Latent Growth Modeling Using Mplus

1. Latent Growth ModelingUsing Mplus Friday Harbor Psychometrics Workshop Richard N. Jones1,2, Frances M. Yang1,2, Douglas Tommet1 1Institute for Aging Research, Hebrew SeniorLife and Beth Israel Deaconess Medical Center, Division of Gerontology 2Harvard Medical School jones@hrca.harvard.edu September 1, 2009 Corrected 9/2/2009

2. Acknowledgements • Funded in part by Grant R13AG030995-01A1 from the National Institute on Aging • The views expressed in written conference materials or publications and by speakers and moderators do not necessarily reflect the official policies of the Department of Health and Human Services; nor does mention by trade names, commercial practices, or organizations imply endorsement by the U.S. Government. 2

3. Session Overview • Other Resources • General Framework • Comparison with Random Effects • Modeling Framework • Special Model Considerations • Some Results from ROS • Detailed Example from ROS • Questions and Discussion

4. Other Resources • What is longitudinal data analysis? • Singer JD & Willett JB. Applied longitudinal data analysis: Modeling change and event occurrence. 2003, New York: Oxford University Press. (Also see worked examples at UCLA ATS) • How do I do latent growth curve modeling? • Duncan TE, Duncan SC, & Strycker LA. An introduction to latent variable growth curve modeling: concepts, issues and applications. Second ed. 2006, Mahwah, New Jersey: Lawrence Erlbaum Associates, Inc. • Tell me more about the math behind latent curve methods • Bollen KA & Curran PJ. Latent curve models: a structural equation perspective. Wiley series in probability and statistics. 2006, Hoboken, N.J.: Wiley-Interscience. • Our workshop • 2009 workshop was LDA, come back in 2011 • http://sites.google.com/site/lvmworkshop for slides, syntax, data

6. Five+ Approaches to LDA in Mplus • Latent growth curve model • Random effects model • Multilevel model • Latent change/Dual change score model • Autoregressive/latent simplex model • Latent Structural Models (Hyperbolic functions for learning data)

7. Random Effects and Latent Growth Curves: Same But Different • Reconceptualize random effects as latent variables • Use multivariate record layout (wide) • Main difference: • RE: time is data • LGC: time is a model parameter …unless it is data

8. HLM and Mixed Effect Modeling Software for highly nested multilevel data better developed Easier to get model fit and diagnostics Use time-varying weights LGC modeling Embed in more complex models Flexible curve shape (time is a parameter and/or data) Modification Indices can help with misspecified models Advantages Note: HLM Hierarchical Linear Modeling

9. Latent Growth Curve Models • Latent Growth Curve (LGC) modeling is just like CFA • Reconceptualize “factors” as “random effects” • Factor loadings are • (usually) not estimated but given by design or data, and • relate to the sequence of repeated observations • The action is in the mean structure part of the model (factor means, item means, factor variances) as opposed to factor loadings

10. Latent Growth Model (Linear Change)

11. Latent Growth Model (Linear Change)

12. Latent Growth Model (Linear Change)

13. Latent Growth Model (Linear Change) “*” Implies parameter freely estimated. All other parameters are held constant to the indicated value.

14. Latent Growth Model (Linear Change)

15. Latent Growth Model (Linear Change)

16. Latent Growth Model (Linear Change)

17. Change in Ordinal Outcome

18. Model a Retest Effect

19. Model a Retest Effectthat is dependent on baseline

20. Regress Change on Baseline “*” Implies parameter freely estimated. All other parameters are held constant to the indicated value.

21. Multiple Indicator Growth Model

22. Growth Mixture Model

23. Alternative Time Bases

24. Model Building (LGC) † unless required to specify the time basis, e.g., age group

26. Post-Estimation Fit Evaluation • Save Factor Scores • Import into stat package • Compute expected scores • Graph Residuals • Empirical r-square

27. Example: PW 2008 • ROS • Change in global cognition (globcog) • Random effects growth mixture model • Time basis: Age centered at baseline mean within age group • Retest effect (occasion basis) • Mixture model part for growth parameters • Covariates: age group dummies (to define age metric)

