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1
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
2
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
3
Trajectories of Physical Aggression
(Child Development, 1999)
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
6 10 11 12 13 14 15
Age
Ph
ysic
al A
gg
ressio
n
Low-actual Mod. desister-actual High desister-actual Chronic-actualLow-pred. Mod. desister-pred. High desister-pred. Chronic-pred
4%
28%
52% 16%
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
4
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
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
5
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
ije
ex
)(
j
i
j
iji YPxYP )()()(
Identification of Distinctive Developmental
Trajectories: An Illustrative Example
age
crime
Pop. Total
Adolescent
onset (50%)
Adolescent
limited (50%)
6
Types of Data
Psychometric scales--Censored Normal
(Tobit) Model
Count Data--Poisson-based Model
Binary Data--Logit-based Model
Linking Age to Behavior
3
3
2
210)log( it
j
it
j
it
jjj
it AgeAgeAge
3
3
2
210
*
it
j
it
j
it
jjj
it AgeAgeAgey
7
Censored Normal (Tobit) Model for Psychometric Data
ymax
y* = 0 + 1Age+ε (note: could include x2 or x3)
y = 0 if y*0
y = y* if 0<yymax
y* = ymax if y*>ymax
age
E(y*)
E(y)
E(y*)=E(y)
Go to Example 1
8
Zero-inflated Poisson Model for Count Data
33
2210
33
2210
1
)ln(
1)(
0)(
3
3
2
210
ageageage
ageageage
e
e
ageageage
yprobabilitwithPoisson
yprobabilitwithxp
01
23
co
nvic
tion
ra
te
1 1.5 2 2.5 3
scaled age
1 69.5% 2 12.4% 3 12.2%
4 5.9%
Criminology (1993, 1995)
Trajectories of Convictions: London Data
9
01
23
co
nvic
tion
ra
te
1 1.5 2 2.5 3
scaled age
1 69.5% 2 12.4% 3 12.2%
4 5.9%
Cambridge Study of Delinquent Development
Trajectories of Convictions
Go to Example 2
10
Logit Model for Binary Data
33
2210
33
2210
1)1(
ageageage
ageageage
e
eyp
where y=1 if yes & y=0 if no
Trajectories of Delinquent Group Membership
(Development & Psychopathology, 2003)
0.2
.4.6
.8
Pro
b o
f g
an
g m
em
bers
hip
1 1.2 1.4 1.6 1.8
scaled age
1 74.3% 2 13.1% 3 12.6%
11
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
Calculation & Use of Posterior
Probabilities of Group Membership
Maximum Probability Group Assignment Rule
j
ji
jii jgroupdatap
jgroupdatapdatajgroupp
ˆ)|(ˆ
ˆ)|(ˆ)|(ˆ
12
Group Profiles
Variable Group
Low High
Never 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)
Go back to example 1
13
Other Uses of Posterior Probabilities
Computing Weighted Averages That Account
for Group Membership Uncertainty
Diagnostics for Model Fit
Matching People with Comparable
Developmental Histories
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
ji
ji
x
x
ije
ex
)(
14
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
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
No R
isk
Low IQ
Broke
n Home
Low E
d Mom
Tee
n M
om
All Risks
pro
bab
ilit
y
Low
Moderate Declining
High Declining
Chronic
15
Go to Example 3
Model Extensions
Entering Covariates into the Trajectory Itself
Joint Trajectory Analysis
Multi Trajectory Modeling
16
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
Rat
e
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
17
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:
2
210)ln( t
j
t
jjj
t AgeAge
t
j
t
j
t
j
t
jjj
t SeparationFailAgeAge 21
2
210
~~~)ln(
The Impact of Grade Retention on the
Rising Trajectory
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
11 12 13 14 15 16 17
Age
Ra
te
No
Retention
Retention
at Age 14
Group
Average
18
Dual Trajectory Analysis: Trajectory of
Modeling of Comorbidity and Heterotypic
Continuity 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
Ph
ys
ica
l A
gg
re
ss
ion
Low
Desisting
High
Trajectories of Adolescent Violent Delinqunecy
from Age 13 to 17
0
1
2
3
4
5
6
7
8
9
13 14 15 16 17
Age
Rate
Low 1
Low 2
Declining
Rising
Chronic
19
Transition Probabilities Linking Trajectories in
Adolescent to Childhood Trajectories
Trajectory in Adolescence
Trajectory
in
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)
20
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
Multi-Trajectory Modeling
21
Linking Trajectories to Later Out Comes—
Trajectories of Physical Aggression from 6 to 15
and Sexual Partners at 17
Adding Subject Attrition to the Model
Probability of Death by
Trajectory Group
.1.2
.3.4
Dro
po
ut p
rob
ab
ility
0 2 4 6 8
Time (weeks)
1 40.2% 2 45.7% 3 14.1%
22
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
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
23
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
2
210 AgeAge