Is Gang Violent Crime More Contagious than Non-Gang Violent Crime?

Abstract

Objectives

Gangs are thought to enhance participation in violence. It is expected then that gang-related violent crimes trigger additional crimes in a contagious manner, above and beyond what is typical for non-gang violent crime.

Methods

This paper uses a multivariate self-exciting point process model to estimate the extent of contagious spread of violent crime for both gang-related and non-gang aggravated assaults and homicides in recent data from Los Angeles. The degree of contagious cross-triggering between gang-related and non-gang violent crime is also estimated.

Results

Gang-related violence triggers twice as many offspring events as non-gang violence and there is little or no cross-triggering. Gang-related offspring events are significantly more lethal than non-gang offspring events, but no more lethal than non-contagious background gang crimes.

Conclusions

Contagious spread of gang-related violent crime is different from contagion in non-gang violence. The results support crime prevention policies that target the disruption of gang retaliations.

Appendix

Parameter Estimation Methods

The models presented above can be estimated directly from data using current machine learning techniques. The specific method we use is a type of Maximum Likelihood Estimation (MLE) known as expectation maximization (EM). The procedures underlying the EM algorithm are very closely related to stochastic declustering (Mohler et al. 2011; Zhuang et al. 2002; Marsan and Lengline 2008). The expectation step of the EM algorithm starts with a random guess for the model parameter values. Given this guess, the probability $\rho_{j}$ that event j is a retaliation caused via self-excitation, or the probability $1 - \rho_{j}$ that it is a background event, is computed using equations analogous to Eqs. (5) and (6) in the main text. The probabilistic expectations for each event are then fed to a maximization step where a new set of parameter values (for iteration k + 1) are determined by maximizing the likelihood with respect to the observed data. This maximization is done for all parameters taking into consideration whether crimes are labeled as gang-related or non-gang. The EM algorithm alternates between the expectation and maximization steps until there is no further change in the parameter values. EM is able to recover true parameter values with a high degree of certainty. Standard errors for the parameter estimates are obtained from the inverse Hessian matrix for the corresponding likelihood function (Ogata 1978). Further technical details for implementation of EM algorithms can be found in (Veen and Schoenberg 2008; Mohler 2013).

Boundary Effects

The available data is restricted to a finite temporal and spatial window. We evaluated the effect of temporal boundaries on the triggering matrix K by progressively reducing the temporal window by 5, 10 and 30 days and re-estimating the model (see Fox et al. 2016). The estimated parameters using reduced data are qualitatively the same as the full data sample (Table 7). Boundary effects are limited because the kernel support has a much shorter temporal length scale (measured in days) relative to the full data (a full year).

Table 7 Triggering parameter estimates using temporal boundary corrections of different durations

Null Model and Convergent-Discriminant Validity

The multivariate point process model suggests a null model for the dynamics of gang-related and non-gang crimes. Consider a hypothetical case where all violent crimes are drawn from the same random process with a common background rate μ and common triggering productivity K. Individual crimes may differ from one another in their spatial and temporal characteristics simply as a result of random variation. As a population of events, however, they are well-described by a single stochastic process. Imagine that we then randomly label a fraction $p_{1}$ of the crimes as gang-related and $1 - p_{1}$ as non-gang. The label is independent of any measured characteristics of the crimes. If this arbitrary labeling process holds, then we expect $\left( {1 - p_{1} } \right)k_{11} = p_{1} k_{00}$ and $\left( {1 - p_{1} } \right)k_{01} = p_{1} k_{10}$. That is, we expect both within-label triggering and between-label cross-triggering to be equivalent after scaling by the fraction of crimes labeled as gang-related. In the special case that $p_{1} = 0.5$, meaning that an equal number of crimes are randomly labeled as gang-related and non-gang, then the expectations is that $k_{11} = k_{00} = k_{01} = k_{10}$. Equivalence in retaliatory triggering seems unlikely given the empirical differences between gang-related and non-gang crimes in participant, situational and neighborhood characteristics (Maxson et al. 1985; Rosenfeld et al. 1999; Huebner et al. 2016). The null model is nonetheless valuable for sharpening our thinking on how to interpret parameter estimates.

The null model also suggests that the triggering pathways shown in Fig. 2 can be interpreted in a manner similar to the convergent-discriminant validity tests used by Decker and Pyrooz (2010) (see also Taylor 2015; Piquero 1999). Convergent-discriminant validity tests posit that two statistics that seek to measure the same underlying construct will be more strongly correlated with one another than two statistics that measure different underlying constructs. If gang-related and non-gang crime labels measure different constructs, then we would expect the within-label triggering to be stronger then between-label triggering, which is what we observe in the main model estimates. To evaluate the significance of this observation, we performed 500 Monte Carlo simulations where we shuffled the labels randomly over events and re-estimated the parameters of the multivariate model after each reshuffling. Reshuffling breaks any correlation between the labels and the underlying dynamics. Recall our null hypothesis that the estimates of $k_{11}$ and $k_{00}$ as well as $k_{01}$ and $k_{10}$ should be equivalent after sample-size scaling if the labels are no better than random. We find exactly this to be the case for the shuffled labels (Table 7). When the shuffled parameter estimates $\tilde{k}_{ij}$ are scaled using the empirical fraction $p_{1} = 0.3$ of gang-related labels, the resulting scaled parameters are statistically indistinguishable (Table 8). We take this as evidence supporting a conclusion the labels correspond to different theoretical constructs with different underlying dynamics. However, this test does not test whether the theoretical constructs are biased.

Table 8 Parameter estimates given Monte Carlo reshuffling of gang-related and non-gang crime labels

Runs Tests for Biased Labeling

One potential source of bias in crime labels is a carryover effect where the classification of one crime as gang-related (or non-gang) biases the label applied to the next crime that arrives. We use the Wald–Wolfowitz runs test to assess whether labels are applied independently to each crime as it arrives, or whether they form groups suggestive of biased carryover. We examined three LAPD Basic Car areas within LAPD Patrol Divisions in the South Los Angeles study region. We assume Basic Car are exposed to crime “arrivals” from a relatively stable set of gang-related and non-gang contexts. We also assume that there is a stable set of individuals involved in labeling of crimes within a Basic Car area, allowing for potential bias to develop. The null hypothesis is that labels are applied randomly and independently from one reported crime arrival to another and that the number of runs (i.e., blocks of identically labeled crimes) should not exceed what is expected under this null hypothesis. We fail to reject the null hypothesis in each of the three Basic Car areas examined (Table 9). This suggests that crimes are labeled independently as they arrive. It also suggests that clusters of crimes generated by contagious triggering among gang-related and non-gang crimes, respectively, are not large enough to override a general pattern of random crime arrivals of each type.

Table 9 Wald-Wolfowitz runs tests for randomness in time ordered labeling of crimes