example models for multi-wave data

Example Models for Multi-wave Data

David A. Kenny

December 15, 2013

2

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  

3

Models• Models

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

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

4

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.

5

Trait Model: Univariate

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

6

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

7

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

8

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

9

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.

10

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

11

Growth Curve Model: Univariate: Results

Slope-Intercept r = -.287

Mean VarianceIntercept 5.407 12.491Slope -0.879 4.001Error 0.000 11.472

12

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.

13

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

14

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

15

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

16

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

17

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

18

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.

19

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.

20

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

21

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.