Example Models for Multi-wave Data

David A. Kenny

December 15, 2013

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Example DataDumenci, 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  

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Models• Models

– Trait– Autoregressive– Latent Growth Curve – STARTS– Trait-State-Occasion (TSO)

• Types– Univariate – DA measure (except TSO) – Latent Variable

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Latent Variable Measurement Models

• Unconstrained– 2(74) = 107.72, p = .006– RMSEA = 0.032; TLI = .986

• Equal Loadings– 2(83) = 123.66, p = .003– RMSEA = 0.034; TLI = .985

• The equal loading model has reasonable fit.• All subsequent latent variable models

(except growth curve) are compared to this model.

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Trait Model: Univariate

• Test of Equal Loadings? No• Model Fit: RMSEA = 0.071; TLI = .974

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Trait Model: Latent Variables

• 2(88) = 156.21, p < .001; RMSEA = 0.042; TLI = .983

• More Trait than State Variance• Trait Variance: 12.78• State Variances: 8.17 to 12.48

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Autoregressive Model: Univariate

• Fixed error variances equal.• Good fitting model: 2(2) = 4.98, p = .083; RMSEA = 0.059;

TLI = .982Reliabilities Stabilities

1: .657 1 2: .802 2: .650 2 3: .8473: .597 3 4: .7384: .568

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Autoregressive Model: Latent Variables

• Not a very good fitting model compared to the CFA– 2(3) = 60.08, p < .001• Overall Fit: 2(86) = 183.74, p < .001• RMSEA = 0.051; TLI = .966• Standardized Stabilities

1 2: .636 2 3: .6593 4: .554

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Growth Curve Models• Unlike other models it fits the means and so

results are directly comparable to other models.

• Scaling of Time: -0.75, -0.25, 0.25, & 0.75; Time 0 is the midpoint of the study.

• Null model of zero correlations and equal means.

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Growth Curve Model: Univariate

• Test of equal error variances: 2(3) = 0.60, p = .896

• Equal variance assumed• Fit: 2(8) = 16.46, p = .036; RMSEA

= 0.049; TLI = .981

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Growth Curve Model: Univariate: Results

Slope-Intercept r = -.287

Mean VarianceIntercept 5.407 12.491Slope -0.879 4.001Error 0.000 11.472

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Growth Curve Model: Latent

VariablesFit of CFA with Latent Means2(92) = 157.93, p < .821, RMSEA = 0.041; TLI = .977Test of Equal Latent Error Variances in the LGC

2(3) = 0.92, p = .821Equal Error Variance assumed.

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Growth Curve Model: Latent

VariablesFit: 2(100) = 170.84, p < .001, RMSEA = 0.040; TLI = .984Slope-Intercept r = -.297

Mean VarianceIntercept 5.404 13.307Slope -0.847 3.934Error 0.000 8.913

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Trait State Occasion Model

• Standard TSO does not have correlated errors, but they are added.

• Fit: 2(90) = 153.92, p < .001; RMSEA = 0.040; TLI = .979

• Variances: Trait 11.139 & State 11.788• Autoregressive coefficient = .198

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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.• Data appear to be stationary, no changes in

variance

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Univariate STARTS Results

• Fit: 2(89) = 15.44, p = .009, RMSEA = 0.069; TLI = .975

• Variances – Seasonality 0.79 (p = .003)– ART 17.32 (p < .001)– State 4.93 (p < .001)

• AR coefficient: .826, r14 = .8263 = .563

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Latent Variable STARTS

• Fit: 2(89) = 136.86, p < .001, RMSEA = 0.035; TLI = .984

• Variances– Seasonality 0.79 (p = .003)– ART 17.32 (p < .001)– State 4.93 (p < .001)

• AR coefficient: .826, r14 = .8263 = .563

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TSO vs. STARTS• Trait factor in TSO becomes the

ART factor in STARTS• The State factor with a low AR

coefficient in TSO becomes the State factor in STARTS with a zero coefficient

• STARTS also has a Seasonality Factor.

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Summary of Fit: Univariate

RMSEA TLITrait 0.071 .974Autoregressive 0.059 .982Growth Curvea 0.049 .981STARTS 0.069 .975

aGrowth Curve Model also explains the means.

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Summary of Fit: Latent Variables

RMSEA TLINo Model 0.034 .985No Model (LGC) 0.041 .979Trait 0.042 .983Autoregressive 0.051 .966Growth Curve 0.040 .984TSO 0.040 .979STARTS 0.035 .984

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Best Model?• While debatable, it would appear

that the Latent Growth Curve Model is the most sensible model to retain.

• The Latent STARTS model has a good fit, but the absence of a Trait factor and the post hoc Seasonal factor make it less than desirable.