Journal of Quantitative Criminology

, Volume 25, Issue 2, pp 129–153

A Developmental Approach for Measuring the Severity of Crimes

Authors

  • Rajeev Ramchand
    • RAND Corporation
    • Department of CriminologyUniversity of Pennsylvania/RAND Corporation
  • Amelia Haviland
    • RAND Corporation
  • Andrew R. Morral
    • RAND Corporation
Original Paper

DOI: 10.1007/s10940-008-9061-7

Cite this article as:
Ramchand, R., MacDonald, J.M., Haviland, A. et al. J Quant Criminol (2009) 25: 129. doi:10.1007/s10940-008-9061-7

Abstract

There is widespread agreement in criminology that some crimes are more severe than others, but precise definitions of crime severity and straightforward methods for measuring it have been elusive. Public perceptions of crime severity and economic estimates of crime costs to society or willingness to pay offer a variety of metrics for the public’s perceptions of severity, but they may not accurately describe severity as reflected in offender preferences. The behavior of offenders is critical for understanding developmental progressions in criminal careers, as one may assume that typically more severe offenses are not undertaken until less severe crimes have been committed. In the present paper we propose an alternative metric of crime severity, drawing on findings from developmental criminology that indicate that more severe crimes occur after less severe crimes in the criminal life course, and a method for estimating crime severity that uses the generalized Bradley–Terry model of multiple paired comparisons. We demonstrate this approach on two samples of youthful offenders: the National Youth Survey and the RAND Adolescent Outcomes Project. The results suggest that sample-specific estimates of crime severity can be derived, that these estimates provide insight into the developmental progression of crime, and that they correspond well to crime severity rankings produced by the public.

Keywords

Crime severityDevelopmental criminologyBradley–Terry

Introduction

Criminology has an historic interest in measuring the severity of crimes and how crimes rank relative to each other.1 Typically, criminologists have relied on perceptual studies or normative rankings in which respondents sort a set of offenses by their perceived severity. Sellin and Wolfgang’s (1964) classic study on the measurement of delinquency developed the most widely used method for weighting the average severity of a variety of criminal offenses. While the technical merits of normative rankings and the appropriate types of analyses to use have been debated (see Rose 1966; Meithe 1982; Cullen et al. 1985; Collins Mark 1988; Lynch and Danner 1993), the Sellin-Wolfgang method remains the conventional approach for estimating crime severity. In addition to normative rankings, others have assessed the relative severity of criminal offenses using calculated economic costs of harm done to crime victims using jury awards, social and economic costs associated with a victimization, and willingness to pay to reduce risk of exposure to different crimes (see Cohen 2000; Cohen et al. 2004).

The severity of a crime can be viewed as a multidimensional construct that considers the harms done to society, personal harms experienced by victims, and the likely consequences for the offender, dimensions typically tapped by normative rankings and economic valuations. These methods used to measure crime severity, however, chiefly concern public perceptions, which may be subject to sudden shifts (e.g., moral panics about marijuana use) that may be quite distinct from the influences that direct individual offending behaviors. The implicit perspective of the offender, which may be revealed in criminal career development, taps another dimension of crime severity which to this point has not been well documented.

Understanding crime severity as it influences offending behavior may be particularly important for applications that seek to understand developmental progressions in criminal offending. Although most adolescent offenders desist from offending, a small but significant proportion continue to offend into adulthood and their criminal actions tend to escalate in severity over time (Farrington 1986; Blumstein et al. 1988; Le Blanc and Loeber 1998; Laub and Sampson 2001). An important research goal in criminology is to identify significant developmental steps in the escalation of offending, and what crimes signify a foray into more severe offending behaviors. However, if we were to apply the public’s perception of the severity of offenses in trying to achieve this goal, we may reach misleading conclusions about where such important developmental steps exist. For example, using Wolfgang’s collection of the public’s judgments (Wolfgang et al. 1985) about crime severity we may identify marijuana use as an important and large incremental step toward more severe criminal offending among a group of youths who previously only engaged in theft. If, however, in the youths’ experience, marijuana use is actually quite a conventional activity and not at all a severe crime, then efforts to understand the factors leading to the “escalation” to marijuana use will have failed to “carve nature at its joints,” as Plato said. Stated differently, the local norms and preferences of a sample of youth for which marijuana use is quite common may differ in important ways from that of the public at large, or the average views of the many segments of society making up the public view captured by surveys. In the same way, understanding the influences on offenders could be important in cross-cultural comparisons of crime progression where cultures have different views of the marginal severity of crimes. If desecrating a religious text is wrong in one culture, then engaging in that behavior represents a violation of local norms that is significant in a criminological context, even if this act is seen as trivial in other cultures.

In the present study we propose a method for measuring crime severity that builds on a developmental understanding of how offending escalates over time, which is specific to local or population-specific concerns. We refer to the developmental process of offending as the “temporal within-individual changes in offending” (Le Blanc and Loeber 1998, p. 117). We posit that using a developmental approach can provide a generalized model that can aid in the measurement of crime severity across diverse samples of offenders and that is sensitive to local or sample-specific behavioral norms. We approach this problem by examining the sequencing of offending over time in a sample, imposing one key assumption: if for some group an offense, A, is less severe than another, B, then members of that group will be more likely to engage in A before they engage in B, rather than the reverse.2

In the following sections we first discuss the prior literature on measuring crime severity and the developmental patterns of offending. Second, we discuss our developmental approach to assessing crime severity and the use of the generalized Bradley–Terry model to estimate the severity of criminal offenses. Third, we apply the Bradley–Terry model on two samples of adolescent and young-adult offenders and examine its rankings of crime severity. Finally, we discuss our findings in light of the research on measuring crime severity.

Literature Review

Measuring Offense Severity

The empirical literature on measuring crime severity has been generated by both small scale and large scale surveys of public perceptions. Experimental psychologists originally attempted to develop scales of offense severity based on the presumption that psychometrically sound scales of human behavior could be derived from “laws of comparative judgment” (Thurstone 1927). Clinical psychologists were interested in developing offense severity scales as a method for assessing the relative psychopathology of offending behaviors, or to “provide a basis for differentially diagnosing the extent of general predisposition toward asociality” (Durea 1936, p. 413). In this line of research, professionals (e.g., psychologists, judges, etc.) were asked to rank order a list of criminal offenses according to how severe they perceived offenses to be (Clark 1922; Thurstone 1927; Durea 1933, 1936; Durea and Pataky 1937; Gorsuch 1938). Results from these early studies suggested that a scale of offense severity could be developed with sound psychometric properties (see Durea 1936).

