Estimating Risks of Arrest and Criminal Populations: Regression Adjustments to Capture

Abstract

The size of criminal populations is unknown, and policy decisions are typically based only on the number of offenses and offenders that come to the attention of the criminal justice system. However, the size of criminal populations may follow different trends than what is observed in official data. We use a regression-adjusted capture–recapture model to estimate the number of people at risk of arrest for offenses involving amphetamine-type stimulants (ATS) from arrests and rearrests occurring in Quebec, Canada, controlling for year of first arrest, age, and gender. The 4,989 individuals arrested were the visible part of an estimated 42,541 [36,936, 48,145] individuals otherwise at risk of arrest (12%). Additional results show that trends in criminal populations and risks of arrest vary across offense type and drug classifications.

Keywords

capture–recapture methods size of criminal populations risks of arrest drugs amphetamines

Introduction

The size of the criminal population refers to the number of offenders who commit one or more offense(s) during an observation period (e.g., year), including those “at risk,” but never arrested, charged, and/or otherwise incarcerated (Visher & Roth, 1986). Because the true size of this population is unknown, criminal justice policy decisions, such as police resource allocation or program evaluations, are strictly based on the number of offenses and offenders that are reported by the police. However, the size of the criminal population may follow a different trend than police-reported crime rates. Nothing prevents a stable population of offenders from increasing their (collective) offending frequency in any given year, especially if they stumble upon an especially profitable opportunity for crime (Tremblay, 2010). When those offenders are among the heaviest contributors to crime statistics, such changes cause an upward shift in crime trends. Similarly, increases in the size of criminal populations may not coincide with increases in frequency of offending if, for example, offenders lack the human and social capital needed to sustain a successful criminal career (Loughran, Nguyen, Piquero, & Fagan, 2013; Lussier, Bouchard, & Beauregard, 2011).

The effectiveness of policy interventions often hinges on knowing the size of the target population. The most extensive body of research in this area considers the number of problem drug users (e.g., Bouchard & Tremblay, 2005; Brecht & Wickens, 1993; Choi & Comiskey, 2003), which informs the size of facilities, staffing, and funding needed to subsidize treatment programs. The same logic applies to the allocation of police resources. Knowing the number of offenders targeted by policy changes, such as increases in the number and/or capacity of police, can better inform whether interdictions produce any measurable deterrent effect by reducing criminal participation (Levitt, 2004). Considering that the criminal justice system is based on a deterrence logic that places a premium on certainty of punishment as an incentive to encourage law-abiding behavior, the size of the criminal population is arguably the most undervalued outcome measure for criminal justice policy. Because the size of the criminal population is unknown, scholars rely on indirect methods that adopt a series of hard-to-verify assumptions to compare the risk of arrest across offense classifications. Blumstein, Cohen, Roth, and Visher (1986), for example, used victimization and self-report surveys revealing co-offending patterns across offense classifications to provide the first detailed consideration of the differential risk of arrest that offenders face in the United States. This method, however, has limitations: Offense definitions in victimization surveys and police data do not always align, and these surveys do not cover “victimless crimes,” such as drug offenses (e.g., possession, selling). Estimating the size of criminal populations means that risk of arrest can be calculated by dividing the number of arrests for a specific type of offense by the total number of offenders who committed that offense—including those who are never arrested. This number provides a means to assess criminal justice policies beyond the conventional raw numbers and official narratives (Bouchard & Tremblay, 2005).

Notwithstanding a few exceptions (e.g., Bouchard, 2007; Collins & Wilson, 1990; Greene & Stollmack, 1981; Rossmo & Routledge, 1990), mainstream criminologists have been slow to explore methods and models to estimate the size of criminal populations. One of the first studies to consider the size of criminal populations and their implications for the criminal justice system was Tillman’s (1987) examination of the arrest distributions of adults aged 29 years and younger across the state of California. Whereas the generalizability of previous cohort studies may have been limited to young offenders residing in cities sharing similar characteristics (e.g., composition, density, geography, socioeconomic factors), Tillman (1987) was able to draw valid inferences as to the prevalence of arrest and frequency of offending at the population level. But such rich data still do not say anything about the number of criminals who were not arrested.

Since Tillman’s (1987) study, an entire methodology has been developed to “size” elusive and/or hidden populations, some of the most promising of which are capture–recapture methods (Bouchard, 2007; Collins & Wilson, 1990; Greene, 1984; Greene & Stollmack, 1981; Rossmo & Routledge, 1990; Tillman, 1987; Van Der Heijden, Cruyff, & Böhning, 2014). Not unlike other estimation methods for partially hidden populations, it relies on patterns found in the observed part of the population to make an inference about the unobserved part.

There are many variations of capture–recapture methodology, each with different assumptions about the capture (count) distribution and the behavior of the population of interest pre- and post-capture. Zelterman’s (1988) truncated Poisson estimator is a moment estimator that has been used in prior work to estimate the number of criminals who were at risk, but not arrested from a right-truncated Poisson distribution of arrests and rearrests (Bouchard, 2007; Collins & Wilson, 1990; Rossmo & Routledge, 1990):

\hat{N} = \frac{N}{1 - e^{(- 2 \times \frac{n_{2}}{n_{1}})}},

where $\hat{N}$ is the estimate of the unobserved population, N is the number of subjects arrested in an observation period, n₁ is the number of subjects arrested once, and n₂ is the number of subjects arrested twice. Zelterman’s (1988) logic was that the probability of capture for unobserved subjects should more closely “resemble” the probability for subjects captured only once or twice (i.e., people who were not arrested are more alike people arrested only once or twice than people arrested many times).¹ The unobserved population $\hat{N}$ is calculated from the Poisson parameter, a constant term assigned to all “captured” or arrested people from a population with m arrests that is used to calculate the probability of not being arrested. If the number of arrests and rearrests follow the Poisson distribution specified by Zelterman (1988), the missing cell in the distribution should estimate the number of people who were at risk, but never arrested. Böhning and Van Der Heijden (2009) later recognized that Zelterman’s (1988) estimator has a binary outcome (i.e., arrested once or twice) and could be generalized into a maximum likelihood estimator under the assumptions of logistic regression. By adding other variables into the equation, such as age and gender, bias is reduced by adjusting the Poisson parameter upward or downward across subgroups of arrestees, depending upon whether people with certain observed attributes are arrested more or less than people with other observed attributes.

