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|style="background-color: #EEE"|[[Image:owwnotebook_icon.png|128px]]<span style="font-size:22px;"> analyzing pooled sequenced data with selection</span>
 
|style="background-color: #EEE"|[[Image:owwnotebook_icon.png|128px]]<span style="font-size:22px;"> analyzing pooled sequenced data with selection</span>
|style="background-color: #F2F2F2" align="center"|<html><img src="/images/9/94/Report.png" border="0" /></html> [[{{#sub:{{FULLPAGENAME}}|0|-11}}|Main project page]]<br />{{#if:{{#lnpreventry:{{FULLPAGENAME}}}}|<html><img src="/images/c/c3/Resultset_previous.png" border="0" /></html>[[{{#lnpreventry:{{FULLPAGENAME}}}}{{!}}Previous entry]]<html>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</html>}}{{#if:{{#lnnextentry:{{FULLPAGENAME}}}}|[[{{#lnnextentry:{{FULLPAGENAME}}}}{{!}}Next entry]]<html><img src="/images/5/5c/Resultset_next.png" border="0" /></html>}}
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==Notes from Meeting==
 
Consider a single lineage for now.
 
 
<math>X_j</math> = frequency of "1" allele at SNP j in the pool (i.e. the true frequency of the 1 allele in the pool)
 
 
*'''Data:'''
 
<math> (n_j^0, n_j^1) </math> = number of "0", "1" alleles at SNP j (<math> n_j = n_j^0 + n_j^1 </math>)
 
 
 
*'''Normal approximation'''
 
<math> n_j^1</math> ~ <math>Bin(n_j, X_j) \approx N(n_jX_j, n_jX_j(1-X_j))</math> Normal approximation to binomial
 
 
<math> \frac{n_j^1}{n_j} \approx N(X_j, \frac{X_j(1-X_j)}{n_j}) </math>
 
The variance of this distribution results from error due to binomial sampling.
 
 
To simplify, we just plug in <math>\hat{X_j} = \frac{n_j^1}{n_j}</math> for <math> X_j </math>
 
 
<math> \implies \frac{n_j^1}{n_j} | X_j \approx N(X_j, \frac{\hat{X_j}(1-\hat{X_j})}{n_j}) </math>
 
 
*'''notation'''
 
 
<math>f_{i,k,j} = </math> frequency of reference allele in group i, replicate and SNP j.
 
 
<math> \vec{f_{i,k}} =  </math> vector of frequencies
 
 
Without loss of generality, we assume that the putative selected site is site <math> j = 1 </math>
 
 
* '''Model'''
 
We assume a prior on our vector of frequencies based on our panel of SNPs <math> (M) </math> of dimension <math> 2mxp </math>
 
 
<math> \vec{f_{i,k}} </math> ~ <math> MVN(\mu, \sum) </math>
 
 
<math> \mu = (1-\theta)f^{panel} + \frac{\theta}{2} 1 </math>
 
 
<math> \sum = (1-\theta)^2 S + \frac{\theta}{2}(1 - \frac{\theta}{2})I </math>
 
 
where <math> S_{i,j} = \sum_{i,j}^{panel}</math> if i = j or <math> e^{-\frac{\rho_{i,j}}{2m} \sum_{i,j}^{panel}} </math> if i not equal to j
 
 
<math> \theta = \frac{(\sum_{i=1}^{2m-1} \frac{1}{i})^{-1}}{2m + (\sum_{i=1}^{2m-1} \frac{1}{i})^{-1}} </math>
 
 
 
* '''at selected site'''
 
<math> log \frac{f_{i,k,1}}{1-f_{i,k,1}} = \mu + \beta g_i + \epsilon_{i,k} </math>
 
 
* '''conditional distribution'''
 
<math> (f_{i,k,2}, .... , f_{i,k,p}) | f_{i,k,1}, M </math> ~ <math> MVN(\bar{\mu}, \bar{\sum}) </math>
 
The conditional distribution is easily obtained when we use a result derived [http://openwetware.org/wiki/User:Hussein_Alasadi/Notebook/stephens/2013/10/14 here].
 
 
let <math> X_2 = (f_{i,k,2}, .... , f_{i,k,p}) </math> and <math> X_1 = f_{i,k,1} </math>
 
 
<math> X_2 | X_1, M </math> ~ <math> N(\vec{\mu_2} + \sigma_{21} \sigma_{11}^{-1} (x_1 - \mu_1), \sigma_{22} - \sigma_{21}\sigma_{11}^{-1}\sigma_{12}) </math>
 
 
 
 
 
 
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Latest revision as of 22:27, 26 September 2017

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