CHAPTER CONCEPTS Dynamic factor model Model specification Model identification Model estimation Model testing Model interpretation
A class of SEM applications that involve stationary and non-stationary latent variables across time with lagged (correlated) measurement error has been called dynamic factor analysis (Hershberger, Molenaar, & Corneal, 1996). A character-istic of the SEM dynamic factor model is that the same measurement instruments are administered to the same subject on two or more occasions. The purpose of the analysis is to assess change in the latent variable between the ordered occasions due to some event or treatment. When the same measurement instruments are used over two or more occasions, there is a tendency for the measurement errors to correlate (autocorrelation). For example, a specific sequence of correlated error, where error at Time 1 correlates with error at Time 2, and error at Time 2 corre-lates with error at Time 3, is called an ARIMA model in econometrics (Wheaton, Muthén, Alwin, & Summers, 1977).
MODEL SPECIFICATION
Educational research has indicated that anxiety increases the level of student achievement and performance. Psychological research in contrast indicates that anxiety has a negative effect upon individuals, and thus should interfere with or have a decreasing impact on the level of achievement and performance. Is it pos-sible that both academic areas of research are correct?
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MODEL IDENTIFICATION
A dynamic factor model was hypothesized that indicated student achievement and performance measures at three equal time points two weeks apart (time 1, time 2, and time 3). The student data indicate achievement (A1) and performance (P1) at time 1, achievement (A2) and performance (P2) at time 2, and achievement (A3) and performance (P3) at time 3. The errors at time 1 were hypothesized to correlate with errors at time 2 and errors at time 2 were hypothesized to correlate with errors at time 3, indicating an ARIMA model. Time 1 predicts time 2, and then time 2 predicts time 3. Recall, a minimum of three time points is required to estimate the slope, and hence to test whether change has occurred. The dynamic factor model is diagrammed in Figure 10.1. The model has df = 2, so we consider it an identified model.
MODEL ESTIMATION
The data set contains 600 students who were measured on the same achievement and performance measures at three different points in time. The two variables, achievement and performance, defined the factor, time. Thus, the latent variable, time, was represented as time 1, time 2, and time 3, with two indicator variables at each time point. How well students did at time 2 was predicted by the time 1 latent variable. Likewise, how well students did at time 3 was predicted by the time 2 latent variable. Students were given a high level of anxiety by having to meet deadlines, take frequent quizzes, and turn in extra assignments.
error error error error error error
A3 P2
A2 P1
Time 1 Time 2 Time 3
error error
A1 P3
Figure 10.1: DYNAMIC FACTOR MODEL (AMOS)
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The errors are lagged to indicate the ARIMA type time series model. The factor loading estimates on the same instruments at the three time points will indicate the validity and stability of the measures. The estimation of the structure coefficients for time 1 predicting time 2, and time 2 predicting time 3 are the key purpose beyond the dynamic factor model analysis. We are estimating five error variances, six factor loadings, and two structure coefficients.
MODEL TESTING
The dynamic factor model results indicated an acceptable model fit (χ2 = 2.76, df = 2, and p = .25). The structural equations indicated the prediction across the three time intervals for the latent variable, time. Time 1 was a statistically signifi-cant predictor of time 2; coefficient was statistically signifisignifi-cant (t = 12.36) and R2 = .47. Time 2 was a statistically significant predictor of time 3 (t = – 9.66;
R2 = .20); however, results indicated a negative coefficient (–.82). The computer output is shown as:
Structural Equations
Time2 = 0.68*Time1, Errorvar.= 0.53 , R² = 0.47
(0.055) (0.071)
12.36 7.50
Time3 = –0.82*Time2, Errorvar.= 0.80, R² = 0.20
(0.085) (0.12)
–9.66 6.52
MODEL INTERPRETATION
The dynamic factor model tested the significance of the latent variable predic-tion across time. Our results would be interpreted as follows: anxiety increased the level of student achievement and performance from time 1 to time 2, but then decreased the level of student achievement and performance from time 2 to time 3. Anxiety increased levels of achievement and performance, but only for a certain amount of time, then it had a negative effect. So, it appears educational research-ers and psychologists are both correct to some extent. The dynamic factor model clarifies how anxiety affects the level of student achievement and performance, given a time continuum.
The dynamic factor model output with standardized coefficients is shown in Figure 10.2.
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SUMMARY
The dynamic factor model extends the repeated measures design to include latent variables. When using latent variables, more than one observed indicator variable can be used. The dynamic factor model presented in this chapter included two indicator variables. The two observed variables were measured at each of three time points by the same subjects. In addition, the time intervals were the same. It is possible to have time-varying measurements such that they are not at the same interval of measurement. You will find many more examples of dynamic factor modeling in the macro-economic and econometric areas. In education and psych-ology the application has centered more on the latent growth model, which we cover in Chapter 14.
EXERCISE
A sports physician was interested in studying heart rate and muscle fatigue of female soccer players. She collected data after three soccer games over a three-week period. A dynamic factor model was used to determine if heart rate and muscle fatigue were stable across time for the 150 female soccer players.
error error error error error error
A3 P2
A2 P1
Time 1 Time 2 Time 3
error error
A1 P3
.68
1.54 .41 .16 .004
–.82
.29
Figure 10.2: DYNAMIC FACTOR MODEL (STANDARDIZED SOLUTION)
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Create an SEM program to analyze and interpret the dynamic factor model.
Include a diagram of the dynamic factor model. The data set information includ-ing observed variables, covariance matrix, sample size, and latent variables are listed below without the model equations.
Observed Variables: HR1 MF1 HR2 MF2 HR3 MF3 Covariance Matrix
10.75
7.00 9.34
7.00 5.00 11.50
5.03 5.00 7.49 9.96
3.89 4.00 3.84 3.65 9.51
2.90 2.00 2.15 2.88 3.55 5.50 Sample Size: 150
Latent Variables: Time1 Time2 Time3
SUGGESTED READINGS
chow, s. M., nesselrade, J. r., shifren, K., & McArdle, J. J. (2004). dynamic structure of emotions among individuals with Parkinson’s disease. Structural Equation Modeling, 11(4), 560–582.
Kroonenberg, P. M., van dam, M., van uzendoorn, M. H., & Mooijaart, A. (1997, May). dynamics of behaviour in the strange situation: A structural equation approach. British Journal of Psychology, 88, 311–332.
Zuur, A. F., Fryer, r. J., Jolliffe, i. t., dekker, r., & Beukema, J. J. (2003). estimating common trends in multivariate time series using dynamic factor analysis.
Environmetrics, 14(7), 665–685.
REFERENCES
Hershberger, s. l., Molenaar, P. c. M., & corneal, s. e. (1996). A hierarchy of uni-variate and multiuni-variate structural times series models. in Marcoulides, g., &
schumacker, r. e. (eds.). Advanced structural equation modeling: Issues and techniques (pp. 159–194). Mahwah, nJ: lawrence erlbaum Associates.
Wheaton, B., Muthén, B., Alwin, d., & summers, g. (1977). Assessing reliability and stability in panel models. in d. r. Heise (ed.), Sociological Methodology (pp. 84–136). san Francisco: Jossey-Bass.