| Variable |
Measure |
| Outcome variable - from Prison Policy Initiative |
| Rate of imprisonment |
per 100,000 residents |
| Established indicators (demographics and socioeconomic) - from US Census |
| Black residents |
percent |
| Median income |
in thousands |
| Residents with at least a bachelor's degree |
percent |
| Unemployment rate |
percent |
| Rural households |
percent |
| Primary explanatory variables - from Substance Abuse and Mental Health Services Administration |
| Travel time to mental health facility |
in minutes at 9:00 am on a weekday from the center of the tract |
| Distance to mental health facility |
driving miles from the center of the tract |
| Travel time to inpatient facility |
in minutes at 9:00 am on a weekday from the center of the tract |
| Driving miles to inpatient facility |
driving miles from the center of the tract |
| Travel time to inpatient facility w/ crisis intervention |
in minutes at 9:00 am on a weekday from the center of the tract |
| Driving miles to inpatient facility w/ crisis intervention |
driving miles from the center of the tract |
| Tract shapfiles |
geometric vector |
\[\begin{aligned}
\text{Layer 1:} & Y_{ij} | \mu_j, \sigma_y \sim \text{model of how imprisonment varies WITHIN counties } j \\
\text{Layer 2:} & \mu_j | \mu, \sigma_\mu \sim \text{model of how the typical imprisonment $\mu_j$ varies BETWEEN counties.} \\
\text{Layer 3:} & \mu, \sigma_y, \sigma_\mu \sim \text{prior models for shared global parameters} \\
\\
i = & \text{Tract} \\
j = & \text{County} \\
\end{aligned}\]
Established indicators (demographics and socioeconomic factors)
\[\begin{aligned}
Y_{ij} | \beta_{0j}, \beta_1, \beta_2, \beta_3, \beta_4, \beta_5, \sigma_y \sim N(\mu_{ij}, \sigma_y^2) \;\; \text{ with } \;\; \mu_{ij} =& \beta_{0j} + \\
&\beta_1\text{Percent Black}_{ij} + \\
&\beta_2\text{Median Income}_{ij} + \\
&\beta_3\text{Unemployment Rate}_{ij} + \\
&\beta_4\text{Percent with Bachelor's Degree}_{ij} + \\
&\beta_5\text{Urban}_{ij} \\
\\
\beta_{0j} | \beta_0, \sigma_0 \stackrel{ind}{\sim} & N(\beta_0, \sigma_0^2) \\
\beta_{0c} \sim & N(100, 10^2) \\
\beta_1 \sim & N(100, 60) \\
\beta_2 \sim & N(-50, 15) \\
\beta_3 \sim & N(100, 25) \\
\beta_4 \sim & N(-100, 35) \\
\beta_5 \sim & N(100, 25) \\
\sigma \sim & \text{Exp}(l) \\
\sigma_y \sim & \text{Exp}(0.072) \\
\sigma_0 \sim & \text{Exp}(1) \\
\end{aligned}\]
Should I include more factors here? Percent disabled? Percent under poverty with medicaid? Or would that sort of just be the same as median income? Other factors? Maybe geographic mobility or population density or anything else?
Proximity to inpatient/crisis mental health facilities
\[\begin{aligned}
Y_{ij} | \beta_{0j}, \beta_6, \sigma_y \sim N(\mu_{ij}, \sigma_y^2) \;\; \text{ with } \;\; \mu_{ij} =& \beta_{0j} + \\
&\beta_6\text{Driving Miles to Inpatient Facility}_{ij} + \\
Y_{ij} | \beta_{0j}, \beta_5, \beta_6, \sigma_y \sim N(\mu_{ij}, \sigma_y^2) \;\; \text{ with } \;\; \mu_{ij} =& \beta_{0j} + \\
&\beta_5\text{Urban}_{ij} \\
&\beta_6\text{Driving Miles to Inpatient Facility}_{ij} \\
\\
Y_{ij} | \beta_{0j}, \beta_1, \beta_2, \beta_3, \beta_4, \beta_5, \beta_6, \sigma_y \sim N(\mu_{ij}, \sigma_y^2) \;\; \text{ with } \;\; \mu_{ij} =& \beta_{0j} + \\
&\beta_1\text{Percent Black}_{ij} + \\
&\beta_2\text{Median Income}_{ij} + \\
&\beta_3\text{Unemployment Rate}_{ij} + \\
