Nearly two decades have passed since the publication of “Age, Criminal Careers, and
Population Heterogeneity: Specification and Estimation of a Nonparametric Mixed Poisson
Model” by Nagin and Land (1993). In that article Nagin and Land laid out a statistical
method that has come to be called group-based trajectory modeling. The principle objective
of the paper was to address issues related to the “hot topic” of the time—the criminal
career debate—not to lay out a new statistical methodology. As described in the paper’s
abstract, these issues were: “First, is the life course of individual offending patterns
marked by distinctive periods of quiescence? Second, at the level of the individual,
do offending rates vary systematically with age? In particular, is the age-crime curve
single peaked or flat? Third, are chronic offenders different from less active offenders?
Do offenders themselves differ in systematic ways?”
Figure 1 reports Nagin’s (2005) updated version of the trajectories reported in Nagin
and Land (1993). The analysis is based on the classic dataset assembled by Farrington
and West (1990), which includes data on convictions from age 10 to 32 in a sample
of over 400 males from a poor neighborhood in London, England. A four group model,
analyzed using the zero-inflated Poisson modeling option, was found to best fit the
data. The largest trajectory group accounted for 69.5% of the population, and was
composed of individuals who generally had no convictions. The three offending trajectories
included an adolescent-limited group (12.4% of the population), which peaked sharply
in late adolescence, and then declined to a near zero rate of offending by age 20,
a high chronic trajectory (5.9% of the population) with a high-humped shaped trajectory
and a low rate chronic trajectory that accounted for the remaining 12.2% of the population.
Also, shown in the figure are 95% confidence intervals around each trajectory.
Fig. 1
Trajectories of convictions (London data)
The dominant legacy of Nagin and Land (1993), however, was not its answers to the
specific questions listed in the abstract but the methodology itself. A review of
applications of group-based trajectory modeling (GBTM) by Piquero (2008) identified
more than 80 applications in criminology. There are even larger numbers of applications
of GBTM outside of criminology. As noted by Bushway and Weisburd (2006), GBTM is one
the few examples of a statistical method with origins in criminology that has come
to be widely used by other disciplines. In a recent review of GBTM for clinical psychologists
(Nagin and Odgers 2010) the authors documented a rapid rise in the application of
trajectory based models in clinical research; a PSYC INFO literature review indicated
that between 2000 and 2008 the application of GBTMs increased from 8 to 80 publications
per year in clinically relevant journals such as the Journal of Clinical and Consulting
Psychology, Child Development, Addiction and the Journal of Abnormal Child Psychology.
Within this area, trajectory models have been applied to understand the etiology and
developmental course of a number of different types of disorders, including: depression
(Dekker et al. 2007; Mora et al. 2009), inattention/hyperactivity (Jester et al. 2008),
post-traumatic stress disorder (Orcutt et al. 2004), substance abuse (Hu et al. 2008)
and conduct disorder (Odgers et al. 2008b). More recently, group-based models have
been extended to capture heterogeneity in treatment responses to clinical and randomized
trials (Brown et al. 2008; Peer and Spaulding 2007) and have been leveraged to facilitate
causal inference in epidemiological observational studies where randomization to treatment
conditions is not possible (Haviland et al. 2007; Haviland et al. 2008; Odgers et
al. 2008a). There have also been numerous applications of GBTM in medical journals
including the New England Journal of Medicine, Archives of General Psychiatry, Pediatrics,
Journal of Adolescent Psychology and Psychiatry that address not only the developmental
course of psychiatric disorders but also target biomarkers such as body mass index
(Mustillo et al. 2003), cortisol levels (Van Bokhoven et al. 2005; Van Ryzin et al.
2009), as well as indicators of disability in elderly populations (Gill et al. 2010).
