Example Models for Multi-wave Data David A. Kenny December 15, 2013

2 Example Data Dumenci, L., & Windle, M. (1996). Multivariate Behavioral Research, 31, 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 Models –Trait –Autoregressive –STARTS –Trait-State-Occasion (TSO) –Latent Growth Curve Types –Univariate (except TSO) -- DA –Latent Variable

4 Latent Variable Measurement Models Unconstrained –  2 (74) = , p =.006 – RMSEA = 0.032; TLI =.986 Equal Loadings –  2 (83) = , p =.003 – RMSEA = 0.034; TLI =.986 The equal loading model has reasonable fit. All latent variable models (except growth curve) are compared to this model.

5 Trait Model: Univariate Test of Equal Loadings: No Model Fit: RMSEA = 0.071; TLI =.974

6 Trait Model: Latent Variables Model with just the trait factor does not fit as well as the saturated model:  2 (74) = 1xx.81 More Trait than State Variance Trait Variance: State Variance 10.39

7 Autoregressive Model: Univariate Fixed error variances equal. Good fitting model:  2 (2) = 4.98, p =.083 Reliabilities Stabilities 1:  2:.802 2:  3:.847 3:  4:.738 4:.568

8 Autoregressive Model: Latent Variables Not a very good fitting model compared to the CFA –  2 (3) = 60.08, p <.001 Overall Fit:  2 (xx) = 1.81, p <.0xx, RMSEA = 0.0xx; TLI =.9xx Stabilities 1  2:.xxx 2  3:.xxx 3  4:.xxx

9 Growth Curve Model: Univariate Unlike other models it fits the means. Fit:  2 (74) = 1xx.81, p <.0xx, RMSEA = 0.0xx; TLI =.9xx Intercept Slope Mean Variance

10 Growth Curve Model: Latent Variables Fit:  2 (74) = 1xx.81, p <.0xx, RMSEA = 0.0xx; TLI =.9xx Intercept Slope Mean Variance

11 Trait State Occasion Model Standard TSO does not have correlated errors, but they are added. Fit:  2 (74) = 1xx.81, p <.0xx, RMSEA = 0.0xx; TLI =.9xx –Variances –Trait –State

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13 STARTS Univariate Difficulty in finding trait factor. None of the models converged. Trait factor as Seasonality: Loadings in the Fall are 1 and in the Spring are -1 Models converged.

14 Univariate STARTS Results Fit:  2 (74) = 1xx.81, p <.0xx, RMSEA = 0.0xx; TLI =.9xx Variances –Seasonality –ART –State AR coefficient:

15 Latent Variable STARTS Fit:  2 (74) = 1xx.81, p <.0xx, RMSEA = 0.0xx; TLI =.9xx Variances –Seasonality –ART –State AR coefficient:

16 Summary of Fit: Univariate Trait Autoregressive Growth Curve STARTS

17 Summary of Fit: Latent Variables RMSEA TLI No Model Trait Autoregressive Growth Curve TSO STARTS