Supplemental Material
eMethods 1. Description of ABM using the ODD protocol
eTable 1. Agent and neighborhood parameters, values, data sources, and update rules
eTable 2. Agent-based model initialization parameters and default values
eFigure 1. Flow diagram illustrating steps in model initialization
eFigure 2. Flow diagram illustrating processes occurring at each step of the model
eMethods 2. Pseudo-code for agent-based model
eTable 3. Estimates of annual violence, PTSD and other parameters from agent-based model
(ABM) and other data sources
eSensitivity analyses
eFigure3a. Ratio of prevalence of violent victimization under each intervention scenario
compared to no intervention, by level of neighborhood influence
eFigure3b. Ratio of prevalence of violence-related PTSD under each intervention scenario
compared to no intervention, by level of neighborhood influence
eFigure 4a. Ratio of prevalence of violent victimization under each intervention scenario
compared to no intervention, by size of radius in which potential perpetrator could search for
victims
eFigure 4b. Ratio of prevalence of violence-related PTSD under each intervention scenario
compared to no intervention, by size of radius in which potential perpetrator could search for
victims
eFigure 5a. Ratio of prevalence of violent victimization under each intervention scenario
compared to no intervention, by size of radius in which violent act could be witnessed by other
individuals
eFigure 5b. Ratio of prevalence of violence-related PTSD under each intervention scenario
compared to no intervention, by size of radius in which violent act could be witnessed by other
individuals
eFigure 6a. Ratio of Prevalence of Violent Victimization Under Each Intervention Scenario
Compared to No Intervention, by Alternate Coefficients in Models Predicting Probabilities of
Victimization and Perpetration
1
eFigure 6b. Ratio of Prevalence of Violence-Related PTSD Under Each Intervention Scenario
Compared to No intervention, by Alternate Coefficients in Models Predicting Probabilities of
Victimization and Perpetration
eFigure 7a. Ratio of Prevalence of Violent Victimization Under Each Intervention Scenario
Compared to No Intervention, by Proportion of Re-born Agents with History of Violent
Victimization, Violent Perpetration, and PTSD
eFigure 7b. Ratio of Prevalence of Violence-Related PTSD Under Each Intervention Scenario
Compared to No intervention, by Proportion of Re-born Agents with History of Violent
Victimization, Violent Perpetration, and PTSD
eFigure 8a. Ratio of Prevalence of Violent Victimization Under Each Intervention Scenario
Compared to No Intervention, by Additional Cell Radius in which Spillover Benefits of Targeted
Policing Could Occur
eFigure 8b. Ratio of Prevalence of Violence-Related PTSD Under Each Intervention Scenario
Compared to No intervention, by Additional Cell Radius in which Spillover Benefits of Targeted
Policing Could Occur
References
2
eMethods 1. Description of ABM using the ODD protocol
A detailed description of the ABM is provided below, following the ODD (Overview,
Design concepts, Details) protocol.1,2
Purpose
The purpose of this ABM was to simulate and compare the effects of primary and tertiary
prevention interventions on population levels of violence-related PTSD in an urban area, where
hot-spot policing and cognitive behavioral therapy (CBT) were chosen as the interventions of
interest.
Entities, state variables, and scales
The model consists of four types of entities: agents, neighborhoods, police officers, and
police patrol areas. Individual agents are characterized by the static and time-varying variables
listed in Table 1, including age, sex, race/ethnicity, marital status, educational attainment,
household income, and duration of residence, as well as variables indicating their location in the
physical space. Individual behaviors include aging, mortality, movement to a new
neighborhood, violent perpetration, violent victimization, witnessing violence, and development
and resolution of posttraumatic stress disorder (PTSD) symptoms.
The model physical environment consists of a rectangular 400 × 625 grid of cells divided
into 42 neighborhoods representing the United Hospital Fund (UHF) neighborhoods in New
York City.3 Each neighborhood is characterized by its x- and y- boundaries, location on the grid,
and list of resident agents.
Police officers are characterized by their location on the grid and the distance over which
they can prevent violence; their only behavior is preventing the occurrence of violence when a
potential perpetrator encounters a potential victim.
3
Police patrol areas were 9 × 9 cell squares characterized by their x- and y- boundaries,
coordinates of their center cell, and presence of a police officer.
Each time step of the model represents one year. Simulations were run for forty years,
with the first ten years discarded as a “burn-in period,”4 during which the agent population
accumulated a history of violence and PTSD but other agent characteristics (e.g., age, location)
remained unchanged. The duration of this burn-in period was selected so that the initial
prevalence of violent victimization, perpetration, and PTSD among agents matched expected
distributions based on samples of similarly-aged adults.
Process overview and scheduling
The model proceeded in discrete annual time steps. Within each time step, eleven
modules were processed in the following order: (1) aging, (2) death and rebirth, (3) resolution of
PTSD symptoms from the previous time step, (4) movement to a new location, (5) assignment of
police officer locations, (6) potential violent victimization and perpetration, (7) actual violent
incidents and witnessed violence, (8) other traumatic events, (9) development of PTSD
symptoms, (10) treatment of PTSD symptoms, and (11) updates to neighborhood characteristics
(see Figure 2 for a flow diagram depicting the processes at each step of the model, and Appendix
2 for pseudo-code for the model). Within each module, agents and neighborhoods were
processed in sequential order, except for the occurrence of actual violent incidents, for which
potential perpetrators were randomly ordered when seeking out potential victims to ensure that
the same perpetrators were not dominating the violent incidents in the landscape.
Design concepts
The model implemented several key features of agent-based models, including
emergence, learning, sensing, interaction, stochasticity, and collectives. Specifically, emergence
4
was present in that population levels of violence and PTSD emerged from the behaviors and
interactions of agents, which in turn were influenced by the characteristics of their
neighborhoods and the presence of police officers nearby. Adaptation was modeled in that
traumatic event exposure and PTSD, once experienced, increased an agent’s probability of future
traumatic events and PTSD during subsequent time steps; decisions about moving were also
based on experiences of violence.
