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. 2005. Group-Based Modeling of Development, Harvard University Press

4. Montreal Data 1037 Caucasian, francophone, nonimmigrant males First assessment at age 6 in 1984 Most recent assessment at age 17 in 1995 Data collected on a wide variety of individual, familial, and parental characteristics, behaviors, and psychopathologies

5. 4% 28% 52% 16%

6. Important Capabilities of Group- based Trajectory Modeling Identify Rather Than Assume Groups of Distinctive Developmental Trajectories-- Avoids over/under-fitting of data Estimation of Proportion of Population by Trajectory Group Identification of Distinctive Characteristics of Trajectory Groups

7. 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

8. Trajectory Estimation Software Easy-to-Use, STATA and SAS-based Procedure Handles Missing Data (including exposure time for count data) Handles Sample Weights Does not Require Regular Time Spacing of Measurements Accommodates over-lapping cohort designs Provides confidence intervals on trajectory estimates Conducts Wald tests of coefficient differences Available www.andrew.cmu.edu/user/bjones/index.htm

9. The Likelihood Function

10. Identification of Distinctive Developmental Trajectories: An Illustrative Example age crime Pop. Total Adolescent onset (50%) Adolescent limited (50%)

11. Types of Data Psychometric scales--Censored Normal (Tobit) Model Count Data--Poisson-based Model Binary Data--Logit-based Model

12. Linking Age to Behavior

13. Zero-inflated Poisson Model for Count Data

16. Logit Model for Binary Data where y=1 if yes & y=0 if no

17. Trajectories of Delinquent Group Membership (Development & Psychopathology, 2003)

18. Topics for Discussion Profiling Trajectory Group Members Measuring the Effect of Individual Characteristics on Probability of Trajectory Group Membership Adding Covariates to the Trajectory Itself Dual Trajectory Modeling Groups as an approximation Group-based Modeling v. Growth Curve Modeling

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

20. Group Profiles

21. Other Uses of Posterior Probabilities Computing Weighted Averages That Account for Group Membership Uncertainty Diagnostics for Model Fit Matching People with Comparable Developmental Histories

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

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

25. Model Extensions Entering Covariates into the Trajectory Itself Joint Trajectory Analysis Multi Trajectory Modeling

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

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

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

30. Dual Trajectory Analysis: Trajectory of Modeling of Comorbidity and Heterotypic Continuity

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

32. Transition Probabilities Linking Trajectories in Adolescent to Childhood Trajectories Trajectory in Adolescence Trajectory in Childhood Low 1&2 RisingDecliningChronic Low.889.092.019.000 Declining.707.136.128.029 High.422.215.206.158

33. Multi-Trajectory Modeling

34. Linking Trajectories to Later Out Comes— Trajectories of Physical Aggression from 6 to 15 and Sexual Partners at 17

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

36. Implications of Using Groups to Approximate a More Complex Underlying Reality 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 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

37. Group-Based Trajectory Modeling Compared to Conventional Growth Curve Modeling (HLM) Common point of departure: both model individual level trajectories by a polynomial equation in age or time: Point of departure: how to model individual- level differences in developmental trajectories (e.g., population heterogeneity)  HLM use normally distributed random effects  Group-based trajectory modeling approximates an unknown distribution of individual differences with groups