A Conditional Likelihood Model of the Relationship Between Officer Features and Rounds Discharged in Police Shootings

Abstract

Objectives

Assess whether the number of rounds fired in an officer-involved shooting is related to police officer features.

Data

The data come from 55 member agencies in the Major Cities Chiefs Association. The full dataset describes 2574 officers involved in 1600 shootings between 2010 and 2018 but only incidents involving multiple officers provide information. Our final dataset included 317 shooting incidents involving 849 officers and 5026 rounds.

Methods

We match officers on the scene of a shooting incident and develop a conditional truncated Poisson model that eliminates confounding due to time, place, and environment. We use a permutation test to formally assess the strength of the relationship between officer features and shooting rate.

Results

We find no officer feature strongly predicts shooting rate. Age at recruitment, age at the time of the shooting, and years of experience all had relative rates nearly equal to 1.0. There was no statistical relationship with an officer being female (p = 0.27), black (p = 0.64), or Hispanic (p = 0.39). Having prior involvement in shootings, prior force complaints, and special assignments appear to elevate the relative rate of shooting, but all confidence intervals included 1.0.

Conclusions

Officer features appear to have little or no relationship with shooting rate. These findings correspond with police scholars’ supposition that duty assignment may be more responsible for explaining differences in police use of force than individual officer characteristics. It contrasts with some prior research suggesting that officer race, age at recruitment, and prior performance affect shooting risk. In doing so, these results also lend support to theoretical frameworks emphasizing the role of organizational features and other system-level factors over individual-level explanations for police use of force. The proposed methodology addresses bias due to confounding, but demands a large number of shootings. Expanded participation in multi-agency data collections and including data on all non-shooting officers at the scene of the incident can increase precision.

Appendix A: Efficient Computation of the Conditional Log-Likelihood

A.1 Derivation of Recursive Computation

Estimating the maximum likelihood estimator requires repeated evaluation of the log conditional likelihood function shown in (11). Let

$$Q_{si} \left( \rho \right) = \frac{1}{\rho !}{ \exp }\left( {\left( {\rho - r_{si} } \right)\beta^{{\prime }} {\mathbf{x}}_{si} } \right)$$

(13)

In (13) we drop the $r_{si} !$ that had appeared in (11) since these are constants that can factor out. Equation (10) demonstrates this most clearly as the 144 can factor out of the denominator.

Let

$$A_{s} \left( {k,n_{r} } \right) = \mathop \sum \limits_{{\mathop \sum \limits_{i = k}^{{n_{s} }} \rho_{i} = n_{r} ,\rho_{i} > 0}} \mathop \prod \limits_{i = 1}^{{n_{s} }} Q_{si} \left( {\rho_{i} } \right)$$

(14)

where $n_{s}$ is the number of officers involved in shooting s and $n_{r} = \mathop \sum \limits_{i = 1}^{{n_{s} }} r_{si}$ is the total number of rounds fired in shooting s.

So that the log conditional likelihood is

$$\log L\left( \beta \right) = - \mathop \sum \limits_{s = 1}^{S} { \log }\left( {A_{s} \left( {1,n_{r} } \right)} \right)$$

(15)

We gain efficiency by factoring out $Q_{s,1} \left( {\rho_{1} } \right)$ in (14) so that we can define $A_{s} \left( {k,n_{r} } \right)$ recursively.

$$A_{s} \left( {1,n_{r} } \right) = \mathop \sum \limits_{{\rho_{1} = 1}}^{{n_{r} - n_{s} + 1}} \left[ {Q_{s,1} \left( {\rho_{1} } \right)\mathop \sum \limits_{{\mathop \sum \limits_{i = 2}^{{n_{s} }} \rho_{i} = n_{r} - \rho_{1} ,\rho_{i} > 0}} \mathop \prod \limits_{i = k + 1}^{{n_{s} }} Q_{si} \left( {\rho_{i} } \right)} \right]$$

(16)

$$= \mathop \sum \limits_{{\rho_{1} = 1}}^{{n_{r} - n_{s} + 1}} Q_{s,1} \left( {\rho_{1} } \right)A_{s} \left( {2,n_{r} - \rho_{1} } \right)$$

