matrix containing the first order interactions as $\Omega_X = A^T \Omega A$. Notice that we could define $\Omega$ as a diagonal matrix and we would still estimate pairwise interactions between the regressors, further details are given in *Sections 2.3* and *2.5*.

In epidemiology studies, it is of interest to include interactions between chemical exposures and demographic covariates. The covariates are often binary variables, like *race* or *sex*, or continuous variables that are non-normally distributed, like *age*. Hence, we do not want to assume a latent normal structure for the covariates. Letting $Z_i = (z_{i1}, \cdots, z_{iq})^T$ be a vector of covariates, we modify model $(1)$ to include a main effect for $Z_i$ and an interaction term between $Z_i$ and the latent factor $\eta_i$: $$\begin{aligned}
& y_i = \eta_i^T \omega + \eta_i^T \Omega \eta_i + Z_i^T \alpha + \eta_i^T \Delta Z_i + \epsilon_{y,i},  \quad \epsilon_{y,i} \sim N(0,\sigma^2) \nonumber,  \\ 
& X_i = \Lambda \eta_i+ \epsilon_i,   \quad  \epsilon_i \sim N_p(0,\Psi), \\
& \eta_i \sim N_k(0,I), \nonumber\end{aligned}$$ where $\Delta$ is a $k \times q$ matrix of interaction coefficients between the latent variables and the covariates, and $\alpha = (\alpha_1, \cdots, \alpha_q)$ are main effects for the covariates. Following *Proposition 1* we have that 