analyzing pooled sequenced data with selection Main project page
Next entry

## 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 $\bar{\mu} = \vec{\mu_2} + \Sigma_{21} \Sigma_{11}^{-1} (x_1 - \mu_1), \bar{\Sigma} = \Sigma_{22} - \Sigma_{21}\Sigma_{11}^{-1}\Sigma_{12}$

• 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