Consider the data model described in "data":
$$Y_{i} = \alpha_{0} + \sum_{t=1}^{q}\alpha_{t}Clin_{it} + \sum_{m=1}^{k}\theta_{m}E_{im} + \sum_{j=1}^{p} \big(U_{ij}^\top\beta_{j}\big) +\epsilon_{i},$$
where the main and interaction effects of the \(j\)th genetic variant is corresponding to the coefficient vector \(\beta_{j}=(\beta_{j1}, \beta_{j2},\ldots,\beta_{jL})^\top\).
When structure="sparsegroup" (default setting), selection will be conducted on both individual and group levels (bi-level selection):
Group-level selection: by determining whether \(||\beta_{j}||_{2}=0\), we can know if the \(j\)th genetic variant has any effect at all.
Individual-level selection: investigate whether the \(j\)th genetic variant has main effect, G\(\times\)E interaction or both, by determining which components in \(\beta_{j}\) has non-zero values.
If structure="group", only group-level selection will be conducted on \(||\beta_{j}||_{2}\). If structure="individual", only individual-level selection will be conducted on each \(\beta_{jl}\), (\(l=1,\ldots,L\)).
When sparse=TRUE (default), spike--and--slab priors are imposed on individual and/or group levels to identify important main and interaction effects. Otherwise, Laplacian shrinkage will be used.
When robust=TRUE (default), the distribution of \(\epsilon_{i}\) is defined as a Laplace distribution with density
\(
f(\epsilon_{i}|\nu) = \frac{\nu}{2}\exp\left\{-\nu |\epsilon_{i}|\right\}
\), (\(i=1,\dots,n\)), which leads to a Bayesian formulation of LAD regression. If robust=FALSE, \(\epsilon_{i}\) follows a normal distribution.
Both \(X\) and \(E\) will be centered before the generation of interaction terms, in order to prevent the multicollinearity between main effects and interaction terms.
Users can modify the hyper-parameters by providing a named list of hyper-parameters via the argument `hyper'.
The list can have the following named components
a0, b0: shape parameters of the Beta priors (\(\pi^{a_{0}-1}(1-\pi)^{b_{0}-1}\)) on \(\pi_{0}\).
a1, b1: shape parameters of the Beta priors (\(\pi^{a_{1}-1}(1-\pi)^{b_{1}-1}\)) on \(\pi_{1}\).
c1, c2: the shape parameter and the rate parameter of the Gamma prior on \(\nu\).
d1, d2: the shape parameter and the rate parameter of the Gamma priors on \(\eta\).
Please check the references for more details about the prior distributions.