User:Hussein Alasadi/Notebook/stephens/2013/10/13

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analyzing pooled sequenced data with selection Main project page
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Intro to Wen & Stephens in 2D

Suppose we have only summary-level data for haplotypes h1,h2,...,h2n. Specifically let the summary-level data be denoted by y = (y_1, y_2)' = \frac{1}{2n} \sum_i^{2n} h_i. We assume in this two locus model, the first locus is typed and the second locus is untyped. We hope to predict what the allele frequency is at the untyped SNP using information from panel data (perhaps this can be interpreted as our prior). Formally, let y1 denote the allele frequency at the typed SNP and y2 the allele frequency at the untyped SNP. We assume that h1,h2,...,h2n are independent and identically distributed from P(M) (our prior).

We assume that y = N(μ,Σ). By properties of bi-variate normal distributions y2 / y1,M ~ N(\mu_2 + \rho \frac{\sigma_2}{\sigma_1}(y_1 - u_1), (1-\rho^2)\sigma_1^2) where \rho = \frac{E[y_1y_2]}{\sigma_1 \sigma_2} (for any partition of y into y1,y2 ). The genius of Wen & Stephens lies in the idea that the distribution of y2 (assign as the vector of untyped SNPs) is a function of both the panel data (μ2) and the typed SNPs (y1).

Li & Stephens in 2D

We describe the Li & Stephens haplotype copying model: Let h1,h2,...,hk denote the k sampled haplotypes at 2 loci. Thus there are 4 possible haplotypes. The first haplotype is randomly chosen with equal probability from the four possible haplotypes.

Consider now the conditional distribution of hk + 1 given h1,h2,...,hk. Recall the intuition is that hk + 1 is a mosaic of h1,h2,..,hk.

Let Xj denote which hapolotype hk + 1 copies at site j (so X_j \in {1,2,..,k}).

We model Xj as a markov chain on 1,..,k with P(x_1 =x) = \frac{1}{k}. The transition probabilities are:

P(X_{j+1}=x'/X_j = x) = e^{-\frac{\rho_jd_j}{k}} + (1-e^{-\rho_jd_j})(1/k) if x' = x and

(1-e^{-\frac{\rho_jd_j}{k}})(1/k) otherwise. ρj and dj denote recombination and physical distances, respectively.

Now in a hidden markov model, there is also the transmission process. To mimic the effects of mutation, the copying process may be imperfect. P(h_{k+1,j} =a / X_j = x, h_1,..,h_k) = \frac{k}{k+\theta} + \frac{\theta}{2(k+\theta)} if hx,j = a and \frac{\theta}{2(k+\theta)} otherwise. \theta = (\sum_{m=1}^{n-1} \frac{1}{m})^{-1}, where the motivation is the more haplotypes the less frequent mutation occurs.

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