Appendix B — Model parameters and priors
This appendix contains full information on the parameters of each model.
| Parameter name | Symbol | Prior | Dimension |
|---|---|---|---|
| Overall intercept | \(\alpha_0\) | \(\mathcal{N}\left(0,\sigma^2=10^5\right)\) | 1 |
| Overall slope | \(\beta_0\) | \(\mathcal{N}\left(0,\sigma^2=10^5\right)\) | 1 |
| Regional intercept | \(\alpha_{1r}\) | \(\mathcal{N}\left(0,{\sigma_{\alpha_{1r}}}^2\right)\) | 9 |
| Regional intercept standard deviation | \(\sigma_{\alpha_{1r}}\) | \(\mathcal{U}\left(0,2\right)\) | 1 |
| District intercept | \(\alpha_{1d}\) | \(\mathcal{N}\left(0,{\sigma_{\alpha_{1d}}}^2\right)\) | 314 |
| District intercept standard deviation | \(\sigma_{\alpha_{1d}}\) | \(\mathcal{U}\left(0,2\right)\) | 1 |
| MSOA intercept | \(\alpha_{1s}\) | \(\mathcal{N}\left(0,{\sigma_{\alpha_{1s}}}^2\right)\) | 6791 |
| MSOA intercept standard deviation | \(\sigma_{\alpha_{1s}}\) | \(\mathcal{U}\left(0,2\right)\) | 1 |
| Regional slope | \(\beta_{1r}\) | \(\mathcal{N}\left(0,{\sigma_{\beta_{1r}}}^2\right)\) | 9 |
| Regional slope standard deviation | \(\sigma_{\beta_{1r}}\) | \(\mathcal{U}\left(0,2\right)\) | 1 |
| District slope | \(\beta_{1d}\) | \(\mathcal{N}\left(0,{\sigma_{\beta_{1d}}}^2\right)\) | 314 |
| District slope standard deviation | \(\sigma_{\beta_{1d}}\) | \(\mathcal{U}\left(0,2\right)\) | 1 |
| MSOA slope | \(\beta_{1s}\) | \(\mathcal{N}\left(0,{\sigma_{\beta_{1s}}}^2\right)\) | 6791 |
| MSOA slope standard deviation | \(\sigma_{\beta_{1s}}\) | \(\mathcal{U}\left(0,2\right)\) | 1 |
| Age group intercept | \(\alpha_{2a}\) | \(\mathcal{N}\left(\alpha_{2,a-1},{\sigma_{\alpha_{2a}}}^2\right)\) | 18 |
| Age group intercept standard deviation | \(\sigma_{\alpha_{2a}}\) | \(\mathcal{U}\left(0,2\right)\) | 1 |
| Age group slope | \(\beta_{2a}\) | \(\mathcal{N}\left(\beta_{2,a-1},{\sigma_{\beta_{2a}}}^2\right)\) | 18 |
| Age group slope standard deviation | \(\sigma_{\beta_{2a}}\) | \(\mathcal{U}\left(0,2\right)\) | 1 |
| Age group MSOA interaction | \(\xi_{as}\) | \(\mathcal{N}\left(0,\sigma_\xi^2\right)\) | 19 x 6791 |
| Age group MSOA interaction standard deviation | \(\sigma_\xi\) | \(\mathcal{U}\left(0,2\right)\) | 1 |
| MSOA random walk over time | \(\nu_{st}\) | \(\mathcal{N}\left(\nu_{s,t-1},\sigma_\nu^2\right)\) | 6791 x 17 |
| MSOA random walk over time standard deviation | \(\sigma_\nu\) | \(\mathcal{U}\left(0,2\right)\) | 1 |
| Age group random walk over time | \(\gamma_{at}\) | \(\mathcal{N}\left(\gamma_{a,t-1},\sigma_\gamma^2\right)\) | 19 x 17 |
| Age group random walk over time standard deviation | \(\sigma_\gamma\) | \(\mathcal{U}\left(0,2\right)\) | 1 |
| Overdispersion parameter | \(r\) | \(\mathcal{U}\left(0,50\right)\) | 1 |
| Parameter name | Symbol | Prior | Dimension |
|---|---|---|---|
| Overall intercept | \(\alpha_0\) | \(\mathcal{N}\left(0,\sigma^2=10^5\right)\) | 1 |
| Overall slope | \(\beta_0\) | \(\mathcal{N}\left(0,\sigma^2=10^5\right)\) | 1 |
| District intercept | \(\alpha_{1d}\) | \(\mathcal{N}\left(0,{\sigma_{\alpha_{1d}}}^2\right)\) | 33 |
| District intercept standard deviation | \(\sigma_{\alpha_{1d}}\) | \(\mathcal{N}^+(1)\) | 1 |
| MSOA intercept | \(\alpha_{1m}\) | \(\mathcal{N}\left(0,{\sigma_{\alpha_{1m}}}^2\right)\) | 983 |
