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Atks: Redirected page to Genotype Likelihood based Allele Frequency

2013-06-04T17:38:41Z

Redirected page to Genotype Likelihood based Allele Frequency

← Older revision		Revision as of 13:38, 4 June 2013
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	~~=== Estimation of Genotype Frequencies without assuming HWE ===~~		#REDIRECT [[Genotype_Likelihood_based_Allele_Frequency]]

	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:

	~~<math>~~
	~~\begin{align}~~
	~~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)}}}~~
	~~\end{align}~~
	~~</math>~~

	~~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>.~~
	~~The initial genotype probability is given by:~~

	~~<math>~~
	~~\begin{align}~~
	~~P(G_{i,j})^{(0)} = f_{i,j}^{(0)} = \frac{2}{n(n+1)}~~
	~~\end{align}~~
	~~</math>~~

	~~The E step equates the expectation of the genotype <math>G_{i,j}</math> for individual k as:~~

	~~<math>~~
	~~\begin{align}~~
	~~E[G_{i,j}\|R_{k}]^{(l)}=P(G_{i,j}\|R_{k})^{(l)}~~
	~~\end{align}~~
	~~</math>~~

	~~The M step estimates the genotype frequency using the individual expected genotype counts:~~

	~~<math>~~
	~~\begin{align}~~
	~~P(G_{i,j})^{(l)} = f_{i,j}^{(l)} = \frac{1}{N}\sum_{k}{E[G_{i,j}\|R_{k}]}^{(l)}~~
	~~\end{align}~~
	~~</math>~~

	~~This is repeated till the appropriate convergence criteria is achieved.~~

	~~=== Estimation of Genotype Frequencies assuming HWE ===~~

	~~In order to estimate allele frequencies under HWE assumption, the E step estimates the individual expected posterior allele count for each individual.~~

	~~<math>~~
	~~\begin{align}~~
	~~E[I\|R_{k}]^{(l)}=P(G_{i,i}\|R_{k})^{(l)} + 0.5P(G_{i,j}\|R_{k})^{(l)}~~
	~~\end{align}~~
	~~</math>~~

	~~In the M step, the posterior genotype frequencies are derived from the computed genotype allele frequencies obtained in the E step assuming HWE.~~

	~~<math>~~
	~~\begin{align}~~
	~~P(I)^{(l)} = \frac{1}{N}\sum_{k}{E[I\|R_{k}]}^{(l)}~~
	~~\end{align}~~
	~~</math>~~

	~~<math>~~
	~~P(G_{i,j})^{(l)} = \begin{cases}~~
	~~(P(I)^{(l)})^2, & \text{if }i=j \\~~
	~~2P(I)^{(l)}P(J)^{(l)}, & \text{if }i \ne j~~
	~~\end{cases}~~
	~~</math>~~

	~~This is repeated till the appropriate convergence criteria is achieved.~~

	~~=== Used in ===~~

	[[~~HWEP\|Hardy-Weinberg Likelihood Test statistic]~~] ~~and [[FIC\| Inbreeding Coefficient]]~~

	~~=== Derivation ===~~

	~~Adrian with much help from Hyun.~~

	~~=== Maintained by ===~~

	~~This page is maintained by [mailto:atks@umich.edu Adrian~~].

Atks: /* Derivation by */

2013-04-11T18:04:14Z

Derivation by

← Older revision		Revision as of 14:04, 11 April 2013
Line 67:		Line 67:
	[[HWEP\|Hardy-Weinberg Likelihood Test statistic]] and [[FIC\| Inbreeding Coefficient]]		[[HWEP\|Hardy-Weinberg Likelihood Test statistic]] and [[FIC\| Inbreeding Coefficient]]

	=== Derivation by ===		=== Derivation ===

	Adrian with much help from Hyun.		Adrian with much help from Hyun.

Atks: /* Derivation by */

2013-04-11T18:03:37Z

Derivation by

← Older revision		Revision as of 14:03, 11 April 2013
Line 62:		Line 62:

	This is repeated till the appropriate convergence criteria is achieved.		This is repeated till the appropriate convergence criteria is achieved.