This study considers the number of people who were at risk of arrest, but never arrested for possession, selling, and/or other offenses involving amphetamine-type stimulants (ATS), which include methamphetamine, ecstasy, and other synthetic drugs like amphetamine sulfate (i.e., speed) in Quebec, Canada, over the 11 years spanning 1999-2009. Year of first arrest, age effects, and gender were regressed on truncated counts of one and two arrests across four observation periods: 1999-2001, 2002-2004, 2005-2007, and 2008-2009. Zelterman’s maximum likelihood estimator was used to calculate point and interval estimates of the population at risk of arrest. The external validity of our results is considered by exploiting a change in police reporting of offenses involving ATS that occurred at the peak in the upward trend in arrests over the study period: Beginning in 2008, police across Quebec began reporting whether an arrest involved methamphetamine and/or ecstasy instead of simply the “other controlled drugs and substances” classification when possible (see Ouellet & Morselli, 2014). Estimates from these 2 years are compared with estimates drawn from general population surveys of past-month and past-year drug use by a multiplier method used in prior work (Bouchard & Tremblay, 2005; MacCoun & Reuter, 2001).

Our data run through a decade where trends in arrest (see Figure 1) for possession more than doubled and arrest for trafficking increased more gradually, before leveling off and/or gradually declining over the next 7 to 9 years. Trends show that arrests for possession of ATS became more common than arrests for possession and trafficking of cocaine (shown for comparison) by 2004 and continued to rise until 2007, whereas arrests for ATS trafficking rose to similar rates as cocaine possession and trafficking by 2005, before declining in 2008. This increase in arrests happened during a time when some argued about Canada’s place in global ATS production after several large international seizures (Kirby & MacDonald, 2009; United Nations Office on Drugs and Crime, 2009). This claim was never really substantiated and was perhaps exaggerated (see Kilmer & Pacula, 2009; Morselli et al., 2016), although there is evidence of large supply and demand for ATS in Quebec: (a) the small number of local lab seizures that happened during the study period nonetheless produced a large quantity of ATS (Morselli et al., 2016), (b) there were few arrests for importation–exportation of ATS during the study period, and (c) available data on past-month, past-year, and lifetime drug use from general population surveys show consistently high rates of methamphetamine and ecstasy use in Quebec year-after-year compared with other Canadian provinces. Our results show that rising numbers of arrests were not just a police response to (mis)perceptions about Canada’s role in global ATS production and trafficking and Quebec’s high rates of ATS use, but rather a growing number of users, sellers, and people involved in production and importation–exportation that coincided with growth in other illegal drug markets in Quebec (see Bouchard & Dion, 2009). Risk of arrest ranged 8% to 12% and was greater for offenses involving methamphetamine (8%-12%) than ecstasy (1%-3%).

Method

Capture–recapture requires complete arrest records for a particular offense or offense(s) during a specified observation period, such that the initial arrests and subsequent rearrests for everyone arrested are recorded. Our data are official arrest records from the Module d’Information Policières records system maintained by the Sûreté du Quebéc, Quebec’s Provincial Police, of adults (18 years of age and older) across the province of Quebec from 1999 to 2009.² Our data included 17,747 arrests for possession, trafficking, importation and/or exportation, and production (i.e., “cooking”) of ATS, including methamphetamine, ecstasy, and “other” drugs (e.g., lysergic acid diethylamide [LSD], prescription pills, speed). People’s identities were concealed for confidentiality reasons, although everyone who was arrested was marked with a unique identification number, which allowed us to count his or her arrests across the observation period. Arrest on an ATS-related charge may or may not have been the primary offense for which people were arrested; however, inclusion of anyone arrested for an offense involving ATS, whether or not it was the most serious charge, was the most important criterion for a person’s inclusion in this sample (Bouchard, 2007; Gallupe, Bouchard, & Caulkins, 2011).

The use of arrest records means that what we estimated was the number of people “at risk of arrest” during the observation period(s). It would have been useful to compare estimates derived from arrest records against those drawn from conviction records, but conviction records were not made available to us; however, the most accurate estimates of the size of a criminal population are drawn directly from arrest records because of the loss of data that occurs as arrests and charges are processed by the courts (see Blumstein & Cohen, 1979). Arrest as an indicator of criminal participation produces Type I errors, caused when people are arrested without sufficient cause and, thus, not charged and/or convicted of an offense, but are nonetheless included in a sample when they should arguably be excluded, whereas conviction as an indicator causes Type II errors, caused when people are arrested, and guilty, but not convicted (Maltz, 1984).

Our only other inclusion criterion was that rearrests had to occur at least 5 days after the initial or previous arrest to be included in the sample. We followed previous work on recidivism in establishing the 5-day threshold (Gallupe et al., 2011), but concede the arbitrary nature of this decision. This procedure removed 1.4% of arrests, the majority of which were multiple (same day) entries of the same capture. These entries may have had some internal logic (e.g., separate deals, separate labs), but they refer to the same “capture” for our purposes. Although it is possible that other close-range arrests occurring outside of the 5-day threshold are also related, we felt that such a threshold also allowed for a bona fide recapture process to happen—for example, if a seller was arrested on a Sunday, incarcerated 24 hr, released, went back to selling on the following Friday, and was arrested again on a similar charge.

The arrest data included three separate classifications of charges (any offense, possession, and selling [or possession for purpose of trafficking])³ for three controlled substances (all ATS [e.g., amphetamine, methamphetamine, ecstasy⁴], methamphetamine, and ecstasy),⁵ although police across Quebec only began reporting whether an arrest involved methamphetamine and/or ecstasy specifically instead of simply the “other controlled drugs and substances” classification when possible, beginning in 2008. Arrests for a given charge for a given substance were not mutually exclusive. People may be considered at risk of arrest for more than one offense if they were arrested for possession and selling. For example, someone arrested on a possession charge at t₁ (e.g., 2008) and on a selling charge at t₂ (e.g., 2009) would not only be recorded as rearrested in the “all ATS” arrest distribution but also be found in the arrest distributions for possession and selling. Similarly, people arrested for both methamphetamine and ecstasy offenses were captured in both arrest distributions; therefore, simply adding the populations at risk of arrest for possession to those at risk of arrest for selling would exaggerate the true size of the criminal population, although estimates are assumed to be valid when considered individually. The “any offense” and “all ATS” arrest distributions were most useful to obtain a general idea of the ceiling across offense types and drug markets.

