Difference between revisions of "User:Timothee Flutre/Notebook/Postdoc/2011/11/10"

From OpenWetWare
Jump to: navigation, search
(Bayesian model of univariate linear regression for QTL detection: add conditional posterior of B)
(Bayesian model of univariate linear regression for QTL detection: start posterior of tau)
Line 21: Line 21:
  
  
* '''Likelihood''':
+
* '''Likelihood''': <math>\forall i \in \{1,\ldots,N\}, \; y_i = \mu + \beta_1 g_i + \beta_2 \mathbf{1}_{g_i=1} + \epsilon_i \text{ with } \epsilon_i \overset{i.i.d}{\sim} \mathcal{N}(0,\tau^{-1})</math>
 
 
<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>.
 
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>.
Line 31: Line 27:
 
Let's now write in matrix notation:
 
Let's now write in matrix notation:
  
<math>Y = X B + E</math>
+
<math>Y = X B + E \text{ where } B = [ \mu \; a \; d ]^T</math>
 
 
where <math>B = [ \mu \; a \; d ]^T</math>
 
  
 
which gives the following conditional distribution for the phenotypes:
 
which gives the following conditional distribution for the phenotypes:
Line 88: Line 82:
  
 
<math>B | Y, X, \tau \sim \mathcal{N}(\Omega X^TY, \tau^{-1} \Omega)</math>
 
<math>B | Y, X, \tau \sim \mathcal{N}(\Omega X^TY, \tau^{-1} \Omega)</math>
 +
 +
 +
* '''Posterior of <math>\tau</math>''':
 +
 +
Similarly to the equations above:
 +
 +
<math>\mathsf{P}(\tau | Y, X) \propto \mathsf{P}(\tau) \mathsf{P}(Y | X, \tau)</math>
 +
 +
But now, to handle the second term, we need to integrate over <math>B</math>, thus effectively taking into account the uncertainty in <math>B</math>:
 +
 +
<math>\mathsf{P}(\tau | Y, X) \propto \mathsf{P}(\tau) \int \mathsf{P}(B | \tau) \mathsf{P}(Y | X, \tau, B) \mathsf{d}B</math>
  
 
<!-- ##### DO NOT edit below this line unless you know what you are doing. ##### -->
 
<!-- ##### DO NOT edit below this line unless you know what you are doing. ##### -->

Revision as of 09:46, 21 November 2012

Owwnotebook icon.png Project name <html><img src="/images/9/94/Report.png" border="0" /></html> Main project page
<html><img src="/images/c/c3/Resultset_previous.png" border="0" /></html>Previous entry<html>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</html>Next entry<html><img src="/images/5/5c/Resultset_next.png" border="0" /></html>

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 the (quantitative) phenotypes (e.g. expression level at a given gene), and the genotypes at a given SNP (as allele dose, 0, 1 or 2).


  • Goal: we want to assess the evidence in the data for an effect of the genotype on the phenotype.


  • 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:

where is in fact the additive effect of the SNP, noted from now on, and is the dominance effect of the SNP, .

Let's now write in matrix notation:

which gives the following conditional distribution for the phenotypes:


  • Priors: conjugate


  • Joint posterior:


  • Conditional posterior of B:

Here and in the following, we neglect all constants (e.g. normalization constant, , etc):

We use the prior and likelihood and keep only the terms in :

We expand:

We factorize some terms:

Let's define . We can see that , which means that is a symmetric matrix. This is particularly useful here because we can use the following equality: .

This now becomes easy to factorizes totally:

We recognize the kernel of a Normal distribution, allowing us to write the conditional posterior as:


  • Posterior of :

Similarly to the equations above:

But now, to handle the second term, we need to integrate over , thus effectively taking into account the uncertainty in :