A significant milestone in measuring crime severity occurred when Sellin and Wolfgang (1964) elaborated on the expert rating method by presenting scenarios for 141 different criminal events (taken from a random sample of Philadelphia police offense reports) to samples of university students, juvenile court judges, and police officers. Respondents were asked to rate the “magnitude of severity” of each offense scenario relative to a baseline offense (stealing a bicycle) that was anchored at a value of 10.3 Scores could range from a minimum of 0 to very large numbers. From these magnitude scores weights were estimated through regression analysis to the common criminal offense of theft of $1. Early objections to this approach included Rose (1966), who suggested that Sellin and Wolfgang’s conclusions that there was no substantial occupational group differences in the relative ratings of crime severity and that one could add scores together from separate crimes to yield meaningful severity scores were assumptions not subjected to appropriate statistical tests (p. 421). Nevertheless, this method of using normative judgments became a mainstay of criminology. A number of replications of this work followed, with many studies confirming that similar rankings were found in diverse samples of raters (see e.g., Wellford and Wiatrowski 1975; Figlio 1975).

Research from this tradition indicates a high degree of consensus across society in the perceived severity of criminal offenses. Rossi et al. (1974), for example, sampled ratings of offense severity made by Black and White adults living in Baltimore, and found high concordance between the views of Blacks and Whites, and between males and females. They also found that a small number of offense characteristics accounted for a large share of the variation from crime to crime rating (e.g., crimes against persons vs. crimes against property). Wolfgang et al. (1985) provided the most comprehensive societal assessment of crime severity in their National Survey of Crime Severity (NSCS). The NSCS was conducted as a supplement to the National Crime Survey in 1977 and asked 51,623 adult household members to rate the severity of offenses against a reference crime (theft of a bicycle). A total of 204 criminal offenses were rated, and severity scores were developed from a regression adjusted ratio to the crime of theft of $1 (values ranged from 0.2 to 72.1) (Wolfgang et al. 1985). Overall, the results from the NSCS suggested general agreement across regions, gender, income, and race/ethnic groups on the perceived rankings of crimes. However, NSCS results did indicate some differences in the relative scaling of offenses (e.g., Whites compared to Blacks tend to give offenses higher magnitude scores).

The technical merits of using magnitude weights to rank the relative severity of crimes have been debated, and a number of other approaches that rely on public perceptions have been offered (Parton et al. 1991). Gottfredson et al. (1980), for example, found that the assumption that separate components of criminal events (e.g., level of monetary loss) could be added incrementally to any offense to describe its level of severity did not hold uniformly across offense types. Specifically, they found that monetary loss had less influence on ratings of violent crimes (robbery and rape) than on property crimes (e.g., theft and vandalism). Lynch and Danner (1993) suggested that scenario-based methods for rating crime severity lead to oversimplified judgments about severity, and consequently to measurement error and distorted scales. They argued that simple scenarios require respondents to engage in too much speculation about the unreported details of the crime, like what it would feel like to be victimized, that are important to provide valid judgments about severity. Instead, Lynch and Danner (1993) suggested using a hedonic rating scale, whereby crime victims themselves rate the relative severity of crimes they have experienced. All of these studies share the common use of normative judgments as a method for ranking the severity of criminal offenses.

Another approach for ranking the severity of crimes has been developed from the environmental economics literature. This research has rated crime severity on the basis of the costs crimes impose on victims or society. These cost calculations are derived from victim reports, jury awards, or willingness to pay estimates. Early attempts to rank the severity of crimes according to their economic costs relied on calculating the costs of crimes according to victimization reports collected as part of the National Crime Survey (Bureau of Justice Statistics 1984; Cohen 1988). These cost estimates included both the direct (e.g., value of any lost, stolen, or damaged goods, lost wages, and medical expenses) and indirect costs (monetary values associated to pain and suffering and risk of death) (Cohen 1988). The first approaches for calculating the costs associated with pain and suffering and risk of death used jury settlement awards from comparable injuries due to accidents (Cohen 1988). Later, the monetary value of pain and suffering was calculated by applying jury awards for injuries resulting from individual crimes after weighting those costs by the probability of each type of injury being associated with the crime (Miller et al. 1996). Studies using these approaches found crime severity rankings that were similar to those derived through the public perception methods pioneered by Sellin and Wolfgang.

An additional economic approach for estimating the severity of crimes is to calculate what individuals are willing to pay for specific crime control programs. The most noteworthy of these, Cohen et al. (2004) developed a telephone survey in which respondents were told that a hypothetical crime control program successfully prevented one in every ten specific crime types, and then asked if they would be willing to pay a predetermined dollar amount per year to continue the program, where the dollar amount was randomized between $25 and $225 (in $25 intervals). Using this approach, the authors found in a representative sample of the U.S. population that among the crimes of burglary, armed robbery, serious assaults, rape, and murder, people were willing to pay the least to reduce burglary ($104 per household for a 10% reduction) and the most to reduce murder ($146 per household for a 10% reduction).

In general, the literature on measuring crime severity suggests agreement in both public perceptions and economic cost estimates as to what constitutes the most severe offenses, with some differences in the ranking of mid to low range offenses. There is clearly value to both of these approaches. Both the public’s perception of what constitutes the most severe offenses and the estimated economic costs of offenses provide important points of comparison to the appropriateness of contemporary sentencing statutes or crime prevention strategies. Ideally, that is, sentencing guidelines reflect the public’s perceptions of crime severity, and the public’s perceptions track closely with the real burdens different crimes impose on victims and society.

There are, however, limitations to both public perceptions and economic costs for assessing the relative severity of crimes. Public perceptions of crime severity, for instance, can be influenced by singular events that are widely publicized, by moral considerations that may not track well with an offense’s cost to society, or to the views of subpopulations following different moral compasses. Historical evidence suggests important geographic or temporal patterns for social norms regarding crime and social threat. Kai Erikson notes the fear surrounding the diminishing importance of the church generated the heightened concern with witchcraft and the subsequent trials and hangings in colonial era Salem, Massachusetts (Erikson 1966). These fears, however, quickly dissipated and witchcraft was no longer treated as a severe crime. In more recent times, some have argued that social panic over the perceived dangerousness of drugs has resulted in a heightened public concern and draconian criminal justice policies for drug-related offenses that far exceed their actual social threat or societal cost (Tonry 1995; Caplow and Simon 1999). A similar pattern of perceived social threat is evident with the contemporary example of the coming wave of juvenile “super-predators;” where academics forecasted a looming number of serious juvenile offenders that were going to be more prone to violence than those in the past (Wilson 1995; DiIulio 1995; Fox 1996) and there was an increased push from policy makers to treat juvenile offenders in adult courts (Feld 1999). These forecasts, however, turned out to be inaccurate as trends in juvenile violence decreased significantly (Zimring 1998).

