Consider a clinical trial with three discrete covariates $B_1,B_2,B_3$ collected for each subject. Let $Y$ denote the response and $Z$ denote treatment arm. Let $G$ denote a subgroup defined by a combination of $B_1,B_2,B_3$. Let \[\theta_g=E(Y|Z=1,G=g)-E(Y|Z=0,G=g).\] The details of the available statistical models are given below:

##### Model: No subgroup effect

This model assumes there is no covariate effects; thus, no subgroup effects. The mathematical model is as follows:
\begin{align*} \theta_g &\sim N(\tau, \sigma_g^2) \\ \tau &\sim N(0, 1000) \\ \end{align*}

##### Model: Full stratification

This model assumes there is no information sharing among subgroups. That is, the subgroups are fully stratified. The mathematical model is as follows:
\begin{align*} \theta_g &\sim N(\tau_g, \sigma_g^2) \\ \tau_g &\sim N(0, 1000) \\ \end{align*}

##### Model: Simple regression

This model assumes that subgroup effects are only main effects of covariates. The mathematical model is as follows:
\begin{align*} \theta_g &\sim N(\tau_g, \sigma_g^2) \\ \tau_g &= \tau + \gamma_1 B_1 + \gamma_2 B_2 + \gamma_3 B_3 \\ \tau, \gamma_1,\gamma_2, \gamma_3 &\sim N(0,1000) \\ \end{align*}

##### Model: Simple shrinkage

The mathematical model is as follows:
\begin{align*} \theta_g &\sim N(\tau_g, \sigma_g^2) \\ \tau_g &= \tau + \phi_g \\ \tau &\sim N(0, 1000) \\ \phi_g &\sim N(0, \omega^2) \\ \omega^2 &\sim HalfN(0,1) \\ \end{align*}

##### Model: Regression and shrinkage

The mathematical model is as follows:
\begin{align*} \theta_g &\sim N(\tau_g, \sigma_g^2) \\ \tau_g &= \tau + \gamma_1 B_1 + \gamma_2 B_2 + \gamma_3 B_3 + \phi_g \\ \tau, \gamma_1,\gamma_2, \gamma_3 &\sim N(0,1000) \\ \phi_g &\sim N(0, \omega^2) \\ \omega^2 &\sim HalfN(0,1) \\ \end{align*}

##### Model: Dixon and Simon

The mathematical model is as follows:
\begin{align*} \theta_g &\sim N(\tau_g, \sigma_g^2) \\ \tau_g &= \tau + \gamma_1 B_1 + \gamma_2 B_2 + \gamma_3 B_3 \\ \tau &\sim N(0,1000) \\ \gamma_1,\gamma_2, \gamma_3 &\sim N(0, \omega^2) \\ \omega^2 &\sim HalfN(0,1) \\ \end{align*}

##### Model: Extended Dixson and Simon

The mathematical model is as follows:
\begin{align*} \theta_g &\sim N(\tau_g, \sigma_g^2) \\ \tau_g &= \tau + \gamma_1 B_1 + \gamma_2 B_2 + \gamma_3 B_3 + \delta_1 B_1B_2 + \delta_2 B_1B_2 \\ & \hspace{1cm} + \delta_3 B_2B_3 + \alpha B_1B_2B_3 \\ \tau &\sim N(0,1000) \\ \gamma_1,\gamma_2, \gamma_3 &\sim N(0, \omega_1^2) \\ \delta_1,\delta_2, \delta_3 &\sim N(0, \omega_2^2) \\ \alpha &\sim N(0, \omega_3^2) \\ \omega_i^2 &\sim HalfN(0,1) \\ \end{align*}