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

Growth Curve Models (being revised)

Thanks due to Betsy McCoach

David A. KennyAugust 26, 2011

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

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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)

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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.

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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.

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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• Default model in Amos is wrong!• 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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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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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

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Intercept Factor

PeerAlcohol Use

Intercept

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P1

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P2

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P3

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err2

0,

err31

0,

err11

1

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Intercept Factor with Loadings

PeerAlcohol Use

Intercept

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P1

0

P2

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P3

1

0,

err2

0,

err31

0,

err11

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1

1

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Slope Factor

PeerAlcohol Use

Intercept

PeerAlcohol Use

Slope

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P1

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P2

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P3

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err2

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err31

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err11

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Slope Factor with Loadings

PeerAlcohol Use

Intercept

PeerAlcohol Use

Slope

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P1

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P2

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P3

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err2

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err31

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err11

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Estimates1.30, 2.42

PeerAlcohol Use

Intercept

.56, .40

PeerAlcohol Use

Slope

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P1

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P2

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P3

1.00

1.00

2.00

0, 1.24

err2

0, 1.49

err31

-.37

0, .60

err11

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.00

1.00

1.00

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Parameter Estimates

Estimate SE CRMEANS

Intercept 1.304 .091 14.395Slope 0.555 .050 11.155

VARIANCESIntercept 2.424 .300 8.074 Slope 0.403 .132 3.051Error1 0.596 .244 2.441Error2 1.236 .143 8.670Error3 1.492 .291 5.132

COVARIANCE*Intercept-Slope -0.374 .163 -2.297

*Correlation = -.378

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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

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Model Fit2(1) = 4.98, p = .026

RMSEA = .105

CFI = (442.49 – 5 – 4.98 + 1)/ (442.49 – 5) = .991

Conclusion: Good fitting model. (Remember that the RMSEA with small df

can be misleading.)

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NonlinearityLatent 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.

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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.

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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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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.

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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, 288-299.

• 389 Dutch Adults after College Graduation• 5 Waves Every Six Months• Distress Measure

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Distress at Five Times-.04, .17

Slope

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PD11

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err111

3.28, 6.56

Intercept

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PD21

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err211

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PD31

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err311

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PD41

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err411

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PD51

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err511

-.46

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Parameter Estimates

Estimate SE CRMEANS

Intercept 3.276 .156 20.946Slope -0.043 .040 -1.079

VARIANCESIntercept 6.558 .707 9.272 Slope 0.170 .052 3.250

All error variances statistically significantCOVARIANCE*

Intercept-Slope -0.458 .156 -2.926

*Correlation = -.433

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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.

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Model Fit2(10) = 110.37, p < .001

RMSEA = .161

CFI = (895.35 – 14 – 110.35 + 10)/ (895.35 – 14)

= .886

Conclusion: Poor fitting model.

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Alternative Options for Error Variances

• Force error variances to be equal across time.

– 2(4) = 19.1 (not helpful)

• Non-independent errors

– errors of adjacent waves correlated

• 2(4) = 10.4 (not much help)

– autoregressive errors (err1 err2 err3)

• 2(4) = 10.5 (not much help)

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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.

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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.

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

AdolescentAlcohol Use

Intercept

AdolescentAlcohol Use

Slope

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T1

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T2

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T3

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E2

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E31

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E11

Age

GD

COA

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V

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Effects of Exogenous Variables

Variable Intercept Slope

Age .606* .057

Gender -.113 .527*

COA .462 .705*

R2 .101 .054

2(4) = 4.9

Intercept: Older children start out higher.

Slope: More change for Boys and Children of Alcoholics.

(Trimming ok here.)

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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

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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

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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.

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Interpretation• Main effects of the covariate.

– Path: .504 (p < .001)– 2(3) = 8.44, RMSEA = .071– Peer “affects” own drinking

• Covariate by Time interaction– Chi square difference test: 2(2) = 4.24, p = .109– No strong evidence that the effect of peer

changes over time.

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Time Varying Covariates

P1

P2

P3

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T2

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T1

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F11

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F21

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F31

OwnIntercept

OwnSlope

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a1

a2

a3

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Results• Main effects model

• Interaction model– Changes the intercept at each time. – Covariate acts like a step function.

