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## Latent Growth Modeling

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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
- 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 + t3 + t
- i = i + i11 + i22 + i
- i = s + s11 + s22 + s

Note: i=individual, t = time, 1 and 2 = time-invariant covariates, 3 = time-variant covariate. i andI 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

Latent Growth Curve Modeling within SEM Framework

- Data—wide format

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
- Time scores: factor loadings of the slope factor

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…

Experimental Group

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

- 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?

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

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

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