What accounts for the rapid adoption of GBTM in such diverse settings? In many ways,
it is not surprising that GBTM has been embraced by clinical researchers interested
in the developmental course of psychiatric and physical disorders. GBTM maps closely
on how researchers conceptualize the growth and development of a wide range of phenomena;
provide an empirical means of identifying clusters of individuals following both typical
and atypical courses of development; and offer a new set of tools for evaluating individual
variation in response to interventions and randomized trials. With respect to theory-method
fit, there is a long tradition of group-based theorizing about both normal and pathological
development in psychology. Examples include theories of personality development (Caspi
1998), learning (Holyoak and Spellman 1993), language and conceptual development (Markman
1989), mental disorders such as depression (Kasen et al. 2001), eating disorders (Tyrka
et al. 2000), alcoholism (Cloninger 1987) conduct disorder and delinquency (Loeber
1991; Moffitt 1993; Patterson et al. 1989) and anxiety (Cloninger 1986) as well as
the development of prosocial behaviors such as conscience (Kochanska 1997).
But what accounts for its widespread use among non-clinical researchers, most specifically
by criminologists? In part the application of GBTMs within this context reflects the
influence of developmental psychopathology research on what has come to be called
developmental criminology. Leading researchers in this tradition—David Farrington,
Magda and Rolf Loeber, and Terrie Moffitt—straddle criminology and developmental psychology
and have been instrumental in encouraging the field to develop and test theories related
to the developmental course of antisocial behavior and crime across the lifespan.
To this end, GBTM is ideally suited for analyzing the influential taxonomic theories
of antisocial and delinquent behavior of Moffitt (1993) and Patterson et al. (1989,
1998).
We conjecture, however, that GBTMs widespread use in criminology is affected by more
than just the influence of imminent psychologists who also double as criminologists.
A hallmark of modern longitudinal studies is the variety and richness of measurements
that are made about the study’s subjects and their circumstances. Less often acknowledged
is the fact that this abundance of information is accompanied by a difficult companion—complexity—and
the desire among researchers to disentangle population heterogeneity and move beyond
a ‘one size fits all’ approach to describing development across the lifespan. Commonly,
researchers are confronted with the dilemma of how best to explore and test theories
of development within these rich sets of measurements without increasing the analytical
complexity to the point where the lessons to be learned from the data are lost on
them and their audience. By segmenting the data into trajectory groups, the group-based
approach to studying development, provides an empirical means of summarizing large
amounts of data in an easily comprehensible fashion and for testing long standing
developmental theories with a taxonomic dimension.
Table 1 illustrates how trajectory models can be leveraged to summarize large amounts
of data collected across development and test theories regarding the origins of antisocial
behavior for each of the four trajectory groups shown in Fig. 1. For example, the
high chronics, on average, were most likely to have a low IQ, to have had at least
one parent with a criminal record, to have had poor parenting, and to have engaged
in risky activities. Conversely, the rare group was lowest on these risk factors.
The contrasting characteristics of the chronic and rare groups are strongly consistent
with much prior research. The adolescent-limited and low-chronic groups fall in between
but the differences between these two groups form a more complex pattern. The low
chronics have a higher incidence of low IQ than the adolescent-limited group but have
a lower incidence of parental criminality and risk-taking behavior. This suggests
the possibility of a difference in the etiology that underlies the criminality of
these two groups. The profiles illustrate the utility of GBTM in two important functions
that transcend specific subject matter: (1) communication of research findings in
an easily interpretable format and (2) identification of subtle but significant variations
across trajectory groups in their predictors and outcomes.
Table 1
Trajectory group profiles (London data)
Variable
Group
Rare
Adolescent limited
Low chronic
High chronic
Low IQ (%)
16.3
23.5
34.8
43.5
Poor parenting (%)
18.4
29.4
30.4
47.8
High risk taking (%)
21.2
47.1
37.0
69.5
Parents with criminal record (%)
18.0
43.5
33.3
60.9
Extensions to of the basic model also lend themselves to achieving these two objectives.
These extensions include modeling predictors of probability of trajectory group membership,
estimating the possible influence of other covariates beyond age or time on the trajectory
of the phenomenon under study, and assessing the inter-relationship of two or more
trajectories of distinct but related outcomes. Examples of the last modeling capacity
is modeling the comorbidity of trajectories of delinquency, drug use, and sexual activity
or the interconnection of trajectories of childhood physical aggression and trajectories
of adolescent violent delinquency (Nagin 2005). For an extended discussion of these
modeling capabilities see Nagin (2005) or Nagin and Odgers (2010; forthcoming) and
for discussion of the software capabilities for estimating such models see Jones et
al. (2001) and Jones and Nagin (2007).