&\beta_4\text{Percent with Bachelor's Degree}_{ij} + \\
&\beta_5\text{Urban}_{ij} \\
&\beta_6\text{Driving Miles to Inpatient Facility}_{ij} \\
\\
\beta_{0j} | \beta_0, \sigma_0 \stackrel{ind}{\sim} & N(\beta_0, \sigma_0^2) \\
\beta_{0c} \sim & N(100, 10^2) \\
\beta_1 \sim & N(100, 60) \\
\beta_2 \sim & N(-50, 15) \\
\beta_3 \sim & N(100, 25) \\
\beta_4 \sim & N(-100, 35) \\
\beta_5 \sim & N(100, 25) \\
\beta_6 \sim & N(100, 25) \\
\sigma \sim & \text{Exp}(l) \\
\sigma_y \sim & \text{Exp}(0.072) \\
\sigma_0 \sim & \text{Exp}(1) \\
\end{aligned}\]
Capacity of nearest facilities
\[\begin{aligned}
Y_{ij} | \beta_{0j}, \beta_7, \sigma_y \sim N(\mu_{ij}, \sigma_y^2) \;\; \text{ with } \;\; \mu_{ij} =& \beta_{0j} + \\
&\beta_7\text{Capacity of Nearest Inpatient Facility}_{ij} + \\
Y_{ij} | \beta_{0j}, \beta_5, \beta_7, \sigma_y \sim N(\mu_{ij}, \sigma_y^2) \;\; \text{ with } \;\; \mu_{ij} =& \beta_{0j} + \\
&\beta_5\text{Urban}_{ij} \\
&\beta_7\text{Capacity of Nearest Inpatient Facility}_{ij} \\
\\
Y_{ij} | \beta_{0j}, \beta_1, \beta_2, \beta_3, \beta_4, \beta_5, \beta_7, \sigma_y \sim N(\mu_{ij}, \sigma_y^2) \;\; \text{ with } \;\; \mu_{ij} =& \beta_{0j} + \\
&\beta_1\text{Percent Black}_{ij} + \\
&\beta_2\text{Median Income}_{ij} + \\
&\beta_3\text{Unemployment Rate}_{ij} + \\
&\beta_4\text{Percent with Bachelor's Degree}_{ij} + \\
&\beta_5\text{Urban}_{ij} \\
&\beta_7\text{Capacity of Nearest Inpatient Facility}_{ij} \\
\\
\beta_{0j} | \beta_0, \sigma_0 \stackrel{ind}{\sim} & N(\beta_0, \sigma_0^2) \\
\beta_{0c} \sim & N(100, 10^2) \\
\beta_1 \sim & N(100, 60) \\
\beta_2 \sim & N(-50, 15) \\
\beta_3 \sim & N(100, 25) \\
\beta_4 \sim & N(-100, 35) \\
\beta_5 \sim & N(100, 25) \\
\beta_7 \sim & N(100, 25) \\
\sigma \sim & \text{Exp}(l) \\
\sigma_y \sim & \text{Exp}(0.072) \\
\sigma_0 \sim & \text{Exp}(1) \\
\end{aligned}\]
Both (Interaction?)
\[\begin{aligned}
Y_{ij} | \beta_{0j}, \beta_6, \sigma_y \sim N(\mu_{ij}, \sigma_y^2) \;\; \text{ with } \;\; \mu_{ij} =& \beta_{0j} + \\
&\beta_6\text{Driving Miles to Inpatient Facility}_{ij} + \\
&\beta_7\text{Capacity of Nearest Inpatient Facility}_{ij} + \\
Y_{ij} | \beta_{0j}, \beta_5, \beta_6, \sigma_y \sim N(\mu_{ij}, \sigma_y^2) \;\; \text{ with } \;\; \mu_{ij} =& \beta_{0j} + \\
&\beta_5\text{Urban}_{ij} \\
&\beta_6\text{Driving Miles to Inpatient Facility}_{ij} + \\
&\beta_7\text{Capacity of Nearest Inpatient Facility}_{ij} \\
\\
Y_{ij} | \beta_{0j}, \beta_1, \beta_2, \beta_3, \beta_4, \beta_5, \beta_6, \sigma_y \sim N(\mu_{ij}, \sigma_y^2) \;\; \text{ with } \;\; \mu_{ij} =& \beta_{0j} + \\
&\beta_1\text{Percent Black}_{ij} + \\
&\beta_2\text{Median Income}_{ij} + \\
&\beta_3\text{Unemployment Rate}_{ij} + \\
&\beta_4\text{Percent with Bachelor's Degree}_{ij} + \\
&\beta_5\text{Urban}_{ij} \\
&\beta_6\text{Driving Miles to Inpatient Facility}_{ij} + \\
&\beta_7\text{Capacity of Nearest Inpatient Facility}_{ij} \\
\\
\beta_{0j} | \beta_0, \sigma_0 \stackrel{ind}{\sim} & N(\beta_0, \sigma_0^2) \\
\beta_{0c} \sim & N(100, 10^2) \\
\beta_1 \sim & N(100, 60) \\
\beta_2 \sim & N(-50, 15) \\
\beta_3 \sim & N(100, 25) \\
\beta_4 \sim & N(-100, 35) \\
\beta_5 \sim & N(100, 25) \\
\beta_6 \sim & N(100, 25) \\
\beta_7 \sim & N(100, 25) \\
\sigma \sim & \text{Exp}(l) \\
\sigma_y \sim & \text{Exp}(0.072) \\
\sigma_0 \sim & \text{Exp}(1) \\
\end{aligned}\]