An Overview of Two Recent Advances in Trajectory Modeling

1
An Overview of Two Recent Advances in Trajectory
Modeling

• Daniel S Nagin

2
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 datasetrich
  measurements about the characteristics and
  behaviors of participants

5
Annual Assessments of Violent Delinquency and
Gang Membership

• Violent Delinquencyfrequency 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
(No Transcript)
7
Cochrans Advice on how to proceed How should
the study be conducted if it were possible to do
it by controlled experimentation?

• Well defined treatmentwhat 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
  controlsprovided by propensity score matching

8
Treatment, Covariates, Outcomes
Responses to gang status at 14Outcomes
Outcomes-violence at 14 and beyond
Treatment compliance-gang status at 15 and
beyond
Time
Treatment Assignment-1st-time gang status at 14
Time0
Time -
Baseline covariatesFixed and time
varying Including violence prior to age 14
9
Baseline Measurements Trajectories of Violent
Delinquency from Age 11 to 13 for Sub-sample with
NO Gang Involvement over this Period
31 of Chronics Join Gangs at Age 14
15 of Decliners Join Gangs at Age 14
7 of Lows Join Gangs at Age 14
10
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 balanceMatch 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

13
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

16
Balance before and after matching for selected
variables
17
Standardized differences across the 15 variables
used in matching
18
Intent to Treat Effects of First-time Gang
Membership at 14 on Violence at age 14 to 17
Age Group Significance Level
14 Low Declining .008 .033
15 Low Declining .034 .086
16 Low Declining .044 .753
17 Low Declining .070 .530
19
Effects of First-time Gang Membership at 14 on
Violence at 14 to 17
20
Concluding Observations on Strengths of this
Approach

• Trajectory Group Specific Effects
• Transparency
• Weaknesses Open to View
• Keeping Time in Order

21
Extending Group-Based Trajectory Modeling to
Account for Subject Attrition

• Daniel S. Nagin
• Carnegie Mellon University
• Bobby Jones
• Carnegie Mellon University
• Amelia Haviland
• Rand Corporation

22
Trajectories Based on 1979 Dutch Conviction
Cohort
23
Missing Data

• Two Types
• Intermittent missing assessments (y1, y2 , . ,y4,
  . ,y6)
• Subject attrition where assessments cease
  starting in period t (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

24
Some Notation
Tnumber of assessment periods
ti period t in which subject i drops out
Probability of Drop out in group j in period t
25
(No Transcript)
26
The Dropout Extended Likelihood for Group j
27
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

28
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

29
Simulation 1 Two Group Model With Different
Drop Probabilities and Small Initial Separation
10
10
E(y)
E(y)
No dropout Slope.5
Time
Time
10
10
E(y)
E(y)
Time
Time
30
Simulation Results Group 1 and Group 2 Initially
not Well Separated
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 Without Dropout Model With Dropout Model With Dropout Model With Dropout
Group 1 Per Period Dropout Probability Expected Group 1 Assessment Periods Group 1 Prob. Est. (p1) Percent Bias Group 1 Prob. Est. (p1) 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
31
Simulation 2 Projecting to the Population Level
from Model Parameter Estimates
32
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

33
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

34
(No Transcript)
35
(No Transcript)
36
(No Transcript)
37
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?

38
Testing Strategy and Results

• Specify group membership probability (pj ) 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

39
Projecting the reduction in population average
ADL count from a 25 reduction in the incidence
of the life threatening disease at baseline
Projected Reduction in Population Average ADL
Count
Year 1998 2000 2002 2005
Reduction () 3.0 2.2 1.5 .7
40
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