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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
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| <math> \vec{f_{i,k}} </math> ~ <math> MVN(\mu, \sum) </math>
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| <math> \mu = (1-\theta)f^{panel} + \frac{\theta}{2} 1 </math>
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| <math> \sum = (1-\theta)^2 S + \frac{\theta}{2}(1 - \frac{\theta}{2})I </math>
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| where <math> S_{i,j} = \left \sum_{i,j}^{panel} i = j \\ exp(-\frac{\rho_{i,j}}{2m} \sum_{i,j}^{panel}) i \neqq j \right </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}</math> ~ <math> MVN(\bar{\mu}, \bar{\sum}) </math>
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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