Economic costs to crime victims also suffer from the potential effect of sample selection bias. Jury awards, for example, may provide inflated costs since they are unlikely to be representative of common criminal events. There are also considerable limitations to willingness to pay estimates. In these types of studies, respondents do not actually have to forgo any money when assessing what they would pay for a reduction in specific crimes. While some respondents may provide well thought out and informed answers to these types of questions, there is also potential for error. For example, respondents may report being willing to pay large amounts of money to reduce all crimes, which reveals a preference for a general reduction in crime but are less useful for distinguishing between the different crime types. Respondents may also report being willing to pay an amount because they feel compelled to respond in this way, but may display quite different behaviors had they actually been required to forgo their own earnings.

Sensitivity analyses can be used to moderate the potential volatility of normative judgments and economic valuations for assessing crime severity. A more fundamental issue with either public perceptions or economic costs is that they treat each criminal event as an independent occurrence and ignore the developmental processes that occur among offenses over time. For example, studies indicate that escalation from less to more severe offenses over time appears to be common among criminally active individuals (Le Blanc and Loeber 1998). Viewing crime severity through a developmental progression in offending behaviors does not replace these other methods, but it could complement them by providing a measure of crime severity based on offending behavior, a perspective that could be particularly useful in our understanding of criminal careers (Blumstein et al. 1986). We turn now to discussing support for this assertion.

The Developmental Process of Offending

By nearly any definition of severity, individuals’ criminal actions rarely jump to high severity crimes without first engaging in less severe crimes (Farrington 1986; Blumstein et al. 1988; Le Blanc and Loeber 1998). People who engage in severe crimes typically start off engaging in petty crimes, and before engaging in petty crimes they misbehaved in less severe ways at home and in schools. Offenses also appear to occur in clusters such that “the onset of one type of deviancy is often associated with the onset of another type of deviancy” (Le Blanc and Loeber 1998, pp. 133–136). Take, for instance, the common offending pattern of progressing from petty thefts in early adolescence (e.g., ages 12–14) to more severe forms of larceny and burglary in late adolescence (e.g., ages 16–18). Clearly, the behavioral process of this type of offending pattern suggests a developmental progression in the severity of criminal offenses. A number of longitudinal studies that rely on official or self-reported offending reveal this pattern where the onset of criminal behavior escalates in severity over time (Blumstein et al. 1986; Farrington 1986; Elliot et al. 1989; Elliot 1994). For example, Elliott et al. (1989) found in the National Youth Survey that on average minor delinquent offenses (e.g., alcohol use) tended to precede more severe forms of delinquency (e.g., use of hard drugs like cocaine, assault, robbery) (see LeBlanc and Loeber, 1998 for a review of this research).

There are competing explanations for this progression in criminal offending. Some criminologists see it as evidence of some existence of state-dependency; whereby prior criminal behavior has an independent behavioral effect on future criminal behavior through changing opportunity structures or influencing inhibitions (Nagin and Paternoster 1991). Other criminological perspectives argue that the development of crime from less-severe to more-severe offending behavior is a function of adult socialization and peer influence, age-specific changes in opportunities, and/or physical maturation (Akers 1998; Le Blanc and Loeber 1998; Moffitt 1993; Sampson and Laub 1993). Regardless of the causal factors driving the sequencing of offending, it is widely accepted that on average, individuals’ offending is sequenced, such that those who engaged in more severe crimes were earlier engaged in less severe ones.

A Developmental Approach to Measuring Crime Severity

The empirical literature suggests general agreement on what constitutes the most severe offenses across cultures and social class positions. There is less agreement across these methods of ranking crime severity on what constitutes medium range offense severity, such as the relative severity of burglary and motor vehicle theft. Clearly, differences in survey settings, age groups interviewed, and time periods all can influence how offenses get ranked relative to how severe they are perceived to be. If criminal offending on average follows a developmental process in which less severe offenses typically occur before more severe ones, it is possible to develop an alternative method for assessing the relative severity of crimes that can be estimated by observing the behaviors of any group of offenders.

In the present study we build on previous research by developing a method for assessing crime severity that is tied to the observed developmental progression in offender behavior. Using this approach and applying it to two different cohorts of youth, we derive severity estimates for two samples that rank crimes from the least to most severe and that can be viewed as developmental steps from less to more severe offenses in the course of an individual’s observed criminal career.

Methods

Analytic Assumption to Model Offense Severity

Recognizing the development process of crimes over time within individuals allows us to state our basic assumption for measuring crime severity:

Assumption 1

If within some population group, the severity of crime A is generally less than the severity of crime B, then the probability of an individual from that group committing crime A before he commits crime B is greater than the probability he commits crime B before he commits crime A. This assumption can be written formally as:

$$ {\text{Severity}}\left( {\text{A}} \right) < {\text{ Severity}}\left( {\text{B}} \right){\text{ if and only if P}}\left( {\text{A before B}} \right) > {\text{ P}}\left( {\text{B before A}} \right) $$
Support for this assumption is found in the developmental criminology literature (Farrington 1986; Le Blanc and Loeber 1998).

The Formal Model

Based on this proposition we can estimate a Bradley–Terry model of crime severity using longitudinal data on offending behavior. This model dates back to Thurstone’s (1927) method of paired comparisons and was extended to compare outcomes in ranking experiments that involve block treatment designs or multiple paired comparisons of treatments (see Bradley and Terry 1952; Davidson 1970; Lancaster and Quade 1983). Specifically, the Bradley–Terry model involves a generalization of the binomial model that can handle multiple comparisons, and has been applied to a number of research areas including publishing (e.g., Stigler 1994), sports (e.g., Agresti 1990), and machine learning (e.g., Huang et al. 2006; see Bradley and Terry 1952 or Agresti 1990 for further information on the model).

