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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

Latent Growth Modeling

Chongming Yang

Research Support Center

FHSS College

objectives
Objectives
  • Understand the basics of LGM
  • Learn about some applications
  • Obtain some hands-on experience
limitations of traditional repeated anova manova glm
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
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
Run Linear Regression for each case
  • yit = i + iT + it
    • i = individual
    • T = time variable
model intercepts and slopes
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
Unconditional Growth Model--Growth Model without Covariates

yt =  + T + t

= i + i (i = intercept here)

= s + s

conditional growth model growth model with covariates
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
Limitations of Multilevel/Mixed Modeling
  • No latent variables
  • Growth pattern has to be specified
  • No indirect effect
  • No time-variant covariates
specific measurement models
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
Unconditional Latent Growth Model

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

five parameters to interpret
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
Interchangeable Concepts
  • Intercept = initial level = overall level
  • Slope = trajectory = trend = change rate
  • Time scores: factor loadings of the slope factor
growth pattern specification slope factor loadings
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…

slide28

Control Group 

Experimental Group 

slide29

Cohort 1

Cohort 2

Cohort 3

two part growth model for data with floor effect or lots of 0
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

mixture growth modeling
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
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?
advantage of sem approach
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
model specification growth of observed variable
Model Specification growth of observed variable

ANALYSIS:

MODEL:

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

specify growth model of factors with continuous indicators
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
Why fix intercepts at 0 ?
  • Y = 1 + F1
  • F1 = 2 + Intercept
  • Y = (1 = 2 =0) + Intercept
specify growth model of factors with categorical indicators
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
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
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 standardized estimates
Unstandardized or StandardizedEstimates?
  • Report unstandardized If the growth in observed variable is modeled,
  • If latent construct measured with indicators are , report standardized
resources
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
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