Contrasts

Whether from the fast approximation approach or from MCMC plus BIDET, the output from the analysis at any level in the hierarchy gives us a multivariate non-central t-distribution (equation 16). As in the frequentist framework we can ask questions about linear combinations (or contrasts) $c^T\beta_g$ of the parameters in $\beta _g$.

If $c$ is a $P\times 1$ vector representing a t-contrast, we can use equation 16 to give us the univariate non-central t-distribution over $c^T\beta_g$

$$p(c^T\beta_g\vert Y)=\mathcal{T}(c^T\beta_g;c^T\mu_{\beta_g},\sigma_{\beta_g}^2c^T\Sigma_{\beta_g}c,\nu_{\beta_g})$$ (20)

We can then look at the $p(c^T\beta_g>0\vert Y)$. Note that this is equal to the probability of getting a t-value greater than the t-statistic:

$$t = c^T\mu_{\beta_g}/\sqrt{c^T\sigma_{\beta_g}^2\Sigma_{\beta_g}c}$$ (21)

under a central t-distribution with degrees of freedom $\nu_{\beta_g}$.