The underlying theory of the Bradley–Terry model is that each in a set of items being compared has an underlying “ability” score, denoted as π, such that:
$$ {\text{pr}}\left( {i{\text{ beats }}j{\text{ when }}i{\text{ and }}j{\text{ are compared}}} \right) = {{\pi_{i} } \mathord{\left/ {\vphantom {{\pi_{i} } {\left( {\pi_{i} + \pi_{j} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\pi_{i} + \pi_{j} } \right)}} $$
(1)
which can also be expressed as:
$$ {\text{Odds}}\left( {i{\text{ beats }}j{\text{ when }}i{\text{ and }}j{\text{ are compared}}} \right) = {{\pi_{i} } \mathord{\left/ {\vphantom {{\pi_{i} } {\pi_{j} }}} \right. \kern-\nulldelimiterspace} {\pi_{j} }} $$
(2)
where the odds is defined as the probability that i beats j divided by the probability that j beats i.

Sports rankings provide an illustrative example to highlight the strengths of the Bradley–Terry model. Assume a sports league with any number of teams in it. The model assumes that each team in the league has an underlying strength or ability, π, which can be estimated based on the number of times it has won and lost against each of its opponents. Team rankings are thus based on listing the values of π, with higher values indicating the “best” team, and results are all interpreted relative to a reference team with πi = 1.0. Further, π for each of two given teams i and j can be estimated even if the teams have not played each other based on the games each has played against the other teams in the league (Agresti 1990).

Our extension of the Bradley–Terry model to crime severity is analogous to the sports ranking example. One can imagine that each crime has an underlying value of π, which in our application will be referred to as severity scores. With the proposition given above, competitions between crimes occur over time within individuals, whereby crime i “beats” crime j if it occurred for the first time at any point after crime j first occurred. In this model ties are ignored. It is important to note that an individual who commits crime i but never commits crime j provides no direct information on the ability score of crime j. Thus, although our method is based upon the assumption of temporality between pairs of crimes, it is not sensitive to proportion of young offenders who do not commit more severe crimes in the future.

In addition, even though the empirical evidence suggests that many individuals de-escalate or desist from severe offending over time (see Sampson and Laub 1993; Laub and Sampson 2001), this would have no impact on our method of measuring crime severity, which exclusively considers the first instance of each crime type in a person’s criminal career. As long as on average individuals in the sample commit less severe crimes earlier than more severe crimes, the rank ordering of the severity scores will be correct.

Estimating π using maximized likelihood can be achieved under two assumptions: (1) competitions between specific crimes i and j are independent between individuals, and (2) competitions between unique pairings of crimes are also independent. Under these conditions, estimating πi requires a simple iterative procedure that maximizes the log likelihood of the model:
$$ L({{\mathbf{a}}}|\pi ) = \mathop {\mathop {\prod \prod }\limits_{i < j} \left( {\begin{array}{*{20}c} {n_{ij}} \\ {a_{ij}} \\ \end{array} } \right)}\limits_{{}} \frac{{\mathop \pi \nolimits_{i}^{a_{ij}} \mathop \pi \nolimits_{j}^{n_{ij} - a_{ij}} }}{{\mathop {\left( {\pi_{i} + \pi_{j}} \right)}\nolimits^{n_{ij}} }} $$
(3)
In Eq. 3, nij represents the number of times crimes i and j are compared while aij represents the number of times i occurs before j, a = {aij}, and πi represents the severity score of crime i (Lancaster and Quade 1983). The same methods used to estimate variability in maximum likelihood logistic regression models can be used to estimate the variability of each crime’s ability score.
In the current analysis, we present estimates of severity scores for crimes relative to a reference crime of burglary (i.e., π (burglary) = 1.0). By fixing the estimate of crime severity to burglary, the severity score for each crime (πi) can be interpreted as the odds of the given crime occurring for the first time after burglary first occurred and πi/(πi + 1) can be interpreted as the probability of a given crime occurring for the first time after burglary first occurred. For ease of interpretation, we also present results from the Bradley–Terry model on the logit scale, where:
$$ \lambda_{\text{i}} = {\text{logit}}\left( {\frac{\pi_{i}}{1 + \pi_{i}}} \right) = \log \pi_{i} {\text{ for all }}i $$
(4)

This approach to ranking the severity of crimes is advantageous analytically because it can be approximated to any cohort of offenders and is not constrained by period effects or survey setting. Rather, it permits a measurement of crime severity that is internally consistent to a given sample. We demonstrate this methodological approach on two different samples of youthful offenders and show similar results, despite the fact that the samples are under very different constraints.4

Sensitivity Analysis and Hypothesis Testing

There is the possibility that one severity score estimate may be lower than the next highest score, but there exists no statistically significant difference between the two. We used two procedures to test the sensitivity of our severity score estimates and determine whether there was a statistically significant difference between estimates. First, we conducted a bootstrap procedure with 1,000 iterations and present the median values of π to discern whether there were any changes in crime rankings. Next, we estimated a t*t matrix (where t represents the total number of crimes and the lower diagonal is left empty) containing the relative log odds of each crime being more severe than each of the other crimes (i.e., λi − λj).5 We then conducted a bootstrap procedure of 1,000 iterations for each of the λi − λj estimates to construct a hypothesis test of paired crimes, and were particularly interested in those instances in which the range of λi − λj estimates between the 0.025 and 0.975 quantiles included the null value (=0). If the range of estimates included the null value, it suggests that we could not reject the null hypothesis that the pair of crimes’ severity was the same.6 All analyses were performed using R (R Development Core Team, available at http://www.r-project.org/) using the ‘BradleyTerry’ package (Firth 2005).

Samples

Data used in this study come from two sources: The National Youth Survey (NYS) and the RAND Adolescent Outcomes Project (AOP). The NYS is a longitudinal study of 1,725 adolescents aged 11 to 17 at baseline. Respondents were selected based on a probability sample designed to be representative of the adolescent U.S. population in 1976. Although the respondents continue to be followed to the present, we examine data from the first five waves of follow-up (i.e., 1976–1980), which occurred at equally-spaced intervals one year apart (see Elliot et al. 1989). The number of youth assessed at each follow-up time period and for which offense data were reported was 1,655 at wave 2, 1,626 at wave 3, 1,543 at wave 4, at 1,494 at wave 5, yielding a final retention rate of 87% at wave 5. The RAND Adolescent Outcomes Project (AOP) was designed to examine the effectiveness of a substance abuse treatment program serving youth referred from the criminal justice system (Morral et al. 2004; Morral et al. 2003). The AOP sample was selected to be representative of youths sent to large residential treatment programs by the Los Angeles Probation Department, the largest juvenile court system in the country. The original purpose of the AOP study was to examine the outcomes of youth on probation attending one long-term residential substance abuse program (Phoenix Academy) versus youth placed in other traditional community settings (e.g., group homes). All participants were recruited in February 1999 and for 15 months thereafter from the three juvenile detention facilities in Los Angeles County while they awaited their community probation. Participants were between the ages of 13 and 17 years and provided written informed consent and parental notification of enrollment in the study. Of 574 deemed eligible to participate in the AOP, a total of 449 were interviewed at intake (baseline). Most of those who did not participate in the study were placed in a detention facility before they could be interviewed, though some did not participate because they did not speak English, their parents refused to allow them to participate, or other miscellaneous reasons. Respondents were interviewed again at 3-months (n = 406), 6-months (n = 410), 12-months (n = 408), and 72-months (n = 365) after baseline. Thus, retention was over 90% for the first 3 waves of follow-up, and 83% at the most recent (72-month) assessment.

