Difference between revisions of "User:Timothee Flutre/Notebook/Postdoc/2011/11/10"
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− | == | + | ==Bayesian model of univariate linear regression for QTL detection== |
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+ | ''See Servin & Stephens (PLoS Genetics, 2007).'' | ||
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+ | * '''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). | ||
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+ | * '''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. | ||
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+ | * '''Assumptions''': the relationship between genotype and phenotype is linear; the individuals are not genetically related; there is no hidden confounding factors in the phenotypes. | ||
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+ | * '''Likelihood''': | ||
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+ | <math>\forall i \in \{1,\ldots,N\}, \; y_i = \mu + \beta_1 g_i + \beta_2 \mathbf{1}_{g_i=1} + \epsilon_i</math> | ||
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+ | with: <math>\epsilon_i \overset{i.i.d}{\sim} \mathcal{N}(0,\tau^{-1})</math> | ||
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+ | 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>. | ||
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+ | Let's now write in matrix notation: | ||
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+ | <math>Y = X B + E</math> | ||
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+ | where <math>B = [ \mu \; a \; d ]^T</math> | ||
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+ | which gives the following conditional distribution for the phenotypes: | ||
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+ | <math>Y | X, B, \tau \sim \mathcal{N}(XB, \tau^{-1} I_N)</math> | ||
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+ | * '''Priors''': conjugate | ||
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+ | <math>\tau \sim \Gamma(\kappa/2, \, \lambda/2)</math> | ||
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+ | <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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Revision as of 08:38, 21 November 2012
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Bayesian model of univariate linear regression for QTL detectionSee Servin & Stephens (PLoS Genetics, 2007).
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \forall i \in \{1,\ldots,N\}, \; y_i = \mu + \beta_1 g_i + \beta_2 \mathbf{1}_{g_i=1} + \epsilon_i} with: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \epsilon_i \overset{i.i.d}{\sim} \mathcal{N}(0,\tau^{-1})} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \beta_1} is in fact the additive effect of the SNP, noted Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle a} from now on, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \beta_2} is the dominance effect of the SNP, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle d = a k} . Let's now write in matrix notation: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle Y = X B + E} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle B = [ \mu \; a \; d ]^T} which gives the following conditional distribution for the phenotypes: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle Y | X, B, \tau \sim \mathcal{N}(XB, \tau^{-1} I_N)}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \tau \sim \Gamma(\kappa/2, \, \lambda/2)} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\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)} |