An Overview of Two Recent Advances in Trajectory Modeling Daniel S Nagin.

An Overview of Two Recent Advances in Trajectory Modeling

Daniel S Nagin

Combining Propensity Score Matching and Group-Based Trajectory Analysis in an Observational Study (Psychological Methods, 2007) (Also, Developmental Psychology, 2008)Amelia Haviland, RAND Corporation

Daniel S. Nagin, Carnegie Mellon University

Paul R. Rosenbaum, University of Pennsylvania

3

Problem Setting Inferring the “treatment (aka causal) effect” of an

important life event or a therapeutic intervention with non-experimental longitudinal data

Overcoming severe selection problem whereby treatment probability depends heavily upon prior trajectory of the outcome-- Boys with high prior violence levels are more likely to join gangs

Dealing with feedback effects--violence and gang membership may be mutually reinforcing

Treatment effect may also depend upon prior trajectory of the outcome

Measuring effect of gang membership is prototypical example of a large set of important inference problems in psychopathology Divorce and depression Drug treatment and drug abuse

4

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 including self-reported violent delinquency and gang membership from age 11 to 17

Prototypical modern longitudinal dataset—rich measurements about the characteristics and behaviors of participants

5

Annual Assessments of Violent Delinquency and Gang Membership Violent Delinquency—frequency in last year of:

Gang fighting Fist fighting Carrying/Using a Deadly Weapon Threatening or Attacking Someone Throwing an object at someone

Gang Membership: In the past year have you been part of a group or gang that committed reprehensible acts?

6

00.5

11.5

22.5

33.5

4

violence age 14

violence age 13

violence age 12

violence age 11

The Selection Problem: Violent Delinquency from Age 11 to 14 of Gang Members at Age 14

Gang member age 14

Non-gang member age 14

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Cochran’s Advice on how to proceed: “How should the study be conducted if it were possible to do it by controlled experimentation?” Well defined treatment—what is the effect of first-time

gang membership at age 14 on violence at age 14 and beyond?

Good baseline measurements on the treated (gang members at 14) and controls (non-gang members at 14)—provided by trajectory groups

Randomize treatment to create comparability (i.e. balance) on all covariates between treated and controls—provided by propensity score matching

8

Treatment, Covariates, & Outcomes

TreatmentAssignment-1st-time gang status at 14

Baseline covariates—Fixed and time varyingIncluding violence prior to age 14

Responses to gang status at 14—Outcomes

Outcomes-violence at 14 and beyond

“Treatment compliance”-gang status at 15 and beyond

Time=0

Time= -

Time=+

9

Baseline Measurements: Trajectories of Violent Delinquency from Age 11 to 13 for Sub-sample with NO Gang Involvement over

this Period

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

11 12 13

Age

Delin

qu

en

t V

iole

nce

31% of ChronicsJoin Gangsat Age 14

15% of Decliners Join Gangs at Age 14

7% of LowsJoin Gangsat Age 14

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Trajectory Groups as Baseline Measurements Allows test of whether facilitation effect of

gang membership depends on developmental history

Aids in controlling for selection effects by comparing gang and nongang members with comparable histories of violence that are uncontaminated by the effects of prior gang membership

11

Creating balance with propensity score matching Propensity score relates probability of treatment

to specified covariates By matching on propensity score, treated and

controls are balanced on the covariates in the propensity score

Imbalance may remain on other covariates

12

Creating balance—Match first-time gang joiners at 14 with one or more “comparable” non-gang joiners Match within trajectory group

Group-specific treatment effect estimates Helps to balance prior history of violence

Within Group Matching based on: Propensity score for gang membership at age 14 Covariates in the propensity score include:

Self reported violence at ages 10-13 plus teacher and peer ratings of aggression

Posterior probability of trajectory group membership Many risk factors for violence-gang membership such as

low iq and having a teen mother, hyperactivity and opposition

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Twelve Covariates Comparing Gang Joiners at 14 with Potential Controls

14

Propensity for gang joining by trajectory group (before matching)

15

Matching Strategy

21 gang joiners in low trajectory matched with 105 (out of 276) non-gang joiners from that trajectory Number of matches range 2 to 7

38 gang joiners in declining trajectory matched with 114 (out of 216) non-gang joiners from that trajectory Number of matches range from 1 to 6

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Balance before and after matching for selected variables

17

Standardized differences across the 15 variables used in matching

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“Intent to Treat” Effects of First-time Gang Membership at 14 on Violence at age 14 to 17 Age Group Significance Level

14 LowDeclining

.008

.033

15 LowDeclining

.034

.086

16 LowDeclining

.044

.753

17 Low Declining

.070

.530

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Effects of First-time Gang Membership at 14 on Violence at 14 to 17

Low Trajectory: Violence at Ages 14-17 by Gang Status at Age 14

0

0.5

1

1.5

2

violenceage 14

violenceage 15

violenceage 16

violenceage 17

Gang member age 14

Non-gang member age14

Declining Trajectory: Violence at ages 14 to 17 by Gang Status at Age 14

0

0.5

11.5

2

2.5

3

violenceage 14

violenceage 15

violenceage 16

violenceage 17

Gang member age 14

Non-gang member age14

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Concluding Observations on Strengths of this Approach Trajectory Group Specific Effects Transparency Weaknesses Open to View Keeping Time in Order

Extending Group-Based Trajectory Modeling to Account for Subject Attrition

Daniel S. NaginCarnegie Mellon UniversityBobby Jones Carnegie Mellon UniversityAmelia HavilandRand Corporation

