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.