1. Confirmatory Factor Analysis of Longitudinal Data David A. Kenny December 23. 2013

2. 2 Task Same set of measures that form a latent variables are measured at two or more times on the same sample.

3. 3 Example Data Dumenci, L., & Windle, M. (1996). Multivariate Behavioral Research, 31, 313-330. Depression with four indicators (CESD) PA: Positive Affect (lack thereof) DA: Depressive Affect SO: Somatic Symptoms IN: Interpersonal Issues Four times separated by 6 months 433 adolescent females Age 16.2 at wave 1

4. 4 Equal Loadings Over Time Want to test that the factor loadings are the same at all times. If the loadings are the same, then it becomes more plausible to argue that one has the same construct at each time. Many longitudinal models requires temporally invariant loadings.

5. 5 Correlated Measurement Error Almost always with longitudinal data, the errors of measurement of an indicator (technically called uniquenesses) should be correlated. To be safely identified, at least, three indicators are needed. Identified with just two indicators, but must assume the loadings are equal (i.e., both set to one).

6. 6 Equal Error Variances Another possibility is that error variance of the same measure at different times are equal. As in the example with four indicators at four times, each indicator would have one error variance for each of the four times, a total of 12 constraints.

7. 7 Equal Loadings Correlated Errors Equal Error Variances

8. 8 Latent Variable Measurement Models Model ²² dfRMSEA  ² diff df diffp Comparison Model I No Correlated Errors 856.729980.135 II Correlated Errors (CE) 107.718740.032 749.01024>.001I III CE and Equal Loadings (EL) 123.657830.034 15.938 9.068II IV CE, EL, and Equal Error Variances 143.645950.034 19.99812.067III

9. 9 Conclusion Definitely need correlated errors in the model (something that will almost always be the case). Forcing equal loadings, while worsening the fit some, seems reasonable in this case. Equal error variances is also reasonable.

10. 10 Means of a Latent Variable Fix the intercept of the marker variable at each time to zero. Free the other intercepts but set them equal over time; a total of (k – 1)(T – 1) constraints. –9 constraints for the example dataset Free factor means and see if the model fits.

11. 11 Equality of the Means of a Latent Variable Assuming good fit of a model with latent means, fix the factor means (m1 = m2 = m3 = m4) to be equal to test the equality of factor means; T – 1 df.

12. 12 Equal Intercepts

13. 13 Example Means: Latent Variable, Base Model Model with No Constraints on the Means –  2 (83) = 123.66, p =.003 – RMSEA = 0.034; TLI =.985 Model with Latent Means and Constraints on the Intercepts –  2 (92) = 157.49, p =.003 – RMSEA = 0.041; TLI =.979 Fit is worse with the constraints, but the model fit (RMSEA and TLI) are acceptable. Can test if means differ.

14. 14 Example Means: Latent Variable The four means: 25.34, 25.82, 21.72, 20.09 Base Model –  2 (92) = 157.49, p =.003 – RMSEA = 0.041; TLI =.979 Equal Latent Means –  2 (95) = 182.94, p <.001 – RMSEA = 0.046; TLI =.973 Test of the null hypothesis of equal variance:  2 (3) = 25.44, p <.001 Conclusion: Means differ.

15. 15 Equal Variance: Latent Variable Fix the T factor variances to be equal (s1 = s2 = s3 = s4). Compare this model to a model in which factor variances are free to vary with T – 1 df.

16. 16

17. 17 Example: Latent Variable The four latent variances: 25.34, 25.82, 21.72, 20.09 Base Model –  2 (83) = 123.66, p =.003 – RMSEA = 0.034; TLI =.985 Equal Variances –  2 (86) = 133.43, p =.001 – RMSEA = 0.036; TLI =.984 Test of the null hypothesis of equal variance: –  2 (3) = 9.76, p =.021 Variances significantly different, but model fit is not all that different from the base model.