			=== Used in ===

			[[HWEP\|Hardy-Weinberg Likelihood Test statistic]] and [[FIC\| Inbreeding Coefficient]]

	=== Derivation by ===		=== Derivation by ===

Atks: /* Estimation of Genotype Frequencies without assuming HWE */

2013-04-11T18:02:22Z

Estimation of Genotype Frequencies without assuming HWE

← Older revision		Revision as of 14:02, 11 April 2013
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	</math>		</math>

	The E step equates the expectation of the genotype $G_{i,j}$ for individual k as:		The E step equates the expectation of the genotype <math>G_{i,j}</math> for individual k as:

	<math>		<math>

Atks: /* Estimation of Genotype Frequencies without assuming HWE */

2013-04-11T18:01:56Z

Estimation of Genotype Frequencies without assuming HWE

← Older revision		Revision as of 14:01, 11 April 2013
Line 9:		Line 9:
	</math>		</math>

	where $G_{i,j}$ denotes the genotype composed of alleles $i$ and $j$. $k$ indexes the individuals from $1$ to $N$.		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>.
	The initial genotype probability is given by:		The initial genotype probability is given by:

Atks at 17:34, 11 April 2013

2013-04-11T17:34:07Z

← Older revision		Revision as of 13:34, 11 April 2013
Line 63:		Line 63:
	This is repeated till the appropriate convergence criteria is achieved.		This is repeated till the appropriate convergence criteria is achieved.

			=== Derivation by ===

			Adrian with much help from Hyun.

	=== Maintained by ===		=== Maintained by ===

	This page is maintained by [mailto:atks@umich.edu Adrian].		This page is maintained by [mailto:atks@umich.edu Adrian].
	~~with much help from Hyun.~~

Atks: /* Estimation of Genotype Frequencies without assuming HWE */

2013-04-11T15:05:43Z

Estimation of Genotype Frequencies without assuming HWE

← Older revision		Revision as of 11:05, 11 April 2013
Line 1:		Line 1:
	=== Estimation of Genotype Frequencies without assuming HWE ===		=== Estimation of Genotype Frequencies without assuming HWE ===

	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 ($R_k$) for the $l$th iteration is given by: \\		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:

	<math>		<math>

Atks: Created page with '=== Estimation of Genotype Frequencies without assuming HWE === We propose an EM algorithm to estimate the genotype frequencies without assuming HWE. The posterior probability …'

2013-04-11T14:30:54Z

Created page with '=== Estimation of Genotype Frequencies without assuming HWE === We propose an EM algorithm to estimate the genotype frequencies without assuming HWE. The posterior probability …'

New page

=== Estimation of Genotype Frequencies without assuming HWE ===

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 ($R_k$) for the $l$th iteration is given by: \\

<math>
\begin{align}
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)}}}
\end{align}
</math>

where $G_{i,j}$ denotes the genotype composed of alleles $i$ and $j$. $k$ indexes the individuals from $1$ to $N$.
The initial genotype probability is given by:

<math>
\begin{align}
P(G_{i,j})^{(0)} = f_{i,j}^{(0)} = \frac{2}{n(n+1)}
\end{align}
</math>

The E step equates the expectation of the genotype $G_{i,j}$ for individual k as:

<math>
\begin{align}
E[G_{i,j}|R_{k}]^{(l)}=P(G_{i,j}|R_{k})^{(l)}
\end{align}
</math>

The M step estimates the genotype frequency using the individual expected genotype counts:

<math>
\begin{align}
P(G_{i,j})^{(l)} = f_{i,j}^{(l)} = \frac{1}{N}\sum_{k}{E[G_{i,j}|R_{k}]}^{(l)}
\end{align}
</math>

This is repeated till the appropriate convergence criteria is achieved.

=== Estimation of Genotype Frequencies assuming HWE ===

In order to estimate allele frequencies under HWE assumption, the E step estimates the individual expected posterior allele count for each individual.

<math>
\begin{align}
E[I|R_{k}]^{(l)}=P(G_{i,i}|R_{k})^{(l)} + 0.5P(G_{i,j}|R_{k})^{(l)}
\end{align}
</math>

In the M step, the posterior genotype frequencies are derived from the computed genotype allele frequencies obtained in the E step assuming HWE.

<math>
\begin{align}
P(I)^{(l)} = \frac{1}{N}\sum_{k}{E[I|R_{k}]}^{(l)}
\end{align}
</math>

<math>
P(G_{i,j})^{(l)} = \begin{cases}
(P(I)^{(l)})^2, & \text{if }i=j \\
2P(I)^{(l)}P(J)^{(l)}, & \text{if }i \ne j
\end{cases}
</math>

This is repeated till the appropriate convergence criteria is achieved.

=== Maintained by ===

This page is maintained by [mailto:atks@umich.edu Adrian].
with much help from Hyun.