1 / 44

Latent Growth Modeling

Latent Growth Modeling. Chongming Yang Research Support Center FHSS College. Objectives. Understand the basics of LGM Learn about some applications Obtain some hands-on experience. Limitations of Traditional Repeated ANOVA / MANOVA / GLM. Concern group-mean changes over time

keira
Download Presentation

Latent Growth Modeling

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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…

  21. Parallel Growths

  22. Cross-lagged Model

  23. Parallel Growth with Covariates

  24. Antecedent and Subsequent (Sequential) Processes

  25. Control Group  Experimental Group 

  26. Cohort 1 Cohort 2 Cohort 3

  27. Piecewise Growth Model Slope2 Slope1

  28. 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

  29. 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

  30. Growth Mixtures

  31. 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?

  32. 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

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

  34. 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 ;

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

  36. 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 ;

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

  38. 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

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

  40. 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

  41. 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

More Related