Latent Growth Modeling

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# Latent Growth Modeling - PowerPoint PPT Presentation

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

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## Latent Growth Modeling

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### Latent Growth Modeling

Chongming Yang

Research Support Center

FHSS College

Objectives
• Understand the basics of LGM
• Obtain some hands-on experience
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
Recent Approaches Individual changes
• Multilevel/Mixed /HL modeling
• Generalized Estimating Equations (GEE)
• Structural equation modeling (latent growth (curve) modeling)
Run Linear Regression for each case
• yit = i + iT + it
• i = individual
• T = time variable
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

Unconditional Growth Model--Growth Model without Covariates

yt =  + T + t

= i + i (i = intercept here)

= s + s

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.

Limitations of Multilevel/Mixed Modeling
• No latent variables
• Growth pattern has to be specified
• No indirect effect
• No time-variant covariates
Specific Measurement Models
• y1= 1 + 1 + 1
• y2= 2 + 2 + 2
• y3= 3 + 3 + 3
• y4= 4 + 4 + 4

 = i+ i

 = s+ s

Unconditional Latent Growth Model

y =  +  +   y = 0 + 1*i + s + 

Five Parameters to Interpret
• Mean & Variance of Intercept Factor (2)
• Mean & Variance of Slope Factor (2)
• Covariance /correlation between Intercept and Slope factors (1)
Interchangeable Concepts
• Intercept = initial level = overall level
• Slope = trajectory = trend = change rate
• Linear:

Time Scores = 0, 1, 2, 3 … (0, 1, 2.5, 3.5…)

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…

Control Group 

Experimental Group 

Cohort 1

Cohort 2

Cohort 3

Continuous Indicators

Original Rating 0-4

Categorical Indicators

Dummy- Coding 0-1

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
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?
• 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
Model Specification growth of observed variable

ANALYSIS:

MODEL:

I S | y1@0 y2@1 y3 y4 ;

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 ;

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

Practical Tip
• Specify a growth trajectory pattern to ensure the model runs
• Examine sample and model estimated trajectories to determine the best pattern
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
Unstandardized or StandardizedEstimates?
• Report unstandardized If the growth in observed variable is modeled,
• If latent construct measured with indicators are , report standardized
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
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