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## Growth Curve Model Using SEM

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**Growth Curve Model Using SEM**David A. Kenny**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 • 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**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 • 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