Modeling Developmental Trajectories: A Group-based Approach

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Modeling Developmental Trajectories A
Group-based Approach

• Daniel S. Nagin
• Carnegie Mellon University

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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
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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)

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Trajectory Estimation Software

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

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52
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Antisocial Behavior Trajectories (N526 males)
Conduct Problems Scale
7 9 11 13 15
18 21
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Age
Odgers, Caspi et al., Arch Gen Psychiatry, 2007
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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

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The Likelihood Function
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Using Groups to Approximate an Unknown
Distribution
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Implications of Using Groups to Approximate a
More Complex Underlying Reality

• Trajectory Groups are latent strataindividuals
  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

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Calculation Use of Posterior Probabilities of
Group Membership
Maximum Probability Group Assignment Rule
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Group Profiles
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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)

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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

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Risk Factors for Physical Aggression Trajectory
Group Membership

• Broken Home at Age 5
• Low IQ
• Low Maternal Education
• Mother Began Childbearing as a Teenager

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Does School Grade Retention and Family Break-up
Alter Trajectories of Violent Delinquency
Themselves?(Nagin, 2005 Development and
Psychopathology 2003)
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The Overall Model
Z1 Z2 Z3
Z4 Z5 . . Zm
Probability of Trajectory Group Membership
Trajectory 1 Trajectory 2 Trajectory
3 Trajectory 4
X1t X2t X3tXlt
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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


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Dual Trajectory Analysis Trajectory of Modeling
of Comorbidity and Heterotypic Continuity (Nagin
and Tremblay, 2001 Nagin (2005)
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Modeling the Linkage Between Trajectories of
Physical Aggression in Childhood and Trajectories
of Violent Delinquency in Adolescence
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Transition Probabilities Linking Trajectories in
Adolescent to Childhood Trajectories
Trajectory in Adolescence
Low 12 Rising Declining Chronic
Low .889 .092 .019 .000
Declining .707 .136 .128 .029
High .422 .215 .206 .158
Trajectory in Childhood
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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)

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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 .00 .18 .46
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Multi-Trajectory Modeling
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Linking Trajectories to Later Out
ComesTrajectories of Physical Aggression from 6
to 15 and Sexual Partners at 16
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Accounting for Non-random Subject Attrition
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Accounting for Non-random Subject Attrition
(cont.)
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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,
  43873-904.
• 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.

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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.