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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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

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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

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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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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

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The Likelihood Function
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Identification of Distinctive Developmental
Trajectories An Illustrative Example
Adolescent onset (50)
Adolescent limited (50)
crime
Pop. Total
age
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Types of Data

• Psychometric scales--Censored Normal (Tobit)
  Model
• Count Data--Poisson-based Model
• Binary Data--Logit-based Model

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Linking Age to Behavior
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Zero-inflated Poisson Model for Count Data
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Logit Model for Binary Data
where y1 if yes y0 if no
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Trajectories of Delinquent Group
Membership(Development Psychopathology, 2003)
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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

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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
• Diagnostics for Model Fit
• Matching People with Comparable Developmental
  Histories

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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

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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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Model Extensions

• Entering Covariates into the Trajectory Itself
• Joint Trajectory Analysis
• Multi Trajectory Modeling

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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
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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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Multi-Trajectory Modeling
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Linking Trajectories to Later Out
ComesTrajectories of Physical Aggression from 6
to 15 and Sexual Partners at 17
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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

• 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

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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