Outcome: Truncated Counts of Arrests and Rearrests

The outcome variables consisted of truncated counts of arrests and rearrests for possession, selling, and any and/or all offenses involving ATS across the four study periods. The arrest distributions presented in Table 1 show that 92% to 96% of people are arrested only once and that selling offenses typically produce more rearrests than possession offenses.

Table 1.

The Distributions of Arrests Across Offense Classifications and Observation Periods.

	ATS all offenses	ATS possession	ATS selling
1999-2001
N arrests	2,169	1,263	955
1	1,996	1,203	882
2	130	48	50
3	25	10	8
4	6	1	4
5+	12	1	11
Mean year at first arrest (SD)	2000.03 (0.78)	2000.02 (0.79)	2000.01 (0.77)
Mean age at window period (SD)	28.00 (9.33)	27.46 (8.61)	28.82 (10.14)
Female = 1 (%)	0.13	0.12	0.15
2002-2004
N arrests	2,836	1,746	1,120
1	2,687	1,691	1,068
2	128	47	46
3	16	7	2
4	4	0	4
5+	1	1	0
Mean year at first arrest (SD)	2003.12 (0.82)	2003.12 (0.81)	2003.11 (0.83)
Mean age at window period (SD)	27.13 (9.00)	26.36 (8.33)	28.21 (9.78)
Female = 1 (%)	0.11	0.10	0.13
2005-2007
N arrests	6,225	3,620	2,791
1	5,768	3,457	2,637
2	391	141	131
3	47	19	21
4	11	2	2
5+	8	1	0
Mean year at first arrest (SD)	2006.08 (0.81)	2006.08 (0.81)	2006.08 (0.80)
Mean age at window period (SD)	28.36 (9.81)	27.66 (9.08)	29.16 (10.50)
Female = 1 (%)	0.13	0.13	0.13
2008-2009
N arrests	4,989	2,945	2,192
1	4,603	2,796	2,045
2	321	134	125
3	48	14	14
4	12	0	7
5+	5	1	1
Mean year at first arrest (SD)	2008.46 (0.50)	2008.44 (0.50)	2008.46 (0.50)
Mean age at window period (SD)	28.90 (10.06)	28.29 (9.49)	29.57 (10.71)
Female = 1 (%)	0.13	0.13	0.14

Note. ATS = amphetamine-type stimulants.

An important question for capture–recapture models like Zelterman’s (1988) is whether the arrest distributions presented in Table 1 follow the Poisson distribution. We tested whether our data follow the Poisson distribution using the ratio-plot test—a chi-square test designed specifically for capture–recapture analysis that gives the optimal point (k) at which you should truncate the distribution (Böhning, Baksh, Lerdsuwansri, & Gallagher, 2013). Truncated are people whose number of arrests exceeded their expected value. This test statistic follows a $χ_{k - 1}^{2}$ distribution, where the null hypothesis is a homogeneous Poisson distribution and k = 1 indicates a “perfect” fit, with no degrees of freedom. Our data showed that k = 1 across all combinations of substance and offense classifications across the four observation periods, except for arrests involving selling ATS in 2002-2004 where k = 2 (see the appendix). Because the distributions of arrests are largely one-time arrests with few rearrests and fewer people arrested three or more times, we estimated the population at risk of arrest from the truncated distribution described by Zelterman (1988).

Zelterman’s (1988) estimator may still derive estimates that are too large if there is one-inflation, or “too many ones,” in the distributions of arrests.⁶ The concern would be that the unobserved population at risk of arrest is overestimated because of the large number of people arrested only once compared with the relatively few people arrested two or more times—thought to be the result of people’s learned behavior about how to avoid arrest (Ouellet & Bouchard, 2017). The argument in favor of adjusting for one-inflation is that the unobserved population at risk of arrest is likely smaller than the number of people arrested once, so the number of “zeros” relative to the number of “ones” in the distributions of arrest should reflect what is observed in the Poisson distribution. We considered this possibility using Godwin and Bohning’s (2017) one-inflated Poisson regression—a lower bound estimator than Zelterman’s (1988), without the underestimation bias of more simple estimators (Böhning & Van Der Heijden, 2009). However, the point estimates of the unobserved populations at risk of arrest derived by a one-inflation regression showed not to be a good fit to our data.⁷

Control Variables

The arrest records included only three variables of use. Descriptive statistics are presented in Table 1 for offenses involving all ATS across the four observation periods and in Table 2 for offenses involving methamphetamine and ecstasy in 2008-2009, specifically.

Table 2.

The Distributions of Arrests and Descriptive Statistics Across Offenses Involving Methamphetamine and Ecstasy, 2008-2009.

	All offenses	Possession	Selling
Methamphetamine
N arrests	900	544	374
1	851	526	351
2	44	18	22
3	4	0	1
4	1	0	0
5+	0	0	0
Year at first arrest (SD)	2008.63 (0.48)	2008.61 (0.49)	2008.68 (0.47)
Mean age at window period (SD)	29.43 (10.25)	28.22 (8.93)	31.01 (11.65)
Female = 1 (%)	0.15	0.13	0.18
Ecstasy
N arrests	359	191	164
1	353	187	163
2	6	4	1
3	0	0	0
4	0	0	0
5+	0	0	0
Year at first arrest (SD)	2008.75 (0.44)	2008.72 (0.45)	2008.77 (0.42)
Mean age at window period (SD)	27.20 (8.54)	27.38 (8.83)	27.01 (8.24)
Female = 1 (%)	0.11	0.16	0.06

Year of first arrest is perhaps our most important control variable: The timing of the first arrest affects the likelihood of a second arrest, and so on, which at least partially offsets the bias of a shorter observation period for people arrested for the first time in the second and/or third year.⁸ By default, a person arrested in 2009 has a lower probability of rearrest by year’s end than a person first arrested in 2008—a particularity of a research design where estimates were derived from a cross section of arrest records that accumulated over time.