Regarding sensing, individual agents were assumed to know their own characteristics
(e.g., age, sex), which influenced their behaviors. They were also assumed to know the
characteristics of the neighborhoods in the model, both in terms of influencing their behaviors
and guiding their selection of a new neighborhood when moving. Agents with the potential to
perpetrate violence could also detect the nearby presence of potential victims, and some agents
were aware of violent incidents occurring near them and thus became witnesses to that violence.
Interaction was critical to the model dynamics and outcomes, in that violence occurred
through the direct interaction of a potential victim and potential perpetrator in the physical space.
Interactions between police officers and potential victims and perpetrators were also capable of
preventing violence from occurring.
Stochasticity was used in assigning agent characteristics and behaviors, at model
initialization and throughout the model runs. Specifically, all agent demographic and behavioral
parameters were interpreted as probabilities, with characteristics and behaviors assigned by
drawing a random number between 0 and 1 and comparing the selected number to the agent’s
calculated probability; this allowed the model population’s characteristics and behaviors to
match expected distributions. Parameters that were not probabilities (e.g., amount of symptom
resolution among treated cases of PTSD) were drawn from normal distributions so that average
5
values for the population matched expected estimates but some variability existed in the
population. As a result, the population composition varied slightly across model runs but
population patterns of movement, violence, PTSD, and mental health service utilization
demonstrated expected frequencies and correlates.
Collectives were present in the model in the form of agents grouped together in
neighborhoods and police patrol areas. Characteristics of all agents located within the
boundaries of each neighborhood or patrol area were averaged to derive the average levels of
income and violence.
Finally, to allow observation for model testing, the values of agent and neighborhood
parameters were recorded for each unit at each time step. For model analysis, only populationlevel variables were recorded at each time step (e.g., percent of agents who were victimized). To
account for the stochastic nature of the model, each model scenario was run 50 times, with the
mean, 2.5th percentile, and 97.5th percentile reported from across the 50 runs.
Initialization
At model initialization, the agent population consisted of 60,000 individuals aged 18
years and older with socio-demographic characteristics assigned to match distributions of the
adult population of NYC according to the 2000 U.S. Census 5 (see Tables 1 and 2 and Figure 1).
The grid representing the physical space was divided into 42 areas reflecting the NYC UHF
neighborhoods, with sizes proportional to UHF land areas and locations consistent with
adjoining UHF borders.3 Agents were assigned to each neighborhood at initialization on the basis
of age, sex, race/ethnicity, and household income so that the composition of the area matched the
Census data for the respective UHF neighborhood, including proportionate population size.6 The
total number of police officers in the model was based on a 1% sample of the average police
6
force in New York City from 1990-1993, the years before the police force was increased as part
of the order-maintenance policing strategy championed by Police Commissioner William Bratton
and Mayor Rudolph Guiliani in 1994. At initialization, the number of police officers assigned to
each neighborhood was proportional to the neighborhood population size; within the
neighborhood, police officers were assigned to random locations.
Other parameters set during the initialization of the model are listed in Table 2.
Given previous evidence for the influence of neighborhood characteristics on exposure to
violence,7-10 we allowed five percent of individual agents’ probabilities of violent victimization
and perpetration to be determined by their neighborhood characteristics. The radius within
which perpetrators searched for victims was set at initialization to 15 cells. The radius within
which police officers could prevent violence was set at 4 cells, and the radius within which
agents could witness violence was set to 2 cells. These parameters were varied across a range of
values in sensitivity analyses to ensure the observed results were robust to alternate initialization
scenarios (see eSensitivity Analyses).
Input data
The environment is assumed to be constant, so no input data were needed to represent
time-varying processes.
Submodels
The eleven modules implemented at each time step are described in greater detail below,
including the specific equations and data sources used to calculate behavioral probabilities.
(1) Aging: Following the burn-in period, each agent aged by one year at each time step.
(2) Death and rebirth: Some agents died at each time step, with probabilities of all-cause
7
mortality assigned to agents based on their age, sex, and race/ethnicity so that mortality rates in
the agent population matched those in the NYC adult population in the year 2000.11 Each agent
who died was replaced with an 18-year-old agent with the same characteristics and neighborhood
location as the deceased agent, thus maintaining a constant population size and composition in
the model, except for age structure. We also repeated the model runs allowing specified
proportions of the “re-born” agents to re-enter the model with a history of violent victimization,
perpetration, and/or PTSD (see eSensitivity Analyses).
(3) Resolution of PTSD symptoms from the previous time step: From a meta-analysis of
PTSD treatment effects on symptoms, we used estimates from waiting list conditions to
determine the resolution of PTSD symptoms among untreated agents.12 PTSD symptom
resolution was greater for PTSD cases undergoing cognitive behavioral therapy (CBT) than for
untreated cases, with the reduction in symptoms for treated cases estimated from a meta-analysis
of the effects of group CBT on PTSD symptoms.13
In particular, Bradley and colleagues reported an average effect size of 0.35 (95% CI
0.19-0.51) for pre- versus post-treatment symptoms from studies of waiting list conditions;
translating this effect size for our PTSD symptom scale, we selected the amount of symptom
resolution for untreated agents from a normal distribution with mean 1.05 and standard deviation
0.24. There were 15 studies included in the meta-analysis from which this parameter was
derived, wherein the wait-list condition was comprised of groups assigned to receive minimal
contact or no contact.
For PTSD treatment, Barrera and colleagues reported an average effect size of 1.13 (95%
CI 0.69-1.56) for pre- versus post-treatment symptoms from studies of group CBT; again
translating this effect size for our PTSD symptom scale, we selected the amount of symptom
8
resolution for treated agents from a normal distribution with mean 3.39 and standard deviation
0.65. There were 12 studies included in the meta-analysis from which this parameter was
derived, wherein group CBT consisted of between 6 and 30 weekly sessions of 90 minutes or
more.
