# Appendix A: The CAR Hierarchical Models

## 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 CAR Hierarchical Model

The CAR 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 = "binomial"`,

\[ \begin{split} Y_{i} &\sim \text{Binomial}(n_{i}, \lambda_{i}) \\ \theta_{i} &= \text{Logit}(\lambda_{i}) \\ \end{split} \]

For models using `method = "poisson"`, \[ \begin{split} Y_{i} &\sim \text{Poisson}(n_{i} \lambda_{i}) \\ \theta_{i} &= \text{Log}(\lambda_{i}) \\ \end{split} \]

For both models,

\[ \begin{split} \theta_{i} &\sim \text{Normal}(\beta_{j} + Z_{i}, \tau^2), \\ i &=\{1,...,N_{s}\},\ j =\{1,...,N_{is}\} \\ p(\beta_{j}) &\propto 1 \\ Z &\sim \text{CAR}(\sigma^2) \\ \sigma^2 &\sim \text{InvGamma}(a_\sigma,b_\sigma) \\ \tau^2 &\sim \text{InvGamma}(a_\tau,b_\tau) \end{split} \]

## The MCAR Hierarchical Model

The MCAR model used by RSTr is based on the model developed by Gelfand and Vounatsou (2003):

For models using `method = "binomial"`,

\[ \begin{split} Y_{ik} &\sim \text{Binomial}(n_{ik}, \lambda_{ik}) \\ \theta_{ik} &= \text{Logit}(\lambda_{ik}) \\ \end{split} \]

For models using `method = "poisson"`,

\[ \begin{split} Y_{ik} &\sim \text{Poisson}(n_{ik}, \lambda_{ik}) \\ \theta_{ik} &= \text{Log}(\lambda_{ik}) \\ \end{split} \]

For both models,

\[ \begin{split} \theta_{ik} &\sim \text{Normal}(\beta_{jk} + Z_{ik}, \tau_k^2), \\ i &=\{1,...,N_s\}, k =\{1,...,N_{g}\}, j=\{1,...,N_{is}\} \\ p(\beta_{jk}) &\propto 1 \\ Z &\sim \text{CAR}(G) \\ G &\sim \text{InvWishart}(\nu,G_0) \\ \tau^2 &\sim \text{InvGamma}(a_\tau,b_\tau) \end{split} \]

## 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 = "binomial"`,

\[ \begin{split} Y_{ikt} &\sim \text{Binomial}(n_{ikt}, \lambda_{ikt}) \\ \theta_{ikt} &= \text{Logit}(\lambda_{ikt}) \\ \end{split} \]

For models using `method = "poisson"`,

\[ \begin{split} Y_{ikt} &\sim \text{Poisson}(n_{ikt} \lambda_{ikt}) \\ \theta_{ikt} &= \text{Log}(\lambda_{ikt}) \\ \end{split} \]

For both models,

\[ \begin{split} \theta_{ikt} &\sim \text{Normal}(\beta_{jkt} + Z_{ikt}, \tau_k^2), \\ i &=\{1,...,N_s\},\ k =\{1,...,N_g\},\ t=\{1,...,N_t\},\ j=\{1,...,N_{is}\} \\ p(\beta_{j}) &\propto 1 \\ Z &\sim \text{MSTCAR}(\mathcal{G}, \mathcal{R}), \ \mathcal{G}=\{G_1,...,G_{N_t}\}, \ \mathcal{R}=\{R_1,...,R_{N_g}\} \\ G_t &\sim \text{InvWishart}(A_G, \nu) \\ A_G &\sim \text{Wishart}(A_{G_0}, \nu_0) \\ R_k &= \text{AR}(1,\rho_k) \\ \rho_k &\sim \text{Beta}(a_{\rho}, b_{\rho}) \\ \tau_k^2 &\sim \text{InvGamma}(a_\tau,b_\tau) \end{split} \]