The effects of age were included in our regressions, using people’s age at the start of the window period (e.g., January 1, 2002), which has been shown to be one of the best predictors of differential risks of arrests in capture–recapture studies of criminal populations (Bouchard & Lussier, 2015; Collins & Wilson, 1990).⁹ The mean age at the window period for ATS across offense classifications and by observation period is shown in Table 1, whereas Table 2 shows the age distributions for ecstasy and methamphetamine in 2008-2009. The mean age at the window period ranged in the late 20s, but the corresponding age–arrest curves (not shown) peak in the early 20s, with the largest cross-sample age group consisting of people aged 20 to 24 years (27.73%), and people arrested for possession are typically younger than those arrested for selling. Quadratic effects of age were also included in our regressions to try and better estimate the higher probability of arrest among this 20- to 24-year-old age group, although the results were no better when estimating regressions with quadratic age effects.¹⁰

The effects of gender were captured with a dummy indicator, where females were coded 1 and males coded 0, as risk of arrest is higher for males than females. Tables 1 and 2 show that more than 80% of the people arrested are males. The highest proportions of females are found in 2008-2009, when methamphetamine and ecstasy are considered separately (see Table 2).

Capture–Recapture Assumptions

Capture–recapture methodology comes with several assumptions about the population of interest. The first assumption is that there are complete records of arrests, with no errors or undercounts of arrests and rearrests that would downward- or upward-bias the estimated population at risk of arrest. The high degree of convergence in the number of arrests provided to us by the Sûreté du Quebéc and the number of arrests reported to the Uniform Crime Reporting (UCR) survey (r > .90) means that we have near-complete coverage of arrests. This list of arrest records is assumed to be a random and representative sample of the population at risk of arrest. Whether our data adhere to this assumption or not is hard to verify. The arrestee population most likely oversamples career criminals (Blumstein, Canela-Cacho, & Cohen, 1993) while excluding some of the most serious offenders who were sentenced to prison for periods longer than our observation period and less likely to be found in the group of offenders rearrested (Bouchard & Lussier, 2015). However, the age and gender demographics of the arrested population within and across the observation periods are consistent with patterns from general population surveys on drug use: The prevalence of methamphetamine and ecstasy use is much higher in the population under 25 years of age than the population aged 25 years and older, the prevalence of methamphetamine use is highest among males aged 25 years and older, and the prevalence of ecstasy use is higher among young females than young males.

The second assumption is that the population at risk of arrest has constant, equal, and independent probability of arrest and rearrest across the study period (i.e., population homogeneity and independence), which assumes there are no behavioral changes in the population after an arrest (Ouellet & Bouchard, 2017), police do not target people or subgroups after arrest, and/or there are no cohort effects that increase or decrease the probability of arrest (Tillman, 1987). Without rap sheets and court records, we cannot determine the outcome of arrests, particularly how and the speed at which people progress through the criminal justice system, which affects their exposure to arrest and rearrest: The year-long observation periods may not be sufficient for people to get arrested, sentenced, perhaps spend some time incarcerated, and be released for long enough to be rearrested. We cannot provide a fully satisfying solution to this problem, but our control for year of first arrest at least partially offsets any bias that would otherwise skew estimates, although a true exposure time variable (see Piquero et al., 2001) would be preferable. The choice of estimator also reduces the effects of sample heterogeneity: Zelterman’s (1988) estimator reduces sample heterogeneity in the outcome variable by discounting the minority of people with three or more arrests. Zelterman’s (1988) logic was that a good estimator should consider only necessary parameters and outcomes closest to the zeros or the unobserved population. The generalization of Zelterman’s (1988) estimator into a regression model (Böhning & Van Der Heijden, 2009) also enabled us to control for observed heterogeneity in the data and partially corrected for bias that would otherwise exist using simpler extrapolation procedures.

The third assumption is that there is no growth or displacement in the criminal population, an assumption which is only likely to be true if the study period is short.¹¹ Zelterman’s maximum likelihood estimator is part of a capture–recapture methodology that assumes a “closed” population.¹² The most serious threat to the closure assumption is that people arrested at some point during the study period desist from drug use, selling, and/or other offenses associated with ATS (e.g., import–export, production). The other threat to closure is migration within and across the province of Quebec. The arrest records shared with us by the Sûreté du Quebéc provide province-wide coverage, which greatly reduces the possibility of people being excluded from the sample simply because they moved to another city, but without record of the city of arrest, we could not provide more substantive evidence regarding the degree to which we do or do not violate this assumption; however, arrests and rearrests for possession, selling, and other offenses involving ATS are likely not as spatially concentrated (e.g., street segments or neighborhood) as arrests for possession and selling of street drugs like crack cocaine and heroin (see Weisburd & Green, 1995). The few rearrests also mean that it is not likely that people were arrested in more than one city across Quebec. Even so, Kendall’s (1999) simulations showed that estimates of wildlife prevalence are only minimally affected when migrations in the population occur randomly and when there is no sign of a massive migration during the observation period. Human migrations are unlikely to be swift and large enough to bias estimates if the window period between observation periods is short enough, so the continuous observation periods across the years for which we have arrest records should satisfy this condition.

Analytic Strategy

The number of offenders at risk of arrest for possession, selling, and/or any offense involving ATS, methamphetamine, and ecstasy in Quebec was estimated across four observation periods covering the 11 years spanning 1999-2009 to test trends in arrest against trends in population growth. Year, age, and gender effects were regressed on the truncated arrest distributions across the three offense classifications involving ATS and four observation periods (1999-2001, 2002-2004, 2005-2007, and 2008-2009).