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Covariate by Time Interaction• Covariate by Time (linear), Phantom

variable approach

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

Disturbance (Phantom Variables)

P1

P2

P3T3

T2

T1

0,

F11

0,

F21

0,

F31

a

a

a

0

V1

0

V2

1

2

b

b

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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.

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Multivariate Growth Curve Model

im, iv

PeerIntercept

sm, sv

PeerSlope

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P1

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P2

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P3

01

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E2

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E3

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0,

E11

1

0

T1

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T2

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T3

0,

F11

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F2

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0,

F3

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im, iv

OwnIntercept

sm, sv

OwnSlope

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2

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11

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1

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Example

• Basic Model: 2(4) = 8.18 – Correlations

• Intercepts: .81• Slopes: .67

• Same Factors: 2(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

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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

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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.

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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

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Bilinear or Piecewise Model

• Inflection point• Two slope factors

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Wave

DV

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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.

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Option 1: Two Rates

Slope1 Slope2 Int

0 0 1

1 0 1

2 0 1

2 1 1

2 2 1

Slope1 Slope2 Int

0 0 1

1 0 1

2 0 1

3 1 1

4 2 1

Option 2: Rate & Deflection

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Piecewise Bilinear Model

Slope1

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PD11

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err111

Intercept

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PD21

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err211

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PD31

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err311

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PD41

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err411

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PD51

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err511

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Slope2

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Results

• Bilinear: 2(6) = 102.91, p < .001– RMSEA = .204

• Piecewise: 2(6) = 102.91, p < .001– RMSEA = .204

• Conclusion: No real improvement of fit for these two different but equivalent methods

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Step Function: Change in Intercept

• Level jumps at some point (e.g., point of intervention)• Two intercept factors

Slope Int1 Int2

0 1 0

1 1 0

2 1 1

3 1 1

4 1 1

Slope

0

PD11

0,

err111

Intercept

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PD21

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err211

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PD31

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err311

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PD41

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err411

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PD51

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err511

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Step

1

1

Note Int2 measures the size of intervention effect for each person.

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Results

• Change in intercept– 2(6) = 98.60– RMSEA = .199

• Conclusion: No real improvement of fit

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Modeling Nonlinearity

• Quadratic Effects

• Seasonal Effects

• Empirically based slopes of any form.

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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– 2(6) = 98.59; RMSEA = .199

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Modeling Seasonal Effects

• Note the alternating positive and negative coefficients for the slope

Slope

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PD11

0,

err111

Intercept

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PD21

0,

err211

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PD31

0,

err311

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PD41

0,

err411

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PD51

0,

err511

-1

1

-1

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Results

2(6) = 65.41, p < .001– RMSEA = .120

• No evidence of Slope Variance (actually estimated as negative!)

• Conclusion: Fit better, but still poor.

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Empirically Estimated Scaling of Time

• Allows for any possible growth model.• Fix one slope loading (usually one).• No intercept factor.

0

PD11

0,

err111

Slope

0

PD21

0,

err211

0

PD31

0,

err311

0

PD41

0,

err411

0

PD51

0,

err511

1

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ResultsCurvilinear 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

2(9) = 62.5, p < .001

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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

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LDS Model

Intercept

Slope

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T1

0

T2

0

T3

0,

E2

0,

E31

1

0,

E11

0

L1

0

L2

0

L3

1

1

1

1

1

0

D2-1

0

D3-2

1

1

a

a

1

1

1

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Relation to a LinearGrowth 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.

AdolescentAlcohol Use

Intercept

AdolescentAlcohol Use

Slope

0

T1

0

T2

0

T3

0,

E2

0,

E31

1

0,

E11

0

L1

0

L2

0

L3

1+a

1+a

1

1

1

1

1

1

1

1

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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

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More Complex Nonlinear Growth

• Sinusoid– Nonzero first and

second order derivative

• Pendulum– dampening

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Estimation Using AR(2) Model

• Negative Exponential1 > a1 > -1 (the rate of change) and a2 = 0

• Sinusoid2 > a1 > 1 and a2 = -1

Cobb formula for period length = /cos-1√a1

• Pendulumdampening factor = 1 - a2

Cobb formula for period length = /cos-1√a1

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