| MSOA intercept standard deviation | \(\sigma_{\alpha_{1m}}\) | \(\mathcal{N}^+(1)\) | 1 |
| LSOA intercept | \(\alpha_{1s}\) | \(\mathcal{N}\left(0,{\sigma_{\alpha_{1s}}}^2\right)\) | 4835 |
| LSOA intercept standard deviation | \(\sigma_{\alpha_{1s}}\) | \(\mathcal{N}^+(1)\) | 1 |
| District slope | \(\beta_{1d}\) | \(\mathcal{N}\left(0,{\sigma_{\beta_{1d}}}^2\right)\) | 33 |
| District slope standard deviation | \(\sigma_{\beta_{1d}}\) | \(\mathcal{N}^+(1)\) | 1 |
| MSOA slope | \(\beta_{1m}\) | \(\mathcal{N}\left(0,{\sigma_{\beta_{1m}}}^2\right)\) | 983 |
| MSOA slope standard deviation | \(\sigma_{\beta_{1m}}\) | \(\mathcal{N}^+(1)\) | 1 |
| LSOA slope | \(\beta_{1s}\) | \(\mathcal{N}\left(0,{\sigma_{\beta_{1s}}}^2\right)\) | 4835 |
| LSOA slope standard deviation | \(\sigma_{\beta_{1s}}\) | \(\mathcal{N}^+(1)\) | 1 |
| Age group intercept | \(\alpha_{2a}\) | \(\mathcal{N}\left(\alpha_{2,a-1},{\sigma_{\alpha_{2a}}}^2\right)\) | 18 |
| Age group intercept standard deviation | \(\sigma_{\alpha_{2a}}\) | \(\mathcal{N}^+(1)\) | 1 |
| Age group slope | \(\beta_{2a}\) | \(\mathcal{N}\left(\beta_{2,a-1},{\sigma_{\beta_{2a}}}^2\right)\) | 18 |
| Age group slope standard deviation | \(\sigma_{\beta_{2a}}\) | \(\mathcal{N}^+(1)\) | 1 |
| Age group LSOA interaction | \(\xi_{as}\) | \(\mathcal{N}\left(0,\sigma_\xi^2\right)\) | 19 x 4835 |
| Age group LSOA interaction standard deviation | \(\sigma_\xi\) | \(\mathcal{N}^+(1)\) | 1 |
| LSOA random walk over time | \(\nu_{st}\) | \(\mathcal{N}\left(\nu_{s,t-1},\sigma_\nu^2\right)\) | 4835 x 17 |
| LSOA random walk over time standard deviation | \(\sigma_\nu\) | \(\mathcal{N}^+(1)\) | 1 |
| Age group random walk over time | \(\gamma_{at}\) | \(\mathcal{N}\left(\gamma_{a,t-1},\sigma_\gamma^2\right)\) | 19 x 17 |
| Age group random walk over time standard deviation | \(\sigma_\gamma\) | \(\mathcal{N}^+(1)\) | 1 |
| Overdispersion parameter | \(\rho\) | \(\exp (0.1)\) | 1 |
| Parameter name | Symbol | Prior | Dimension |
|---|---|---|---|
| Overall intercept | \(\alpha_0\) | \(\mathcal{N}\left(0,10\right)\) | 1 |
| Overall slope | \(\beta_0\) | \(\mathcal{N}\left(0,10\right)\) | 1 |
| District intercept | \(\alpha_{1s}\) | ICAR | 314 |
| District intercept standard deviation | \(\sigma_{\alpha_{1s}}\) | \(\mathcal{N}^+(1)\) | 1 |
| District slope | \(\beta_{1s}\) | ICAR | 314 |
| District slope standard deviation | \(\sigma_{\beta_{1s}}\) | \(\mathcal{N}^+(1)\) | 1 |
| Age group intercept | \(\alpha_{2a}\) | \(\mathcal{N}\left(\alpha_{2,a-1},{\sigma_{\alpha_{2a}}}^2\right)\) | 18 |
| Age group intercept standard deviation | \(\sigma_{\alpha_{2a}}\) | \(\mathcal{N}^+(1)\) | 1 |
| Age group slope | \(\beta_{2a}\) | \(\mathcal{N}\left(\beta_{2,a-1},{\sigma_{\beta_{2a}}}^2\right)\) | 18 |
| Age group slope standard deviation | \(\sigma_{\beta_{2a}}\) | \(\mathcal{N}^+(1)\) | 1 |
| Age group district interaction | \(\xi_{as}\) | \(\mathcal{N}\left(0,\sigma_\xi^2\right)\) | 19 x 314 |
| Age group district interaction standard deviation | \(\sigma_\xi\) | \(\mathcal{N}^+(1)\) | 1 |
| Age group random walk over time | \(\gamma_{at}\) | \(\mathcal{N}\left(\gamma_{a,t-1},\sigma_\gamma^2\right)\) | 19 x 17 |
| Age group random walk over time standard deviation | \(\sigma_\gamma\) | \(\mathcal{N}^+(1)\) | 1 |