The generalized Zelterman maximum likelihood estimator was published as a Stata program (Böhning & Van Der Heijden, 2009).¹³ Point estimates and confidence intervals across offense classifications and observation periods represent the population that was at risk of arrest (for some unknown time) during those 2 or 3 years, which were divided by the number of years spanning the observation period to get a crude average of the number of people at risk of arrest per year of an observation period. We began by estimating fully specified regressions to identify which variables had a statistically significant effect on the probability of arrest, removed nonsignificant effects, and repeated this process until only statistically significant variables were included in our regressions; therefore, we present the “best” regression specifications for all offense classifications. Our decision to present only the best specification is grounded in logic: Adding nonsignificant variables is almost always associated with larger point estimates and confidence intervals, and goodness-of-fit of a regression specification, as judged by Akaike’s information criterion (AIC), favors parsimony by penalizing the inclusion of nonsignificant variables in a regression.

Results

We start by presenting the best regression models across substance and offense classifications for the 2008-2009 observation period, where estimates can be provided for methamphetamine and ecstasy, specifically.

Table 3 shows estimates of the number of people at risk of arrest for possession, selling, and/or all other offenses involving ATS. The population at risk is estimated at 42,541, with a 95% confidence interval of [36,936, 48,145]. Only year of first arrest was statistically significant. This result simply means that people who are first arrested in 2008 have a higher likelihood of being rearrested before the end of the window period (i.e., December 31, 2009) than people first arrested in 2009. The better fit of the covariate-adjusted model controlling only for year of first arrest is a common result for almost all models. The null model for all offenses yielded a lower estimate of 38,323 people, which is still within the lower bound provided by what we consider as the best model.

Table 3.

Population At Risk of Arrest for Offenses Involving ATS, 2008-2009.

Parameters	All offenses		Possession		Selling
Parameters	Null	Adjusted	Null	Adjusted	Null	Adjusted
$\hat{N}$	38,323	42,541	32,221	33,876	19,049	21,304
95% CI	[34,162, 42,483]	[36,936, 48,145]	[26,785, 37,657]	[27,568, 40,183]	[15,729, 22,369]	[16,727, 25,881]
Constant	−2.66*** (0.06)	1,406.63*** (250.52)	−3.04*** (0.09)	954.32** (375.21)	−2.80*** (0.09)	1,165.56** (393.40)
Year of first arrest	—	−0.70*** (0.13)	—	−0.48** (0.19)	—	−0.58** (0.20)
Age at window period	—	—	—	—	—	—
Gender	—	—	—	—	—	0.58** (0.23)
AIC	2,375.54	2,343.73	1,090.53	1,085.76	958.20	946.63
G ²	—	33.81	—	6.78	—	15.57
N	4,989		2,945		2,192

Note. The unstandardized coefficients are shown. Standard errors are given in parentheses. ATS = amphetamine-type stimulants; CI = confidence interval; AIC = Akaike’s information criterion.

p < .05. **p < .01. ***p < .001.

Table 3 also lists the estimated number of people at risk of arrest for possession and selling of ATS. We estimate 33,876 people [27,568, 40,183] at risk of arrest for possession and 21,304 [16,727, 25,881] people at risk of arrest for selling ATS of any kind. The 42,541 people participating across all offenses give the best estimate of the size of the total population at risk of arrest, with approximately 80% of those at risk of arrest for possession only. The covariate-adjusted model estimating the number of people at risk of arrest for selling deserves particular attention, as gender was significantly associated with the probability of (re)arrest—the only instance in which we found a significant gender effect across all regression specifications. The negative effect of gender suggests that risk of rearrest is higher for females.

Tables 4 and 5 present estimates for the number of offenders at risk of arrest for possession, selling, and/or other offenses involving methamphetamine and ecstasy, respectively. In general, results of the regression analyses show that the higher the proportion of rearrests across classifications of offenses and substances, the smaller the estimates for the number of people at risk of arrest. The estimates from the null models are systematically lower than any of the estimates produced by the covariate-adjusted models, which is why we recommend parsimonious models that only include significant predictors of the probability of (re)arrest. The logic of the model implies that adding statistically significant covariates is correcting an arrest rate that was assumed to be too high for a significant portion of the population. In short, an overestimated arrest rate leads to an underestimated population.

Table 4.

Population at Risk of Arrest for Offenses Involving Methamphetamine, 2008-2009.

Parameters	All offenses		Possession		Selling
Parameters	Null	Adjusted	Null	Adjusted	Null	Adjusted
$\hat{N}$	9,161	10,686	8,224	11,061	3,174	—
95% CI	[6,466, 11,857]	[6,770, 14,602]	[4,431, 12,016]	[3,639, 18,483]	[1,856, 4,493]	—
Constant	−2.96*** (0.16)	1,738.40** (627.96)	−3.38*** (0.24)	2,370.10* (1,020.25)	−2.77*** (0.22)	—
Year of first arrest	—	−0.87** (0.31)	—	−1.18* (0.51)	—	—
Age at window period	—	—	—	—	—	—
Gender	—	—	—	—	—	—
AIC	352.91	347.15	160.11	156.28	169.22	—
G ²	—	7.76	—	5.83	—	—
N	900		544		373

Note. The unstandardized coefficients are shown. Standard errors are given in parentheses. CI = confidence interval; AIC = Akaike’s information criterion

p < .05. **p < .01. ***p < .001.

Table 5.

Population at risk of arrest for offenses involving ecstasy, 2008-2009.

Parameters	All offenses		Possession		Selling
Parameters	Null	Adjusted	Null	Adjusted	Null	Adjusted
$\hat{N}$	10,741	—	4,561	13,111	13,448	—
95% CI	[2,151, 19,332]	—	[95, 9,027]	[−12,291, 38,513]	[−12,909, 39,805]	—
Constant	−4.08*** (0.41)	—	−3.85*** (0.51)	−5.08*** (1.00)	−5.09*** (1.00)	—
Year of first arrest	—	—	—	—	—	—
Age at window period	—	—	—	—	—	—
Gender	—	—	—	2.88** (1.17)	—	—
AIC	63.00	—	40.84	35.66	14.19	—
G ²	—	—	—	7.18	—	—
N	359		191		164

Note. The unstandardized coefficients are shown. Standard Errors are given in parentheses. CI = confidence interval; AIC = Akaike’s information criterion.

p < .05. **p < .01. ***p < .001.

