User:Hussein Alasadi/Notebook/stephens/2013/10/03

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analyzing pooled sequenced data with selection Main project page
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Notes from Meeting

Consider a single lineage for now.

Xj = frequency of "1" allele at SNP j in the pool (i.e. the true frequency of the 1 allele in the pool)

  • Data:

 (n_j^0, n_j^1) = number of "0", "1" alleles at SNP j ( n_j = n_j^0 + n_j^1 )

  • Normal approximation

 n_j^1 ~ Bin(n_j, X_j) \approx N(n_jX_j, n_jX_j(1-X_j)) Normal approximation to binomial

 \frac{n_j^1}{n_j} \approx N(X_j, \frac{X_j(1-X_j)}{n_j}) The variance of this distribution results from error due to binomial sampling.

To simplify, we just plug in \hat{X_j} = \frac{n_j^1}{n_j} for Xj

 \implies \frac{n_j^1}{n_j} | X_j \approx N(X_j, \frac{\hat{X_j}(1-\hat{X_j})}{n_j})

  • notation

fi,k,j = frequency of reference allele in group i, replicate and SNP j.

 \vec{f_{i,k}} =  vector of frequencies

Without loss of generality, we assume that the putative selected site is site j = 1

  • Model

We assume a prior on our vector of frequencies based on our panel of SNPs (M) of dimension 2mxp

 \vec{f_{i,k}} ~ MVN(μ,Σ)

 \mu = (1-\theta)f^{panel} + \frac{\theta}{2} 1

 \Sigma = (1-\theta)^2 S + \frac{\theta}{2}(1 - \frac{\theta}{2})I

where  S_{i,j} = \sum_{i,j}^{panel} if i = j or  e^{-\frac{\rho_{i,j}}{2m} \sum_{i,j}^{panel}} if i not equal to j

 \theta = \frac{(\sum_{i=1}^{2m-1} \frac{1}{i})^{-1}}{2m + (\sum_{i=1}^{2m-1} \frac{1}{i})^{-1}}

  • at selected site

 log \frac{f_{i,k,1}}{1-f_{i,k,1}} = \mu + \beta g_i + \epsilon_{i,k}

  • conditional distribution

(fi,k,2,....,fi,k,p) | fi,k,1,M ~  MVN(\bar{\mu}, \bar{\Sigma}) The conditional distribution is easily obtained when we use a result derived here.

let X2 = (fi,k,2,....,fi,k,p) and X1 = fi,k,1

X2 | X1,M ~  N(\vec{\mu_2} + \Sigma_{21} \Sigma_{11}^{-1} (x_1 - \mu_1), \Sigma_{22} - \Sigma_{21}\Sigma_{11}^{-1}\Sigma_{12})

Thus   \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}

And equivalently we could derive the distribution X1 | X2,M which is again f_{i,k,1} | f_{i,k,2}, .... , f_{i,k,p}), M

  • Likelihood for frequency a the test SNP t given all data

let f_{obs} = \prod_{j \not= t} f_{i,k,j}

 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)}

where  f_{i,k,t}^{true}  | M ~ N(μ,σ2Σ) The parameter σ2 allows for over-dispersion

where fobs | M ~ Np − 122Σ22 + ε2I) where ε2 allows for measurement error.

and I don't understand  f_{obs} | f_{i,k,t}^{true}, M . Shouldn't it come from (2.12) and not (2.13) - ask Matthew

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