Appendix A: The MSTCAR Hierarchical Model • RSTr

Overview

In this vignette, we outline the hierarchical models used in the RSTr package, along with the full-conditional distributions used for each update.

The UCAR Hierarchical Model

The UCAR model used by RSTr is based on the model developed by Besag, York, and Mollié (1991) with modifications using inverse transform sampling for restricted informativeness based on Quick, et al. (2021):

For models using method = "binom",

\[ \begin{split} Y_{i} &\sim \text{Binomial}(n_{i}, \theta_{i}) \\ \theta_{i} &\sim \text{LogitNormal}(\beta_{j} + Z_{i}, \tau^2), \\ i &=\{1,...,N_s\}, j =\{1,...,N_{is}\} \end{split} \]

For models using method = "pois",

\[ \begin{split} Y_{i} &\sim \text{Poisson}(n_{i} \theta_{i}) \\ \theta_{i} &\sim \text{LogNormal}(\beta_{j} + Z_{i}, \tau^2), \\ i &=\{1,...,N_{s}\},\ j =\{1,...,N_{is}\} \end{split} \]

For both models,

\[ \begin{split} \beta_{j} &\sim \text{Normal}(0,\sigma_{\beta}^2),\ \sigma_{\beta}^2 \to \infty \\ Z &\sim \text{CAR}(\sigma^2) \\ \sigma^2 &\sim \text{InvGamma}(a_\sigma,b_\sigma) \\ \tau^2 &\sim \text{InvGamma}(a_\tau,b_\tau) \end{split} \]

The MCAR Hierarchical Model

The MCAR model used by RSTr is an extension of the model developed by Besag, York, and Mollié (1991):

For models using method = "binom",

\[ \begin{split} Y_{ik} &\sim \text{Binomial}(n_{ik}, \theta_{ik}) \\ \theta_{ik} &\sim \text{LogitNormal}(\beta_{jk} + Z_{ik}, \tau_k^2), \\ i &=\{1,...,N_s\}, k =\{1,...,N_{g}\}, j=\{1,...,N_{is}\} \end{split} \]

For models using method = "pois",

\[ \begin{split} Y_{ik} &\sim \text{Poisson}(n_{ik}, \theta_{ik}) \\ \theta_{ik} &\sim \text{LogNormal}(\beta_{jk} + Z_{ik}, \tau_k^2), \\ i &=\{1,...,N_s\}, k =\{1,...,N_{g}\}, j=\{1,...,N_{is}\} \end{split} \]

For both models,

\[ \begin{split} \beta_{jk} &\sim \text{Normal}(0,\sigma_{\beta}^2),\ \sigma_{\beta}^2 \to \infty \\ Z &\sim \text{CAR}(G) \\ G &\sim \text{InvWishart}(\nu,G_0) \\ \tau^2 &\sim \text{InvGamma}(a_\tau,b_\tau) \end{split} \]

The MSTCAR Hierarchical Model

The MSTCAR model used by RSTr is based on the model developed by Quick, et al. (2017):

For models using method = "binom",

\[ \begin{split} Y_{ikt} &\sim \text{Binomial}(n_{ikt}, \theta_{ikt}) \\ \theta_{ikt} &\sim \text{LogitNormal}(\beta_{jkt} + Z_{ikt}, \tau_k^2), \\ i &=\{1,...,N_s\},\ k =\{1,...,N_g\},\ t=\{1,...,N_t\},\ j=\{1,...,N_{is}\} \end{split} \]

For models using method = "pois",

\[ \begin{split} Y_{ikt} &\sim \text{Poisson}(n_{ikt} \theta_{ikt}) \\ \theta_{ikt} &\sim \text{LogNormal}(\beta_{jkt} + Z_{ikt}, \tau_k^2), \\ t &=\{1,...,N_t\},\ i =\{1,...,N_s\},\ k=\{1,...,N_g\},\ j=\{1,...,N_{is}\} \end{split} \]

For both models,

\[ \begin{split} \beta_{jkt} &\sim \text{Normal}(0,\sigma_{\beta}^2),\ \sigma_{\beta}^2 \to \infty \\ Z &\sim \text{MSTCAR}(\mathcal{G}, \mathcal{R}), \ \mathcal{G}=\{G_1,...,G_{N_t}\}, \ \mathcal{R}=\{R_1,...,R_{N_g}\} \\ G_t &\sim \text{InvWishart}(A_G, \nu) \\ A_G &\sim \text{Wishart}(A_{G_0}, \nu_0) \\ R_k &= \text{AR}(1,\rho_k) \\ \rho_k &\sim \text{Beta}(a_{\rho}, b_{\rho}) \\ \tau_k^2 &\sim \text{InvGamma}(a_\tau,b_\tau) \end{split} \]

For more information regarding the model and its full-conditional distributions, reference Quick, et al. (2017).