Estimates of the number of people at risk of arrest for possession, selling, and/or all (other) offenses involving methamphetamine across Quebec are shown in Table 4. Estimates for all offenses suggest a population of 10,686 with a 95% confidence interval of [6,770, 14,601]. The estimate for possession is only slightly larger at 11,061, but the much wider confidence interval [3,639, 18,483] suggests the volatility is due to the small number of arrests (526) and rearrests (18) from which the estimates are extrapolated (see Table 4). This situation shows the importance of not putting too much stock in a single number and the importance of preserving confidence intervals as lower and upper bound estimates. Table 4 also shows that only one of our covariates is significantly associated with the risk of rearrest. Year of first arrest was significantly and negatively associated with arrest for possession and all offenses involving methamphetamine: The earlier the year of first arrest, the higher the probability of arrest during the window period.

Table 5 presents estimates for the number of offenders at risk of arrest for possession, selling, and all (other) offenses involving ecstasy across Quebec. Estimates for offenses involving ecstasy provide an interesting demonstration on validity. Only the model estimating the number of people at risk for any and/or all offenses involving ecstasy provides a valid population estimate. The model estimates that a population of 10,741 people were at risk of arrest across Quebec at some point throughout 2008-2009. Yet, it comes with a wide confidence interval [2,151, 19,332], given the small numbers of arrests (353) and rearrests (6) for offenses involving ecstasy (see Table 1). Estimates of people at risk of rearrest for possession or selling have confidence intervals that cross zero, but we preserved these models to demonstrate the volatility of capture–recapture estimates with small numbers of arrests and rearrests. The four rearrests over 187 arrests for possession (see Table 2) and the one rearrest over 163 arrests for selling are simply too small to be reliable raw numbers for capture–recapture purposes. Although we would not put too much stock on the estimate, the ecstasy possession model uncovered a significant effect of recapture for gender similar to the ATS model for selling (Table 3), implying that females, although a minority of all arrests, are more likely to be rearrested for ecstasy possession than males.

An important benefit of having estimates of the total size of a population is to derive a measure of risk of arrest. Figure 2 shows average risks of arrest by drug and offense classification, calculated by dividing the number of arrests and rearrests by the populations derived from the “best” regression specifications from across the time periods. Consistent with the prior literature (Bouchard & Tremblay, 2005; MacCoun & Reuter, 2001), the risk of arrest seems dependent upon the severity of the offenses and harms of the substance: Risk of arrest is greater for selling than for possession (for methamphetamine and ATS more generally, look ahead to Figure 4), and risk of arrest is higher for offenses involving methamphetamine compared with ecstasy.

Trends in Populations At Risk of Arrest Across Observation Periods

An important part of the story of this article concerns the trends in the population at risk of arrest across the four observation periods. First, the trends in our population estimates provide for a highly relevant validity test: Are the trends in our estimates consistent with trends in alternative data? Second, having estimates across multiple observation periods allows us to say something about the growth and/or decline within the population at risk of arrest, as opposed to simply focusing on one number or another from arrests that occurred during just one time period.

The “best” point estimates and confidence intervals across the four observation periods are plotted in Figure 3, which shows upward trends in the population at risk of arrest for ATS across offense classifications.¹⁴ The trends revealed that the number of people at risk of arrest grew more than four times in absolute size over the 11 years we observed. The growing population indicates that police were not simply arresting and rearresting the same people during this decade when arrests were increasing, but that there were perhaps a growing number of people who were using and/or selling ATS, or involved in other offenses such as the production of ATS, and/or police resources were allocated to policing the use, sale, and production of ATS.

Figure 4 presents the risk of arrest by offense classifications and across observation periods and shows that trends in arrests do not necessarily follow trends in the populations at risk of arrest. Rather, as the number of people arrested for ATS offenses was increasing in the mid-2000s (Figure 1), the size of the population at risk of arrest was growing at a faster pace. The result is that the risks of arrest for possession or selling of ATS decreased between 2002 and 2007, before increasing again to the 1999-2001 levels in 2008-2009, just as the number of arrests was leveling off (see Figure 1).

Assessing the External Validity of the Estimates

Although the logic of capture–recapture is sound, the results are simply estimates, and much care must be put into validity assessments of the estimates via alternative data. The external validity of our results is considered by exploiting a change in police reporting that occurred at the peak in the upward trend in arrests over the study period: Beginning in 2008, police across Quebec began reporting whether an arrest involved methamphetamine and/or ecstasy instead of simply the “other controlled drugs and substances” classification when possible (see Ouellet & Morselli, 2014). The point estimates and confidence intervals drawn separately from the distributions of arrests and rearrests involving methamphetamine and ecstasy were compared with estimates drawn from the 2009 Canadian Alcohol and Other Drug Use Monitoring Survey (CADUMS) of past-month and past-year drug use and general census data that captured Quebec’s population aged 15 years and older. Here, we use the “best” regression estimates across the six possible drug and offense classification combinations given in Tables 4 and 5.

The prevalence of past-year ecstasy use reported in Quebec (1.0%) was similar to the reported past-year use across Canada (0.9%), numbers comparable to figures captured by the 2004 Canadian Addiction Survey (CAS) in which 1.1% of participants reported to have used ecstasy at least once in the past year (Adlaf, Begin, & Sawka, 2005). Consistency across the two surveys suggests stability in the prevalence of ecstasy use in the general population. There were no prevalence rates of methamphetamine use in Quebec reported in the 2004 CAS or 2009 CADUMS¹⁵; therefore, we derived an estimate by assuming that the prevalence of methamphetamine users is the same in Quebec as the 0.1% past-year use reported by the 2009 CADUMS across Canada.¹⁶

Table 6 shows estimates of methamphetamine and ecstasy users aged 15 and older in Quebec, along with estimates adjusting for 20% and 50% underreporting. Here, we assume underreporting of street populations, who are disproportionately heavy methamphetamine users and typically underrepresented in general population surveys (Wood, Stoltz, Montaner, & Kerr, 2006; see also Public Health Agency of Canada, 2006, 2009). The general population surveys suggest that the population of methamphetamine users is one tenth the size of the population of ecstasy users. Keeping in mind that methamphetamine users, in particular, should be vastly underestimated from these surveys, we find that the 8,684 methamphetamine users at risk of arrest for possession using the null model (see Table 4) is very close to the total population of users estimated in Table 5, after adjusting for underreporting [7,973, 9,967]. This consistency suggests that risk of arrest is higher for methamphetamine users than ecstasy users or ATS users more generally (see Figure 2). The situation for ecstasy resembles what is typically found for drugs with a larger contingent of recreational users, where the population at risk of arrest is much smaller than the total population of users derived from general drug use surveys (Bouchard & Tremblay, 2005). We find that the 13,111 individuals estimated to be at risk of arrest for possession of ecstasy (see Table 5) would be only a fraction (13.6%-20.4%) of the total population of ecstasy users captured by the 2009 CADUMS [66,444, 99,666].

