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

$\displaystyle \forall i \in \{1,\ldots,N\}, \; y_i = \mu + \beta_1 g_i + \beta_2 \mathbf{1}_{g_i=1} + \epsilon_i$

with: $\displaystyle \epsilon_i \overset{i.i.d}{\sim} \mathcal{N}(0,\tau^{-1})$

where $\displaystyle \beta_1$ is in fact the additive effect of the SNP, noted $\displaystyle a$ from now on, and $\displaystyle \beta_2$ is the dominance effect of the SNP, $\displaystyle d = a k$ .

Let's now write in matrix notation:

$\displaystyle Y = X B + E$

where $\displaystyle B = [ \mu \; a \; d ]^T$

which gives the following conditional distribution for the phenotypes:

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

• Priors: conjugate

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

$\displaystyle B | \tau \sim \mathcal{N}(\vec{0}, \, \tau^{-1} \Sigma_B) \text{ with } \Sigma_B = diag(\sigma_{\mu}^2, \sigma_a^2, \sigma_d^2)$