(17)

Essentially, this recursion works by setting $\rho_{1}$ to a specific value in the summation, then summing over all the other combinations of $\rho_{i}$s that sum to $n_{r} - \rho_{1}$ and multiplying the result by $Q_{s,1} \left( {\rho_{1} } \right)$. Then the sum is over fewer officers. We can repeat the recursion for $A_{s} \left( {2,n_{r} - \rho_{1} } \right)$, setting specific values for $\rho_{2}$ and summing over the remaining $\rho_{i}$s that sum to $n_{r} - \rho_{1} - \rho_{2}$. More generally

$$A_{s} \left( {k,n} \right) = \mathop \sum \limits_{{\rho_{k} = 1}}^{{n_{r} - n_{s} + k}} Q_{s,k} \left( {\rho_{k} } \right)A_{s} \left( {k + 1,n - \rho_{k} } \right)$$

(18)

$$A_{s} \left( {n_{s} ,n} \right) = Q_{{s,n_{s} }} \left( {\rho_{{n_{s} }} } \right)$$

(19)

A.2 Monte Carlo Estimation of $A_{s} \left( {1,n_{r} } \right)$

For shootings that are complex, such as when $\left( {\begin{array}{*{20}c} {n_{r} - 1} \\ {n_{s} - 1} \\ \end{array} } \right) > 10^{8}$, the recursive computation in “Appendix A.1” is slow. We can approximate a shooting’s contribution to the log-likelihood using a Monte Carlo estimate of the expected value of a function of a draw from a multinomial distribution. From (13) and (14) we have

$$A_{s} \left( {1,n_{r} } \right) = \mathop \sum \limits_{{\mathop \sum \limits_{i = 1}^{{n_{s} }} \rho_{i} = n_{r} ,\rho_{i} > 0}} \mathop \prod \limits_{i = 1}^{{n_{s} }} \frac{1}{{\rho_{i} !}}{ \exp }\left( {\left( {\rho_{i} - r_{si} } \right)\beta^{{\prime }} {\mathbf{x}}_{si} } \right)$$

(20)

Rewrite the indices on the summation so that the $\rho_{i}$ start at 0. In this way the $\rho_{i}$ count how many rounds in excess of 1 the ith officer discharges.

$$A_{s} \left( {1,n_{r} } \right) = \mathop \sum \limits_{{\mathop \sum \limits_{i = 1}^{{n_{s} }} \rho_{i} = n_{r} - n_{s} ,\rho_{i} \ge 0}} \mathop \prod \limits_{i = 1}^{{n_{s} }} \frac{1}{{\left( {\rho_{i} + 1} \right)!}}{ \exp }\left( {\left( {\rho_{i} + 1 - r_{si} } \right)\beta^{{\prime }} {\mathbf{x}}_{si} } \right)$$

(21)

$$= \mathop \sum \limits_{{\mathop \sum \limits_{i = 1}^{{n_{s} }} \rho_{i} = n_{r} - n_{s} }} \mathop \prod \limits_{i = 1}^{{n_{s} }} \frac{1}{{\rho_{i} + 1}}\frac{1}{{\rho_{i} !}}\left( {{ \exp }\left( {\beta^{{\prime }} {\mathbf{x}}_{si} } \right)} \right)^{{\rho_{i} }} \left( {{ \exp }\left( {\beta^{{\prime }} {\mathbf{x}}_{si} } \right)} \right)^{{1 - r_{si} }}$$

(22)

$$= \frac{1}{{\left( {n_{r} - n_{s} } \right)!}}\mathop \prod \limits_{i = 1}^{{n_{s} }} \left( {{ \exp }\left( {\beta^{{\prime }} {\mathbf{x}}_{si} } \right)} \right)^{{1 - r_{si} }} \mathop \sum \limits_{{\mathop \sum \limits_{i = 1}^{{n_{s} }} \rho_{i} = n_{r} - n_{s} }} \left( {n_{r} - n_{s} } \right)!\mathop \prod \limits_{i = 1}^{{n_{s} }} \frac{1}{{\rho_{i} + 1}}\frac{1}{{\rho_{i} !}}\left( {{ \exp }\left( {\beta^{{\prime }} {\mathbf{x}}_{si} } \right)} \right)^{{\rho_{i} }}$$