Table 6.

Estimated Number of Methamphetamine and Ecstasy Users (Aged 15+) and Sellers, Quebec, 2009.

Substance	Nonadjusted	Adjusted
Substance	0%^a	20%^a	50%^a
Methamphetamine^b
$\hat{N}$ methamphetamine users	6,644	7,973	9,967
$\hat{N}$ methamphetamine sellers^c	443	933	1,424
Ecstasy
$\hat{N}$ ecstasy users	66,444	79,732	99,666
$\hat{N}$ ecstasy sellers^c	4,429	9,334	14,238
Methamphetamine + ecstasy users	73,088	87,706	109,633
Methamphetamine + ecstasy sellers	4,872	10,267	15,662

Note. Estimates based on past-year prevalence use reported by 2009 CADUMS. CADMUS = Canadian Alcohol and Other Drug Use Monitoring Survey.

Percentage underreporting.

Methamphetamine/Crystal meth.

Here, $\hat{N}$ is the population at risk of arrest estimated by dividing the adjusted and nonadjusted methamphetamine and ecstasy users by a user-to-dealer ratio of 15:1 at 0%, 1:1 at 20%, and 7:1 at 50% underreporting.

We also used the multiplier method (see Bouchard & Tremblay, 2005; MacCoun & Reuter, 2001) to provide crude estimates of the number of offenders at risk of arrest for selling methamphetamine and ecstasy. This method relies on the assumption of the reliability of two indicators: (a) the prevalence of drug users from general population surveys, and (b) the user-to-dealer ratio. Estimates were derived by

\hat{N} = \frac{p}{r},

where $\hat{N}$ is the multiplier estimate, p is the prevalence of methamphetamine or ecstasy users derived from general population surveys, and r is the number of drug users per dealer. We estimated the first indicator (numerator) with adjustments for underreporting in Table 6. The second indicator (denominator) requires firsthand knowledge of the buying and selling patterns found in Quebec’s ATS markets. Lacking this information, we used the smallest (7:1 for heroin) and largest (15:1 for cannabis) user-to-dealer ratios derived from other illegal drug markets in Quebec by Bouchard and Tremblay (2005) to produce the lower and upper bound estimates in Table 6.

The estimated 933 [443, 1,424] people at risk of arrest for selling methamphetamine from the multiplier method is much smaller than the 3,174 [1,856, 4,493] people at risk of arrest for selling offenses involving methamphetamine (see Table 4); however, the 9,334 [4,429, 14,238] multiplier estimate of the number of people at risk of arrest for selling ecstasy is near the 13,448 [–12,209, 39,805] people estimated at risk of arrest for selling offenses involving ecstasy (see Table 5). Which estimates are most plausible? It is hard to know for certain, as the user-to-dealer ratios were not derived from the ATS market. The safest statement that can be made is that the population at risk of arrest for selling methamphetamine ranges from 400 to 6,900, whereas the population at risk of arrest for selling ecstasy ranges from 4,500 to 14,000. But the smaller number of people at risk of arrest for selling methamphetamine compared with ecstasy gives us confidence in the logic underlying the estimation procedures—we would expect fewer people at risk of arrest for selling methamphetamine given that methamphetamine markets have been shown to be more centralized than ecstasy markets (Ouellet & Morselli, 2014).

Conclusion

Methods used to estimate the size of criminal populations have yet to gain traction among criminologists, despite their utility. Although simpler models like multiplier methods may be preferable in certain contexts, regression models control for observed heterogeneity within a sample drawn from a population of interest, which provides substantive context regarding (a) the characteristics of a population that increase or decrease the probability of capture or recapture and (b) the composition of the population that is unobserved but shares the characteristics of the observed sample. Regression models also provide a much higher level of precision than do extrapolation procedures, such as multiplier methods that rely solely upon observed arrest distributions. Crude estimates may be useful, but they imply the use of a single arrest rate. Confidence intervals and goodness-of-fit tests calculated in conjunction with regressions, which are not calculated along with multiplier methods, provide the means to compare nested models and assess the plausibility of estimates. Yet, regression models only partially solve the uncertainty involved in these estimates. Good statistical fit does not imply accuracy.

By our estimates, Quebec’s ATS markets consist of approximately 42,541 [36,936, 48,145] individuals at risk of arrest for any drug-related offense throughout 2008-2009. By the raw numbers, 80% of the estimated population was at risk of arrest only for possession. That more people were estimated to have been at risk of arrest for possession than selling is consistent with arrest data across drug and offense types (Reuter & Kleiman, 1986), which adds further support for our estimates. Although the sheer numbers of arrests for possession were higher than for selling, the relative risk of arrest was greater for selling offenses, suggesting that following an arrest for selling, offenders come under greater scrutiny from law enforcement.

The relatively tight confidence intervals provide an indicator of the quality of the estimates for the number of offenders at risk of arrest for possession, selling, and/or any drug-related charge for all ATS and methamphetamine. The estimates of the number of people at risk for possession and/or selling of ecstasy have much wider confidence intervals, with invalid lower bound estimates that cross zero. The volatility of these estimates has more to do with scarcity of the data than the model itself, as there were few arrests and rearrests for ecstasy relative to methamphetamine and all ATS-related offenses. We were unable to provide estimates of importers, exporters, and manufacturers for these very reasons: These populations were small and arrested offenders have a higher likelihood of being incarcerated for longer periods of time. In such contexts, simply not enough people get rearrested within a reasonable time for many capture–recapture methods to be of use.

