Miles Taylor, Ph.D. Florida State University

1. The Good, the Bad, and the Mean (µ): Limitations and Extensions of Latent Growth Curves in Health Disparities Research Miles Taylor, Ph.D. Florida State University

2. What is Growth Curve Analysis? • The broad category of models includes multiple types of models (from multiple traditions) that are used to analyze individual change using more than 2 time points • Ex’s: latent growth curve analysis, latent trajectory analysis, random effects models, hierarchical linear models, etc. • Note that “curve” does not necessarily mean a nonlinear trend. On the contrary, most of the growth “curves” predicted by these various types of models are linear. • Examples: trajectories of reading ability in children, depressive symptoms across the life course, tumor growth in rats

3. 1999 1982 1984 1989 1994

4. 1999 1982 1984 1989 1994 1984 1989 1994

5. ε3 ε4 ε1 ε2 y3 y4 y1 y2 1 1 1 2 1 1 0 3 β α Unconditional Model • Level 1 model: • Level 2 model: • Combined 1999

6. Structural Equation Models (SEM) • Structural Equation Models (SEM) refer to a broad class of powerful models • Instead of emphasizing cases, SEM emphasizes variances/covariances. • This allows testing whether and how variables are interrelated in a set of linear relationships • The acronym is sometimes switched for simultaneous equation modeling (SEM) since it can handle many interrelated equations that are jointly estimated

7. Why Choose a Structural Equation Modeling (SEM) Approach to Growth Curves? • Various forms of measurement error • Estimators and fit indices for continuous, dichotomous, or ordinal repeated measures • Flexibility in handling time • Statistical packages like Mplus make more complex models possible • Other approaches do have advantages in some instances, such as observations at different time points

8. The Good • Improvement over aggregate change approaches – not Markovian or semi-Markovian • Can incorporate many repeated observations • Can handle time invariant and time variant covariates as well as repeated outcomes • Can be combined in an SEM context • Allow examination of life course developmental processes, testing developmental theories • Can examine whether inequalities or disparities are persistent, increasing, etc. over time both within and across individuals

9. Example of the Good • Valle, G. Thomas, K. & Taylor, M. G. “Parental Incarceration: Influences on Children’s Mental Health during the Transition to Adulthood” Preliminary Findings, Please do not cite without permission

10. Example of the Good • Valle, G. Thomas, K. & Taylor, M. G. “Parental Incarceration: Influences on Children’s Mental Health during the Transition to Adulthood”

11. Why it works • The findings from the alpha and beta (intercept and slope) were meaningful in a life course context (persisting inequality changes to an underlying effect emerging in adulthood) • Individual loadings were freed and then fixed, allowing more complex nonlinearity to be modeled • The outcome is easily thought of as developmental / continuous in nature • The treatment was estimated before W1

12. The Bad (1) People or Patterns are “Missed” • Level 1 equation parameterizes individual trajectories before calculating their variation from the mean • Model specification (linear, quadratic, etc.) is based on the average trajectory specification • Trajectory methodologists acknowledge we should free the loadings but we trade parsimony and therefore fit • What if some collection of the trajectories is nonlinear and meaningful • What if timing of the developmental process is important?

13. 1999 1982 1984 1989 1994

14. 1999 1982 1984 1989 1994 1984 1989 1994

15. Extensions • Group-based modeling strategies can handle this efficiently (latent class analysis of trajectories, finite mixture models, growth mixtures with freed loadings) • Work of Nagin, Land, Muthen • Hybridized models can handle this where “onset” of developmental process varies at random. • Work of Albert & Shih (2003), Taylor (2008; 2010), and Haas & Rohlfson (2010)

16. Group Trajectory Example

17. Why it works • Shows that there is more than one average trajectory and multiple forms of meaningful nonlinearity. • Efficiently models linear trajectories like linear along with a lagged onset, etc. • Referent group is no longer the mean trajectory. It is assumed to be the most prevalent group by default but may be set to any meaningful experience (here: nondisabled over the period) • Covariates are thus used to predict patterns rather than high/low on intercept and slope/s.

