Transcript Growth Curve Model - of David A. Kenny

Growth Curve Model
Using SEM
David A. Kenny
December 15, 2013
Thanks due to Betsy McCoach
Linear Growth Curve Models
• We have at least three time points for each
individual.
• We fit a straight line for each person:
30
Outcome
25
20
15
10
5
0
0
2
4
6
8
10
Time
• The parameters from these lines describe
the person.
• Nonlinear growth models are possible.
3
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.
4
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.”
5
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
6
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.
7
Independence Model in SEM
• No correlations, free variances, and equal means.
• df of T(T + 1)/2 – 1
m, v1
T1
T2
m, v4
m, v3
m, v2
T3
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
9
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
11
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
12
some intercept
variance, and
slope and intercept
being positively
correlated
18
16
14
no intercept
variance
Outcome
12
10
8
6
4
intercept variance,
with slope and
intercept being
negatively
correlated
2
0
1
2
3
4
5
6
Time
13
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
0
0,
1
P1
err1
Peer
Alcohol Use
Intercept
0
0,
1
P2
err2
0
0,
1
err3
P3
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Intercept Factor with Loadings
0
0,
1
1
P1
err1
Peer
Alcohol Use
Intercept
1
0
0,
1
1
P2
err2
0
0,
1
err3
P3
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Slope Factor
0
0,
1
1
P1
err1
Peer
Alcohol Use
Intercept
1
0
0,
1
1
P2
err2
0
0,
Peer
Alcohol Use
Slope
1
err3
P3
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Slope Factor with Loadings
0
0,
1
1
P1
err1
Peer
Alcohol Use
Intercept
1
0
0,
1
1
P2
err2
0
1
0
0,
2
1
err3
Peer
Alcohol Use
Slope
P3
20
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)
21
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 23