1. Latent Growth Modeling Using 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
September 1, 2009
Corrected 9/2/2009

2. Acknowledgements
Funded in part by Grant R13AG A1 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 Our workshop 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
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. Advantages 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
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 Effect that 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 = __ 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 ;
[r] ; ;
i s on cagecat1-cagecat6* ;
y1-y15 *0.1 (theta_1) ;

30. TITLE: ROS GLOBCOG Trajectories 8/26/2009
DATA:
FILE = __ 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 ;
[r] ; ;
i on cagecat1*.714 cagecat2*.661 cagecat3*.224
cagecat5* cagecat6* ;
s on cagecat1*.068 cagecat2*.06 cagecat3*.015
cagecat5* cagecat6* ;
%c#1%
[i* s* r*.187] ;
i*.734 s*.367 ;
i with s *0 ;
y1-y15 *.0375 (theta_1) ;
%c#2%
[i*-.519 s*-.622 r*.187] ;
i*.734 s*.734 ;
y1-y15 *.075 (theta_2) ;
%c#3%
[i*-.83 s* r*.187] ;
i*.734 s* ;
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

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 ;
[y1*] ;
y1* ;
y2 on ;
dy on y1 * ;
dy* ;
[dy*] ; ; ;
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. Advantages Dual Change Score
More flexibility for estimating lagged and leading effects
Latent GC modeling
Better fit (good for descriptive analysis)
Better for sequential patterns

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 Model40

41. Help Coding in Mplus … from Zhiyong Zhang, Ph.D. 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. Questions Discussion