18. Random Onset Model • Taylor, Miles G. 2010. “Capturing Transitions and Trajectories: The Role of Socioeconomic Status in Later Life Disability.” Journals of Gerontology: Social Sciences 65B: 733-743

19. Why it works • A second process (here: first onset) is modeled. Therefore, the growth curves only include nonzero values. • Delayed onset (modeled through a discrete time hazard) captures the meaningful nonlinearity of the disability trajectories. • This means that one can reconcile findings from state based (transition) and developmental trajectory literatures • It also means covariates can predict these simultaneous processes in shared or independent ways

20. The Bad (2) Selection Processes • Selection into the observation window with/without starting the developmental process (meaningful partial left censoring) • Random onset model handles this better than traditional LGC’s

21. Extension: Random Onset • Taylor, Miles G. 2008. “Timing, Accumulation, and the Black/White Disability Gap in Later Life: A Test of Weathering.” Research on Aging: Special Issue on Race,SES, and Health 30: 226-250.

22. Extension: Random Onset • Taylor, Miles G. 2008. “Timing, Accumulation, and the Black/White Disability Gap in Later Life: A Test of Weathering.” Research on Aging: Special Issue on Race, SES, and Health 30: 226-250.

23. Why it works • A second process (here: first onset) is modeled. Therefore, the growth curves only include nonzero values. • Traditional LCG’s returned findings supporting a cumulative disadvantage theory. • Random onset model reveals that in this sample, the disparity lies in the onset process. • Black individuals were more likely to select into the sample with some nonzero level of disability, but their process of accumulation thereafter was not significantly different from whites.

24. The Bad (2) Selection Processes • Selection out of the sample that is meaningful (attrition, mortality selection) • Transition models (survival, etc.) have specific extensions for this (competing risk/multiple decrement) • In traditional LCG’s, the best we get is to “include” those until they drop out or include some kind of “control” for attrition

25. The Bad (2) Selection Processes • With SEM it is possible (just like in the random onset model) to include additional equations to handle this transition (either time variant or no) • This means we can include a parallel joint process (like the random onset model) but this time it is a timing of exit • A.K.A., one can create a sort of competing risk between changes in the developmental process of the outcome over time vs. attrition/death

26. Extension: Attrition Process Taylor, Miles G. and Scott M. Lynch. 2011. “Cohort Differences and Chronic Disease Profiles of Differential Disability Trajectories”. Journals of Gerontology: Social Sciences. 66B: 729-738.

27. Extension: Attrition Process Taylor, Miles G. and Scott M. Lynch. 2011. “Cohort Differences and Chronic Disease Profiles of Differential Disability Trajectories”. Journals of Gerontology: Social Sciences. 66B: 729-738.

28. Why it works • The second process here is mortality, and this I can model jointly with disability. A.K.A they affect one another over time. • Here I was primarily interested in cohort differences, and allowing these covariates to impact both disablement trajectories and death inform findings on the compression of morbidity. • Chronic diseases were also included in later models, and these impacts I could see on disability over the decade net of death and vice versa.

29. Summary • Potential weaknesses of traditional LGC’s: • People or meaningful patterns are missed through misspecification in the level 1 equation • Extensions: • Multiple ways to “disentangle” or unpack the mean growth or important deviations from it • Consider group based trajectories for modeling meaningful nonlinearity efficiently • Inclusion of additional processes (onset, recovery, etc.)

30. Summary • Potential weaknesses of traditional LGC’s: • Differential Selection: into the sample on level of outcome, out of the sample • Extensions: • Random onset as simultaneous process for partial left censoring • Mortality or other meaningful attrition as a simultaneous process

31. Conclusions • Latent Growth Curve (LGC) modeling in an SEM framework is extremely versatile due to the ability to model equations simultaneously • New softwares for SEM/Latent variable modeling (a.k.a. Mplus) allow more flexibility in modeling noncontinuous endogenous/outcome variables • Documentation now exists on replicating standard models like simply discrete-time hazard and finite mixtures/cluster analysis in the SEM context. • It’s time to move beyond the mean, beyond the noise.