Difference between revisions of "User:Hussein Alasadi/Notebook/stephens/2013/10/03"

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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>
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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)
<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>
<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, \Sigma) </math>
<math> \mu = (1-\theta)f^{panel} + \frac{\theta}{2} 1 </math>
<math> \Sigma = (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{\Sigma}) </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>
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>
And equivalently we could derive the distribution <math> X_1 | X_2, M </math> which is again <math> f_{i,k,1} | f_{i,k,2}, .... , f_{i,k,p}),  M </math>
*'''Likelihood for frequency a the test SNP t given all data'''
let <math>f_{obs} = \prod_{j \not= t} f_{i,k,j} </math>
<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>
Confused here, can we just use the expression derived above for <math>P( f_{i,k,t}^{true}  | M, f_{obs})  </math>. Also, isn't <math> f_{i,k,t}^{true} | M </math> ~
<math> N(\mu_1, \Sigma_{11}) </math> and <math> f^{obs} | M </math> ~ <math> N(\mu_2, \Sigma_{22}) </math>. But, how do we then incorporate <math> \beta </math> into the likelihood calculation?
But maybe we want to incorporate dispersion and measurement error parameters
<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
<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.
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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