Growth Curve Model Using SEM

David A. Kenny

December 15, 2013

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.

0

5

10

15

20

25

30

0 2 4 6 8 10

Time

Outcome

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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.

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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.”

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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

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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.

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

m, v1

T1

m, v2

T2

m, v3

T3

m, v4

T4

m, v5

T5

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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

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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).

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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

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Different Choices Result In• Same

– model fit (2 or RMSEA)

– slope mean and variance – error variances

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

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0

2

4

6

8

10

12

14

16

18

1 2 3 4 5 6

Time

Ou

tco

me

no intercept variance

intercept variance, with slope and intercept being

negatively correlated

some intercept variance, and

slope and intercept being positively

correlated

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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

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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

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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.

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Intercept Factor

PeerAlcohol Use

Intercept

0

P1

0

P2

0

P3

0,

err2

0,

err31

0,

err11

1

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Intercept Factor with Loadings

PeerAlcohol Use

Intercept

0

P1

0

P2

0

P3

1

0,

err2

0,

err31

0,

err11

1

1

1

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Slope Factor

PeerAlcohol Use

Intercept

PeerAlcohol Use

Slope

0

P1

0

P2

0

P3

1

0,

err2

0,

err31

0,

err11

1

1

1

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Slope Factor with Loadings

PeerAlcohol Use

Intercept

PeerAlcohol Use

Slope

0

P1

0

P2

0

P3

1

1

2

0,

err2

0,

err31

0,

err11

1

0

1

1

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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)

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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.

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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