(23)

The term inside the sum resembles the components of a multinomial distribution with extra $1/\left( {\rho_{i} + 1} \right)$ terms. We need an additional term to normalize the ${ \exp }\left( {\beta '{\mathbf{x}}_{si} } \right)$ so that they sum to 1.

$$\begin{aligned} & \frac{1}{{\left( {n_{r} - n_{s} } \right)!}}\frac{{\left( {e^{{\beta^{{\prime }} {\mathbf{x}}_{s1} }} + \cdots + e^{{\beta^{{\prime }} {\mathbf{x}}_{{sn_{s} }} }} } \right)^{{n_{r} - n_{s} }} }}{{\left( {e^{{\beta^{{\prime }} {\mathbf{x}}_{s1} }} } \right)^{{r_{s1} - 1}} \cdots \left( {e^{{\beta^{{\prime }} {\mathbf{x}}_{{sn_{s} }} }} } \right)^{{r_{{sn_{s} }} - 1}} }} \\ & \quad \sum\limits_{{\sum\nolimits_{i = 1}^{{n_{s} }} {\uprho_{i} = n_{r} - n_{s} } }} {\frac{1}{{\rho_{1} + 1}} \cdots \frac{1}{{\rho_{{n_{s} }} + 1}}\left( {\begin{array}{*{20}c} {n_{r} - n_{s} } \\ {\uprho_{1} \cdots \uprho_{{n_{s} }} } \\ \end{array} } \right)\left( {\frac{{e^{{\beta^{{\prime }} {\mathbf{x}}_{s1} }} }}{{e^{{\beta^{{\prime }} {\mathbf{x}}_{s1} }} + \cdots + e^{{\beta^{{\prime }} {\mathbf{x}}_{{sn_{s} }} }} }}} \right)^{{\rho_{1} }} \cdots \left( {\frac{{e^{{\beta^{{\prime }} {\mathbf{x}}_{{sn_{s} }} }} }}{{e^{{\beta^{\prime}{\mathbf{x}}_{s1} }} + \cdots + e^{{\beta^{{\prime }} {\mathbf{x}}_{{sn_{s} }} }} }}} \right)^{{\rho_{{n_{s} }} }} } \\ \end{aligned}$$

(24)

Let $p_{i} = e^{{\beta^{\prime}{\mathbf{x}}_{si} }} /\left( {e^{{\beta^{\prime}{\mathbf{x}}_{s1} }} + \cdots + e^{{\beta^{\prime}{\mathbf{x}}_{{sn_{s} }} }} } \right)$.

$$= \frac{1}{{\left( {n_{r} - n_{s} } \right)!}}\frac{1}{{p_{1}^{{r_{s1} - 1}} \cdots p_{{n_{s} }}^{{r_{{sn_{s} }} - 1}} }}\mathop \sum \limits_{{\mathop \sum \limits_{i = 1}^{{n_{s} }}\uprho_{i} = n_{r} - n_{s} }} \frac{1}{{\rho_{1} + 1}} \cdots \frac{1}{{\rho_{{n_{s} }} + 1}}\left( {\begin{array}{*{20}c} {n_{r} - n_{s} } \\ {\uprho_{1} \cdots \uprho_{{n_{s} }} } \\ \end{array} } \right)p_{1}^{{\rho_{1} }} \cdots p_{{n_{s} }}^{{\rho_{{n_{s} }} }}$$

(25)

If $\varvec{\rho}\sim {\text{Multinomial}}\left( {n_{r} - n_{s} ,\varvec{p}} \right)$ then the sum in (25) is $E\left( {\frac{1}{{\left( {\rho_{1} + 1} \right) \cdots \left( {\rho_{{n_{s} }} + 1} \right)}}} \right)$, which can be estimated via Monte Carlo draws from the multinomial distribution.

A.3 Computation

Here we provide R/C++ code for computing the negative log conditional likelihood. The analyst can insert negCLL() into an optimizer (like nlm()) to obtain maximum likelihood estimates of β.