1. Latent Growth Curve Modeling using Mplus
Richard N. Jones, Sc.D.
Hebrew Rehabilitation Center for Aged
Research and Training Institute
Boston, MA
University of Washington
Psychometrics Workshop
September 20-25, 2004
Please send questions and corrections to

2. Overview of this talk Other Resources Statistical model
Implementation in Mplus
Example
Discussion
Limitations
Advantages
Mplus (

3. Other Resources (Don’t take my word for it)
Curran, P. (2000). A latent curve framework for the study of developmental trajectories in adolescent substance use. In J. Rose, L. Chassin, C. Presson & J. Sherman (Eds.), Multivariate Applications in Substance Use Research. Hillsdale, NJ: Lawrence Erlbaum Associates.
Curran, P. J., & Muthén, B. (1996). Testing developmental theories in intervention research: latent growth analysis and power estimation. Los Angeles: Graduate School of Education and Information Studies, University of California, Los Angeles.
Muthén, B., Brown, C. H., Masyn, K., Jo, B., Khoo, S.-T., Yang, C.-C., et al. (2002). General growth mixture modeling for randomized preventive interventions. Biostatistics, 3(4),
Muthén, B., & Curran, P. (1997). General growth modeling in experimental designs: A latent variable framework for analysis and power estimation (CSE Technical Report No. 443): National Center for Research and Evalation, Standards and Student Testing (CRESST)/Graduate School of Education & Information Studies, University of California, Los Angeles.
Muthén, B., & Curran, P. (1997). General longitudinal modeling of individual differences in experimental designs: a latent variable framework for analysis and power estimation. Psychological Methods, 2(4),
Muthén, B. O. ( ). Mplus Technical Appendices. Los Angeles, CA: Muthén & Muthén.
Muthén, L. K., & Muthén, B. O. ( ). Mplus users guide (February 2001, Version 2 ed.). Los Angeles, CA: Muthén & Muthén.
Mplus Short Courses (November 2004, Alexandria VA) see
Also see
Jones, R. N., Allaire, J., McCoy, K., Kleinman, K., Rebok, G., Malloy, P., et al. (2004). A growth curve model of learning acquisition among cognitively normal older adults. Experimental Aging Research, in press.

4. Implementing LGM Models in Mplus
Mplus has capabilities of popular SEM software packages (LISREL, EQS, AMOS)
Mplus has unique features, e.g.
Random effects for individually-varying times of observations
Easy modeling of mean structures (thresholds, means)
Also
Latent class and mixture modeling
Regression based treatment of exogenous variables (covariates)
These special features make modeling change easy

5. Structural Equation Modeling
Includes path analysis, confirmatory factor analysis
Unobserved (latent) variables account for covariation among observed variables
Mplus uses y* variables when y are categorical

6. Understanding the Mplus and LISCOMP Models

7. Longitudinal Data Analysis

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

9. Be Eclectic in Treatment of Time Dimension
Changes over time vs. changes with age
Changes over time/age vs. disease progression
Functional form of changes
Linear
Quadratic
Logarithmic
Piecewise

10. Time: Linear Change

11. Time: Quadratic Change

12. Time: Logarithmic Change

13. Time: Piecewise Change

14. Cohort-Sequential Design

15. Steps in Analysis
Exploratory data analysis (missingness, general shape of trend)
LGM without covariates (a highly constrained confirmatory factor analysis (CFA))
Identify likely model mis-specification of time steps, direct relationships of x’s to y’s (e.g., time-specific or time varying covariates)
Possible multiple-group models or mixture models
Final model estimation (use WLSMV estimator)

16. Using Mplus “Wide” record layout Mplus requires data as raw text file
fixed, free, delimited
stata2mplus module available
“Wide” record layout
Raw text command file
Use Mplus language generator
Have the manual handy
runmplus module (contact me)

17. Mplus Command File Linear Latent Growth Model (without covariates)
Title: Growth Model 2 – Linear Chage
Data: File = EX.DAT;
Variable: Names are av01-av05;
Output: Tech1 Tech8 standardized;
Model:
eta1 by
eta2 by
Alternative Model Statement
MODEL:
eta1 eta2 | av01*0 av02*0.25 av03*0.5 av04*0.75
... using runmplus from within STATA
(and more complete specification of starting values)
runmplus av01-av05, missing(99) ///
ti(growth model 2 - linear change) /// type(missing H1 meanstructure) standardized /// /// eta1 by /// eta2 by ///
av01*0.5 av02*1.3 av03*0.9 av04*1; av05*1.9; ///
[eta1*5.5 eta2*5]; eta1*2 eta2*1;)

