Latent Growth Modeling

1. Latent Growth Modeling Chongming Yang Research Support Center FHSS College

2. Objectives • Understand the basics of LGM • Learn about some applications • Obtain some hands-on experience

3. Limitations of Traditional Repeated ANOVA / MANOVA / GLM • Concern group-mean changes over time • Variances of changes not explicit parameters • List-wise deletion of cases with missing values • Can’t incorporate time-variant covariate

4. Recent Approaches Individual changes • Multilevel/Mixed /HL modeling • Generalized Estimating Equations (GEE) • Structural equation modeling (latent growth (curve) modeling)

5. Long Format Data Layout—Trajectory(T)(for Multilevel Modeling)

6. Run Linear Regression for each case • yit = i + iT + it • i = individual • T = time variable

7. Intercept & Slope

8. Individual Level SummaryLinear Regression

9. Model Intercepts and Slopes = i+ i= s + s IF variance of i = 0, Then = i , starting the same IF variance of s = 0, Then = s, changing the same Thus variances of iand s are important parameters

10. Unconditional Growth Model--Growth Model without Covariates yt =  + T + t = i + i (i = intercept here) = s + s

11. Estimating Different Trajectories

12. Conditional Growth Model--Growth Model with Covariates • yt = i + iT + t3 + t • i = i + i11 + i22 + i • i = s + s11 + s22 + s Note: i=individual, t = time, 1 and 2 = time-invariant covariates, 3 = time-variant covariate. i andI arefunctions of 1,2…n,yit is also a function of 3i.

13. Limitations of Multilevel/Mixed Modeling • No latent variables • Growth pattern has to be specified • No indirect effect • No time-variant covariates

14. Latent Growth Curve Modeling within SEM Framework • Data—wide format

15. Measurement Model of Y y =  +  + 

16. Specific Measurement Models • y1= 1 + 1 + 1 • y2= 2 + 2 + 2 • y3= 3 + 3 + 3 • y4= 4 + 4 + 4  = i+ i  = s+ s

17. Unconditional Latent Growth Model y =  +  +   y = 0 + 1*i + s + 

18. Five Parameters to Interpret • Mean & Variance of Intercept Factor (2) • Mean & Variance of Slope Factor (2) • Covariance /correlation between Intercept and Slope factors (1)

19. Interchangeable Concepts • Intercept = initial level = overall level • Slope = trajectory = trend = change rate • Time scores: factor loadings of the slope factor

20. Growth Pattern Specification(slope-factor loadings) • Linear: Time Scores = 0, 1, 2, 3 … (0, 1, 2.5, 3.5…) • Quadratic: Time Scores = 0, .1, .4, .9, 1.6 • Logarithmic: Time Scores = 0, 0.69, 1.10, 1.39… • Exponential: Time Scores = 0, .172, .639, 1.909, • To be freely estimated: Time Scores = 0, 1, blank, blank…

23. Parallel Growths

24. Cross-lagged Model

25. Parallel Growth with Covariates

26. Antecedent and Subsequent (Sequential) Processes

28. Control Group  Experimental Group 

29. Cohort 1 Cohort 2 Cohort 3

30. Piecewise Growth Model Slope2 Slope1

31. Two-part Growth Model(for data with floor effect or lots of 0) Continuous Indicators Original Rating 0-4 Categorical Indicators Dummy- Coding 0-1

32. Mixture Growth Modeling • Heterogeneous subgroups in one sample • Each subgroup has a unique growth pattern • Differences in means of intercept and slopes are maximized across subgroups • Within-class variances of intercept and slopes are minimized and typically held constant across all subgroups • Covariance of intercept and slope equal or different across groups

33. Growth Mixtures

34. T-scores approach • Use a variable that is different from the one that indicates measurement time to examine individual changes • Example • Sample varies in age • Measurement was collected over time • Research question: How measurement changes with age?

35. Advantage of SEM Approach • Flexible curve shape via estimation • Multiple processes • Indirect effects • Time-variant and invariant covariates • Model indirect effects • Model growth of latent constructs • Multiple group analysis and test of parameter equivalence • Identify heterogeneous subgroups with unique trajectories

36. Model Specification growth of observed variable ANALYSIS: MODEL: I S | y1@0 y2@1 y3 y4 ;

37. Specify Growth Model of Factorswith Continuous Indicators MODEL: F1 BY y11 y12(1) y13(2); F2 BY y21 y22(1) y23(2); F3 BY y31 y32(1) y33(2); (invariant measurement over time) [Y11-Y13@0 Y21-Y23@0 Y31-Y33@0 F1-F3@0]; (intercepts fixed at 0) I S | F1@0 F2@1 F3 F4 ;

38. Why fix intercepts at 0 ? • Y = 1 + F1 • F1 = 2 + Intercept • Y = (1 = 2 =0) + Intercept

39. Specify Growth Model of Factorswith Categorical Indicators MODEL: F1 BY y11 y12(1) y13(2); F2 BY y21 y22(1) y23(2); F3 BY y31 y32(1) y33(2); [Y11$1-Y13$1](3); [Y21$1-Y23$1](4); [Y31$1-Y33$1](5); (equal thresholds) [F1-F3@0]; (intercepts fixed at 0) [I@0]; (initial mean fixed 0, because no objective measurement for I) I S | F1@0 F2@1 F3 F4 ;

40. Practical Tip • Specify a growth trajectory pattern to ensure the model runs • Examine sample and model estimated trajectories to determine the best pattern

41. Practical Issues • Two measurement—ANCOVA or LGCM with variances of intercept and slope factors fixed at 0 • Three just identified growth (specify trajectory) • Four measurements are recommended for flexibility in • Test invariance of measurement over time when estimating growth of factors • Mean of Intercept factor needs to be fixed at zero when estimating growth of factors with categorical indicators • Thresholds of categorical indicators need to be constrained to be equal over time

42. Unstandardized or StandardizedEstimates? • Report unstandardized If the growth in observed variable is modeled, • If latent construct measured with indicators are , report standardized

43. Resources • Bollen K. A., & Curren, P. J. (2006). Latent curve models: A structural equation perspective. John Wiley & Sons: Hoboken, New Jersey • Duncan, T. E., Duncan, S. C., Strycker, L. A., Li, F., & Alpert A. (1999). An introduction to latent variable growth curve modeling: Concepts, issues, and applications. Lawrence Erlbaum Associates, Publishers: Mahwah, New Jersey • www.statmodel.com Search under paper and discussion for papers and answers to problems

44. Practice • Estimate an unconditional growth model • Compare various trajectories, linear, curve, or unknown to determine which growth model fit the data best • Incorporate covariates • Use sex or race as grouping variable and test if the two groups have similar slopes. • Explore mixture growth modeling