(4) Movement to a new location: At each time step, each agent had a certain probability of
moving to a new neighborhood. Probabilities of moving over a one-year period were calculated
from the Detroit Neighborhood Health Study (DNHS),14 according to the following logistic
regression equation:
(Eq. 1)
logit(P_MOVE) = -4.25 + (1.834*DURATION1) + (0.782*DURATION2) + (0.147*DURATION3) + (0.855*INC1) + (0.706*INC2) + (0.597*INC3) +
(1.307*LASTVICT)
where
P_MOVE =
probability of moving to a new neighborhood
DURATION1-DURATION3 = dummy variables indicating duration of residence in
current neighborhood of 0-5 years (DURATION1), 6-10 years
(DURATION2), and 11-20 years (DURATION3) (>20 years is the
referent)
INC1-INC3 = dummy variables indicating household income of <$20,000 (INC1),
$20,000-$39,999 (INC2), $40,000-$74,999 (INC3) (≥$75,000 is the
referent)
LASTVICT = dummy variable indicating whether agent was victimized at last time step
9
The calculated probability was further reduced (by half, on average) if the agent had committed
violent perpetration at the last time step. The final probability of moving was thus calculated as
follows:
REDUCEPROB ~ N(0.50, 0.025) = reduction in probability if perpetrated violence at last
time step
(Eq. 2) PROB_MOVE = (exp(logit(P_MOVE))/(1 + exp(logit(P_MOVE)))) × REDUCEPROB
(5) Assignment of police officer locations: At each time step, police officers were assigned
to new locations in the physical space. If the hot-spot policing intervention was not in place at
the time step in question, police officers were assigned to a random location within the
neighborhood to which they were assigned at baseline (as described above in the Initialization
section). If the hot-spot policing intervention was in effect, all police patrol areas in the physical
space were sorted according to the proportion of agents residing in the area who were victims of
violence at the last time step. One police officer was then assigned to the center cell of each
police patrol area, from highest violence to lowest violence, until no officers were remaining.
(6) Potential violent victimization and perpetration: At each time step, each agent had a
certain probability of committing a violent act and of being a victim of a violent act.
Probabilities of violent perpetration in the past year were calculated from Wave 2 of the National
Epidemiologic Survey of Alcohol and Related Conditions (NESARC), a national study of adult
U.S. residents.15,16 Perpetration was defined as physically hurting someone, including hitting and
using a weapon, or forcing someone to have sex.15,16 Probabilities of violent victimization in the
past year were calculated from the World Trade Center (WTC) Study, a large longitudinal study
of adult residents of the NYC metropolitan area initiated after the September 11th attacks.17
10
Victimization was defined as being attacked with or without a weapon or forced to engage in
unwanted sexual contact. These probabilities were estimated from individual- and
neighborhood-level characteristics, according to the following logistic regression equations.
(Eq. 3)
logit(P_VICTIM1) = -4.45 + (0.6527*MALE) + (-0.2339*AGE2) +
(0.3678*AGE3) + (-0.2139*AGE4) + (-0.7979*AGE5) + (-0.5967*AGE6) +
(0.275*BLACK) + (0.4144*HISP) + (-0.0297*OTHERRACE) + (0.0318*MARRIED) + (0.6621*DIVSEPWID) + (0.0674*HS) + (0.1262*MOREHS) + (0.0588*INC2) + (-0.1788*INC3) + (-0.1977*INC4) +
(1.4864*PRIORVICT) + (0.5073*PRIORPERP) + (0.1957*LASTPTSDSX)
(Eq. 4)
logit(P_PERP1) = -5.25 + (0.6514*MALE) + (0.5406*AGE2) + (0.1349*AGE3)
+ (-0.6045*AGE4) + (-1.3125*AGE5) + (-0.3049*AGE6) + (0.4747*BLACK) +
(-0.094*HISP) + (-0.3272*OTHERRACE) + (-0.2104*MARRIED) +
(0.2251*DIVSEPWID) + (-0.0326*HS) + (-0.3262*MOREHS) + (0.0142*INC2) + (-0.1978*INC3) + (0.0269*INC4) + (1.4566*PRIORVICT) +
(1.3053*PRIORPERP) + (0.0802*LASTPTSDSX)
where
P_VICTIM1 = probability of violent victimization at current time step
P_PERP1 =
probability of violent perpetration at current time step
MALE =
dummy variable indicating male gender (female is referent)
AGE2-AGE6 = dummy variables indicating age 25-34 (AGE2), 35-44 (AGE3), 45-54
(AGE4), 55-64 (AGE5), and ≥ 65 (AGE6) years (18-24 years is referent)
11
BLACK, HISP, OTHERRACE = dummy variables indicating Black, Hispanic, or other
race/ethnicity (White is the referent)
MARRIED, DIVSEPWID = dummy variables indicating married (MARRIED) and
divorced/separated/widowed (DIVSEPWID) marital status (never married
is referent)
HS, MOREHS = dummy variables indicating high school degree or equivalent (HS) or
more than high school education (MOREHS) (less than high school is
referent)
INC2-INC4 = dummy variables indicating household income of $20,000-$39,999
(INC2), $40,000-$74,999 (INC3), and ≥ $75,000 (INC4) (< $20,000 is
referent)
PRIORVICT = dummy variable indicating whether agent was victimized at any prior
time step
PRIORPERP = dummy variable indicating whether agent committed violent perpetration
at any prior time step
LASTPTSDSX = number of PTSD symptoms at last time step
To ensure that model results were not overly sensitive to the data sources chosen to
derive the above equations, we also conducted sensitivity analyses varying the magnitude of
some of the model coefficients, including the estimated effects of the number of PTSD
symptoms (LASTPTSDSX), the oldest age group (AGE6), and prior violent perpetration
(PRIORPERP) (see eSensitivity Analyses).
12
Probabilities calculated from these individual-level models accounted for 95% of the
agent’s final probability, while the remaining 5% was calculated from the following multilevel
logistic regression equation estimated using neighborhood-level exposures from NYSES and
Census data to predict violent victimization from WTC data. Specifically,
(Eq. 5)
logit(P_VICTIM2) = -5.40 + (2.9516*HOODINC1) + (2.3801*HOODINC2) +
(0.0591*HOODVIOL)
where
HOODINC1-HOODINC2 = dummy variables indicating average neighborhood income
<$40,000 (HOODINC1) or $40,000-$59,999 (HOODINC2) (≥$60,000 is
the referent)
HOODVIOL = proportion of agents residing in neighborhood who were victims of
violence at last time step
The equation for violent perpetration (which was not available for NYC specifically and
thus could not be linked easily to relevant neighborhood data) was modified from the
victimization equation. Specifically, the intercept of the equation was decreased to account for
the lower probability of perpetration, but associations between neighborhood characteristics and
perpetration were assumed to be the same as those estimated for victimization.