How has the application of GBTM advanced theory and method development in the field
of criminology? Before attempting an answer to this question, a caveat emptor is in
order. All statistical methods are devices for summarizing data. Grouping longitudinal
data according to trajectories groups is but one form of data summary, which carries
the important benefit of mapping closely to how we conceptualize the development of
a wide range of behaviors, emotions and related phenomenon. Other popular methods
for capturing this type of growth and change over time include grouping by clinical
cut-offs, conventional growth curve modeling, and growth mixture modeling. See Nagin
and Odgers (2010; forthcoming) for a discussion of these alternatives. All of these
methods share the common objective of explaining population differences in the developmental
course of the phenomenon of interest. Thus, no method should have hegemony. Still
grouping by trajectory group does uniquely facilitate addressing some types of issues.
In the discussion which follows we describe three findings that have emerged from
GBTM analysis, that in our judgment, are important to criminology and highlight the
strengths of the method.
The examples featured below all relate to what is perhaps the most influential empirical
regularity in criminology—the age-crime curve. It has been repeatedly demonstrated
that age specific arrest rates rise steady from early adolescence, peak at about age
18 and steadily decline thereafter (Hirschi and Gottfredson 1983). While there are
variations in this pattern by crime type, time and place, the regularity is remarkably
robust (Farrington 1986). We describe this regularity as influential because it is
hard to overstate how much research and theorizing in criminology has been committed
to explaining the rising tide of misbehavior during adolescence and its subsequent
decline from early adulthood onward.
At least with regards to violence, the research collaboration of Nagin and Tremblay
that made extensive use of GBTM challenges the assumption that the onset of violence
begins in adolescences (c.f., Côté et al. 2002; Lacourse et al. 2002; Nagin et al.
2003; Nagin and Tremblay 1999, 2001). Figure 2 from Nagin and Tremblay (1999) reports
trajectories of physical aggression from age 6 to 15 based on a prospective longitudinal
study of about 1,000 white, French-speaking males from low socio-economic status neighborhoods
in Montreal. A group called “lows” is composed of individuals who display little or
no physically aggressive behavior. This group is estimated to compose about 15% of
the sampled population. A second group, composing about 50% of the population, is
best labeled “moderate declining.” At age 6, boys in this group displayed a modest
level of physical aggression, but by age 10 they had largely desisted. A third group,
composing about 30% of the population, labeled “high declining”, starts off scoring
high on physical aggression at age 6 but scores far lower by age 15. Notwithstanding
this marked decline, at age 15 they continue to display a modest level of physical
aggression. Finally, there is a small group of “chronics,” making up less than 5%
of the population, who display high levels of physical aggression throughout the observation
period.
Fig. 2
Trajectories of physical aggression (Montreal data)
These trajectories are notable both for what is present and what is not present. As
for what is present, all of the trajectories are stable or declining from age 6 on.
Thus, over the period from age 6 to 15 there is no evidence of rising physical aggression
even among a small sub-population in these data. As for what is not present, we see
no evidence of late onset-like trajectories of physical aggression, namely a trajectory
that rises from a zero or negligible level at some point between 6 and 15. Because
the trajectories are at their highest at age 6, this suggests that to understand the
developmental origins of physical aggression in these boys we need to look back in
time prior to age 6 rather than forward in time into their adolescence. Indeed much
research confirms this supposition (Tremblay 2010). While developmental psychologists
have long observed elevated mean levels of aggression in early childhood, trajectory
modeling has provided a tool for showcasing the developmental trends in aggression
over time for both the entire population and for key subgroups of children. The absence
of late onset-type trajectories of physical aggression is not unique to these Montreal
males. A follow-up analysis of five additional prospective longitudinal studies—one
more from Canada, two from New Zealand, and two from the US—again found no evidence
of the onset of physical aggression after age 6 (Broidy et al. 2003). Arguably, the
application of GBTMs has helped to extend theorizing about the age-crime (or age-aggression
curve) beyond the typical age period relied on by criminologists, who are working
primarily with official arrest records that by design censor childhood behaviors.
It is important to recognize that the age-crime curve is only a population average.
There are potentially large individual level variations about the population average.