Trajectories Based on 1979 Dutch Conviction Cohort

Missing Data

• Two Types– Intermittent missing assessments (y1,

y2 , . ,y4, . ,y6)– Subject attrition where assessments cease

starting in period τ (y1 , y2 , y3 , . , . , .)• Both types assumed to be missing at random • Model extension designed to account for

potentially non-random subject attrition• No change in the model for intermittent

missing assessments

Some Notation

τi =period t in which subject i drops out

T=number of assessment periods

jt = Probability of Drop out in group j in period t

Probability of Dropout in Period t

Period Probability of Drop Out 1 0 2 3 4 . . . . . . T

No Drop Out

1 – all the above probabilities

The Dropout Extended Likelihood for Group j

).3()1)(;,,0|(),;,|(1

1

jT

t

jtjiitit

jjii i

i

jagewypjageYP

Specification of

• Binary Logit Model• Predictor Variables

– Fixed characteristics of i, – Prior values of outcome,

• If trajectory group was known within trajectory group j dropout would be “exogenous” or “ignorable conditional on observed covariates”

• Because trajectory group is latent, at population level, dropout is “non-ignorable”

jt

ix,...., 21 itit yy

Simulation Objectives

Examine effects of differential attrition rate across groups that are not initially well separated

Examine the effects of using model estimates to make population level projections

Simulation 1: Two Group Model With Different Drop Probabilities and Small Initial Separation

10 10

10 10

No dropoutSlope=.5

Time

E(y)

E(y)

E(y) E(y)

Time

Time Time

Group 1 Per Period

Dropout Probability

Expected Group 1

Assessment Periods

Probability of Group 1

Dropout on or before Period 6

Model Without Dropout

Model With Dropout

Group 1 Prob. Est.

(π1)

Percent Bias

Group 1 Prob. Est.

(π1)

Percent Bias

Dropout Prob.Est.

0 6.0 0 .200 0.0 .200 0.0 .000.05 5.3 .226 .171 -14.5 .199 -0.5 .051.10 4.7 .410 .146 -27.0 .199 -0.5 .099.15 4.2 .556 .122 -39.0 .200 0.0 .150.20 3.7 .672 .100 -50.0 .199 -0.5 .199.25 3.3 .762 .079 -60.5 .200 0.0 .250.30 2.9 .832 .061 -69.5 .199 -0.5 .301.35 2.6 .884 .046 -77.0 .199 -0.5 .350.40 2.4 .922 .034 -83.0 .199 -0.5 .398

Simulation Results: Group 1 and Group 2 Initially not Well Separated

Simulation 2: Projecting to the Population Level from Model Parameter Estimates

Chinese Longitudinal Healthy Longevity Survey (CLHLS) Random selected counties and cities in 22

provinces 4 waves 1998 to 2005 80 to 105 years old at baseline 8805 individual at baseline 68.9% had died by 2005 Analyzed 90-93 years old cohort in 1998

Activities of Daily Living

On your own and without assistance can you: Bath Dress Toilet Get up from bed or chair Eat

Disability measured by count of items where assistance is required

Table 3

Summary Statistic for the Age 90 to 93 CLHLS Cohort at Baseline

Variable N Average ADL 1998 Count 1078 .84 ADL 2000 Count 580 1.05 ADL 2002 Count 335 1.16 ADL 2005 Count 120 1.26

Female 1078 .52 Life Threatening

Disease 1078 .11

Table 4

Predict Population Average ADL counts from the Models With and Without Dropout

Model Without Drop Out

Model With Drop Out

Period Average ADL

Count

Predict ADL

Count

% Error

~1

t ~2

t ~3

t Predicted

ADL Count

% Error

1998 .84 .91 8.3 .201 .586 .213 .93 10.7 2000 1.05 1.19 13.3 .254 .600 .146 1.07 1.9 2002 1.16 1.42 22.4 .309 .593 .097 1.17 .9 2005 1.26 1.89 50.0 .366 .571 .063 1.58 25.4

Adding Covariates to Model to Test the Morbidity Compression v. Expansion Hypothesis • Will increases in longevity compress or expand

disability level in the population of the elderly?• “Had a life threatening disease” at baseline or

prior is positively correlated with both ADL counts at baseline and subsequent mortality rate.

• Question: Would a reduction in the incidence of life threatening diseases at baseline increase or decrease the population level ADL count?

Testing Strategy and Results

• Specify group membership probability (πj ) and dropout probability ( ) to be a function of life threatening disease variable

• Both also functions of sex and dropout probability alone of ADL count in prior period

• Life threatening disease significantly related to group membership in expected way but has no relationship with dropout due to death

• Thus, unambiguous support for compression

jt

Projecting the reduction in population average ADL count from a 25% reduction in the incidence of the life threatening disease at baseline

Year 1998 2000 2002 2005Reduction (%) 3.0 2.2 1.5 .7

Projected % Reduction in Population Average ADL Count

Table 6

Own and Cross Elasticity Estimates (%) for Life Threatening Disease Incidences

Cross Elasticity

Group Own Elasticity

Group 2

Group 3

Total Elasticity

1. Low ()201.1

NA -.033 -.059 -.092

2. Medium ()586.2

.069 NA -.173 -.104

3. High()213.3

.232 -.036 NA .196

Conclusions and Future Research Large differences in dropout rates across

trajectory groups matter Future research

Investigate effects of endogenous selection Compare results in data sets with more modest

dropout rates Further research morbidity expansion and

contraction