(Eq. 6)
logit(P_PERP2) = -6.67 + (2.9516*HOODINC1) + (2.3801*HOODINC2) +
(0.0591*HOODVIOL)
Thus, the final probabilities of agent victimization at each time step were calculated as
follows, with corresponding steps taken to calculate the final probabilities of perpetration:
(Eq.7) P_VICTIM1 = exp(logit(P_VICTIM1))/(1 + exp(logit(P_VICTIM1)))
(Eq.8) P_VICTIM2 = exp(logit(P_VICTIM2))/(1 + exp(logit(P_VICTIM2)))
13
(Eq.9) P_VICTIM = (0.95*P_VICTIM1) + (0.05*P_VICTIM2)
(7) Actual violent incidents and witnessed violence: Once potential perpetrators and victims
are identified in the ABM, an additional process occurs to determine whether a violent incident
actually takes place. Specifically, potential perpetrators search a 15-cell radius around their
location for potential victims who have not already been victimized at that time step. If a police
officer is present within a 4-cell radius of the potential victim, the violent act is prevented;
however, if no police officer is present, the victim falls prey to the perpetrator.
A completed violent encounter can also be “witnessed” by nearby agents. In particular,
for 85% of witnessed events, each agent within a 2-cell radius of the event has a 5% probability
of witnessing the violence occur; for the remaining 15% of events, nearby agents have a 15%
probability of being witnesses. This reflects the occurrence of some violent acts in public places
(e.g., bars) or during daytime hours, with more witnesses present, while the majority of violent
events are unlikely to be witnessed.
(8) Other traumatic events: In addition to violence, agents could also experience other
traumatic events at each time step. Probabilities of exposure to other traumatic events (including
natural disasters, serious accidents, and other situations causing serious injury or fear of death or
serious injury) were calculated from the WTC study referenced above,17 according to the
following logistic regression equations at the individual and neighborhood levels.
(Eq. 10)
logit(P_OTHERTRAUMA1) = -2.327 + (-0.3525*MALE) + (0.0305*AGE2) +
(0.402*AGE3) + (0.1814*AGE4) + (-0.2601*AGE5) + (-0.4182*AGE6) + (0.1346*BLACK) + (-0.1544*HISP) + (0.1842*OTHERRACE) +
(0.0361*MARRIED) + (0.0129*DIVSEPWID) + (0.2641*HS) +
14
(0.3237*MOREHS) + (0.0503*INC2) + (0.2208*INC3) + (-0.1455*INC4) +
(0.7793*PRIORTRAUMA) + (0.1103*LASTPTSDSX)
(Eq. 11)
logit(P_OTHERTRAUMA2) = -1.5547 + (-0.1362*HOODINC1) + (0.5026*HOODINC2) + (0.0556*HOODVIOL)
where
P_OTHERTRAUMA1 = probability of experiencing other traumatic event at current time
step, based on individual-level characteristics
P_OTHERTRAUMA2 = probability of experiencing other traumatic event at current time
step, based on neighborhood-level characteristics
PRIORTRAUMA = dummy variable indicating whether agent had experienced any
traumatic event at any previous time step
(9) Development of PTSD symptoms: Agents who experienced one or more types of
violence or other traumatic event exposure were then at risk for developing PTSD symptoms.
The Poisson regression equation specified below was calculated from the WTC data and used to
assign the number of PTSD symptoms (ranging from 0-17) to each agent exposed to violence or
trauma. From an ROC analysis conducted on the WTC data, we determined that a cutpoint of 7
symptoms was optimal for the identification of probable PTSD cases; therefore, any agent with
more than 7 PTSD symptoms was identified as meeting criteria for PTSD at that time step.
Violence-related PTSD cases in the model were identified as those with more than 7 PTSD
symptoms and exposure to violent victimization, perpetration, or witnessed violence at that time
step.
15
(Eq. 12)
log(CURPTSDSX) = 0.965 + (-0.0256*MALE) + (0.1667*AGE2) +
(0.0886*AGE3) + (0.1036*AGE4) + (0.0006*AGE5) + (-0.0655*AGE6) +
(0.0141*MARRIED) + (0.0481*DIVSEPWID) + (0.2126*HS) +
(0.0930*MOREHS) + (-0.1160*INC2) + (-0.2030*INC3) + (-0.3358*INC4) +
(0.6400*VICTIM) + (0.2846*PERP) + (0.1551*WITNESS) +
(0.1786*OTHERTRAUMA) + (0.04*LASTPTSDSX) + (0.3814*PRIORPTSD)
(Eq. 13)
log(CURPTSDSX2) = 1.5671 + (0.1986*HOODINC1) + (-0.1064*HOODINC2)
+ (0.0876*HOODVIOL)
where
CURPTSDSX1 = number of PTSD symptoms at current time step, based on individuallevel characteristics
CURPTSDSX2 = number of PTSD symptoms at current time step, based on
neighborhood-level characteristics
VICTIM =
dummy variable indicating whether agent was victim of violence at
current time step
PERP =
dummy variable indicating whether agent was violent perpetrator at
current time step
WITNESS = dummy variable indicating whether agent witnessed violence at current
time step
PRIORPTSD = dummy variable indicating whether agent ever met criteria for PTSD at
any previous time step
(10)
Treatment of PTSD symptoms: All agents who met criteria for PTSD were
16
eligible for treatment with cognitive behavioral therapy (CBT). Probabilities of using CBT were
calculated from WTC data, according to the following logistic regression equation.1 As noted
above, PTSD symptom resolution was greater for PTSD cases undergoing CBT than for
untreated cases.
(Eq. 14)
logit(P_CBT) = -1.60 + (-0.2008*MALE) + (-0.5828*BLACK) + (-1.073*HISP)
+ (-0.35*OTHERRACE) + (0.2449*OLDERAGE) + (1.8377*PRIORCBT)
where
P_CBT =
probability of using CBT at current time step
OLDERAGE = dummy variable indicating whether agent is 45 years old or older
PRIORCBT = dummy variable indicating whether agent had ever used CBT at any
previous time step
If the CBT intervention was implemented at the current time step, the probability of CBT
use calculated above was artificially increased by a specified amount (e.g., 50%) for all agents
with PTSD who resided in high-violence neighborhoods.
