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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> </html>}}{{#if:{{#lnnextentry:{{FULLPAGENAME}}}}|[[{{#lnnextentry:{{FULLPAGENAME}}}}{{!}}Next entry]]<html><img src="/images/5/5c/Resultset_next.png" border="0" /></html>}} | | |style="background-color: #F2F2F2" align="center"|[[File:Report.png|frameless|link={{#sub:{{FULLPAGENAME}}|0|-11}}]][[{{#sub:{{FULLPAGENAME}}|0|-11}}|Main project page]]<br />{{#if:{{#lnpreventry:{{FULLPAGENAME}}}}|[[File:Resultset_previous.png|frameless|link={{#lnpreventry:{{FULLPAGENAME}}}}]][[{{#lnpreventry:{{FULLPAGENAME}}}}{{!}}Previous entry]] }}{{#if:{{#lnnextentry:{{FULLPAGENAME}}}}|[[{{#lnnextentry:{{FULLPAGENAME}}}}{{!}}Next entry]][[File:Resultset_next.png|frameless|link={{#lnnextentry:{{FULLPAGENAME}}}}]]}} |
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| ==Notes from Meeting==
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| Consider a single lineage for now.
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| <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)
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| *'''Data:'''
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| <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>)
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| *'''Normal approximation'''
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| <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
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| <math> \frac{n_j^1}{n_j} \approx N(X_j, \frac{X_j(1-X_j)}{n_j}) </math>
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| The variance of this distribution results from error due to binomial sampling.
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| To simplify, we just plug in <math>\hat{X_j} = \frac{n_j^1}{n_j}</math> for <math> X_j </math>
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| <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>
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| *'''notation'''
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| <math>f_{i,k,j} = </math> frequency of reference allele in group i, replicate and SNP j.
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| <math> \vec{f_{i,k}} = </math> vector of frequencies
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| Without loss of generality, we assume that the putative selected site is site <math> j = 1 </math>
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| * '''Model'''
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| We assume a prior on our vector of frequencies based on our panel of SNPs <math> (M) </math> of dimension <math> 2mxp </math>
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| <math> \vec{f_{i,k}} </math> ~ <math> MVN(\mu, \Sigma) </math>
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| <math> \mu = (1-\theta)f^{panel} + \frac{\theta}{2} 1 </math>
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| <math> \Sigma = (1-\theta)^2 S + \frac{\theta}{2}(1 - \frac{\theta}{2})I </math>
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| 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
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| <math> \theta = \frac{(\sum_{i=1}^{2m-1} \frac{1}{i})^{-1}}{2m + (\sum_{i=1}^{2m-1} \frac{1}{i})^{-1}} </math>
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| * '''at selected site'''
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| <math> log \frac{f_{i,k,1}}{1-f_{i,k,1}} = \mu + \beta g_i + \epsilon_{i,k} </math>
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| * '''conditional distribution'''
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| <math> (f_{i,k,2}, .... , f_{i,k,p}) | f_{i,k,1}, M </math> ~ <math> MVN(\bar{\mu}, \bar{\Sigma}) </math>
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| 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].
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| let <math> X_2 = (f_{i,k,2}, .... , f_{i,k,p}) </math> and <math> X_1 = f_{i,k,1} </math>
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| <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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| Thus <math> \bar{\mu} = \vec{\mu_2} + \Sigma_{21} \Sigma_{11}^{-1} (x_1 - \mu_1), \bar{\Sigma} = \Sigma_{22} - \Sigma_{21}\Sigma_{11}^{-1}\Sigma_{12} </math>
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| And equivalently we could derive the distribution <math> X_1 | X_2, M </math> which is again f_{i,k,1} | f_{i,k,2}, .... , f_{i,k,p}), M
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| *'''Likelihood for frequency a the test SNP t given all data'''
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| let <math>f_{obs} = \prod_{j \not= t} f_{i,k,j} </math>
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| <math> L(f_{i,k,t}^{true}) = P(f_{obs} | f_{i,k,t}^{true}, M) = \frac{P( f_{i,k,t}^{true} | M, f_{obs}) P(f^{obs}|M)}{P(f_{i,k,t}^{true} | M)}</math>
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| where <math> f_{i,k,t}^{true} | M </math> ~ <math> N(\mu, \sigma^2 \Sigma) </math> The parameter <math> \sigma^2 </math> allows for over-dispersion
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| where <math> f^{obs}| M </math> ~ <math> N_{p-1} (\mu_2, \sigma^2 \Sigma_{22} + \epsilon^2 I) </math> where <math> \epsilon^2 </math> allows for measurement error.
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| and I don't understand <math> f_{obs} | f_{i,k,t}^{true}, M </math>. Shouldn't it come from (2.12) and not (2.13) - ask Matthew
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| <!-- ##### DO NOT edit below this line unless you know what you are doing. ##### --> | | <!-- ##### DO NOT edit below this line unless you know what you are doing. ##### --> |
| |} | | |} |
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| __NOTOC__ | | __NOTOC__ |
analyzing pooled sequenced data with selection
Main project page
