30
Outcome
25
20
15
10
5
0
0
2
4
6
8
10
Time
Growth Curve Models
Thanks due to Betsy McCoach
David A. Kenny
August 26, 2011
Overview
•
•
•
•
•
Introduction
Estimation of the Basic Model
Nonlinear Effects
Exogenous Variables
Multivariate Growth Models
2
Not Discussed or Briefly Discussed
(see extra slides at the end)
•
•
•
•
•
Modeling Nonlinearity
LDS Model
Time-varying Covariates
Point of Minimal Intercept Variance
Complex Nonlinear Models
3
Two Basic Change Models
• Stochastic
– I am like how I was, but I change randomly.
– These random “shocks” are incorporated into
who I am.
– Autoregressive models
• Growth Curve Models
– Each of us in a definite track.
– We may be knocked off that track, but eventually
we end up “back on track.”
– Individuals are on different tracks.
4
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.
5
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.
6
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.”
7
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
8
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.
9
Independence Model
• Default model in Amos is wrong!
• 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
10
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
11
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).
12
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
13
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
14
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
15
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
16
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
17
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.
18
Example Data
•
•
•
•
•
Curran, P. J. (2000)
Adolescents, ages 10.5 to 15.5 at Time 1
3 times, separated by a year
N = 363
Measure
– Perceived peer alcohol use
– 0 to 7 scale, composite of 4 items
19
Intercept Factor
0
0,
1
P1
err1
Peer
Alcohol Use
Intercept
0
0,
1
P2
err2
0
0,
1
err3
P3
20
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
21
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
22
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
23
Estimates
0
0, .60
1.30, 2.42
1
1.00
P1
err1
Peer
Alcohol Use
Intercept
1.00
0
0, 1.24
1
1.00
-.37
P2
err2
.00
1.00
0
0, 1.49
1
err3
.56, .40
Peer
Alcohol Use
Slope
2.00
P3
24
Parameter Estimates
Estimate SE
MEANS
Intercept
Slope
VARIANCES
Intercept
Slope
Error1
Error2
Error3
COVARIANCE*
Intercept-Slope
*Correlation = -.378
CR
1.304 .091 14.395
0.555 .050 11.155
2.424
0.403
0.596
1.236
1.492
.300
.132
.244
.143
.291
8.074
3.051
2.441
8.670
5.132
-0.374 .163
-2.297
25
Interpretation
• Mean
– Intercept: The average person starts at 1.304.
– Slope: The average rate of change per year is .555
units.
• Variance
– Intercept
• +1 sd = 1.30 + 1.56 = 2.86
• -1 sd = 1.30 – 1.56 = -0.26
– Slope
• +1 sd = .56 + .63 =1.19
• -1 sd = .56 – .63 = -0.07
• % positive slopes P(Z > -.555/.634) = .80
26
Model Fit
c2(1) = 4.98, p = .026
RMSEA = .105
CFI = (442.49 – 5 – 4.98 + 1)/ (442.49 – 5)
= .991
Conclusion: Good fitting model. (Note that
the RMSEA with small df can be
misleading.)
27
Nonlinearity
Latent Basis Model: Some Loadings Free
Fix the loadings for two waves of data to
different nonzero values and free the other
loadings.
Slope
Intercept
0
1
?
1
2
1
In essence
rescales time.
28
Results for Alcohol Data
Wave 1: 0.00
Wave 2: 0.84
Wave 3: 2.00
Function fairly linear as 0.84 is close to 1.00.
29
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.
30
Using Amos
• Must tell the Amos to “Estimate means and
intercepts.”
• Growth curve plug-in
• It names parameters, sets measures’
intercepts to zero, frees slope and intercept
factors’ means and variance, sets error
variance equal over time, fixes intercept
loadings to 1, and fixes slope loadings from
0 to 1.
31
Second Example
• Ormel, J., & Schaufeli, W. B. (1991).
Stability and change in psychological
distress and their relationship with selfesteem and locus of control: A dynamic
equilibrium model. Journal of Personality
and Social Psychology, 60, 288-299.