(11) Updates to neighborhood characteristics: At each time step, the average levels of income
and violent victimization were recalculated for each neighborhood to account for experiences of
violence among neighborhood residents, as well as the changing agent composition of
neighborhoods as individuals move to new locations in the physical space. The average violence
in each police patrol area was also recalculated.
1
CBT use was not directly assessed in the WTC study and was thus approximated by reports of visits to a psychologist or counselor among those
with PTSD, assuming these mental health providers would be more likely to utilize CBT than other providers like psychiatrists.
17
eTable 1. Agent and Neighborhood Parameters, Values, Data Sources, and Update Rules
Parameter
Values
Data source(s)
Update rules
Reference
18-100 (in single years)
Age, sex, and race/ethnicity
Age increases by one year
Census 20005
were jointly assigned based
at each time step.
Male; Female
on joint distributions from the
--
Census 20005
White non-Hispanic;
2000 Census, Summary File
--
Census 20005
Black non-Hispanic;
1.
--
Census 20005
--
Census 20005
Agent characteristics
Age
Sex
Race/ethnicity
Hispanic; Other nonHispanic
Marital status
Never married; Married;
Marital status was assigned
Divorced, separated,
based on age category, sex,
widowed
and race/ethnicity, using data
from the 2000 Census,
Summary File 4.
Educational attainment
< High school; High
Educational attainment was
18
school degree or
assigned based on age
equivalent; > High
category, sex, and
school
race/ethnicity, using data
from the 2000 Census,
Summary File 4.
Household income
--
Census 20005
Initial duration of residence
Duration of residence
Census 20005
was assigned based on age,
increases by one year at
using data from the 2000
each time step. When
Census, Summary File 3.
agent moves to a new
< $20,000; $20,000-
Household income was
$39,999; $40,000-
assigned based on
$74,999; ≥ $75,000
race/ethnicity, using data
from the 2000 Census,
Summary File 3.
Duration of residence in
neighborhood
0-40 (in single years)
neighborhood, duration of
residence is reset to 0.
19
Probability of dying
0-1
Mortality probabilities were
Mortality probabilities are
NYC
assigned based on age
updated when agent
DOHMH,
category, sex, and
moves into an older age
20006
race/ethnicity, based on year
category.
2000 mortality data from the
NYC Department of Health
and Mental Hygiene.
Probability of moving
to a new neighborhood
0-1
Calculated from Detroit
Recalculated at each time
Goldmann et
Neighborhood Health Study,
step.
al., 201114
based on household income,
duration of residence in
current neighborhood, and
violent victimization at last
time step; also adjusted for
violent perpetration at last
time step.
20
Probability of violent
victimization
0-1
Calculated from World Trade
Recalculated at each time
Galea et al.,
Includes: being attacked
Center (WTC) cohort study of
step.
200817
with a gun, knife, or
NYC residents, based on
other weapon; attacked
individual age, sex,
without a weapon but
race/ethnicity, marital status,
with the intent to kill or
education, income, prior
injure; or forced to
history of violence, PTSD
engage in unwanted
symptoms at last time step,
sexual contact through
and neighborhood
the use of physical force
characteristics
or threat of force.
Probability of violent
perpetration
0-1
Calculated from NESARC
Recalculated at each time
Elbogen,
Includes: using a
study of U.S. residents, based
step.
200915
weapon like a stick,
on individual age, sex,
knife, or gun; hitting
race/ethnicity, marital status,
someone hard enough to
education, income, prior
21
require medical
history of violence, PTSD
attention; forcing
symptoms at last time step,
someone to have sex; or
and neighborhood
physically hurting
characteristics.
someone in another way
on purpose.
Witnessed violence
Probability of other
traumatic event
No; Yes
Determined by proximity to
Reassessed at each time
Emerges from
victim of violence.
step.
model
0-1
Calculated from WTC cohort
Recalculated at each time
Galea et al.,
Includes: exposure to a
study of NYC residents,
step.
200817
natural disaster, being in
based on individual age, sex,
a serious accident, being
race/ethnicity, marital status,
seriously injured, and
education, income, prior
other events causing
traumatic events, and PTSD
fear of death or fear of
symptoms at last time step.
serious injury.
22
PTSD symptoms
0-17
Based on DSM-IV
criteria for PTSD
Calculated from WTC cohort
Recalculated at each time
Galea et al.,
study of NYC residents,
step. PTSD symptoms
200817
based on individual age, sex,
from previous time step
marital status, education,
decline according to
income, type of violence or
average symptom declines
other traumatic event, and
among individuals
prior history of PTSD, and
assigned to waiting list
PTSD symptoms at last time
conditions (Bradley et al.,
step.
2005; for untreated
PTSD) or group CBT
(Barrera et al., 2013; for
treated PTSD).
Probability of CBT use
0-1
Calculated from WTC cohort
Recalculated at each time
Galea et al.,
study of NYC residents,
step.
200817
based on individual age, sex,
race/ethnicity, and prior use
23
of CBT.
Neighborhood characteristics
Average household
income
Average violent
victimization
< $25,000; $25,000-
Calculated as average income
Recalculated at each time
Emerges from
$49,999; ≥ $50,000
of neighborhood residents.
step.
model
0-100
Calculated as percent of
Recalculated at each time
Emerges from
neighborhood residents who
step.
model
were victimized at last time
step.
24
eTable 2. Agent-Based Model Initialization Parameters and Default Values
Parameter
Value
Number of agents
60,000
Number of neighborhoods
Neighborhood influence on agent behaviorsa
42
0.05
Cell radius searched by potential perpetrator for potential victims of violence
15
Cell radius in which police officers can prevent violence
4
Cell radius in which agents can witness violence
2
a
Percent of the probability of agent behaviors that is determined by the agent’s neighborhood
characteristics
25
eFigure 1. Flow Diagram Illustrating Steps in Model Initialization
26
eFigure 2. Flow Diagram Illustrating Processes Occurring at Each Step of the Model
27
28
eMethods 2. Pseudo-code for Agent-Based Model
MODEL INITIALIZATION
Set parameters (user-defined or read from a parameter file).
