Confirmatory Factor Analysis of Longitudinal Data David A

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

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

Example Data Dumenci, L. , & Windle, M. (1996). Multivariate Behavioral Research, 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 3

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. 4

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). 5

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. 6

Correlated Errors Equal Error Variances Equal Loadings 7

Latent Variable Measurement Models Model c² df RMSEA df diff c² diff p Comparison Model I No Correlated Errors 856. 729 98 0. 135 II Correlated Errors (CE) 107. 718 74 0. 032 749. 010 24 >. 001 I III CE and Equal Loadings (EL) 123. 657 83 0. 034 15. 938 9 . 068 II IV CE, EL, and Equal Error Variances 143. 645 95 0. 034 19. 998 12 . 067 III 8

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. 9

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. 10

Equality of the Means of a Latent Variable • Assuming good fit of a model with latent means, fix the factor means (m 1 = m 2 = m 3 = m 4) to be equal to test the equality of factor means; T – 1 df. 11

Equal Intercepts 12

Example Means: Latent Variable, Base Model • Model with No Constraints on the Means – c 2(83) = 123. 66, p =. 003 – RMSEA = 0. 034; TLI =. 985 • Model with Latent Means and Constraints on the Intercepts – c 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. 13 • Can test if means differ.

Example Means: Latent Variable • The four means: 25. 34, 25. 82, 21. 72, 20. 09 • Base Model – c 2(92) = 157. 49, p =. 003 – RMSEA = 0. 041; TLI =. 979 • Equal Latent Means – c 2(95) = 182. 94, p <. 001 – RMSEA = 0. 046; TLI =. 973 • Test of the null hypothesis of equal variance: c 2(3) = 25. 44, p <. 001 14 • Conclusion: Means differ.

Equal Variance: Latent Variable • Fix the T factor variances to be equal (s 1 = s 2 = s 3 = s 4). • Compare this model to a model in which factor variances are free to vary with T – 1 df. 15

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Example: Latent Variable • The four latent variances: 25. 34, 25. 82, 21. 72, 20. 09 • Base Model – c 2(83) = 123. 66, p =. 003 – RMSEA = 0. 034; TLI =. 985 • Equal Variances – c 2(86) = 133. 43, p =. 001 – RMSEA = 0. 036; TLI =. 984 • Test of the null hypothesis of equal variance: – c 2(3) = 9. 76, p =. 021 • Variances significantly different, but model fit is not all that different from the base model. 17