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
Trajectories of Physical Aggression(Child Development, 1999)
00.5
11.5
22.5
33.5
44.5
6 10 11 12 13 14 15
Age
Phys
ical A
ggre
ssio
n
Low-actual Mod. desister-actual High desister-actual Chronic-actualLow-pred. Mod. desister-pred. High desister-pred. Chronic-pred
4%
28%
52%16%
Con
duct
Pro
blem
s S
cale
Age
Antisocial Behavior Trajectories (N=526 males)
7 9 11 13 15 18 21 26
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
.)(N
iYPL
PJ(Yi) = probability of Yi given membership in group j
j= probability of membership in group j
ji
ji
x
x
ij eex
)(
j
ij
iji YPxYP )()()(
Using Groups to Approximate an Unknown Distribution
20100
0.10
0.05
0.00
z
f(z)
20100
0.10
0.05
0.00
z
f(z)
Panel A
Panel B
z
z1 z2 z3 z4 z5
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
jji
jii jgroupdatap
jgroupdatapdatajgroupp
ˆ)|(ˆ
ˆ)|(ˆ)|(ˆ
Group Profiles
Variable Group
Low HighNever Desister Desister Chronic
Years of School - Mother 11.1 10.8 9.8 8.4
Years of School - Father 11.5 10.7 9.8 9.1
Low IQ (%) 21.6 26.8 44.5 46.4
Completed 8th Grade 80.3 64.6 31.8 6.5 on Time (%)
Juvenile Record (%) 0.0 2.0 6.0 13.3
# of Sexual Partners at 1.2 1.7 2.2 3.5 Age 17 (Past Year)
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)
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
ji
ji
x
x
ij eex
)(
Risk Factors for Physical Aggression Trajectory Group Membership
Broken Home at Age 5 Low IQ Low Maternal Education Mother Began Childbearing as a Teenager
Impact of Risk Factors on Group Membership Probabilities
00.10.20.30.40.50.60.7
prob
abili
ty
Low
Moderate Declining
High Declining
Chronic
Does School Grade Retention and Family Break-up Alter Trajectories of Violent Delinquency Themselves?
(Nagin, 2005; Development and Psychopathology
2003)Trajectories of Violent Delinquency
0
1
2
3
4
5
6
7
8
9
10
11 12 13 14 15 16 17
Age
Rate
Low 1 (34.8$) Low 2(30.6%) Rising (13.4%)Declining (16.7%) Chronic (4.5%)
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
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:
2210)ln( tj
tjjj
t AgeAge
tj
tj
tj
tjjj
t SeparationFailAgeAge 212
210
~~~)ln(
Dual Trajectory Analysis: Trajectory of Modeling of Comorbidity and Heterotypic Continuity (Nagin and Tremblay, 2001; Nagin (2005)
Panel A-Conventional Approach
Behavior X: X1 X 2 X3 ……………… XT
Comorbidity
Behavior Z: Z1 Z2 Z3 ……………… ZT
Behavior X: X1 X 2 X3
……………… XT
Heterotypic Continuity
Behavior Z: ZT ZT+1 Zt+3 ……………… ZT+K
Panel B-Dual Trajectory Approach
Behavior X: X1 X 2 X3 ……………… XT
Comorbidity
Behavior Z: Z1 Z2 Z3 ……………… ZT
Behavior X: X1 X 2 X3
……………… XT
Heterotypic Continuity
Behavior Z: ZT ZT+1 Zt+3 ……………… ZT+K
Modeling the Linkage Between Trajectories of Physical Aggression in Childhood and Trajectories of Violent Delinquency in Adolescence
Trajectories of Childhood Physical Aggression from Age 6 to 13
0
1
2
3
4
6 8 10 12
Age
Phys
ical
Agg
ress
ion Low
Desisting
High
Trajectories of Adolescent Violent Delinqunecy from Age 13 to 17
0123456789
13 14 15 16 17
AgeR
ate
Low 1
Low 2
Declining
Rising
Chronic
Transition Probabilities Linking Trajectories in Adolescent to Childhood Trajectories
Trajectory in Adolescence
Trajectoryin Childhood
Low
1&2
Rising Declining Chronic
Low .889 .092 .019 .000
Declining .707 .136 .128 .029
High .422 .215 .206 .158
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 .1275th Percentile
.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 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.