# Growth Curve Models (being revised)

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Growth Curve Models (being revised)
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

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)

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.

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.

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.

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.”

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

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.

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

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

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).

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

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

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

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

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

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.

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

Intercept Factor

Slope Factor

Estimates

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

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

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.)

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

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.

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.

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.

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

Distress at Five Times

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

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.

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

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)

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.

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.

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

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.)

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

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

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.

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.

Time Varying Covariates

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

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

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

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

Multivariate Growth Curve Model

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

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

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.

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

Bilinear or Piecewise Model
Inflection point Two slope factors

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.

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

Piecewise Bilinear Model

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

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.

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

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

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

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

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

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

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

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

LDS Model

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.

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

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

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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