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