Variables

Variable	Measure
Data Sources and Variables
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
Reference Data - from American Community Survey
Tract shapfiles	geometric vector

DAG

Multilevel Model

\[\begin{aligned} \text{Layer 1:} & Y_{ij} | \mu_j, \sigma_y \sim \text{model of how imprisonment varies WITHIN counties } j \\ \text{Layer 2:} & \mu_j | \mu, \sigma_\mu \sim \text{model of how the typical imprisonment $\mu_j$ varies BETWEEN counties.} \\ \text{Layer 3:} & \mu, \sigma_y, \sigma_\mu \sim \text{prior models for shared global parameters} \\ \\ i = & \text{Tract} \\ j = & \text{County} \\ \end{aligned}\]

Established indicators (demographics and socioeconomic factors)

\[\begin{aligned} Y_{ij} | \beta_{0j}, \beta_1, \beta_2, \beta_3, \beta_4, \beta_5, \sigma_y \sim N(\mu_{ij}, \sigma_y^2) \;\; \text{ with } \;\; \mu_{ij} =& \beta_{0j} + \\ &\beta_1\text{Percent Black}_{ij} + \\ &\beta_2\text{Median Income}_{ij} + \\ &\beta_3\text{Unemployment Rate}_{ij} + \\ &\beta_4\text{Percent with Bachelor's Degree}_{ij} + \\ &\beta_5\text{Urban}_{ij} \\ \\ \beta_{0j} | \beta_0, \sigma_0 \stackrel{ind}{\sim} & N(\beta_0, \sigma_0^2) \\ \beta_{0c} \sim & N(100, 10^2) \\ \beta_1 \sim & N(100, 60) \\ \beta_2 \sim & N(-50, 15) \\ \beta_3 \sim & N(100, 25) \\ \beta_4 \sim & N(-100, 35) \\ \beta_5 \sim & N(100, 25) \\ \sigma \sim & \text{Exp}(l) \\ \sigma_y \sim & \text{Exp}(0.072) \\ \sigma_0 \sim & \text{Exp}(1) \\ \end{aligned}\]

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?)

\[\begin{aligned} Y_{ij} | \beta_{0j}, \beta_6, \sigma_y \sim N(\mu_{ij}, \sigma_y^2) \;\; \text{ with } \;\; \mu_{ij} =& \beta_{0j} + \\ &\beta_6\text{Driving Miles to Inpatient Facility}_{ij} + \\ &\beta_7\text{Capacity of Nearest Inpatient Facility}_{ij} + \\ Y_{ij} | \beta_{0j}, \beta_5, \beta_6, \sigma_y \sim N(\mu_{ij}, \sigma_y^2) \;\; \text{ with } \;\; \mu_{ij} =& \beta_{0j} + \\ &\beta_5\text{Urban}_{ij} \\ &\beta_6\text{Driving Miles to Inpatient Facility}_{ij} + \\ &\beta_7\text{Capacity of Nearest Inpatient Facility}_{ij} \\ \\ Y_{ij} | \beta_{0j}, \beta_1, \beta_2, \beta_3, \beta_4, \beta_5, \beta_6, \sigma_y \sim N(\mu_{ij}, \sigma_y^2) \;\; \text{ with } \;\; \mu_{ij} =& \beta_{0j} + \\ &\beta_1\text{Percent Black}_{ij} + \\ &\beta_2\text{Median Income}_{ij} + \\ &\beta_3\text{Unemployment Rate}_{ij} + \\ &\beta_4\text{Percent with Bachelor's Degree}_{ij} + \\ &\beta_5\text{Urban}_{ij} \\ &\beta_6\text{Driving Miles to Inpatient Facility}_{ij} + \\ &\beta_7\text{Capacity of Nearest Inpatient Facility}_{ij} \\ \\ \beta_{0j} | \beta_0, \sigma_0 \stackrel{ind}{\sim} & N(\beta_0, \sigma_0^2) \\ \beta_{0c} \sim & N(100, 10^2) \\ \beta_1 \sim & N(100, 60) \\ \beta_2 \sim & N(-50, 15) \\ \beta_3 \sim & N(100, 25) \\ \beta_4 \sim & N(-100, 35) \\ \beta_5 \sim & N(100, 25) \\ \beta_6 \sim & N(100, 25) \\ \beta_7 \sim & N(100, 25) \\ \sigma \sim & \text{Exp}(l) \\ \sigma_y \sim & \text{Exp}(0.072) \\ \sigma_0 \sim & \text{Exp}(1) \\ \end{aligned}\]