# 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 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 (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:

with:

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:

where

which gives the following conditional distribution for the phenotypes:

• Priors: conjugate