Create the grid for agent locations.
Create lists for all agents, neighborhoods, and police patrol areas.
// CREATE AGENTS
For 1 to the defined number of agents (specified by the user or file)
Create a new agent, with all baseline characteristics.
Add agent to list of agents.
// CREATE NEIGHBORHOODS, CREATE CELLS WITHIN NEIGHBORHOODS, AND
// ASSIGN POLICE OFFICERS TO LOCATIONS
For 1 to 42 (i.e., the desired number of neighborhoods)
Create a new neighborhood corresponding to a particular NYC neighborhood.
Add neighborhood to list of neighborhoods.
Specify boundaries of the neighborhood.
Create individual cells within that neighborhood.
Randomly select initial locations of police officers on cells within the neighborhood.
Notify selected cells that police officer is present.
29
// ASSIGN AGENTS TO NEIGHBORHOODS AND SPECIFIC CELL LOCATIONS
For 1 to the defined number of agents
Select a neighborhood for the agent based on agent’s characteristics.
Randomly assign X, Y values for specific location of agent within the neighborhood.
Notify cell that agent is present.
Add agent to list of agents in the neighborhood.
// CALCULATE CHARACTERISTICS OF NEIGHBORHOODS
For 1 to the defined number of neighborhoods
Calculate average neighborhood characteristics by averaging characteristics of all agents
located
in that neighborhood.
Identify neighborhoods with above- and below-average levels of income and violence.
Create police patrol areas within the neighborhood.
Assign boundaries to each police patrol area.
EACH MODEL STEP
// RESET CELL VARIABLES
For 1 to the defined number of cells
Reset cell variables.
// RESET AGENT VARIABLES AND ALLOW DEATH AND MOVEMENT
30
For 1 to the defined number of agents
Increase age by one year.
If agent died at last time step
Reset age and history of violence and PTSD.
Reset movement, victimization, perpetration, and other traumatic event variables.
If agent had PTSD symptoms at last time step
Calculate amount of PTSD symptom resolution based on CBT status.
If number of PTSD symptoms is less than or equal to 7
Agent no longer has PTSD.
Reset agent’s probability of CBT use based on updated characteristics.
Update agent’s probability of dying.
Select random number from 0 to 1.
If random number is less than agent’s probability of death
Agent dies at this time step.
Update agent’s probability of moving to a new neighborhood.
Select random number from 0 to 1.
If random number is less than agent’s probability of moving
Remove agent from list of agents in old neighborhood.
Select a new neighborhood for the agent based on the agent’s characteristics.
Randomly assign X, Y values for specific location of agent within the new
neighborhood.
Notify cell that agent is present.
Add agent to list of agents in the neighborhood.
31
// UPDATE NEIGHBORHOOD CHARACTERISTICS AFTER MOVES
For 1 to the defined number of neighborhoods
Update average neighborhood income.
// RESET POLICE OFFICER LOCATIONS
If policing intervention not implemented at this time step
For 1 to the defined number of neighborhoods
Randomly reassign police officer locations within neighborhood.
Notify selected cells that police officer is present.
Else if policing intervention is implemented at this time step
Increase number of police available, if applicable.
Sort police patrol areas from highest to lowest violence.
For 1 to the defined number of police officers
Assign police officer to patrol area with next highest level of violence.
Place police officer in center cell of selected patrol area.
Notify cell that police officer is present.
// IDENTIFY POTENTIAL PERPETRATORS AND VICTIMS OF VIOLENCE
For 1 to the defined number of agents
Calculate agent’s probability of violent victimization.
Select random number from 0 to 1.
If random number is less than agent’s probability of victimization
32
Identify agent as potential victim.
Notify agent’s cell that potential victim is present.
Calculate agent’s probability of violent perpetration.
Select random number from 0 to 1.
If random number is less than agent’s probability of perpetration
Identify agent as potential perpetrator.
Notify agent’s cell that potential perpetrator is present.
// IDENTIFY ACTUAL PERPETRATORS AND VICTIMS OF VIOLENCE
Shuffle list of agents.
For 1 to the defined number of agents
If the agent is a potential perpetrator
Create a vector containing all cells within the specified perpetration radius of the
potential perpetrator (where the radius is specified by the user or read from
a parameter file).
For 1 to the number of cells in the perpetration vector
If the cell contains a potential victim who has not yet been victimized
Create a vector containing all cells within the specified police
protection
radius of the potential victim (where the radius is specified
by the user or read from a parameter file).
For 1 to the number of cells in the police protection vector
If the cell contains a police officer
33
Violent act is prevented.
If no cells in the police protection vector contain an officer
Identify index agent as a perpetrator.
Identify agent in selected cell as a victim.
Notify cell that actual victim is present.
// IDENTIFY WITNESSES TO VIOLENCE
For 1 to the defined number of agents
If agent is a victim of violence at this time step
Create vector containing all cells within the specified witnessing radius of the
victim
(where the radius is specified by the user or read from a parameter file).
For 1 to the number of cells in the witnessing radius
If the cell contains an agent
Select a random number from 0 to 1.
If random number is less than 0.15
Set probability of witnessing to 0.15.
Else if random number is greater than or equal to 0.15
Set probability of witnessing to 0.05.
Select another random number from 0 to 1.
If random number is less than probability of witnessing
Identify agent as a witness.
Notify cell that witness is present.
34
// IDENTIFY AGENTS EXPOSED TO OTHER TRAUMATIC EVENTS, ASSIGN NUMBER
OF
// PTSD SYMPTOMS, AND DETERMINE CBT (TREATMENT) STATUS
For 1 to the defined number of agents
Calculate agent’s probability of other traumatic event.
Select random number from 0 to 1.
If random number is less than agent’s probability of other traumatic event
Identify agent as exposed to other traumatic event.
If agent was perpetrator, victim, witness or exposed to other traumatic event
Calculate agent’s number of PTSD symptoms.
If number of symptoms is greater than 7
Identify agent as having PTSD.
If agent was perpetrator, victim, or witness of violence
Identify agent as having violence-related PTSD.
