Using Stata a Plug-in to Estimate
Group-Based Trajectory Models
Daniel S. Nagin
Carnegie Mellon University
Installing the Plug-in
Traj can be installed by issuing the
following commands within Stata. An
additional command, trajplot, supports
plotting the results.
. net from
http://www.andrew.cmu.edu/user/bjone
s/traj
. net install traj, replace
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
Calculation & Use of Posterior
Probabilities of Group Membership
pˆ ( group j | datai ) 
pˆ (datai | group j )ˆ j
 pˆ (data | group j)ˆ
i
j
Maximum Probability Group Assignment Rule
j
Group Profiles
Variable
Group
Never
Low
Desister
High
Desister
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
on Time (%)
80.3
64.6
31.8
6.5
Juvenile Record (%)
0.0
2.0
6.0
13.3
# of Sexual Partners at
Age 17 (Past Year)
1.2
1.7
2.2
3.5
Chronic
Trajectories of Convictions
0
1
2
3
Cambridge Study of Delinquent Development
1
1.5
1
4
69.5%
5.9%
2
scaled age
2
12.4%
2.5
3
3
12.2%
Logit Model for Binary Data
p( y  1) 
e
 0  1age   2 age 2   3age 3
1 e
 0  1age   2 age 2   3age 3
where y=1 if yes & y=0 if no
0
.2
.4
.6
.8
Trajectories of Delinquent Group Membership
(Development & Psychopathology, 2003)
1
1.2
1
74.3%
1.4
scaled age
2
13.1%
1.6
1.8
3
12.6%
Statistically Linking Group Membership to Individual
Characteristics (Chapter 6)

Use of Multinomial Logit Model to Create a
Multivariate Probabilistic Linkage
e
 j ( xi ) 
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(tj )  0j  1j Aget  2j Aget2
Trajectory with retention and separation impacts:
~j
~j
~j
ln( )  0  1 Aget  2 Aget2  1j Failt  2j Separationt
j
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



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
.4
Adding Subject Attrition to the Model
.1
.2
.3
Probability of Death by
Trajectory Group
0
2
1 40.2%
4
Time (weeks)
2 45.7%
6
8
3 14.1%