User:Timothee Flutre/Notebook/Postdoc/2011/11/10: Difference between revisions
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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> | |||
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Bayesian model of univariate linear regression for QTL detectionSee Servin & Stephens (PLoS Genetics, 2007).
[math]\displaystyle{ \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]\displaystyle{ \epsilon_i \overset{i.i.d}{\sim} \mathcal{N}(0,\tau^{-1}) }[/math] where [math]\displaystyle{ \beta_1 }[/math] is in fact the additive effect of the SNP, noted [math]\displaystyle{ a }[/math] from now on, and [math]\displaystyle{ \beta_2 }[/math] is the dominance effect of the SNP, [math]\displaystyle{ d = a k }[/math]. Let's now write in matrix notation: [math]\displaystyle{ Y = X B + E }[/math] where [math]\displaystyle{ B = [ \mu \; a \; d ]^T }[/math] which gives the following conditional distribution for the phenotypes: [math]\displaystyle{ Y | X, B, \tau \sim \mathcal{N}(XB, \tau^{-1} I_N) }[/math]
[math]\displaystyle{ \tau \sim \Gamma(\kappa/2, \, \lambda/2) }[/math] [math]\displaystyle{ 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] |