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| === Estimation of Genotype Frequencies without assuming HWE ===
| | #REDIRECT [[Genotype_Likelihood_based_Allele_Frequency]] |
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| We propose an EM algorithm to estimate the genotype frequencies without assuming HWE. The posterior probability of the genotype given the reads for individual k (<math>R_k</math>) for the <math>l</math>th iteration is given by:
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| <math>
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| \begin{align}
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| P(G_{i,j}|R_{k})^{(l)}=\frac{P(R_{k}|G_{i,j})P(G_{i,j})^{(l-1)}}{\sum_{(i,j)}{P(R_{k}|G_{i,j})P(G_{i,j})^{(l-1)}}}
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| \end{align}
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| </math>
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| where <math>G_{i,j}</math> denotes the genotype composed of alleles <math>i</math> and <math>j</math>. <math>k</math> indexes the individuals from <math>1</math> to <math>N</math>.
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| The initial genotype probability is given by:
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| <math>
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| \begin{align}
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| P(G_{i,j})^{(0)} = f_{i,j}^{(0)} = \frac{2}{n(n+1)}
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| \end{align}
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| </math>
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| The E step equates the expectation of the genotype <math>G_{i,j}</math> for individual k as:
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|
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| <math>
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| \begin{align}
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| E[G_{i,j}|R_{k}]^{(l)}=P(G_{i,j}|R_{k})^{(l)}
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| \end{align}
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| </math>
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| The M step estimates the genotype frequency using the individual expected genotype counts:
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|
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| <math>
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| \begin{align}
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| P(G_{i,j})^{(l)} = f_{i,j}^{(l)} = \frac{1}{N}\sum_{k}{E[G_{i,j}|R_{k}]}^{(l)}
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| \end{align}
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| </math>
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| This is repeated till the appropriate convergence criteria is achieved.
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| === Estimation of Genotype Frequencies assuming HWE ===
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| In order to estimate allele frequencies under HWE assumption, the E step estimates the individual expected posterior allele count for each individual.
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| <math>
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| \begin{align}
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| E[I|R_{k}]^{(l)}=P(G_{i,i}|R_{k})^{(l)} + 0.5P(G_{i,j}|R_{k})^{(l)}
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| \end{align}
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| </math>
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| In the M step, the posterior genotype frequencies are derived from the computed genotype allele frequencies obtained in the E step assuming HWE.
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| <math>
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| \begin{align}
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| P(I)^{(l)} = \frac{1}{N}\sum_{k}{E[I|R_{k}]}^{(l)}
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| \end{align}
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| </math>
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| <math>
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| P(G_{i,j})^{(l)} = \begin{cases}
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| (P(I)^{(l)})^2, & \text{if }i=j \\
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| 2P(I)^{(l)}P(J)^{(l)}, & \text{if }i \ne j
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| \end{cases}
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| </math>
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|
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| This is repeated till the appropriate convergence criteria is achieved.
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| === Derivation by ===
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| Adrian with much help from Hyun.
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| === Maintained by ===
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| This page is maintained by [mailto:atks@umich.edu Adrian].
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