User:Timothee Flutre/Notebook/Postdoc/2011/11/10

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Bayesian model of univariate linear regression for QTL detection

See Servin & Stephens (PLoS Genetics, 2007).

  • Data: let's assume that we obtained data from N individuals. We note [math]y_1,\ldots,y_N[/math] the (quantitative) phenotypes (e.g. expression level at a given gene), and [math]g_1,\ldots,g_N[/math] the genotypes at a given SNP (as allele dose, 0, 1 or 2).

  • Goal: we want (i) to assess the evidence in the data for an effect of the genotype on the phenotype, and (ii) estimate the posterior distribution of this effect.

  • Assumptions: the relationship between genotype and phenotype is linear; the individuals are not genetically related; there is no hidden confounding factors in the phenotypes.

  • Likelihood:

[math]\forall i \in \{1,\ldots,N\}, \; y_i = \mu + \beta_1 g_i + \beta_2 \mathbf{1}_{g_i=1} + \epsilon_i[/math]

with: [math]\epsilon_i \overset{i.i.d}{\sim} \mathcal{N}(0,\tau^{-1})[/math]

where [math]\beta_1[/math] is in fact the additive effect of the SNP, noted [math]a[/math] from now on, and [math]\beta_2[/math] is the dominance effect of the SNP, [math]d = a k[/math].

Let's now write in matrix notation:

[math]Y = X B + E[/math]

where [math]B = [ \mu \; a \; d ]^T[/math]

which gives the following conditional distribution for the phenotypes:

[math]Y | X, B, \tau \sim \mathcal{N}(XB, \tau^{-1} I_N)[/math]

  • Priors: conjugate

[math]\tau \sim \Gamma(\kappa/2, \, \lambda/2)[/math]

[math]B | \tau \sim \mathcal{N}(\vec{0}, \, \tau^{-1} \Sigma_B) \text{ with } \Sigma_B = diag(\sigma_{\mu}^2, \sigma_a^2, \sigma_d^2)[/math]