• 389 Dutch Adults after College Graduation
• 5 Waves Every Six Months
• Distress Measure
32
Distress at Five Times
-.04, .17
0
0, 5.32
1
.00
PD11
err11
Slope
1.00
0, 4.85
1
err21
0
2.00
PD21
3.00
0, 3.53
1
err31
0, 3.42
1
err41
0, 3.68
1
err51
0
4.00
-.46
PD31
0
PD41
1.00 1.00
1.00
0
3.28, 6.56
1.00
Intercept
PD51
1.00
33
Parameter Estimates
Estimate SE
CR
MEANS
Intercept
3.276 .156 20.946
Slope
-0.043 .040 -1.079
VARIANCES
Intercept
6.558 .707 9.272
Slope
0.170 .052 3.250
All error variances statistically significant
COVARIANCE*
Intercept-Slope
-0.458 .156 -2.926
*Correlation = -.433
34
Interpretation
Large variance in distress level.
Average slope is essentially zero.
Variance in slope so some are increasing in
distress and others are declining.
Those beginning at high levels of distress
decline over time.
35
Model Fit
c2(10) = 110.37, p < .001
RMSEA = .161
CFI = (895.35 – 14 – 110.35 + 10)/ (895.35 – 14)
= .886
Conclusion: Poor fitting model.
36
Alternative Options for Error Variances
• Force error variances to be equal across
time.
– c2(4) = 19.1 (not helpful)
• Non-independent errors
– errors of adjacent waves correlated
• c2(4) = 10.4 (not much help)
– autoregressive errors (err1  err2  err3)
• c2(4) = 10.5 (not much help)
37
Exogenous Variables
• Often in this context referred to as
“covariates”
• Types
– Person – e.g., age and gender
– Time varying: a different measure at each time
• See “extra” slides.
• Need to center (i.e., remove their mean)
these variables.
– For time-varying use one common mean.
38
Person Covariates
• Center (failing the center makes average slope
and intercept difficult to interpret)
• These variables explain variation in slope and
intercept; have an R2.
• Have them cause slope and intercept factors.
– Intercept: If you score higher on the covariate, do you
start ahead or behind (assuming time 1 is time zero)?
– Slope: If you score higher on the covariate, do you
grow at a faster and slower rate.
• Slope and intercept now have intercepts not
means. Their disturbances are correlated.
39
Three exogenous person variables predict the
slope and the intercept (own drinking)
Adolescent
Alcohol Use
Intercept
1
Age
0
0,
1
E1
T1
0
0,
1
E2
1
1
1
U
T2
GD
0,
0
V
0
0,
1
1
COA
1
E3
T3
2
Adolescent
Alcohol Use
Slope
40
Effects of Exogenous Variables
Variable
Intercept
Slope
Age
.606*
.057
Gender
-.113
.527*
COA
.462
.705*
R2
.101
.054
c2(4) = 4.9
Intercept: Older children start out higher.
Slope: More change for Boys and Children of Alcoholics.
(Trimming ok here.)
41
Extra Slides
•
•
•
•
•
•
•
•
Relationship to multilevel models
Time varying covariates
Multivariate growth curve model
Point of minimal intercept variance
Other ways of modeling nonlinearity
Empirically scaling the effect of time
Latent difference scores
Non-linear dynamic models
42
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 43
Time-Varying Covariates
• A covariate for each time point.
• Center using time 1 mean (or the mean at
time zero.)
• Do not have the variable cause slope or
intercept.
• Main Effect
– Have each cause its measurement at its time.
– Set equal to get the main effect.
• Interaction: Allow the covariate to have a
different effect at each time.
44
Interpretation
• Main effects of the covariate.
– Path: .504 (p < .001)
– c2(3) = 8.44, RMSEA = .071
– Peer “affects” own drinking
• Covariate by Time interaction
– Chi square difference test: c2(2) = 4.24, p = .109
– No strong evidence that the effect of peer
changes over time.
