1. Growth Curve Model Using SEM
David A. Kenny

2. Thanks due to Betsy McCoach

3. Linear Growth Curve Models
We have at least three time points for each individual.
We fit a straight line for each person:
The parameters from these lines describe the person.
Nonlinear growth models are possible.

4. The Key Parameters Slope: the rate of change
Some people are changing more than others and so have larger slopes.
Some people are improving or growing (positive slopes).
Some are declining (negative slopes).
Some are not changing (zero slopes).
Intercept: where the person starts
Error: How far the score is from the line.

5. Latent Growth Models (LGM)
For both the slope and intercept there is a mean and a variance.
Mean
Intercept: Where does the average person start?
Slope: What is the average rate of change?
Variance
Intercept: How much do individuals differ in where they start?
Slope: How much do individuals differ in their rates of change: “Different slopes for different folks.”

6. Measurement Over Time measures taken over time
chronological time: 2006, 2007, 2008
personal time: 5 years old, 6, and 7
missing data not problematic
person fails to show up at age 6
unequal spacing of observations not problematic
measures at 2000, 2001, 2002, and 2006

7. Data Types Raw data Covariance matrix plus means
Means become knowns: T(T + 3)/2
Should not use CFI and TLI (unless the independence model is recomputed; zero correlations, free variances, means equal)
Program reproduces variances, covariances (correlations), and means.

8. Independence Model in SEM
No correlations, free variances, and equal means.
df of T(T + 1)/2 – 1

9. Specification: Two Latent Variables
Latent intercept factor and latent slope factor
Slope and intercept factors are correlated.
Error variances are estimated with a zero intercept.
Intercept factor
free mean and variance
all measures have loadings set to one

10. Slope Factor free mean and variance
loadings define the meaning of time
Standard specification (given equal spacing)
time 1 is given a loading of 0
time 2 a loading of 1
and so on
A one unit difference defines the unit of time. So if days are measured, we could have time be in days (0 for day 1 and 1 for day 2), weeks (1/7 for day 2), months (1/30) or years (1/365).

11. Time Zero Where the slope has a zero loading defines time zero.
At time zero, the intercept is defined.
Rescaling of time:
0 loading at time 1 ─ centered at initial status
standard approach
0 loading at the last wave ─ centered at final status
useful in intervention studies
0 loading in the middle wave ─ centered in the middle of data collection
intercept like the mean of observations

12. Different Choices Result In
Same
model fit (c2 or RMSEA)
slope mean and variance
error variances
Different
mean and variance for the intercept
slope-intercept covariance

13. some intercept variance, and slope and intercept being positively correlated
no intercept variance
intercept variance, with slope and intercept being negatively correlated

14. Identification Need at least three waves (T = 3)
Need more waves for more complicated models
Knowns = number of variances, covariances, and means or T(T + 3)/2
So for 4 times there are 4 variances, 6 covariances, and 4 means = 14
Unknowns
2 variances, one for slope and one for intercept
2 means, one for the slope and one for the intercept
T error variances
1 slope-intercept covariance

15. Model df Known minus unknowns General formula: T(T + 3)/2 – T – 5
Specific applications
If T = 3, df = 9 – 8 = 1
If T = 4, df = 14 – 9 = 5
If T = 5, df = 20 – 10 = 10

16. i.e., the means have a linear relationship with respect to time.
Three-wave Model
Has one df.
The over-identifying restriction is:
M1 + M3 – 2M2 = 0
(where “M” is mean)
i.e., the means have a linear relationship with respect to time.

17. Intercept Factor

18. Intercept Factor with Loadings

19. Slope Factor

20. Slope Factor with Loadings

21. Alternative Options for Error Variances
Force error variances to be equal across time.
Non-independent errors
errors of adjacent waves correlated
autoregressive errors (err1  err2  err3)

22. Trimming Growth Curve Models
Almost never trim
Slope-intercept covariance
Intercept variance
Never have the intercept “cause” the slope factor or vice versa.
Slope variance: OK to trim, i.e., set to zero.
If trimmed set slope-intercept covariance to zero.
Do not interpret standardized estimates except the slope-intercept correlation.

23. Relationship to Multilevel Modeling (MLM)
Equivalent if ML option is chosen
Advantages of SEM
Measures of absolute fit
Easier to respecify; more options for respecification
More flexibility in the error covariance structure
Easier to specify changes in slope loadings over time
Allows latent covariates
Allows missing data in covariates
Advantages of MLM
Better with time-unstructured data
Easier with many times
Better with fewer participants
Easier with time-varying covariates
Random effects of time-varying covariates allowable