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Growth Curve Model Using SEM. David A. Kenny. 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: The parameters from these lines describe the person. Nonlinear growth models are possible.

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Growth Curve Model Using SEM

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Growth Curve Model Using SEM

David A. Kenny

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:

• The parameters from these lines describe the person.

• Nonlinear growth models are possible.

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 in SEM

• 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

Slope Factor

• free mean and variance

• Standard specification (given equal spacing)

• 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

• At time zero, the intercept is defined.

• Rescaling of time:

• 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

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.

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)

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.

Relationship to Multilevel Modeling (MLM)

• Equivalent if ML option is chosen

• Measures of absolute fit

• Easier to respecify; more options for respecification

• More flexibility in the error covariance structure

• Allows latent covariates

• Allows missing data in covariates