45
Time Varying Covariates
Own
Intercept
1
0,
0
1
1
a1
F3
T1
P1
1
0
a2
T2
P2
0,
1
F2
0
0
a3
P3
1
T3
0,
1
F1
2
Own
Slope
46
Results
• Main effects model
• Interaction model
– Changes the intercept at each time.
– Covariate acts like a step function.
47
Covariate by Time Interaction
• Covariate by Time (linear), Phantom
variable approach
48
Partner Drinking as a Time-varying Covariate:
V1 and V2 Are Latent Variables with No
Disturbance (Phantom Variables)
0,
1
a
F3
T1
P1
0
V1
1
b
a
P2
0,
1
T2
F2
0
V2
b
P3
2
a
T3
0,
1
F1
49
Results
• Main Effect of Peer: 0.376 (p = .038)
• Time x Peer: 0.107 (p = .427)
• The effect of Peer increases over time, but not
significantly.
50
Multivariate Growth Curve Model
0,
0
1
E1
im, iv
1
P1
Peer
Intercept
1
0
im, iv
1
Own
Intercept
1
0
0
0,
F1
T1
1
1
1
0,
1
T2
P2
1
0,
0
E2
0
1
0
Peer
Slope
Own
Slope
0, 1
1
0,
0
2
2
P3
F2
sm, sv
1
sm, sv
T3
F3
E3
51
Example
• Basic Model: c2(4) = 8.18
– Correlations
• Intercepts: .81
• Slopes: .67
• Same Factors: c2(13) = 326.30
– One common slope and intercept for both variables.
– 9 less parameters:
• 5 covariances
• 2 means
• 2 covariances
• Much more variance for Own than for Peer
52
Point of Minimal Intercept Variance
• Concept
– The variance of intercept refers to variance in predicted scores a
time zero.
– If time zero is changed, the variance of the intercept changes.
– There is some time point that has minimal intercept variance.
• Possibilities
– Point is before time zero (negative value)
• Divergence or fan spread
• Increasing variance over time
– Point is after the last point in the study
• Convergence of fan close
• Decreasing variance over time
– Point is somewhere in the study
• Convergence and then divergence
• May wish to define time zero as this point
53
Computation
• Should be computed only if there is reliable
slope variance.
• Compute: sslope,intercept/sslope2
• Curran Example
-0.458/0.170 = 1.93
1.93, just before the last wave
Convergence and decreasing variability
Peer perceptions become more
homogeneous across time.
54
More Elaborate Nonlinear Growth Models
• Latent basis model
– fix the loadings for two waves of data
(typically the first and second waves or the
first and last waves) and free the other
loadings
• Bilinear or piecewise model
– inflection point
– two slope factors
• Step function
– level jumps at some point (e.g., treatment
effect)
– two intercept factors
55
Bilinear or Piecewise Model
• Inflection point
• Two slope factors
10
9
8
7
DV
6
5
4
3
2
1
0
1
2
3
4
Wave
5
6
56
Bilinear or Piecewise Model
• OPTION 1: 2 distinct growth rates
– One from T1 to T3
– The second from T3 to T5
• OPTION 2: Estimate a baseline growth plus
a deflection (change in trajectory)
– One constant growth rate from T1 to T5
– Deflection from the trajectory beginning at T3
• Two options are equivalent in term of model
fit.
