1. Growth Curve Models (being revised)
Thanks due to Betsy McCoach
David A. Kenny
August 26, 2011

2. Overview Introduction Estimation of the Basic Model Nonlinear Effects
Exogenous Variables
Multivariate Growth Models

3. Not Discussed or Briefly Discussed
Modeling Nonlinearity
LDS Model
Time-varying Covariates
Point of Minimal Intercept Variance
Complex Nonlinear Models
(see extra slides at the end)

4. 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 (last week)
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.

5. 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.

6. 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.

7. 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.”

8. 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

9. 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.

10. Independence Model Default model in Amos is wrong!
No correlations, free variances, and equal means.
df of T(T + 1)/2 – 1

11. 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

12. 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).

13. 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

14. 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

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

16. 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

17. 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

18. 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.

19. 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

20. Intercept Factor

21. Intercept Factor with Loadings

22. Slope Factor

23. Slope Factor with Loadings

24. Estimates

25. Parameter Estimates Estimate SE CR MEANS Intercept 1.304 .091 14.395
Slope
VARIANCES
Intercept
Slope
Error
Error
Error
COVARIANCE*
Intercept-Slope
*Correlation = -.378

26. Interpretation Mean Variance
Intercept: The average person starts at
Slope: The average rate of change per year is .555 units.
Variance
Intercept
+1 sd = = 2.86
-1 sd = – 1.56 =
Slope
+1 sd = =1.19
-1 sd = .56 – .63 = -0.07
% positive slopes P(Z > -.555/.634) = .80

27. Model Fit c2(1) = 4.98, p = .026 RMSEA = .105
CFI = ( – 5 – )/ ( – 5)
= .991
Conclusion: Good fitting model. (Remember that the RMSEA with small df can be misleading.)

28. Nonlinearity Latent Basis Model: Some Loadings Free
Fix the loadings for two waves of data to different nonzero values and free the other loadings.
In essence rescales time.
Slope
Intercept 1 ? 2

29. 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.

30. 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.

31. 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.

32. Second Example
Ormel, J., & Schaufeli, W. B. (1991). Stability and change in psychological distress and their relationship with self-esteem and locus of control: A dynamic equilibrium model. Journal of Personality and Social Psychology, 60,
389 Dutch Adults after College Graduation
5 Waves Every Six Months
Distress Measure

33. Distress at Five Times

34. Parameter Estimates Estimate SE CR MEANS Intercept 3.276 .156 20.946
Slope
VARIANCES
Intercept
Slope
All error variances statistically significant
COVARIANCE*
Intercept-Slope
*Correlation = -.433

35. 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.

36. Model Fit c2(10) = 110.37, p < .001 RMSEA = .161
CFI = ( – 14 – )/ ( – 14)
= .886
Conclusion: Poor fitting model.

37. Alternative Options for Error Variances
Force error variances to be equal across time.
c2(4) = (not helpful)
Non-independent errors
errors of adjacent waves correlated
c2(4) = (not much help)
autoregressive errors (err1  err2  err3)
c2(4) = (not much help)

38. 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.

39. 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.

40. Three exogenous person variables predict the slope and the intercept (own drinking)

41. Effects of Exogenous Variables
Variable Intercept Slope
Age *
Gender *
COA * R
c2(4) = 4.9
Intercept: Older children start out higher.
Slope: More change for Boys and Children of Alcoholics.
(Trimming ok here.)

42. 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

43. 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

44. 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.

45. 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.

46. Time Varying Covariates

47. Results Main effects model Interaction model
Changes the intercept at each time.
Covariate acts like a step function.

48. Covariate by Time Interaction
Covariate by Time (linear), Phantom variable approach

49. Partner Drinking as a Time-varying Covariate: V1 and V2 Are Latent Variables with No Disturbance (Phantom Variables)

50. Results Main Effect of Peer: 0.376 (p = .038)
Time x Peer: (p = .427)
The effect of Peer increases over time, but not significantly.

51. Multivariate Growth Curve Model

52. Example Basic Model: c2(4) = 8.18 Same Factors: c2(13) = 326.30
Correlations
Intercepts: .81
Slopes: .67
Same Factors: c2(13) =
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

53. 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

54. 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.

55. 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

56. Bilinear or Piecewise Model
Inflection point
Two slope factors

57. 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.

58. Option 2: Rate & Deflection Option 1: Two Rates
Slope1
Slope2
Int 1 2
Slope1
Slope2
Int 1 2 3 4

59. Piecewise Bilinear Model

60. Results Bilinear: c2(6) = 102.91, p < .001
RMSEA = .204
Piecewise: c2(6) = , p < .001
Conclusion: No real improvement of fit for these two different but equivalent methods

61. Step Function: Change in Intercept
Level jumps at some point (e.g., point of intervention)
Two intercept factors
Slope
Int1
Int2 1 2 3 4
Note Int2 measures the size of intervention effect for each person.

62. Results Change in intercept Conclusion: No real improvement of fit
RMSEA = .199
Conclusion: No real improvement of fit

63. Modeling Nonlinearity
Quadratic Effects
Seasonal Effects
Empirically based slopes of any form.

64. 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) = ; RMSEA = .199

65. Modeling Seasonal Effects
Note the alternating positive and negative coefficients for the slope

66. Results
c2(6) = , p < .001
RMSEA = .120
No evidence of Slope Variance (actually estimated as negative!)
Conclusion: Fit better, but still poor.

67. Empirically Estimated Scaling of Time
Allows for any possible growth model.
Fix one slope loading (usually one).
No intercept factor.

68. Results Curvilinear Trend Wave 1: 1.00 Wave 2: 0.74 Wave 3: 0.95
Better Fit, But Not Good Fit
c2(9) = 62.5, p < .001

69. 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

70. LDS Model

71. 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.

72. 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

73. More Complex Nonlinear Growth
Sinusoid
Nonzero first and second order derivative
Pendulum
dampening

74. 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

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