Amelia Haviland Bobby Jones Daniel S. Nagin

1. Extending Group-Based Trajectory Modeling to Account for Subject Attrition(Sociological Methods & Research, 2011) Amelia Haviland Bobby Jones Daniel S. Nagin

2. 4% 28% 52% 16%

3. Trajectories Based on 1979 Dutch Conviction Cohort

4. The Likelihood Function

5. 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

6. Some Notation T=number of assessment periods τi =period t in which subject i drops out = Probability of Drop out in group j in period t

8. The Dropout Extended Likelihood for Group j

9. 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”

10. 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

11. 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

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

13. An Important Distinction from Zhang and Rubin (2003) • Dropout due death • Subject exits population of interest-the living • Data said to be “truncated” • Dropout due termination of study participation • Subject exits the sample but remains in the population • Data said to be censored

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

15. Simulation 2 Continued 12.5 10 Dropout=.2 per period No Dropout

17. 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

18. 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

22. 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?

23. 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

24. 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

25. 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