18. Example ACTIVE cognitive intervention trial pilot study
169 cognitively normal community-dwelling older adults
Dependent variable: learning trials from AVLT

19. Note the scaling of time. Consider the linear change model (2)
Note the scaling of time. Consider the linear change model (2). Since the time scale is arbitrary, time steps are used so that the final time step is 1. This way, the mean of the learning factor can be interpreted as the number of words gained at the final trial, and regressions of the learning trial on covariates can be interpreted as differences in the number of words recalled at the final trial.

20. Table 2. Summary of curve fit model building for AVLT initial recall and learning. ACTIVE pilot sample (N=169). PR
Model c2 df p
AIC
RMSEA
CFI
(1)
Intercept only 13
<.001
0.867
0.581
0.000
(2)
Linear change 10
0.667
0.340
0.733
(3)
Quadratic change
9.185 6
0.163
0.400
0.056
0.996
(4)
Quadratic change - orthogonal
9.409
0.152
0.058
0.995
(5)
Logarithmic change
21.686
0.168
0.083
0.984
(6)
Approximately logarithmic change
9.593 9
0.384
0.600
0.020
0.999
(7)
Time metric freely estimated
5.919 7
0.549
0.467
1.000
Abbreviations: AVLT, Rey Auditory Verbal Learning Test; df, degrees of freedom; AIC, Akaike Information Criteria; RMSEA, Root mean square error of approximation; CFI, Confirmatory Fit Index; Var, variance; PR, Parsimony ratio, defined as df/[.5k(k+1)] where k is the number of observed variables.

22. Example 2: General Growth Mixture Modeling (GGMM)
The Course of Pain Following Fall or Fracture Among Residents of Long-Term Care
RN Jones, A Won
Supported by Harvard Older Americans Independence Center (2P60AG008812),

23. Study Objectives
Examine course of new-onset pain following fall or fracture
Identify sub-populations with qualitatively distinct pain trajectories
Examine the relationship of prescribing patterns on course of pain
Methods
Convenience sample of LTC Residents
All subject characteristics, including fall/fracture and pain from Minimum Data Set (MDS) assessments
Prospectively collected follow-up data, up to 1 year, at two month intervals
Chart review for analgesic administration data

24. Nursing Home Resident Sample
Admitted (availability of data)
Admitted pain free, no recent history of falls, no recent history of fracture
Remained fall/fracture free for at least 4 months
Experienced co-incident fall or fracture and pain during stay
Outcome and Exposure Measures
MDS Pain scale (Fries et al., 2001)
Analgesic (in previous week)
Any non-opioid
Any opioid
Total morphine equivalent for the previous week

25. General Growth Mixture Modeling
Latent growth model part
Captures individual differences in
Intercept
Change over time
Latent class model part
Identify population sub-groups with qualitatively different growth model parameters

26. Analytic Sample 129 Residents matching eligibility criteria identified
17 fractures
123 falls (non-exclusive groups)
727 follow-up assessments

27. General Growth Mixture Model (GGMM) of Pain Trajectories

28. Discussion Advantages Disadvantages
Regressions among random coefficients (growth factors)
Multiple Processes
Multiple Indicators
IRT model for outcome dimension
Growth model for latent trait
Model DIF item difficulty shifts
Multiple Populations
Embed within larger SEM
LGM with discrete time survival model to simultaneously model change and survival
‘Easy’ inclusion of time-varying covariates
Growth Mixture Modeling
‘Simple’ power calculation
Maximum likelihood estimation with missing data
Adjustment for complex sampling design
Modeling with a preponderance of 0’s and/or censored data
Disadvantages
Individually-varying times of observation (Mplus versus other SEM software)

29. Multiple Indicator LGM with Covariate

30. Multiple-Level LGM

31. Parallel Process LGM

32. Multiple Group, Multi-Level LGM To Evaluate a Cluster-Randomized Controlled Trial