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




Mplus




Specialized
SAS based
STATA version in Beta Testing
General Purpose
Its “own platform”
Latent Gold (?)
R-based packages
Physical Aggression
Trajectories of Physical Aggression
(Child Development, 1999)
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
4%
28%
52%
16%
6
Low-actual
Low-pred.
10
11
12
13
14
Mod. desister-actual Age High desister-actual
Mod. desister-pred.
High desister-pred.
15
Chronic-actual
Chronic-pred
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





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
P (Y i )    j ( x i ) P (Y i )
j
j
J
P (Y i) = p ro b ab ility o f Y i given m em b ersh ip in grou p j
 j= p ro b ab ility o f m em b ersh ip in grou p j

j
( xi )  e
x i
j

N
L 

P (Y i ).
e
x i
j
Using Groups to Approximate an Unknown
Distribution
Panel A
f(z)
0.10
0.05
0.00
0
10
20
z
z
Panel B
f(z)
0.10
0.05
0.00
0
z1
10
z2
z3
z
20
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
pˆ ( group j | data i ) 
pˆ ( data i | group j )ˆ j

pˆ ( data i | group j )ˆ j
j
Maximum Probability Group Assignment Rule
Group Profiles
V ariable
G ro up
N ever
Low
D esister
H igh
D esister
Y ears o f S cho o l - M o ther
11.1
10.8
9.8
8.4
Y ears o f S cho o l - Father
11.5
10.7
9.8
9.1
26.8
44.5
46.4
L o w IQ (% )
21.6
th
C hro nic
C o m pleted 8 G rade
o n T im e (% )
80.3
64.6
31.8
6.5
Juvenile R eco rd (% )
0.0
2.0
6.0
13.3
1.7
2.2
3.5
# o f S exual P artners at
A ge 17 (P ast Y ear)
1.2
Other Uses of Posterior Probabilities


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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

j
( xi )  e
x i
j

e
x i
j
Risk Factors for Physical Aggression Trajectory
Group Membership




Broken Home at Age 5
Low IQ
Low Maternal Education
Mother Began Childbearing as a Teenager
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Low
Moderate Declining
High Declining
Ho
me
Lo
w
Ed
M
om
Te
en
M
om
Al
lR
is k
s
IQ
Br
ok
en
Lo
w
sk
Chronic
Ri
No
probability
Impact of Risk Factors on Group Membership
Probabilities
Does School Grade Retention and Family Break-up Alter
Trajectories of Violent Delinquency Themselves?
(Nagin, 2005; Development and Psychopathology 2003)
Trajectories of Violent Delinquency
10
9
8
7
Rate
6
5
4
3
2
1
0
11
12
Low 1 (34.8$)
Declining (16.7%)
13
14
Age
Low 2(30.6%)
Chronic (4.5%)
15
16
Rising (13.4%)
17
The Overall Model
Z1
Z2
Z3
Z4
Z5 ………. …. Zm
Probability
of Trajectory Group
Membership
Trajectory 1 Trajectory 2
X1t
X2t
Trajectory 3 Trajectory 4
X3t……………Xlt
Model of Impact of Grade Retention and
Parental Separation on Trajectory Group j
Model without retention or separation impact:
ln(  )  
j
t
j
0
  1 Age t   Age
j
j
2
2
t
Trajectory with retention and separation impacts:
~
~
~
ln(  )   0   1 Age t   2 Age t   1 Fail
j
t
j
j
j
2
j
  2 Separation
j
t
t
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:
Behavior X:
Z1
X1
Z2
X
2
Z3
X3
………………
………………
ZT
XT
Heterotypic Continuity
Behavior Z:
ZT
ZT+1
Zt+3
………………
ZT+K
Panel B-Dual Trajectory Approach
Behavior X:
X1
X
Behavior Z:
Z1
Z2
Z3
X
X3
2
X3
………………
XT
Comorbidity
Behavior X:
X1
2
………………
………………
ZT
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 Adolescent Violent Delinqunecy
from Age 13 to 17
4
3
Low
2
Desisting
1
High
0
6
8
10
Age
12
Rate
Physical Aggression
Trajectories of Childhood Physical Aggression from
Age 6 to 13
9
8
7
6
5
4
3
2
1
0
Low 1
Low 2
Declining
Rising
Chronic
13
14
15
Age
16
17
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

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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
75th
Percentile
Low
Moderate High
Physical
Physical
Physical
Aggression Aggression 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
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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
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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.