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>  
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Revision as of 11:38, 21 November 2012
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Bayesian model of univariate linear regression for QTL detectionSee Servin & Stephens (PLoS Genetics, 2007).
with: where β_{1} is in fact the additive effect of the SNP, noted a from now on, and β_{2} is the dominance effect of the SNP, d = ak. Let's now write in matrix notation: Y = XB + E where which gives the following conditional distribution for the phenotypes:
