Growth curve model using sem
This presentation is the property of its rightful owner.
Sponsored Links
1 / 23

Growth Curve Model Using SEM PowerPoint PPT Presentation


  • 102 Views
  • Uploaded on
  • Presentation posted in: General

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.

Download Presentation

Growth Curve Model Using SEM

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


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

    • 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.


Intercept Factor


Intercept Factor with Loadings


Slope Factor


Slope Factor with Loadings


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


  • Login