• Neuroscience Researcher @UMich
• Study brain connectivity & psychiatric disorders
• twitter: @dankessler
• web: www.dankessler.me
• I use R for data wrangling and behavioral/fancier analyses (e.g. mixed effects)
• New to Stan and Police Shooting data

• Sociologist/Demographer
• Expert on Stan
• Well-trained Bayesian statistician
• Person of Color
• Police Officer

• Mike Angstadt
• Chandra Sripada
• Kerby Shedden
• Cody Ross
• AARUG & Ann Arbor ASA
• SPARK

• Review Cody Ross's Method & Finding
• Introduce Stan
• Anatomy of a Stan program
• Alternative Model for Police Shootings
• Our very tentative results

The US has recently witnessed a number of high-profile deaths of African-Americans at the hands of police.



There is a pressing need to study this phenomenon quantitatively. Why?

• Case reports are insufficient and could even be misleading
• Better understand the causes
• Help shape solutions

A Multi-Level Bayesian Analysis of Racial Bias in Police Shootings at the County-Level in the United States, 2011-2014Cody T. Ross*

US Police-Shooting Database (USPSD)

• crowd-sourced database
• covers years 2011-2014
• n=721

For each county i, let \(C_{i}\) be the ratio of: \(\frac{P(killed_{unarmed} | black)}{P(killed_{unarmed} | white)}\)

• Step 1: Use Bayesian methods to estimate the \(C_{i}\)'s
• Step 2: Use Bayesian methods to examine the effects of county-level covariates in accounting for variance in the \(C_{i}\)'s

Ross, C. T. (2015). A Multi-Level Bayesian Analysis of Racial Bias in Police Shootings
at the County-Level in the United States, 2011-2014. PLoS ONE, 10(11), e0141854-34

1. Unarmed black civilians are 3.49 times as likely to be shot by police compared to unarmed white civilians
2. There is substantial variation in the county-level risk ratios
   • Miami-Dade 22.88, Los Angeles 10.25, New Orleans 9.29
3. County-level risk ratios are not related to county-level race-specific crime rates
   • This finding is particularly important because black people are overrepresented in violent crime. One might otherwise use this observation to explain away elevated risk ratios

1. We use a newly available Washington Post database
   • likely more comprehensive
   • spans 2015-present
2. We adopt a different modeling framework, while retaining the spirit of Ross's Bayesian methods

1. Are Ross's main results replicated in WashPo database?
   • black/white racial disparity, large spatial variation, disparity unaccounted for by crime rates
2. Is there temporal stability in spatial variation in risk ratios?
   • in other words, will Miami-Dade, Los Angeles, and New Orleans (and other counties with elevated risk) tend to still have higher risk ratios in the WashPo database?

• mc-stan.org
• Bayesian statistical inference w/Markov-Chain Monte Carlo (MCMC) sampling
• Named for Stanislaw Ulam: co-inventor of Monte Carlo methods
• R Bindings (plus other languages)

• Treat parameters as random with underlying distribution (prior)
• Interest: Distribution of parameter conditional on observed data (posterior)
• Parameters can depend on hyper-parameters, and those in turn…
• MCMC: Procedurally sample from posterior of parameters
• Powerful approach enables computation to handle o/w difficult math

• Organized into blocks
• Lines are either declarations (real Counts100];) or statements (x = 5;)
• data (what you read from environment, declarations ONLY)
• transformed data (mix declarations & statements)
• parameters (declarations only)
• transformed parameters (declarations & statements; define parameters in terms of one another)
• model (sampling statements / log probability incrementers)
• generated quantities (values to store/extract later)

• Covers 2015+
• Under version-control on github
• Only fatal shootings by on-duty police

• Ross's approach is based on a binomial (sum of bernoulli trials)
• But, data is captured at an aggregate count level
• Discrete counts of rare events -> Poisson distribution (\(\lambda\))
• \(\lambda\) changes based on covariates
• Poisson Regression
• \(log(E[Y \mid x)) = \theta'x\) or \(E(Y \mid x) = e^{\theta'x}\)
• Relative Risk: Covariates that interact with race give us RR
• \(\frac{\lambda_{Black}}{\lambda_{White}} = \frac{e^{\theta_{1}'x_1 + \theta_{2}'x}}{e^{\theta_{2}'x}}\)
• \(RR_{Black:White} = e^{\theta_{i}}\) for any \(\theta_i\) coded to interact with race

• Are regional disparities in shooting persistent over time?
• Introduce a random variable \(\beta_{i,d}\), for i'th county in d'th dataset
• \(\vec{\beta_{i}} = \begin{bmatrix} \beta_{i,1} & \beta_{i,2} \end{bmatrix}\)
• \(\vec{\beta_{i}} \sim N(0,\Sigma)\)
• Introduce two such \(\beta\), one interacts with race (random slope) other doesn't (random intercept)
• Off-diagonal of \(\Sigma\) for random slope indicates persistent racial disparity in shooting

• Step 1: Shooting ~ Race | Population(Offsets), Dataset
• Step 2: Shooting ~ Race | Population(Offsets), Dataset, B:W Ratio
• Step 3: Shooting ~ Race | Population(Offsets), Dataset, B:W Ratio, Arrest Rate Ratio
• Very limited data for Step 3 (for now)

Let \(C_x\) be an observed count of shootings with associated predictors \(x\).\(C_x \sim \text{Poisson}(\lambda_x)\)\(\lambda_x = e^{\theta'x}\)\(\theta\) is the vector of coefficients for the GLM.Let \(\theta\) have block structure as\(\theta = \begin{bmatrix} \theta_{Race:Demo} & \theta_{Offset} & \theta_{Race} & \theta_{County:Time} & \theta_{Race:County:Time} \end{bmatrix}\)In most cases the elements of \(\theta_{*}\) are simply one or more beta coefficients, which unless otherwise specified have uninformative priors.Introduce two additional random variables:\(\vec{\beta}_{County:Time}^{i} = \begin{bmatrix} \beta_{\textit{D1, County:Time}}^i & \beta_{\textit{D2, County:Time}}^i \end{bmatrix}\)

\(\vec{\beta}_{Race:County:Time}^{i} = \begin{bmatrix} \beta_{\textit{YD, Race:County:Time}}^i & \beta_{\textit{D2, Race:County:Time}}^i \end{bmatrix}\)

\(\vec{\beta}_{County:Time}^{i} \sim N(0,\Sigma_1)\)

\(\vec{\beta}_{Race:County:Time}^{i} \sim N(0,\Sigma_2)\)

• Step 1

  RR Black:White = 4.13
  RR Var = 1.31
  RR Cov(D1,D2)  = .225

• Step 2:

  RR Black:White = 2.70
  RR Var = 1.17
  RR Cov(D1,D2) = .204

• Step 3:

  RR Black:White = 2.66
  RR Var = 1.29
  RR Cov(D1,D2) = .265