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Growth Curve Model Using SEM

David A. Kenny

Thanks due to Betsy McCoach

- 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.

- 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.

- 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.”

- Mean

- 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

- 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.

- No correlations, free variances, and equal means.
- df of T(T + 1)/2 – 1

- 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

- 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).

- 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

- 0 loading at time 1 ─ centered at initial status

- Same
- model fit (c2 or RMSEA)
- slope mean and variance
- error variances

- Different
- mean and variance for the intercept
- slope-intercept covariance

some intercept variance, and slope and intercept being positively correlated

no intercept variance

intercept variance, with slope and intercept being negatively correlated

- 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

- 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

- 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.

- Force error variances to be equal across time.
- Non-independent errors
- errors of adjacent waves correlated
- autoregressive errors (err1 err2 err3)

- 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.

- 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