Measurement

In both samples youth were interviewed in successive waves and asked if they had committed a series of crimes during a specific reference period. From the NYS we selected 21 offenses that were asked consistently across all 5 waves of interviews for the referenced previous 12 months. From the AOP sample we selected 14 crimes that were asked consistently over the first 5 waves of interviews. These items asked respondents about the number of times they had engaged in each offense during the 90 days prior to the interview. The specific criminal offenses and the precise wording of each question are presented in Table 1. Our analytic strategy therefore misses any offending behavior that occurred earlier than 1-year prior to the baseline assessment of the NYS or earlier than 90-days prior to the baseline assessment of the AOP sample.
Table 1

Criminal offenses assessed in the NYS and AOP samples

National youth survey (NYS)

RAND adolescent outcomes project (AOP)

Label

Verbatim question text

Label

Verbatim question text

 

How many times in the last year have you…

 

During the past 90 days, how many times have you…

Assault

Attacked someone with idea of seriously hurting or killing him/her

Aggravated assault

Hurt someone badly enough they needed bandages or a doctor

Burglary

Broken or tried to break into a building or vehicle to steal something

Armed robbery

Used a knife or gun or some other thing (like a club) to get something from a person

Car theft

Stolen (or tried to steal) a MOTOR VEHICLE, such as a car or motorcycle

Arson

Intentionally set a building, car or other property on fire

Carried weapon

Carried a hidden weapon other than a plain pocket knife

Burglary

Broken into a house or building to steal something or just to look around

Damaged property (Others)

Purposely damaged or destroyed OTHER PROPERTY that did not belong to you (not counting family or school property)

Car theft

Taken a car that didn’t belong to you

Damaged property (Parents)

Purposely damaged or destroyed property belonging to your PARENTS or other FAMILY MEMBERS

DUI

Driven a vehicle while under the influence of alcohol or illegal drugs

Damaged property (School)

Purposely damaged or destroyed property belonging to a SCHOOL

Drug sales

Sold, distributed or helped to make illegal drugs

Force others (Robbery)

Used force (strong-arm methods) to get money or things from OTHER PEOPLE (not students or teachers)

Fraud/theft

Passed bad checks, forged (or altered) a prescription or took money from an employer

Force students (Robbery)

Used force (strong-arm methods) to get money or things from other STUDENTS

Larceny/theft

Taken something from a store without paying for it

Force teachers (Robbery)

Used force (strong-arm methods) to get money or things from a TEACHER or other adult at school

Murder

Been involved in the death or murder of another person (including accidents)

Gang fights

Been involved in gang fights

Simple assault

Hit someone or got into a physical fight

Hard drug sales

Sold hard drugs such as heroin, cocaine, and LSD

Stealing

Other than from a store, taken money or property that didn’t belong to you

Possession of stolen goods

Knowingly bought, sold or held stolen goods (or tried to do any of these things)

Vandalism

Purposely damaged or destroyed property that did not belong to you

Prostitution

Been paid for having sexual relations with someone

Weapon (Robbery)

Used a weapon, force, or strong-arm methods to get money or things from a person

Runaway

Run away from home

  

Sold marijuana

Sold marijuana or hashish (pot”, “grass”, “hash”)

  

Steal <$5

Stolen (or tried to steal) things worth $5 or less

  

Steal $5–$50

Stolen (or tried to steal) things worth between $5 and $50

  

Steal >$50

Stolen (or tried to steal) something worth more than $50

  

Steal school

Stolen or tried to steal something at SCHOOL

  

Throw cars/people

Thrown objects (such as rocks, snowballs, or bottles) at cars or people

  

Results

The results are divided into four sections. The first section describes diagnostics of the model estimation procedure. The second section displays the prevalence of each of the self-reported crimes across each time period for both samples. The third section identifies the trends in crime progression and the severity of crimes based on the Bradley–Terry model for each sample of adolescent offenders. The fourth describes whether there is evidence to reject the null hypothesis of no difference between crimes ranked close to each other and how the estimates identify the severity of crimes across the AOP and NYS samples.

Model Estimation

As previously discussed, the Bradley–Terry model ranks crimes by estimating a “severity score” πi for each i of t crimes in an estimation procedure based on the number of times i occurs for the first time before and after the first occurrence of each of the other crimes. In our study, because we use two separate sources of data, we selected the common reference category of “burglary,” which is comparably defined across both samples (see Table 1). For the current analyses, the items t represent the self-reported crimes listed above. Thus, for the NYS sample, there were 21 crimes representing a total of 420 comparisons and for the AOP sample there were 14 crimes and a total of 182 comparisons.

Prevalence of Crimes, across Waves

The proportion of youth who endorsed each crime type, over time and for both NYS and AOP samples, is presented in Table 2. It is clear that these two samples on average represent different spectrums of the youthful offending population. For instance, the prevalence rates for assault-related offenses reported in the AOP sample are almost twice that observed in the NYS sample. In the NYS sample the most frequently reported criminal behavior is throwing things at cars or people, which close to half of the sample reported doing at baseline. Less than 1% of NYS respondents reported using force to get money or things from a teacher in any wave. For the majority of offenses, the prevalence declines over time, with the exception of selling marijuana, selling harder drugs, and car thefts (see Lauritsen 1998 for a discussion of missing data). In the AOP sample, the prevalence of criminal activity declines across all crimes between 0 and 3 months, most likely a function of institutionalization during this period at a residential program that may have constrained criminal behavior. However, there is a general increase in the prevalence of most crimes after 3-months until 12-months. Between 12 and 72 months, some crimes increase (e.g., DUI increases from 8% to 22% and drug sales increases from 10% to 14%), some decrease (e.g., theft decreases from 11.5% to 7%) and some remain relatively stable (e.g., vandalism is 14% at 12-months and 13% at 72-months).
Table 2

