Latent transition analysis (LTA) for modeling discrete change in longitudinal data

1. Latent transition analysis (LTA)for modeling discrete change in longitudinal data Stephanie T. Lanza The Methodology Center The Pennsylvania State University UCLA May 1, 2006

2. The Methodology Center • A group of social scientists and statisticians working together • Work on statistical methods and applications with direct relevance to important scientific questions • Primarily motivated to work on methods relevant to study of substance use and abuse

3. The Methodology Center • Examples of research topics: • Latent class, latent transition analysis • Missing data, theory and applications • Adaptive interventions • Optimal design of behavioral interventions • Analysis of data from intensive data collection methods • Economic cost-effectiveness analysis • Risk assessment

4. Bethany Bray Hwan Chung Linda M. Collins David Lemmon Tammy Root Joseph L. Schafer Recent collaborators on LTA

5. Outline • Overview of what LCA and LTA can do • LTA • Advanced topics and future directions • Some research questions you might address using LTA

6. Outline • Overview of what LCA and LTA can do • LTA • Advanced topics and future directions • Some research questions you might address using LTA

7. Ideas underlying LCA • Individuals can be divided into subgroups, or latent classes, based on unobserved construct • Subgroups are mutually exclusive and exhaustive • True class membership in unknown

8. Ideas underlying LCA • Measurement of that construct typically based on several categorical indicators • There may be error associated with the measurement of the latent classes • Like confirmatory factor analysis (specify number of classes), but latent variable is categorical

9. Parameters estimated in LCA • Latent class membership probabilities • e.g. probability of membership in Advanced Substance Use latent class • Item-response probabilities • e.g. probability of reporting marijuana use given membership in Advanced Substance Use latent class

10. Example of LCA:Depression in adolescence Lanza, S. T., Flaherty, B. P., & Collins, L. M. (2003). Latent class and latent transition analysis. J. A. Schinka, & W. F. Velicer (Eds.), Handbook of Psychology: Vol. 2. Research Methods in Psychology (pp. 663-685). Hoboken, NJ: Wiley. Eight indicators of adolescent depression: Sad • Couldn’t shake blues • Felt depressed • Felt lonely • Felt sad Disliked • People unfriendly • Disliked by people Failure • Life was failure • Life not worth living

11. Example of LCA:Depression in adolescence Five latent classes of depression:

12. Ideas underlying LTA • LTA is a longitudinal extension of latent class models • Some development can be represented as movement through discrete categories or stages • There may be error associated with the measurement of the discrete categories • Different people may take different paths • This heterogeneity may be unobserved (latent)

13. Compare this approach to growth curve approach Ideas underlying LTA

14. Ideas underlying LTA • LTA provides a way of fitting models with these characteristics: • Change is stage sequential • Longitudinal • Measurement error • Developmental heterogeneity

15. Example 1:Depression in adolescence Five stages of depression, two times:

16. Example 2: Substance use over time, effect of pubertal timing Eight stages of substance use, two times:

17. Example 3: Substance use and delinquency over time

18. Outline • Overview of what LCA and LTA can do • LTA • Advanced topics and future directions • Some research questions you might address using LTA

19. Latent transition analysis (LTA) • An extension of latent class theory to longitudinal data • Provides a way of estimating and testing models of stage-sequential development in longitudinal data • In LCA, latent classes are static • In LTA, latent statuses (stages) are dynamic

20. LTA • A multiple‑indicator latent Markov model • Estimates prevalence of stages and incidence of transitions between stages adjusted for measurement error

21. We will use LTA to: • Fit a stage-sequential model of substance use onset in seventh-grade females • Include pubertal timing as grouping variable • Examine the following: • Proportion of girls with early timing • Group differences in substance use at Grade 7 • Group differences in advancement in substance use from Grade 7 to Grade 8 From Lanza & Collins (2002) Prevention Science

22. A model of substance use onset

23. Study participants • From Waves I and II of The National Longitudinal Study of Adolescent Health, known as Add Health (Resnick et al., 1997) • Used only females in Grade 7 at Wave 1 • N = 966

24. Indicators of substance use ALCOHOL Have you had a drink of beer, wine or liquor – not just a sip or taste of someone else’s drink – more than 2 or 3 times in your life? • 1=no, 2=yes CIGARETTES Have you ever tried cigarette smoking, even just 1 or 2 puffs? • 1=no, 2=yes 5+ DRINKS Over past 12 months, on how many days did you drink five or more drinks in a row? • 1=never, 2=one or two days in past 12 months, or more DRUNK Over past 12 months, on how many days have you gotten drunk or “very, very high” on alcohol? • 1=never, 2=one or two days in past 12 months, or more MARIJUANA How old were you when you tried marijuana for the first time? • 1=never tried, 2=all other ages

