BEANS: Statistical Models
Consider a scenario with three discrete covariates \(B_1,B_2,B_3\). Let
\(Y\) denote the response. Let \(G\) denote a subgroup defined by the
combination of \(B_1,B_2,B_3\).
\begin{align*}
Y|G=g &\sim N(\theta_g, \sigma^2) \\
\theta_g &\equiv \tau \\
\tau &\sim N(0, 1000) \\
\sigma^2 &\sim \mbox{Gamma}(0.001,0.001)
\end{align*}
Full stratification
\begin{align*}
Y| G=g &\sim N(\theta_g, \sigma^2) \\
\theta_g &\sim N(0, 1000) \\
\sigma^2 &\sim \mbox{Gamma}(0.001,0.001)
\end{align*}
\begin{align*}
Y|B_1,B_2,B_3 & \sim N(\theta_g,\sigma^2) \\
\theta_g &= \beta_0 + \beta_1 B_1 + \beta_2 B_2 + \beta_3 B_3 \\
\beta_k &\sim N(0,1000) \qquad k=1,\ldots,3\\
\sigma^2 &\sim \mbox{Gamma}(0.001,0.001)
\end{align*}
\begin{align*}
Y|B_1,B_2,B_3 & \sim N(\theta_g,\sigma^2) \\
\theta_g &= \tau + \gamma_1 B_1 + \gamma_2 B_2 + \gamma_3 B_3 + \psi_g$
\beta_k &\sim N(0,1000) \qquad k=1,\ldots,3\\
\sigma^2 &\sim \mbox{Gamma}(0.001,0.001)
\end{align*}
\begin{align*}
&\theta_g = \tau + \gamma_1 B_1 + \gamma_2 B_2 + \gamma_3 B_3 + \psi_g \\
&\tau, \gamma_1,\gamma_2,\gamma_3 \sim N(0, 10^6)\\
&\psi_g \sim N(0, w^2)\\
&w \sim Half N(1)
\end{align*}
\begin{align*}
\theta_g &= \tau + \gamma_1 B_1 + \gamma_2 B_2 + \gamma_3 B_3 \\
\tau &\sim N(0, 10^6) \\
\gamma_k &\sim N(0, w^2)\\
w &\sim Half N(1)
\end{align*}
\begin{align*}
\theta_g &= \tau + \gamma_1 B_1 + \gamma_2 B_2 + \gamma_3 B_3 + \delta_1
B_1B_2 + \delta_2 B_1 B_3 + \delta_3 B_2B_3 + \alpha B_1B_2B_3 \\
\tau &\sim N(0, 10^6) \\
\gamma_k &\sim N(0, w_1^2)\\
\delta_k &\sim N(0, w_2^2)\\
\alpha &\sim N(0,w_3^2) \\
w_l & \sim Half N(1)
\end{align*}