The STARTS ModelThe STARTS Model

David A. Kenny

December 15, 2013

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OverviewOverview STARTS Model Stationarity Assumption Multivariate Generalization

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The STARTS ComponentsThe 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

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Over-Time Correlations Over-Time Correlations Assuming Equal VariancesAssuming Equal Variances

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Correlation

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Over-Time Correlations Large Over-Time Correlations Large Stable Trait VarianceStable Trait Variance

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Over-Time Correlations Large Over-Time Correlations Large ART VarianceART Variance

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Correlation

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Over-Time Correlations Large Over-Time Correlations Large State VarianceState Variance

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The STARTS ModelThe STARTS Model

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b b

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Complexity Mixed Complexity Mixed with Simplicitywith Simplicity

ComplexityMore latent variables (11) than variances and covariances (10)

SimplicityOnly 5 parameters (regardless of the number of waves)

4 variances: ST, ART, S, and U1 path: ART pathall loadings fixed to 1

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Ensuring StationarityEnsuring StationarityVariance 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.

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Unequally Spaced Unequally Spaced MeasurementMeasurement

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).

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IdentificationIdentificationSee 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.

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Problems in EstimationProblems in Estimationif 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)

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Stability of the Stability of the ART ComponentART 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).

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Relaxing the Stationarity Relaxing the Stationarity AssumptionAssumption

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.

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Differential VariancesDifferential Variances

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Multivariate Multivariate GeneralizationGeneralization

TSO Model (Trait, State, and Occasion) of Cole, Martin, and Steiger

Create a latent variable for each timeTwo factors cause the latent variable

Stable Trait (Trait)Autoregressive Trait (Occasion)

State: Error Variance of each measureReally a START not a STARTS modelCan be estimated with 3 waves.

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Multivariate STARTSMultivariate STARTSImplemented 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.

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