Prevalence of self-reported offending

National youth survey

Year 1 (%)

Year 2 (%)

Year 3 (%)

Year 4 (%)

Year 5 (%)

Assault

6.1

3.9

4.1

5.5

4.6

Burglary

4.1

3.9

2.5

3.2

2.4

Car theft

0.9

0.9

1.0

1.3

1.3

Carried weapon

6.5

6.9

6.3

6.4

7.1

Damaged property (Others)

17.7

8.2

14.2

10.7

10.0

Damaged property (Parents)

15.2

7.1

7.0

4.5

4.1

Damaged property (School)

15.8

7.8

11.4

8.4

6.8

Force others (Robbery)

3.0

1.8

1.9

1.7

1.0

Force students (Robbery)

3.0

2.8

2.5

1.5

1.1

Force teachers (Robbery)

0.7

0.2

0.3

0.3

0.1

Gang fights

12.4

10.3

7.9

7.9

5.5

Hard drug sales

0.8

0.8

1.5

2.0

2.3

Possession of stolen goods

9.7

8.4

8.9

6.7

6.4

Prostitution

1.0

0.5

0.5

0.5

0.6

Runaway

5.9

5.5

4.8

4.9

3.7

Sold marijuana

4.4

7.3

10.3

10.4

10.3

Steal <$5

17.6

17.7

14.5

10.0

9.9

Steal $5–$50

5.6

5.7

5.7

5.1

4.1

Steal >$50

2.2

2.2

2.6

2.9

3.1

Steal school

6.5

6.3

4.7

4.0

3.3

Throw cars/people

46.5

23.8

35.1

24.8

19.3

Adolescent outcomes project

Baseline (%)

3-Months (%)

6-Months (%)

12-Months (%)

72-Months (%)

Aggravated assault

16.5

6.7

7.3

8.6

13.2

Armed robbery

4.5

3.7

3.4

3.4

2.5

Arson

3.6

0.5

0.7

1.5

1.4

Burglary

10.9

4.2

2.9

4.2

2.5

Car theft

13.4

3.2

4.6

5.9

3.6

DUI

10.2

4.7

4.9

8.8

21.9

Drug sales

25.2

6.9

8.3

9.6

13.7

Fraud/theft

3.3

0.5

1.0

1.7

2.2

Larceny/theft

34.1

7.9

10.2

11.5

7.1

Murder

2.2

1.0

2.0

1.5

0.5

Simple assault

45.2

17.2

16.8

20.3

29.6

Stealing

20.0

8.4

10.0

9.1

7.7

Vandalism

25.6

11.1

11.5

14.0

12.9

Weapon (Robbery)

6.2

3.2

3.7

3.7

1.9

Bradley–Terry Model Estimates

The empirically derived rankings for both the NYS and AOP samples are presented in Tables 3 and 4, respectively. We present the severity score (π) and its corresponding 95% confidence interval, the probability (π/(1 + π)) that each crime occurs after burglary (the reference crime category) as well as the median bootstrap estimates and the bootstrapped 95% confidence interval (estimated as the 2.5% and 97.5% quantiles).
Table 3

Bradley–Terry estimates: NYS sample

Crime

π

95% CI

Probability π/(1 + π)

Bootstrap

Median π

95% CI

Throw cars/people

0.02

0.02, 0.03

0.02

0.02

0.02, 0.03

Steal <$5

0.12

0.11, 0.13

0.11

0.12

0.10, 0.15

Damaged property (Other)

0.14

0.13, 0.15

0.12

0.14

0.11, 0.17

Damage property (School)

0.18

0.17, 0.19

0.15

0.18

0.14, 0.22

Damage property (Parents)

0.24

0.22, 0.26

0.19

0.24

0.19, 0.29

Gang fights

0.25

0.24, 0.27

0.20

0.25

0.20, 0.31

Possession of stolen goods

0.28

0.26, 0.31

0.22

0.28

0.23, 0.34

Sold marijuana

0.39

0.36, 0.42

0.28

0.39

0.32, 0.47

Carried weapon

0.41

0.38, 0.45

0.29

0.41

0.33, 0.51

Runaway

0.52

0.48, 0.56

0.34

0.51

0.41, 0.65

Steal $5–$50

0.53

0.49, 0.57

0.35

0.53

0.43, 0.64

Steal school

0.53

0.49, 0.57

0.35

0.53

0.43, 0.66

Assault

0.68

0.63, 0.73

0.40

0.67

0.54, 0.85

Burglary

1.00

1.00, 1.00

0.50

1.00

1.00, 1.00

Force students (Robbery)

1.37

1.26, 1.49

0.58

1.37

1.08, 1.73

Steal >$50

1.59

1.46, 1.74

0.61

1.59

1.29, 1.97

Force others (Robbery)

1.74

1.59, 1.90

0.64

1.74

1.32, 2.25

Hard drug sales

3.71

3.35, 4.11

0.79

3.73

2.79, 5.06

Car theft

4.04

3.63, 4.49

0.80

4.04

3.02, 5.66

Prostitution

5.65

5.03, 6.33

0.85

5.73

3.97, 8.79

Force teachers (Robbery)

10.60

9.20, 12.22

0.91

10.72

6.87, 19.06

Table 4

Bradley–Terry estimates: AOP sample

Crime

π

95% CI

Probability π/(1 + π)

Bootstrap

Median π

95% CI

Simple assault

0.06

0.05, 0.07

0.06

0.06

0.04, 0.08

Stealing

0.17

0.14, 0.19

0.15

0.16

0.11, 0.22

Vandalism

0.17

0.15, 0.20

0.15

0.18

0.13, 0.24

Drug sales

0.21

0.18, 0.24

0.17

0.21

0.15, 0.29

Larceny/theft

0.31

0.27, 0.36

0.24

0.31

0.22, 0.43

Aggravated assault

0.42

0.36, 0.48

0.30

0.40

0.29, 0.54

DUI

0.43

0.37, 0.49

0.30

0.44

0.33, 0.60

Car theft

0.68

0.58, 0.79

0.40

0.67

0.46, 0.96

Burglary

1.00

1.00, 1.00

0.50

1.00

1.00, 1.00

Armed robbery

1.93

1.62, 2.31

0.66

1.91

1.26, 2.94

Weapon (Robbery)