28. Random Effects Mixture Model

29. TITLE: ROS GLOBCOG SINGLE CLASS 8/16/2009 DATA: FILE = __000001.dat ; VARIABLE: NAMES = y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12 y13 y14 y15 t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13 t14 t15 cagecat1 cagecat2 cagecat3 cagecat5 cagecat6 projid ; MISSING ARE ALL (-9999) ; IDVARIABLE = projid ; TSCORES = t1-t15 ; ANALYSIS: TYPE = random ; COVERAGE = .02 ; MODEL: i s | y1-y15 AT t1-t15 ; r by y2-y15 @1 ; [r] ; r@0 ; i s on cagecat1-cagecat6* ; y1-y15 *0.1 (theta_1) ;

30. TITLE: ROS GLOBCOG Trajectories 8/26/2009 DATA: FILE = __000001.dat ; VARIABLE: NAMES = y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12 y13 y14 y15 t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13 t14 t15 cagecat1 cagecat2 cagecat3 cagecat5 cagecat6 projid ; MISSING ARE ALL (-9999) ; IDVARIABLE = projid ; TSCORES = t1-t15 ; CLASSES = c(3) ; ANALYSIS: TYPE = mixture random ; COVERAGE = .02 ; STARTS = 0 ; PROCESSORS=2 ; ALGORITHM=integration ema ; INTEGRATION = montecarlo ; MCONVERGENCE = 0.01 ; SAVEDATA: FILE = c:\work\ros\posted\data\derived\gmm3class10AUG2009.dat ; SAVE = fscores cprob ; RESULTS = c:\work\ros\posted\data\derived\gmm3class10AUG2009_results.dat ;

31. MODEL: %OVERALL% i s | y1-y15 AT t1-t15 ; r by y2-y15 @1 ; [r] ; r@0 ; i on cagecat1*.714 cagecat2*.661 cagecat3*.224 cagecat5*-.397 cagecat6*-.764 ; s on cagecat1*.068 cagecat2*.06 cagecat3*.015 cagecat5*-.058 cagecat6*-.071 ; %c#1% [i*-.31125 s*-.27805 r*.187] ; i*.734 s*.367 r@0 ; i with s *0 ; y1-y15 *.0375 (theta_1) ; %c#2% [i*-.519 s*-.622 r*.187] ; i*.734 s*.734 r@0 ; i with s *0 ; y1-y15 *.075 (theta_2) ; %c#3% [i*-.83 s*-1.245 r*.187] ; i*.734 s*1.468 r@0 ; i with s *0 ; y1-y15 *.15 (theta_3) ;

32. Figure 1. Cognitive Change Trajectories by Class and Age Group at First Observation as implied by Mixture Model Parameter Estimates 63% 27% 11% Education (and race/ethnicity, baseline mental status) associated with class Membership. But not age, not sex.

33. Figure 2. Burden of Amyloid and Tangle Neuropathology by Class Membership (N=326)

34. Trajectory Classes and Reserve • Neuropathology at autopsy does not perfectly account for membership in one of two population sub-groups experiencing substantial cognitive decline • Education, a proxy for cognitive reserve, may buffer the functional consequences of neuropathology:

35. A Different SEM model for Change The Latent Change Score Model 35

36. Latent Change Change Score Model TITLE: LCSM DATA: FILE = BLAH.dat ; VARIABLE:NAMES = y1 y2 ; MODEL: dy by y2 @1 ; [y1*] ; y1* ; y2 on y1 @1 ; dy on y1 * ; dy* ; [dy*] ; [y2@0] ; y2 @0 ; NB: As parameterized, just identified or saturated model = zero degrees of freedom. Just as many knowns as estimated parameters. 36

37. ex03-08.inp 37

38. Dual Change Score More flexibility for estimating lagged and leading effects Latent GC modeling Better fit (good for descriptive analysis) Better for sequential patterns Advantages

39. Extensions to the LCSM • Dual Change Score Model • Bivariate CSM • Two outcomes are of interest • Multiple Indicator LCSM • change in a latent variable • Multiple Indicator Dual Change Score Model 39

40. Dual Change Change Score Model 40

41. Help Coding in Mplus … from Zhiyong Zhang, Ph.D. http://www.psychstat.org/us/sort.php/7.htm%20-%20Programs 41

42. FHL 2009 Potential Topics • Replicate ROS analysis in MAP, and/or • Domain-specific REMM • Change-point model (LGCMM or REMM) • Reserve proxies associated with when cognitive decline starts and how fast decline occurs.

43. QuestionsDiscussion