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

From OpenWetWare
< User:Timothee Flutre‎ | Notebook‎ | Postdoc‎ | 2011‎ | 11
Revision as of 09:38, 21 November 2012 by Timothee Flutre (talk | contribs) (→‎Entry title: first version)
Jump to navigationJump to search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.
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 [math]\displaystyle{ y_1,\ldots,y_N }[/math] the (quantitative) phenotypes (e.g. expression level at a given gene), and [math]\displaystyle{ 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]\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]


  • Priors: conjugate

[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]


Retrieved from "https://openwetware.org/mediawiki/index.php?title=User:Timothee_Flutre/Notebook/Postdoc/2011/11/10&oldid=657984"