1. Latent Growth Curve Models
Patrick Sturgis,
Department of Sociology, University of Surrey

2. Overview Random effects as latent variables Growth parameters
Specifying time in LGC models
Linear Growth
Non-linear growth
Explaining Growth
Fixed and time-varying predictors
Benefits of SEM framework

3. SEM for Repeated Measures
The SEM framework can be used on repeated measured data to model individual growth trajectories.
For cross-sectional data latent variables are specified as a function of different items at the same time point.
For repeated measures data, latent variables are specified as a function of the same item at different time points.

4. A Single Latent Variable ModelLV
X11
X12
X13
X14 E1 1 E2 E3 E4
same item at 4 time points
4 different items
Constrain factor loadings
Estimate factor loadings
Estimate mean and variance of trajectory of change over time
Estimate mean and variance of underlying factor

5. Repeated Measures & Random Effects
We have average (or ‘fixed’) effects for the population as a whole
And individual variability (or ‘random’) effects around these average coefficients

6. Random Effects as Latent Variables
In LGC:
The mean of the latent variable is the fixed part of the model.
It indicates the average for the parameter in the population.
The variance of the latent variable is the random part of the model.
It indicates individual heterogeneity around the average.
Or inter-individual difference in intra-individual change.

7. Growth Parameters The earlier path diagram was an over-simplification.
In practice we require at least two latent variables to describe growth.
One to estimate the mean and variance of the intercept.
And one to estimate the mean and variance of the slope.

8. Specifying Time in LGC Models
In random effects models, time is included as an independent variable:
In LGC models, time is included via the factor loadings of the latent variables.
We constrain the factor loadings to take on particular values.
The number of latent variables and the values of the constrained loadings specify the shape of the trajectory.

9. A Linear Growth Curve Model
Constraining values of the intercept to 1 makes this parameter indicate initial status
Constraining values of the slope to 0,1,2,3 makes this parameter indicate linear change 1 1 2 3

10. Quadratic Growth Add additional latent variables with factorX1 X2 X3 X4
Add additional latent
variables with factor
loadings constrained
to powers of the linear
slope 1 1 2 3 9 4 1
ICEPT
SLOPE
QUAD

11. File structure for LGC
For random effect models, we use ‘long’ data file format.
There are as many rows as there are observations.
For LGC, we use ‘wide’ file formats.
Each case (e.g. respondent) has only one row in the data file.

12. A (made up) Example
We are interested in the development of knowledge of longitudinal data analysis.
We have measures of knowledge on individual students taken at 4 time points.
Test scores have a minimum value of zero and a maximum value of 25.
We specify linear growth.

13. Linear Growth Example mean=11.2 (1.4) p<0.0012 3
mean=11.2 (1.4) p<0.001
variance =4.1 (0.8) p<0.001
mean=1.3 (0.25) p<0.001
variance =0.6 (0.1) p<0.001

14. Interpretation
The average level of knowledge at time point one was 11.2
There was significant variation across respondents in this initial status.
On average, students increased their knowledge score by 1.2 units at each time point.
There was significant variation across respondents in this rate of growth.
Having established this descriptive picture, we will want to explain this variation.

15. Explaining Growth
Up to this point the models have been concerned only with describing growth.
These are unconditional LGC models.
We can add predictors of growth to explain why some people grow more quickly than others.
These are conditional LGC models.
This is equivalent to fitting an interaction between time and predictor variables in random effects models.

16. Time-Invariant Predictors
Does initial status influence rate of growth? 1 1 2 3
Do men have a different initial status than women?
Do men grow at a different rate than women?
Gender
(women = 0; men=1)

17. Time-varying predictors of growth

18. Why SEM?
Most of this kind of stuff could be done using random/fixed effects.
SEM has some specific advantages which might lead us to prefer it over potential alternatives:
SPSS linear mixed model
HLM
MlWin
Stata (RE, FE)

19. Fixed Effects/Unit Heterogeneity
A fixed effects specification removes ‘unit effects’
This controls for all observed and unobserved invariant unit characteristics
Highly desirable when one’s interest is in the effect of time varying variables on the outcome
This is done by allowing the random effect to be correlated with all observed covariates
Downside=no information about effect of time invariant variables, possible efficiency loss
SEM allows various hybrid models which fall between the classic random and fixed effect specifications

20. Random effect model b b b b

21. Fixed effect model b b b b

22. Hybrid model Remove equality constraint on beta weights
Allow correlated
errors Z
Introduce Time-Invariant
Covariate that has indirect
Effect on X

23. Multiple Indicator LGC Models
Single indicators assume concepts measured without error
Multiple indicators allow correction for systematic and random error
Reduced likelihood of Type II errors (failing to reject false null)
Tests for longitudinal meaning invariance
Allows modeling of measurement error covariance structure

24. Multiple Indicator LGC Models

25. Other Benefits of SEM Global tests and assessments of model fit
Full Information Maximum Likelihood for missing data
Decomposition of effects – total, direct and indirect
Probability weights
Complex sample data

26. Multiple Process Models
SLOPE1
ICEPT1
X1t1
X1t2
X1t3
X1t4 E1 1 E2 E3 E4 2 3
ICEPT2
SLOPE2
Y1t1
Y1t2
Y1t3
X1t4 E5 1 E6 E7 E8 2 3
Does rate of growth on one variable influence rate of growth on the other?
Does initial status on one variable influence development on the other?