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

From OpenWetWare

(Difference between revisions)
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 13:46, 21 November 2012

Project name Main project page
Previous entry      Next entry

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 y_1,\ldots,y_N the (quantitative) phenotypes (e.g. expression level at a given gene), and g_1,\ldots,g_N 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: \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})

where β1 is in fact the additive effect of the SNP, noted a from now on, and β2 is the dominance effect of the SNP, d = ak.

Let's now write in matrix notation:

Y = X B + E \text{ where } B = [ \mu \; a \; d ]^T

which gives the following conditional distribution for the phenotypes:

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


  • Priors: conjugate

\tau \sim \Gamma(\kappa/2, \, \lambda/2)

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)


  • Joint posterior:

\mathsf{P}(\tau, B | Y, X) = \mathsf{P}(\tau | Y, X) \mathsf{P}(B | Y, X, \tau)


  • Conditional posterior of B:

\mathsf{P}(B | Y, X, \tau) = \mathsf{P}(B, Y | X, \tau)

\mathsf{P}(B | Y, X, \tau) = \frac{\mathsf{P}(B, Y | X, \tau)}{\mathsf{P}(Y | X, \tau)}

\mathsf{P}(B | Y, X, \tau) = \frac{\mathsf{P}(B | \tau) \mathsf{P}(Y | X, B, \tau)}{\int \mathsf{P}(B | \tau) \mathsf{P}(Y | X, \tau, B) \mathsf{d}B}

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

\mathsf{P}(B | Y, X, \tau) \propto \mathsf{P}(B | \tau) \mathsf{P}(Y | X, \tau, B)

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

\mathsf{P}(B | Y, X, \tau) \propto exp(B^T \Sigma_B^{-1} B) exp((Y-XB)^T(Y-XB))

We expand:

\mathsf{P}(B | Y, X, \tau) \propto exp(B^T \Sigma_B^{-1} B - Y^TXB -B^TX^TY + B^TX^TXB)

We factorize some terms:

\mathsf{P}(B | Y, X, \tau) \propto exp(B^T (\Sigma_B^{-1} + X^TX) B - Y^TXB -B^TX^TY)

Let's define \Omega = (\Sigma_B^{-1} + X^TX)^{-1}. We can see that ΩT = Ω, which means that Ω is a symmetric matrix. This is particularly useful here because we can use the following equality: Ω − 1ΩT = I.

\mathsf{P}(B | Y, X, \tau) \propto exp(B^T \Omega^{-1} B - (X^TY)^T\Omega^{-1}\Omega^TB -B^T\Omega^{-1}\Omega^TX^TY)

This now becomes easy to factorizes totally:

\mathsf{P}(B | Y, X, \tau) \propto exp((B^T - \Omega X^TY)^T\Omega^{-1}(B - \Omega X^TY))

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

B | Y, X, \tau \sim \mathcal{N}(\Omega X^TY, \tau^{-1} \Omega)


  • Posterior of τ:

Similarly to the equations above:

\mathsf{P}(\tau | Y, X) \propto \mathsf{P}(\tau) \mathsf{P}(Y | X, \tau)

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

\mathsf{P}(\tau | Y, X) \propto \mathsf{P}(\tau) \int \mathsf{P}(B | \tau) \mathsf{P}(Y | X, \tau, B) \mathsf{d}B


Retrieved from "http://openwetware.org/wiki/User:Timothee_Flutre/Notebook/Postdoc/2011/11/10"
Personal tools