1. Modeling Developmental Trajectories: A Group-based Approach
Daniel S. Nagin
Carnegie Mellon University

2. What is a trajectory?
A trajectory is “the evolution of an outcome over age or time.” (p.1)
Nagin Group-Based Modeling of Development, Harvard University Press

4. Types of Trajectory Modeling
Grow Curve Modeling
Grow Mixture Modeling (GMM)-Muthén and colleagues
Group-Based Trajectory Modeling (GBTM)-Nagin and colleagues
For a recent discussion of differences see Nagin and Odgers (2010)

5. Trajectory Estimation Software
Proc Traj
Specialized
SAS based
STATA version in Beta Testing
Mplus
General Purpose
Its “own platform”
Latent Gold (?)
R-based packages

7. 4%
28%
52%
16%

8. Antisocial Behavior Trajectories (N=526 males)
Conduct Problems Scale
The model with the best empirical fit contained 4-classes.
We saw a class that started in childhood, persistent throughout adolescence and while they decreased slightly were still exhibiting ASB in adulthood.
An adolescent-onset class also emerged, low in childhood, CP emerge in adolescence, but you will notice that they did not desist as anticipated on these measures of ASB.
There was also a low group, majority of the population which was anticipated…and, a group that was not anticipated, a
Childhood-limited subgroup who could not be distinguished from the LCP class in childhood, but desisted rapidly and by the adulthood could not be distinguished form the low (cohort-norm) on ASB.
The trajectories included: a life-course-persistent class (10.5% of the cohort) who initiated antisocial behavior early and persisted into adulthood; an adolescent-onset class (19.6%) whose conduct problems emerged during adolescence; a childhood-limited class who demonstrated conduct problems in childhood but subsequently desisted (24.3%) and; a ‘low’ class (45.6%) characterized by low levels of conduct problems (Figure 1).
Age
Odgers, Caspi et al., Arch Gen Psychiatry, 2007

9. Motivation for Group-based Trajectory Modeling
Testing Taxonomic Theories
Identifying Distinctive Developmental Paths in Complex Longitudinal Datasets
Capturing the Connectedness of Behavior over Time
Transparency in Efficient Data Summary
Responsive to Calls for “Person-based Methods of Analysis

10. The Likelihood Function

11. Using Groups to Approximate an Unknown Distribution
Panel A
Panel B

12. Implications of Using Groups to Approximate a More Complex Underlying Reality
Trajectory Groups are latent strata—individuals following approximately the same developmental course of the outcome variable
Groups membership is a convenient statistical fiction, not a state of being
Individuals do not actually belong to trajectory groups
Trajectory group “members” do not follow the group-level trajectory in lock-step
Groups are not immutable
# of groups will depend upon sample size and particularly length of follow-up period
Search for the True Number of Groups is a Quixotic exercise

13. Calculation & Use of Posterior Probabilities of Group Membership
Maximum Probability Group Assignment Rule

14. Group Profiles

15. Other Uses of Posterior Probabilities
Computing Weighted Averages That Account for Group Membership Uncertainty (Nagin (2005; Section 5.6)
Diagnostics for Model Fit (Section 5.5)
Matching People with Comparable Developmental Histories (Haviland, Nagin, and Rosenbaum, 2007)

16. Moving Beyond Univariate Contrasts
Statistically Linking Group Membership to Individual Characteristics (Chapter 6)
Moving Beyond Univariate Contrasts
Group Identification is Probabilistic not Certain
Use of Multinomial Logit Model to Create a Multivariate Probabilistic Linkage

17. Risk Factors for Physical Aggression Trajectory Group Membership
Broken Home at Age 5
Low IQ
Low Maternal Education
Mother Began Childbearing as a Teenager

19. Does School Grade Retention and Family Break-up Alter Trajectories of Violent Delinquency Themselves? (Nagin, 2005; Development and Psychopathology 2003)

20. of Trajectory Group Membership
The Overall Model
Z Z Z Z Z5 ………. …. Zm
Probability
of Trajectory Group Membership
Trajectory 1 Trajectory Trajectory 3 Trajectory 4
X1t X2t X3t……………Xlt

21. Model of Impact of Grade Retention and Parental Separation on Trajectory Group j
Model without retention or separation impact:
Trajectory with retention and separation impacts:

23. Dual Trajectory Analysis: Trajectory of Modeling of Comorbidity and Heterotypic Continuity (Nagin and Tremblay, 2001; Nagin (2005)

24. Modeling the Linkage Between Trajectories of Physical Aggression in Childhood and Trajectories of Violent Delinquency in Adolescence

25. Transition Probabilities Linking Trajectories in Adolescent to Childhood Trajectories
Trajectory in Adolescence
Low
1&2
Rising
Declining
Chronic
.889
.092
.019
.000
.707
.136
.128
.029
High
.422
.215
.206
.158
Trajectory
in Childhood

26. The Dual-Trajectory Model Generalized to Include Predictors of Conditional Probabilities
Are drug use and family break-up at age 12 predict the conditional probabilities linking childhood physical aggression trajectories with adolescent violent delinquency trajectories?
Answer: yes for drug use but no family break-up
Conditional probabilities specified to follow a “constrained” multinomial logit function (see section 8.7 of Nagin)

27. Probability of Transition to Chronic Trajectory Depending on Drug Use at Age 12 and Childhood Physical Aggression Trajectory
Drug Use at age 12
Low Physical Aggression
Moderate Physical Aggression
High Physical Aggression
None
.00
.02
.12
75th Percentile
.18
.46

28. Multi-Trajectory Modeling

29. Linking Trajectories to Later Out Comes—Trajectories of Physical Aggression from 6 to 15 and Sexual Partners at 16

30. Accounting for Non-random Subject Attrition

31. Accounting for Non-random Subject Attrition (cont.)

32. Recommended Readings
Nagin, D.S. and C.L. Odgers “Group-based trajectory modeling in clinical research.” In S. Nolen-Hoekland, T. Cannon, and T. Widger (eds.), Annual Review of Clinical Psychology. Palo Alto, CA: Annual Reviews.
Nagin, D. S Group-based Modeling of Development. Cambridge, MA.: Harvard University Press.
Nagin, D.S. and R. E. Tremblay “Developmental Trajectory Groups: Fact or a Useful Statistical Fiction?.” Criminology, 43:
Nagin, D. S., and R. E. Tremblay “Analyzing Developmental Trajectories of Distinct but Related Behaviors: A Group-based Method.” Psychological Methods, 6(1):
Nagin, D. S “Analyzing Developmental Trajectories: A Semi-parametric, Group-based Approach.” Psychological Methods, 4:
Nagin, D.S., Pagani, L.S., Tremblay, R.E., and Vitaro, F “Life Course Turning Points: The Effect of Grade Retention on Physical Aggression.” Development and Psychopathology, 15:

33. Suggested Readings Continued
Jones, B., D.S. Nagin. And K. Roeder “A SAS Procedure Based on Mixture Models for Estimating Developmental Trajectories.” Sociological Research and Methods, 29:
Jones, B. and D.S. Nagin “Advances in Group-based Trajectory Modeling and a SAS Procedure for Estimating Them,” Sociological Research and Methods, 35:
Haviland, A., Nagin D.S., and Rosenbaum, P.R “Combining Propensity Score Matching and Group-Based Trajectory Modeling in an Observational Study” Psychological Methods, 12: