{\prod _{k}P(R_k|\text{g})}\\ & =\frac{\prod _k\sum _{i,j}P(R_k,G_{i,j}|\text{p})}{\prod _k\sum _{i,j}P(R_k,G_{i,j}|\text{g})}\\ & =\frac{\prod _k\sum _{i,j}P(R_k|G_{i,j})P(G_{i,j}|\text{p})}{\prod _k\sum _{i,j}P(R_k|G_{i,j})P(G_{i,j}|\text{g})}\\ \end{array}$

The likelihood ratio test statistic is as follows with $v$ degrees of freedom where $n$ is the number of alleles.

$\begin{array}{r}-2logL(R|g)\sim X_v^2,v=\frac{n(n-1)}{2}\end{array}$

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