2.51

2.09, 3.02

0.72

2.52

1.70, 3.77

Fraud/theft

4.18

3.43, 5.10

0.81

3.81

2.45, 5.69

Arson

4.31

3.54, 5.25

0.81

4.76

2.99, 7.16

Murder

5.76

4.65, 7.13

0.85

6.17

3.78, 10.49

The Bradley–Terry estimates from the NYS sample have considerable face validity as measures of crime severity. For instance, stealing or trying to steal something worth less than $5 is less severe than trying to steal something worth $5–$50, which itself is less severe than trying to steal something worth more than $50. Additionally, stealing or trying to steal a car is ranked more severe than the other theft items. Selling marijuana is also ranked less severe than selling harder drugs such as heroin, cocaine, or LSD. The ranking of the robbery items is also consistent with an escalation in offense severity. Using force to get something from students is less severe than using force to get something from other people; whereas using force to get something from one’s teacher ranks as the most severe crime in this sample. One can imagine the typical offense scenario involving using force to get someone’s “lunch money” would be less severe than forcing a teacher to hand over personal property. Finally, the severity rankings remain unchanged after bootstrapping, suggesting that the model is stable and not overly sensitive to the inclusion or exclusion of particular individuals who may engage in rare crimes.

While the severity scores can be used to rank crimes, they have a practical interpretation as well. Consider the crime of selling hard drugs. The odds that the first occurrence of selling hard drugs occurs after an individual first reports committing burglary is 3.71. Therefore, the probability that selling hard drugs occurs after committing a burglary is estimated as 0.79 (3.71/(1.00 + 3.71) = 0.79). Now consider selling marijuana. The severity score for selling marijuana is 0.39, which indicates that selling marijuana tends to occur before committing a burglary. For ease of interpretation, the odds that a burglary occurs after selling marijuana can be converted to an estimate of 2.56 (1.0/0.39), and the probability that a burglary occurs after selling marijuana is therefore 0.72. Finally, we might be interested in the odds or probability that selling hard drugs occurs after selling marijuana. The estimated odds of selling hard drugs occurring after selling marijuana is 9.51 (3.71/0.39) and the corresponding probability is 0.90 (3.71/(3.71 + 0.39)).

It is more difficult to examine the face validity of crimes across the AOP sample, since there is not the same degree of within-crime type variation (i.e., stealing things with escalating worth) in these crimes. However, other suggestions of face validity are present. For example, being involved in someone else’s death (murder) is ranked as the most severe crime. The simple assault offense (hitting someone or getting into a fight) is also ranked as less severe than aggravated assault (injuring somebody enough to warrant medical attention). In this sample of serious adolescent offenders, drug sales are not as severe as some other crimes, such as car theft, burglary, and robbery (armed or with a weapon). In addition, fraud/theft (defined as passing a bad check, forging or altering a prescription, or taking money from an employer) was ranked as the third most severe crime.

Relative Differences in Severity

We used a bootstrap procedure to calculate λi − λj for each pair of i and j crimes and thereby formally test whether crime severity scores were significantly different across each of the different crime pairings. To ease in the interpretation of these results, we depict graphically the log of the severity scores (λ) and their corresponding confidence intervals in Figs. 1 and 2, respectively. Crime pairings for which there was insufficient evidence to reject the null will be roughly the same as those crimes whose log confidence intervals overlap. Matrices that present the median relative odds of each crime occurring after each of the other crimes for both NYS and AOP samples are presented in the Appendix Tables 5 and 6.
https://static-content.springer.com/image/art%3A10.1007%2Fs10940-008-9061-7/MediaObjects/10940_2008_9061_Fig1_HTML.gif
Fig. 1

Severity of crimes in the NYS

https://static-content.springer.com/image/art%3A10.1007%2Fs10940-008-9061-7/MediaObjects/10940_2008_9061_Fig2_HTML.gif
Fig. 2

Severity of crimes in the AOP

The majority of adjacent crimes pairings for which there was insufficient evidence to reject the null are found in the “middle” of the rank order. For instance, in the NYS sample, we cannot reject the null hypothesis that the severity scores for stealing less than $5 (2nd least severe crime) and purposely damaging other peoples’ property (3rd least severe) are equivalent. In addition, purposely damaging one’s parent’s property and being in a gang fight are not significantly different from each other, nor are possessing stolen goods and being in a gang fight or carrying a weapon and selling marijuana. There is also sometimes insufficient evidence that triads of crimes are significantly different from each other. For instance, there is insufficient evidence that the severity of running away, stealing something worth between $5 and $50, and stealing something at school are significantly different from each other. Two other such triads exist: using force on students to get something, stealing more than $50, and using force on others to get something; and a triad of crimes that includes hard drug sale, car theft, and prostitution. For the AOP sample, there was evidence of significant difference in rank between every pair of crimes.

Discussion

In the present study we develop a new method for measuring crime severity that was predicated on the developmental criminology perspective that, on average, individuals engage in less severe crimes before they undertake more severe ones. Using the Bradley–Terry model of multiple paired comparison tests we found that crime severity estimates could be derived from two separate samples of youthful offenders. In general, the results suggest that this method provides estimates of offense severity that are not just face valid, but which also have a developmental interpretation that is specific to the population under study. The offense severity estimates across both samples tend to rank violent crimes as more severe than property and drug-related offenses. Interestingly, the estimates suggest that drug dealing offenses are not as severe as those reported in public perceptions surveys. For example, the National Survey of Crime Severity estimated that selling marijuana had a value of 8.53 whereas common burglary offenses had values ranging from 3.08 to 3.14 (all relative to theft of $1) (see Wolfgang et al. 1985, p. 48). In contrast, the Bradley–Terry model estimates from NYS rank the severity of selling marijuana below that of burglary. Similarly, the estimates from the AOP rank selling drugs in general below burglary. These differences are most likely due to the fact that the Bradley–Terry model estimates crime severity based on offenders’ revealed preferences to engage in drug selling earlier than the commission of a burglary. Given the high prevalence of marijuana use among adolescents, it is easy to imagine that selling marijuana is not a particularly severe stepping stone in the progression of offenders’ criminal career trajectories.

