The STARTS Model

1. The STARTS Model David A. Kenny December 15, 2013

2. Overview STARTS Model Stationarity Assumption Multivariate Generalization

3. The STARTS Components Stable Trait or ST (trait) Unchanging component Autocorrelations of one Autoregressive Trait or ART (state) Slow-changing component State or S (error) Fast-changing, random component

4. Over-Time Correlations Assuming Equal Variances State Autoregressive Trait Stable Trait

5. Over-Time Correlations Large Stable Trait Variance State Autoregressive Trait Stable Trait

6. Over-Time Correlations Large ART Variance State Autoregressive Trait Stable Trait

7. Over-Time Correlations Large State Variance State Autoregressive Trait Stable Trait

8. The STARTS Model U2 U3 U4 1 1 1 b b b ART1 ART3 ART4 ART2 1 1 1 1 S4 S1 S2 S3 1 1 1 1 SE1 SE2 SE3 SE4 1 1 1 1 ST

9. Complexity Mixed with Simplicity Complexity More latent variables (11) than variances and covariances (10) Simplicity Only 5 parameters (regardless of the number of waves) 4 variances: ST, ART, S, and U 1 path: ART path all loadings fixed to 1

10. Ensuring Stationarity Variance of ART at time 2 equals Var(ART2) = b2[Var(ART1)] + Var(U2) Note for the ART variances to be stationarity, it follows that: Var(U2) = Var(ART1)[1 - b2] This nonlinear constraint must be made and an SEM program is needed to do so. Thus the total number of parameters for STARTS is four, regardless what the number of waves are.

11. Unequally Spaced Measurement Assume age at each wave is denoted as At. ART Model for time t-1 to t: ARTt = b(At-At-1)ARTt-1 + Ut For the self-esteem study, we can use in the actual ages and set the time unit for b as one year (autocorrelation for one year).

12. Identification See Cole, Martin, and Steiger (Psychological Methods, 2005). Four waves is the very minimum, but many more (perhaps at least six) are necessary. Estimation is much better with many waves and large N.

13. Problems in Estimation if the AR coefficient is too small (looks like State) or too large (looks like Stable Trait) if a variance component small (explains less than 10% of the variance)

14. Stability of the ART Component There can be a high one-year stability of the ART component but the stability over a long period of time. For example, if the .766 is the year to year stability, the correlation across 11 years is only .053 (.76611).

15. Relaxing the Stationarity Assumption All of the equality assumptions require that the variances of the measures not change over time. Seems rather implausible. Model can be modified to allow for latent stationarity with T – 1 parameters and so T + 3 parameters in total.

16. Differential Variances U2 U3 U4 ART1 ART3 ART4 ART2 S4 S1 S2 S3 V3 V4 V1 V2 1 ? ? ? SE1 ST SE2 SE3 SE4

17. Multivariate Generalization TSO Model (Trait, State, and Occasion) of Cole, Martin, and Steiger Create a latent variable for each time Two factors cause the latent variable Stable Trait (Trait) Autoregressive Trait (Occasion) State: Error Variance of each measure Really a START not a STARTS model Can be estimated with 3 waves.

19. Multivariate STARTS Implemented by Donnellan et al. in a study of self-esteem. Add the true State Factor (S). Correlate errors of the same indicator at different times. Requires at least four waves of data.