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

What is a trajectory? A trajectory is “the evolution of an outcome over age or time.” (p.1) Nagin. 2005. Group-Based Modeling of Development, Harvard University Press

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)

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

4% 28% 52% 16%

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). 7 9 11 13 15 18 21 26 Age Odgers, Caspi et al., Arch Gen Psychiatry, 2007

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

The Likelihood Function

Using Groups to Approximate an Unknown Distribution Panel A Panel B

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

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

Group Profiles

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)

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

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

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

of Trajectory Group Membership The Overall Model Z1 Z2 Z3 Z4 Z5 ………. …. Zm Probability of Trajectory Group Membership Trajectory 1 Trajectory 2 Trajectory 3 Trajectory 4 X1t X2t X3t……………Xlt

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:

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

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

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

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)

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

Multi-Trajectory Modeling

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

Accounting for Non-random Subject Attrition

Accounting for Non-random Subject Attrition (cont.)

Recommended Readings Nagin, D.S. and C.L. Odgers. 2010. “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. 2005. Group-based Modeling of Development. Cambridge, MA.: Harvard University Press. Nagin, D.S. and R. E. Tremblay. 2005. “Developmental Trajectory Groups: Fact or a Useful Statistical Fiction?.” Criminology, 43:873-904. Nagin, D. S., and R. E. Tremblay. 2001. “Analyzing Developmental Trajectories of Distinct but Related Behaviors: A Group-based Method.” Psychological Methods, 6(1): 18-34. Nagin, D. S. 1999. “Analyzing Developmental Trajectories: A Semi-parametric, Group-based Approach.” Psychological Methods, 4: 139-177. Nagin, D.S., Pagani, L.S., Tremblay, R.E., and Vitaro, F. 2003. “Life Course Turning Points: The Effect of Grade Retention on Physical Aggression.” Development and Psychopathology, 15: 343-361.

Suggested Readings Continued Jones, B., D.S. Nagin. And K. Roeder. 2001. “A SAS Procedure Based on Mixture Models for Estimating Developmental Trajectories.” Sociological Research and Methods, 29: 374-393. Jones, B. and D.S. Nagin. 2007. “Advances in Group-based Trajectory Modeling and a SAS Procedure for Estimating Them,” Sociological Research and Methods, 35: 542-571. Haviland, A., Nagin D.S., and Rosenbaum, P.R. 2007. “Combining Propensity Score Matching and Group-Based Trajectory Modeling in an Observational Study” Psychological Methods, 12: 247-267.