The model should ideally be applied to cohorts followed from early childhood where offending behaviors are captured over a long period of time, whereas the current application only observes offending over a limited time period and may miss early developmental steps. Additionally, this model could be applied to offending data in international settings where the choices of crime may differ for socio-culture reasons, and may necessitate a different conceptualization of offense severity. Applying this method to multiple samples could provide a metric for comparing crime ranks across a diverse array of populations and be used to develop an aggregate ordering of offense severity. For example, in a supplemental analysis that examined whether our crime severity estimates differed by gender, we stratified the NYS sample by sex. In general the order of crime severity rankings were the same. The only difference was the severity scores for carrying a weapon and running away from home. In the total sample, carrying a weapon is ranked ninth and running away from home is ranked tenth, where higher ranks indicate more severe crimes. When stratified by sex, for males carrying a weapon was ranked eighth and running away from home was ranked thirteenth. For females, however, running away from home was ranked sixth and carrying a weapon was ranked twelfth. These findings suggest that running away from home is less severe at an early age for females as compared to males, a finding that is consistent with research indicating that running away from home for a variety of reasons (e.g., sexual abuse) is a precursor to more severe crimes (e.g., gang-related violence) for girls (Chesney-Lind and Pasko 2004). Using this method to estimate crime severity with self-report and official records across different populations could also aid in understanding differences in offense severity based on official offense charges versus those self reported by offenders.

We estimated the Bradley–Terry model for crime severity among two different cohorts of youth: a sample representative of the adolescent population in 1976 (NYS) and a sample of youth within Los Angeles juvenile justice system (AOP) in 1999–2000. Although these samples represent different spectrums of the offending population, there are consistencies in the rankings estimated across the samples. For instance, stealing is the second least severe crime in the AOP sample, which asks whether the youth took money or property that didn’t belong to him or her. This compares to stealing something less than five dollars, which is also ranked the second least severe crime in the NYS. Damaging property in the NYS, which aligns with vandalism in the AOP, is also ranked as the third least severe crime in both samples, and drug sales, which does not distinguish between selling marijuana and selling other drugs in the AOP, is ranked as fourth in that sample and occurs in a similar place with respect to the crimes to which it is more severe in the NYS. This may imply that among youth who offend, these crimes are not much more severe or developmentally significant than vandalism.

Rankings between the samples diverge for more severe crimes. Car theft, for example, is the third most severe crime in the NYS, but occurs before burglary in the AOP data. This may reflect different subpopulation experiences between the two samples, and detecting these differences is an inherent strength of the Bradley–Terry modeling approach. However, we cannot discount the possibility that this reflects different ways in which the questions are worded and asked, or differences in the range of crimes for which severity scores are being estimated.

The results from the Bradley–Terry model also provide insight into the development of criminal offending. It is important to know at what points in time youth move from one level of offending to another. Normative crime rankings and economic valuations may be less appropriate metrics for identifying developmental steps in criminal offending trajectories. If we use normative rankings, we may incorrectly identify when important steps are taken, and will be looking at the wrong associated events and precursors. If, for instance, we see sales of marijuana as more severe than burglary, we will view marijuana sales as a major escalation of criminal offending, even though from the choices that youthful offenders make selling marijuana does not appear to be a particularly severe offense choice. From a developmental perspective, we may be more interested in those thresholds of behavior that the youth and his or her peer group previously chose not to engage in, but now begin to commit. These behavioral thresholds are explicitly defined in the current approach.

There is, however, one major limitation to the current methodology. One must assume that opportunities to offend are ubiquitous throughout the entire duration of observation. If some crimes uniformly become more opportune later in the observation period, then they would automatically be ranked as more severe. If, for instance, opportunities for stealing cars are not as available at an earlier age compared to burglary, then including these offense choices in an Bradley–Terry model would likely rank car theft as more severe than burglary. Debate remains in criminology on the relative ubiquity of specific crime opportunities (see Sampson and Laub 1993). In our samples, car theft holds a constant prevalence of around 1% throughout the entire 5-year observation period of the NYS study and actually decreases in the AOP sample from 13.4% at baseline to 3.6% at 72-months, which provides some indication that car theft is not a simple function of age. Fraud/theft, which one might suspect to become more opportune later in offenders’ lives, also decreases over the course of the observation period. On the other hand, in the AOP sample, DUI increases from 10.2% at baseline to 21.9% at 72-months. If this does represent an increased opportunity to commit this offense later in the study participants’ lives, this may indicate that DUI is actually less severe than the results from our model indicate.

In conclusion, we think that the Bradley–Terry model provides a perspective on crime severity thus far absent in the criminology literature, namely a developmental perspective based on the temporal sequencing of criminal behaviors. Such a perspective may be attractive to other applications that seek to understand the escalation in offenders’ behaviors. In particular, we propose that there may be crimes that mark forays into more severe criminal offending, and that the behavior of the offender offers a unique method to identify such steps.

Footnotes
1

We use the term “crime severity” though acknowledge that prior research has often used the term “crime seriousness” when referring to the same relative construct.

 
2

Revealed preference theory in economics suggests that preferences of consumers can be revealed by purchasing habits (Samuelson 1938). This concept can be applied to the temporal order in offenses, from the perspective that the sequence of offenses individuals select to engage in reveals preferences that are tied to the escalation in crime severity.

 
3

Half of the sample was asked to rank the categorical limits for the severity of the offense they were presented with values ranging from 1 ‘least serious to 11 ‘most serious.’ Sellin and Wolfgang (1964) found similarity in the rankings of offenses between both methods after adjusting for a number of covariates (race, rating group, age of offender, etc.). They chose to rely on the magnitude scores because of the larger range of values.

 
4

Item Response Theory or Guttman scaling methods would be the most analogous approach to the Bradely–Terry model (see Raudenbush et al. 2003). Item Response Theory (IRT) and Guttman scaling approaches, however, would deem the severity of crimes by their endorsed prevalence, such that the average likelihood of committing any offense is highest for those who report committing the least prevalent offenses. The least prevalent offenses would be termed the most severe––analogous to item difficulty scores in an IRT model. The advantage of the Bradley–Terry approach used here is that the severity of crimes is estimated from the sequencing of offenses over time and there is no explicit assumption that the order of severity is related to prevalence.

 
5

λi − λj is equivalent to the log of Eq. 2.

 
6

The complete matrices for these estimates are presented in the Appendix Tables 5 and 6.

 

Acknowledgments

This research was supported in part by grants from the Centers for Disease Control and Prevention (CDC) (grant R49CE000574) and the National Institute on Drug Abuse (NIDA) (grant R01DA16722). The opinions expressed in this article are those of the authors and do not represent the official positions of the CDC, NIDA, the RAND Corporation, or any of its clients. The authors would like to thank David McDowall, Ray Paternoster, Greg Ridgeway, and the anonymous reviewers for their helpful suggestions. All errors and omissions remain those of the authors.

Copyright information

© Springer Science+Business Media, LLC 2008