Just as important individual level trajectories may follow markedly different time
paths. The trajectory groups are properly understood as latent strata in longitudinal
data; that is collections of individuals following approximately the same developmental
course. From this perspective, GBTM is a useful methodology for identify groups of
individual following markedly different trajectories of offending.
To illustrate, Fig. 3 from Eggleston et al. (2004) reports trajectories of arrests
based on the Glueck and Glueck (1950) archive and the follow-up data described in
Sampson and Laub (1993) and Laub and Sampson (2003). While the total sample is composed
of 500 juvenile delinquent males selected from two reform schools in Massachusetts
and 500 matched non-delinquent males selected from the Boston public school system,
this analysis focuses on the delinquent sample. Eggleston et al. found that the six
group model shown in Fig. 3 best represented the trajectories of arrest from age 7
to 70. The most striking feature of the model is the wide variation across trajectory
groups in the peak rates of offending as measured by arrest. While all trajectories
follow a pattern of rise and then fall, only two trajectories—the classic desisters
and the moderate-rate desisters—representing only half of the sample reach their peak
rate of offending as teenagers. One small group called the high rate chronics reaches
their peak offending rate at about age 40. Just as important, a sizable proportion
of the sample is offending at elevated rates well past age 30. The Eggleston et al.
analysis illustrates that there is no one age-crime curve to be explained, a finding
that has been repeatedly documented by other applications of GBTM (cf. Bushway et
al. 2003; Brame et al. 2001; Blokland et al. 2005; Odgers et al. 2008b; Piquero et
al. 2001) including the original analysis by Nagin and Land (1993). This collection
of diverse results also illustrates the importance of understanding sample selection
when applying GBTMs and exercising caution when comparing trajectory solutions across
normative versus high-risk or delinquent samples. That is, one should not expect the
numbers, size and shapes of trajectory groups to remain constant across samples from
different populations.
Fig. 3
Official arrest trajectories—Glueck Males—Ages 7–70. (Source: Sampson and Laub 2003,
p. 582, Fig. 11)
The third important finding that has emerged from GBTM is ironic because it has also
been the source of the criticism of GBTM that trajectory groups imply predestined
paths of behavior. To the contrary, GBTM lends itself to demonstrating that past is
not necessarily prologue to the future. An important aim of the Eggleston et al. (2004)
analysis was to demonstrate this point. Their demonstration involved comparing models
based on successively longer periods of follow-up. The model for the shortest period
of follow-up was from age 7 to 24 and the longest was for age 7–70 as reported in
Fig. 3. The models nicely demonstrate that over time trajectory groups can split off.
For example, as shown in Fig. 3 up to age 20 the high rate chronics (3.2%), moderate
rate chronics (18.4%), and moderate rate desisters (18.4%) are indistinguishable.
Only after age 20 do the trajectories of the moderate rate desisters and the moderate
rate chronics progressively split off from the small group of high rate chronics.
In total these three groups compose an estimated 47.7% of the population yet there
was only a .067 (=.032/.477) chance of their combined membership following the high
chronic trajectory. What better way of showing that past is not necessarily prologue
to the future than to isolate the points where trajectory paths diverge over time?
We believe that GBTM offers criminology a valuable statistical tool for the longitudinal
study of crime phenomena. Looking forward, there are a number of intriguing possibilities
for the further application of GBTM in criminology. Some examples include applications
of GBTMs to answer longstanding substantive questions related to heterogeneity in
the ebb and flow of self-control and involvement with delinquent peers with age. There
is also the potential to begin unpacking questions related to the “co-morbidity” of
trajectories of based on official records with those based on self reports using dual
trajectory modeling. One of the most recent, and perhaps most exciting, extensions
of these models has been the combination of GBTMs with propensity score modeling to
facilitate causal inference in longitudinal studies where randomization to treatment
condition is not possible—as is the case in the majority of criminological studies
(see Haviland et al. 2007). To maximize the impact of these applications, additional
methodological work, such as that in Brame et al. (2006), is also required to refine
tests of model fit and selection.
The appeal of GBTMs for the future of criminological research lies in the potential
for the innovative application of trajectory models—on their own, in conjunction with
other statistical methods or embedded within creative study designs—while carefully
considering the perils and pitfalls inherent in the use of any methodology.