If agent has PTSD not related to violence and is not currently in CBT treatment
Select random number from 0 to 1.
If random number is less than agent’s probability of CBT use
Identify agent as receiving CBT treatment.
If agent has violence-related PTSD and is not currently in CBT treatment
If CBT intervention is implemented at this time step
If agent lives in high-violence neighborhood
Increase agent’s probability of CBT use by designated amount.
35
Select random number from 0 to 1.
If random number is less than agent’s probability of CBT use
Identify agent as receiving CBT treatment.
// UPDATE NEIGHBORHOOD CHARACTERISTICS
For 1 to the defined number of neighborhoods
Update average neighborhood violence.
Identify neighborhood as above or below average in violence.
// UPDATE POLICE PATROL AREA CHARACTERISTICS
For 1 to the defined number of police patrol areas
Update average violence in patrol area.
36
eTable 3. Estimates of Annual Violence, PTSD and Other Parameters From Agent-Based Model (ABM) and Other Data
Sources
ABM estimatesa
NYC estimatesb
Published
Sources of published
estimates
estimates
Violence
Norris, 1992;18 Potter et al.,
2009;19 Simon et al., 2008;20
Violent victimizationc
3.95 (3.86-4.01)%
1.4 - 7.0%
2.4 - 8.0%
Vaughn et al., 201021
Corrigan & Watson, 2005;22
Elbogen & Johnson, 2009;15
Silver & Teasdale, 2005;23
Violent perpetrationc
0.79 (0.78-0.80)%
0.45%
0.97 - 3.2%
Swanson, 199324
Witnessed violencec
2.82 (2.75-2.86)%
2.9 - 7.2%
na
--
Posttraumatic stress disorder (PTSD)
37
Prevalence of PTSDc
3.77 (3.65-3.82)%
4.3 - 14.7%
3.5%
Kessler et al., 200525
Prevalence of violence-related PTSDc
3.57 (3.46-3.62)%
0.93 - 3.9%
na
--
Incidence of violence-related PTSDd
13.55 (13.30-13.81)%
3.6 - 33.3%
11.3%
Norris, 199218
Duration of violence-related PTSDd
3.58 (3.51-3.63) yrs
na
3 - 5.33 yrs
Kessler et al., 199526
7.17 (7.12-7.22)
6.43 - 7.89
na
--
15.99 (15.52-16.32)%
17.4 - 17.9%
na
--
2.53
2.76
na
--
6.98 (6.92-7.04)%
7.8%
na
--
Average number of PTSD symptomsd
Cognitive behavioral therapy (CBT) use\e
Density of police officersf
Movement to new residencec
38
Note: na – not available
a
Mean and 95% confidence interval over 50 runs of the ABM.
b
Estimates were calculated from the World Trade Center cohort study,17 as well as the National Epidemiologic Survey on Alcohol
and Related Conditions (violent perpetration16), the 2003 NYC Community Health Survey (CBT use27), New York Police Department
(NYPD) data (density of police officers28), and the 2005-2009 American Community Survey (movement29).
c
Calculated among the total sample.
d
Calculated among those who experienced violence in the past year.
e
Calculated among those with PTSD in the past year.
39
eSensitivity Analyses
A summary of the sensitivity analyses that were conducted to ensure robustness of the
model inferences to alternate specifications of initial conditions is included below.
Sensitivity analysis methods
We considered six sets of sensitivity analyses, in which we tested the robustness of the
results to alternate specifications of the model. First, we varied the level of the neighborhood
influence on individual agent behaviors (from 1% to 9%, with a default value of 5%). Second,
we varied the size of the radius in which potential violent perpetrators could search for potential
victims (from 5 to 25 cells, with a default value of 15 cells). Third, we varied the size of the
radius in which violent acts could be witnessed by nearby agents (from 1 to 4 cells, with a
default value of 2 cells). Fourth, we varied key model parameter values to ensure that patterns of
violence were not unduly influenced by the data sources used to estimate the risk of violent
victimization and perpetration; in particular, coefficient estimates for the influence of older age,
previous violent perpetration, and current PTSD symptoms were varied when calculating the
probabilities of violent victimization and perpetration at each time step. Fifth, since violence and
PTSD symptoms are not infrequent occurrences during childhood,30 we assigned a prior history
of violence and PTSD to specified proportions of “re-born” agents, rather than assuming all reborn agents enter the model without such a history. Finally, we considered the possibility that
the hot-spot policing intervention results in spillover benefits in surrounding areas.31
As for the primary analyses, all sensitivity analyses were run 50 times, with the median,
2.5th percentile, and 97.5th percentile reported from across the 50 simulations. Each set of
40
sensitivity analyses was run under the no intervention scenario, as well as different combinations
of the CBT and hot-spot policing interventions.
Sensitivity analysis results
Sensitivity analyses indicated that the results of the primary analyses were robust across
alternate specifications of the model conditions. In particular, as expected, levels of violent
victimization and violence-related PTSD changed when changes were made to neighborhood
influence, the radii within which violence and witnessed violence could occur (e.g., violent
victimization increased when potential perpetrators could search a larger area for potential
victims), and when more re-born agents had a history of violence. However, the study
conclusions in terms of the influence of the CBT and hot-spot policing interventions on violent
victimization and violence-related PTSD, relative to the no intervention scenario, remained
largely unchanged regardless of the initial conditions and other assumptions of the model. For
example, Figure 3a presents the relative prevalence of violent victimization under each
intervention (compared to no intervention), for different levels of neighborhood influence, while
Figure 3b presents the relative prevalence of violence-related PTSD. Similar results are
presented in Figures 4a and 4b for different size radii of location victims, and in Figures 5a and
5b for different size radii of witnessing violence. Figures 6a and 6b demonstrate the invariance
of model results when modifying the magnitude of associations between key variables and the
risk of violent victimization and perpetration. Similarly, Figures 7a and 7b indicate that the
relative effects of each intervention scenario on violent victimization and violence-related PTSD
were consistent when different proportions of re-born agents were assigned a history of violent
41
victimization, perpetration, and/or PTSD, indicating that the assumption of no such history
among re-born agents did not influence study results.