57
Option 1: Two
Rates
Option 2: Rate &
Deflection
Slope1 Slope2 Int
Slope1 Slope2 Int
0
0
1
0
0
1
1
0
1
1
0
1
2
0
1
2
0
1
2
1
1
3
1
1
2
2
1
4
2
1
58
Piecewise Bilinear Model
0
0,
1
0
PD11
err11
Slope1
1
0
0,
1
err21
2
PD21
2
0
0,
1
err31
2
Slope2
PD31
1
0
0,
1
err41
1
PD41
1
err51
2
1
0
0,
1
1
Intercept
PD51
1
59
Results
• Bilinear: c2(6) = 102.91, p < .001
– RMSEA = .204
• Piecewise: c2(6) = 102.91, p < .001
– RMSEA = .204
• Conclusion: No real improvement of fit for
these two different but equivalent methods
60
Step Function: Change in Intercept
• Level jumps at some point (e.g., point of intervention)
• Two intercept factors
Slope Int1
Int2
0
0,
1
0
PD11
err11
Slope
1
0
1
0
0
0,
1
err21
2
PD21
3
1
1
0
0
0,
1
err31
4
Step
PD31
1
2
3
1
1
1
1
0
0,
1
err41
1
PD41
1
err51
1
1
0
0,
1
1
Intercept
PD51
1
4
1
1
Note Int2 measures the size of
intervention effect for each person. 61
Results
• Change in intercept
– c2(6) = 98.60
– RMSEA = .199
• Conclusion: No real improvement of fit
62
Modeling Nonlinearity
• Quadratic Effects
• Seasonal Effects
• Empirically based slopes of any form.
63
Add a Quadratic Factor
• Add a second (quadratic) slope factor (0, 1,
4, 9 …)
• Correlate with the other slope and intercept
factor.
• Adds parameters
– 1 mean
– 1 variance
– 2 covariances (with intercept and the other
slope)
• No real better fit for the Distress Example
– c2(6) = 98.59; RMSEA = .199
64
Modeling Seasonal Effects
• Note the alternating positive and negative
coefficients for the slope
0
0,
1
1
PD11
err11
Slope
-1
0
0,
1
err21
1
PD21
-1
0
0,
1
err31
PD31
0
0,
1
err41
1
PD41
1
1
1
0
0,
err51
1
1
Intercept
PD51
1
65
Results
 c2(6) = 65.41, p < .001
– RMSEA = .120
• No evidence of Slope Variance (actually
estimated as negative!)
• Conclusion: Fit better, but still poor.
66
Empirically Estimated
Scaling of Time
• Allows for any possible growth model.
• Fix one slope loading (usually one).
• No intercept factor.
0,
0,
PD11
PD21
err51
1
PD31
0
1
PD41
err41
0,
0
1
err31
0,
0
1
err21
0,
0
1
err11
0
1
Slope
PD51
67
Results
Curvilinear Trend
Wave 1: 1.00
Wave 2: 0.74
Wave 3: 0.95
Wave 4: 0.83
Wave 5: 0.87
Better Fit, But Not Good Fit
c2(9) = 62.5, p < .001
68
Latent Difference Score Models
•
•
•
•
Developed by Jack McArdle
Creates a difference score of each time
Uses SEM
Traditional linear growth curve models are
a special case
• Called LDS Models
69
LDS Model
Intercept
1
0
0
0,
1
1
E1
L1
T1
a
1
0
0,
0
1
1
T2
E2
0
1
D2-1
L2
a
1
0,
E3
T3
1
1
1
1
0
0
0
L3
D3-2
1
Slope
70
Relation to a Linear
Growth Curve Model
• The same if a = 0
• If a not equal to zero, the model can be
viewed as a blend of growth curve and
autoregressive models.
Adolescent
Alcohol Use
Intercept
0,
0
1
E1
0
1
T1
L1
1
1
1+a
0
0,
1
1
1
T2
E2
0
L2
1+a
0,
E3
0
1
T3
0
1
L3
1
1
Adolescent
Alcohol Use
Slope
71
Nonlinear Growth: Negative
Exponential
• One Unit Moving Through Time
• Constant Rate of Change (no error)
• The Force Pulling the Score to the Mean
Is a Constant
• The First Derivative Is a Constant
72
More Complex Nonlinear Growth
• Sinusoid
– Nonzero first and
second order
derivative
• Pendulum
– dampening
73
Estimation Using AR(2) Model
• Negative Exponential
1 > a1 > -1 (the rate of change) and a2 = 0
• Sinusoid
2 > a1 > 1 and a2 = -1
Cobb formula for period length = p/cos-1√a1
• Pendulum
dampening factor = 1 - a2
Cobb formula for period length = p/cos-1√a1
74
Go to the next SEM page.
Go to the main SEM page.
75