Modeling Developmental Trajectories A Groupbased Approach Daniel S

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

4% 28% 16% 52%

Antisocial Behavior Trajectories (N=526 Conduct Problems Scale males) 7 9 11 13 15 18 21 26 Age Odgers, Caspi et al. , Arch Gen Psychiatry, 2007

Motivation for Group-based Trajectory Modeling n n n 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 n n n 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 q Individuals do not actually belong to trajectory groups q Trajectory group “members” do not follow the group-level trajectory in lock-step Groups are not immutable q # of groups will depend upon sample size and particularly length of follow-up period q 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 n n n 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)

Statistically Linking Group Membership to Individual Characteristics (Chapter 6) n n n 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 n n Broken Home at Age 5 Low IQ Low Maternal Education Mother Began Childbearing as a Teenager

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

The Overall Model Z 1 Z 2 Z 3 Z 4 Z 5 ………. …. Zm Probability of Trajectory Group Membership Trajectory 1 Trajectory 2 X 1 t X 2 t Trajectory 3 Trajectory 4 X 3 t……………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 Trajectory in Childhood Low 1&2 Rising Declining Chronic . 889 . 092 . 019 . 000 Declining. 707 . 136 . 128 . 029 High . 215 . 206 . 158 Low . 422

The Dual-Trajectory Model Generalized to Include Predictors of Conditional Probabilities n n n 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 None 75 th Percentile Low Moderate High Physical Aggression. 00. 02. 12. 00. 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 30

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

Recommended Readings n n n 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: 873904. 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 Semiparametric, 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 n n n 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.