Finally, Figures 8a and 8b present similar results when spillover benefits of the targeted
policing intervention are allowed to occur in an additional cell radius adjoining the targeted
patrol areas. The CBT interventions were not affected by this change; however, the results
indicate that the overall reductions in violent victimization and violence-related PTSD could be
larger if such spillover benefits occurred. However, the overall conclusion of the study would
still hold, namely that interventions combining both treatment and prevention strategies will
most effectively reduce violence-related PTSD.
Complete results of all sensitivity analyses are available from the authors upon request.
42
eFigure 3a. Ratio of Prevalence of Violent Victimization Under Each Intervention Scenario
Compared to No Intervention, by Level of Neighborhood Influence
Ratio of violent victimization prevalence
compared to no intervention
1.10
1.00
0.90
0.80
0.70
Increase in CBT by 200% for 10 years
Increase in CBT by 300% for 30 years
Increase in police by 0% for 10 years
Increase in police by 15% for 30 years
Increase in CBT by 50% and police by 0% for 5 years
Increase in CBT by 300% and police by 15% for 30 years
0.60
0.50
0.01
0.03
0.05
0.07
Percentage of neighborhood influence on individual behaviors
0.09
eFigure 3b. Ratio of Prevalence of Violence-Related PTSD Under Each Intervention Scenario
Compared to No Intervention, by Level of Neighborhood Influence
Ratio of violence-related PTSD prevalence
compared to no intervention
1.10
1.00
0.90
0.80
0.70
Increase in CBT by 200% for 10 years
Increase in CBT by 300% for 30 years
Increase in police by 0% for 10 years
Increase in police by 15% for 30 years
Increase in CBT by 50% and police by 0% for 5 years
Increase in CBT by 300% and police by 15% for 30 years
0.60
0.50
0.01
0.03
0.05
0.07
Percentage of neighborhood influence on individual behaviors
0.09
43
eFigure 4a. Ratio of Prevalence of Violent Victimization Under Each Intervention Scenario
Compared to No Intervention, by Size of Radius in Which Potential Perpetrator Could Search for
Victims
Ratio of violent victimization prevalence
compared to no intervention
1.10
1.00
0.90
0.80
0.70
Increase in CBT by 200% for 10 years
Increase in CBT by 300% for 30 years
Increase in police by 0% for 10 years
Increase in police by 15% for 30 years
Increase in CBT by 50% and police by 0% for 5 years
Increase in CBT by 300% and police by 15% for 30 years
0.60
0.50
5.00
10.00
15.00
20.00
Cell radius in which violent perpetration could occur
25.00
eFigure 4b. Ratio of Prevalence of Violence-Related PTSD Under Each Intervention Scenario
Compared to No Intervention, by Size of Radius in Which Potential Perpetrator Could Search for
Victims
Ratio of violence-related PTSD prevalence
compared to no intervention
1.10
1.00
0.90
0.80
0.70
Increase in CBT by 200% for 10 years
Increase in CBT by 300% for 30 years
Increase in police by 0% for 10 years
Increase in police by 15% for 30 years
Increase in CBT by 50% and police by 0% for 5 years
Increase in CBT by 300% and police by 15% for 30 years
0.60
0.50
5.00
10.00
15.00
20.00
Cell radius in which violent perpetration could occur
25.00
44
eFigure 5a. Ratio of Prevalence of Violent Victimization Under Each Intervention Scenario
Compared to No Intervention, by Size of Radius in Which Violent Act Could be Witnessed by
Other Individuals
Ratio of violence-related PTSD prevalence
compared to no intervention
1.10
1.00
0.90
0.80
0.70
Increase in CBT by 200% for 10 years
Increase in CBT by 300% for 30 years
Increase in police by 0% for 10 years
Increase in police by 15% for 30 years
Increase in CBT by 50% and police by 0% for 5 years
Increase in CBT by 300% and police by 15% for 30 years
0.60
0.50
5.00
10.00
15.00
20.00
Cell radius in which violent perpetration could occur
25.00
eFigure 5b. Ratio of Prevalence of Violence-Related PTSD Under Each Intervention Scenario
Compared to No intervention, by Size of Radius in Which Violent Act Could be Witnessed by
Other Individuals
Ratio of violence-related PTSD prevalence
compared to no intervention
1.00
0.95
0.90
0.85
0.80
0.75
0.70
0.65
Increase in CBT by 200% for 10 years
Increase in CBT by 300% for 30 years
Increase in police by 0% for 10 years
Increase in police by 15% for 30 years
Increase in CBT by 50% and police by 0% for 5 years
Increase in CBT by 300% and police by 15% for 30 years
0.60
0.55
0.50
1.00
2.00
3.00
Cell radius in which witnessed violence could occur
4.00
45
eFigure 6a. Ratio of Prevalence of Violent Victimization Under Each Intervention Scenario
Compared to No Intervention, by Alternate Coefficients in Models Predicting Probabilities of
Victimization and Perpetration (Note that Eq. 3 refers to the equation predicting victimization,
and Eq. 4 to the equation predicting perpetration)
eFigure 6b. Ratio of Prevalence of Violence-Related PTSD Under Each Intervention Scenario
Compared to No intervention, by Alternate Coefficients in Models Predicting Probabilities of
Victimization and Perpetration (Note that Eq. 3 refers to the equation predicting victimization,
and Eq. 4 to the equation predicting perpetration)
46
eFigure 7a. Ratio of Prevalence of Violent Victimization Under Each Intervention Scenario
Compared to No Intervention, by Proportion of Re-born Agents with History of Violent
Victimization, Violent Perpetration, and PTSD
eFigure 7b. Ratio of Prevalence of Violence-Related PTSD Under Each Intervention Scenario
Compared to No intervention, by Proportion of Re-born Agents with History of Violent
Victimization, Violent Perpetration, and PTSD
47
eFigure 8a. Ratio of Prevalence of Violent Victimization Under Each Intervention Scenario
Compared to No Intervention, by Additional Cell Radius in which Spillover Benefits of Targeted
Policing Could Occur
eFigure 8b. Ratio of Prevalence of Violence-Related PTSD Under Each Intervention Scenario
Compared to No intervention, by Additional Cell Radius in which Spillover Benefits of Targeted
Policing Could Occur
48
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