25. Indicators of pubertal timing • Breast size relative to grade school 1 = On-time/late timing (same size/little bigger) 2 = Early timing (a lot bigger) • Body becomes curvy 1 = On-time/late timing (as curvy, somewhat curvy) 2 = Early timing (a lot more curvy)

26. LTA notation • Y represents an array of cells of the contingency table • Cells are formed by crosstabulating: • Indicators of the dynamic latent variable measured at two or more times • Indicator(s) of grouping variable • S refers to number of latent statuses (stages) • a = 1, … S at Time 1, b = 1, …S at Time 2

27. Parameters in LTA models • = probability of being in group c (e.g. the probability of being in the early pubertal timing group) • = probability of being in latent status a at Time 1 given membership in group c (e.g. the probability of being in the No Use latent status at Time 1 given membership in the early pubertal timing group)

28. Parameters in LTA models • = probability of membership in latent status b at Time t+1 given membership in latent status a at Time t and membership in group c (e.g. the probability of being in the Advanced Substance Use latent status at Time 2, given membership in the No Use latent status at Time 1 and membership in the early pubertal timing group)

29. Parameters in LTA models Time 2 Time 1 • parameters arranged in transition probability matrix

30. Parameters in LTA models • The parameters are item-response probabilities e.g. the probability of a particular response to an item (such as reporting drunkenness) given - time - latent status membership - group membership • These parameters allow you to name the latent statuses, test measurement invariance

31. For two times: The LTA model (one term for each manifest item)

32. Estimation • LTA models can be estimated using WinLTA • This program is available free of charge on our web site http://methodology.psu.edu/

33. Overview of the LTA procedure • LTA is a confirmatory procedure • You tell the program some things about the model: • number of groups • number of latent statuses • number of times • number of manifest items • number of response categories per item • And some instructions about estimation • The program then estimates the parameters

34. Overview of the LTA procedure • WinLTA uses the EM algorithm • Handles missing data, makes MAR assumption

35. Overview of the LTA procedure • LTA computes expected response pattern proportions according to the model and estimated parameters • These expected response pattern proportions are compared to the observed response pattern proportions. • This comparison is expressed in the likelihood ratio statistic G2

36. Parameter restrictions • The LTA user has three options for estimation of EACH parameter: • Free estimation • Constraining a group of parameters to be equal • Fixing the parameter to a pre-specified value (such as 0)

37. Reasons for choosing parameter restrictions • To help improve identification by reducing the number of parameters to be estimated • To express features of the model you wish to test

38. Examples of parameter restrictions • Constraining parameters equal across times • Measurement invariance across time • This assures that the latent statuses can be interpreted the same way across times • Constraining parameters equal across groups • Measurement invariance across groups • Fixing elements of the transition probability matrix • This expresses a model of development

39. A model of no backsliding Time 2 Time 1 Examples of parameter restrictions

40. Examples of parameter restrictions • A model of no change Time 2 Time 1

41. Response probabilities conditional on latent status membership ( ) Based on these parameters, what would YOU name the stages?

42. Prevalence of substance use stages given pubertal timing ( )

43. Advancement in substance use from Grade 7 to Grade 8 ( ) • Note: Early timing probabilities in bold • Early-timing group more likely to advance from No Use

44. Advancement in substance use from Grade 7 to Grade 8 • Summing across certain cells: • 40% early-developing females increase in use • 26% of on-time or late-developing females increase in use • Early-maturing females are 1.5 times more likely to advance in substance use regardless of their level of use in seventh grade (p < .05)

45. Drawbacks and limitations of LTA • Not suitable for small samples • Hypothesis testing flexible but not easy • No consensus on best approach for model selection

46. Outline • Overview of what LCA and LTA can do • LTA • Advanced topics and future directions • Some research questions you might address using LTA

47. Advanced topics andfuture directions • LCA for repeated measures • Data augmentation (DA) • Associative LTA (ALTA) • LCA with covariates

48. LCA for repeated measures • LTA examines pairs of times • Here, each latent class represents trajectory through stage sequence across 3 or more times • Advantage over growth curve models: no smooth function of time necessary; discontinuous development modeled • Example: Lanza & Collins (in press). Journal of Studies on Alcohol.

49. A mixture model of discontinuous development in heavy drinking from ages 18 to 30:The role of college enrollment

50. A mixture model of discontinuous development in heavy drinking from ages 18 to 30:The role of college enrollment