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?

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7 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 Treatment Assignment -1 st -time gang status at 14 Baseline covariates—Fixed and time varying Including 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 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 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 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 AgeGroupSignificance Level 14Low Declining Low Declining Low Declining Low Declining

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

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

Trajectories Based on 1979 Dutch Conviction Cohort

Missing Data Two Types – Intermittent missing assessments (y 1, y 2,.,y 4,.,y 6 ) – Subject attrition where assessments cease starting in period τ (y 1, y 2, y 3,.,.,.) 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 τ i =period t in which subject i drops out T=number of assessment periods = Probability of Drop out in group j in period t

The Dropout Extended Likelihood for Group j

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”

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 No dropout Slope=.5 Time E(y ) 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. ( π 1 ) Percent Bias Group 1 Prob. Est. ( π 1 ) Percent Bias Dropout Prob. Est 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 to 105 years old at baseline 8805 individual at baseline 68.9% had died by 2005 Analyzed 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

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 ( π j ) 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

Projecting the reduction in population average ADL count from a 25% reduction in the incidence of the life threatening disease at baseline Year Reduction (%) Projected % Reduction in Population Average ADL Count

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