Adoption and application of capture–recapture methods is a relatively low-cost extension of our current efforts at monitoring crime trends. Capture–recapture rarely, if ever, involves supplementary data collection; existing data sources like police arrest records and/or hospital records can be used in conjunction with one another if linking multiple record systems is necessary, although some methods can be used with a single data source, as shown in this study. We used Zelterman’s (1988) model and its maximum likelihood estimator (Böhning & Van Der Heijden, 2009) because of its demonstrated validity for sparse crime data (Bouchard, 2007; Bouchard & Tremblay, 2005; Collins & Wilson, 1990; Rossmo & Routledge, 1990), although criminologists should consider the full range of options available as different models may be more reliable depending upon the biases of different data sources. Crime severity, rates of detection, and population size are three interrelated factors that should influence the choice of model. The wider use of capture–recapture methods in criminology also has practical application. Ideally, resource allocation decisions should be based on estimates of the true size of the underlying populations, as opposed to sole reliance on people who already interact with the criminal justice system. As shown in this study, trends in the size and composition of populations at risk of arrest may not coincide perfectly with trends of arrests. Estimates of the size of the criminal populations improve our assessments of why there is growth and/or decline in trends of incidents and arrests and whether criminal justice policy interventions actually effect the number of people at risk of arrest.

Footnotes

Appendix

The Ratio-Plot Cut Point (k) Test on the Distributions of Arrests by Offense Classification Across Observation Periods.

k	All ATS offenses			ATS possession			ATS selling
k	$χ_{k}^{2}$	p	$\hat{N}$	$χ_{k}^{2}$	p	$\hat{N}$	$χ_{k}^{2}$	p	$\hat{N}$
1999-2001
1	0.00	1.000	17,492^a	0.00	1.000	16,338^a	0.00	1.000	8,734^a
2	3.88	0.000	14,836	33.92	0.000	13,207	1.25	0.000	7,584
3	1.17	0.000	14,128	48.67	0.000	12,932	1.36	0.000	6,877
4	1.11	0.000	13,529	48.76	0.000	12,941	2.76	0.000	6,159
5+	1.12	0.000	13,550	41,455.39	0.000	12,426	2.81	0.000	6,181
2002-2004
1	0.00	1.000	31,039^a	0.00	1.000	32,166^a	0.00	1.000	13,518
2	2.43	0.000	27,717	2.84	0.000	27,302	0.30	0.583	13,260^a
3	1.14	0.000	26,608	2.87	0.000	27,405	3.32	0.000	11,575
4	3.68	0.000	26,275	3.21	0.000	26,336	—	—	—
5+	—	—	—	—	—	—	—	—	—
2005-2007
1	0.00	1.000	48,785^a	0.00	1.000	45,999^a	0.00	1.000	29,353^a
2	3.39	0.000	44,728	4.07	0.000	40,311	4.06	0.000	25,268
3	1.44	0.000	43,256	6.68	0.000	39,655	5.69	0.000	24,894
4	9.31	0.000	42,387	4.52	0.000	39,162	—	—	—
5+	1.09	0.000	41,771	—	—	—	—	—	—
2008-2009
1	0.00	1.000	37,992^a	0.00	1.000	32,115^a	0.00	1.000	18,920^a
2	4.74	0.000	33,825	1.61	0.000	29,372	1.11	0.001	17,389
3	1.80	0.000	32,430	1.64	0.000	29,498	1.80	0.000	16,149
4	6.49	0.000	31,852	3.42	0.000	29,077	2.94	0.000	15,979
5+	1.67	0.000	31,686	—	—	—	—	—	—

The χ² test statistic is calculated along with a population estimate extrapolated from an arrest distribution that is derived from an adjusted Turing estimator.

Acknowledgements

The authors wish to thank the anonymous reviewers and the editor for their comments on an earlier version of this manuscript.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Partial funding for the research was received from the National Institute of Justice (2010-IJ-CX-020) and Public Safety Canada.

ORCID iD

Martin Bouchard

Notes

Author Biographies

Martin Bouchard is a professor of Criminology at Simon Fraser University, where he leads the Crime and Illicit Networks Laboratory. His work focuses on the size and organization of illicit markets, as well as on examining the impact of social networks on various outcomes of criminological interest, including the dynamics of gang conflicts.

Carlo Morselli is a professor at the École de criminologie and Director of the International Centre for Comparative Criminology, Université de Montréal. His expertise is primarily in the areas of organized crime, crime networks, illegal markets, and social process theories of crime.

Mitch Macdonald is a PhD student in Criminology at Simon Fraser University. His research interests revolve around social networks and crime, methods to estimate the size of illegal populations, as well as the economics of organized crime.

Owen Gallupe is an associate professor in the Department of Sociology and Legal Studies at the University of Waterloo. His research focuses mainly on peer influence, social networks, and offending. His recent work has been published in venues such as Justice Quarterly, Journal of Research in Crime and Delinquency, and Journal of Quantitative Criminology.

Sheldon Zhang is professor of Criminology and Justice studies at the University of Massachusetts Lowell. He has published on transnational organized crime and correctional reentry program evaluation.

David Farabee is professor Emeritus in Department of Psychiatry and Biobehavioral Sciences at the University of California, Los Angeles and research professor at the NYU Department of Population Health. He has published in the areas of substance use disorder, adult and juvenile crime, correctional mental health services, and recidivism-reduction programs.

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Estimating Risks of Arrest and Criminal Populations: Regression Adjustments to Capture–Recapture Models

Abstract

Keywords

Introduction

Method

Outcome: Truncated Counts of Arrests and Rearrests

Control Variables

Capture–Recapture Assumptions

Analytic Strategy

Results

Trends in Populations At Risk of Arrest Across Observation Periods

Assessing the External Validity of the Estimates

Conclusion

Footnotes

Appendix

Acknowledgements

Declaration of Conflicting Interests

Funding

ORCID iD

Notes

Author Biographies

References