Abstract
A class of four simultaneous component models for the exploratory analysis of multivariate time series collected from more than one subject simultaneously is discussed. In each of the models, the multivariate time series of each subject is decomposed into a few series of component scores and a loading matrix. The component scores series reveal the latent data structure in the course of time. The interpretation of the components is based on the loading matrix. The simultaneous component models model not only intraindividual variability, but interindividual variability as well. The four models can be ordered hierarchically from weakly to severely constrained, thus allowing for big to small interindividual differences in the model. The use of the models is illustrated by an empirical example.
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References
Anderson, T.W. (1963). The use of factor analysis in the statistical analysis of multiple time series.Psychometrika, 28, 1–25.
Bijleveld, C.C.J.H. (1989).Exploratory linear dynamic systems analysis. Leiden, Netherlands: DSWO Press.
Bijleveld, C.C.J.H., & Bijleveld, F.D. (1997). A comparison of the applicability of two approaches to linear dynamic systems analyze for N subjects.Kwantitatieve Methoden, 55, 37–57.
Bro, R., Andersson, C.A., & Kiers, H.A.L. (1999). PARAFAC2-Part II. Modeling chromatographic data with retention time shifts.Journal of Chemometrics, 13, 295–309.
Bro, R., & Smilde, A.K. (2003). Centering and scaling in component analysis.Journal of Chemometrics, 17, 16–33.
Browne, M.W., & Cudeck, R. (1992). Alternative ways of assessing model fit.Sociological Methods & Research, 21, 230–258.
Carroll, J.D., & Chang, J.J. (1970). Analysis of individual differences in multidimensional scaling via anN-way generalization of Eckart-Young decomposition.Psychometrika, 35, 283–319.
Cattell, R.B. (1952).Factor analysis. New York, NY: Harper.
Cattell, R.B. (1963). The structuring of change by P-technique and incremental R-technique. In C.W. Harris (Ed.),Problems in measuring change. Madison, WI: The University of Wisconsin Press.
Cliff, N. (1966). Orthogonal rotation to congruence.Psychometrika, 31, 33–42.
Engle, R., & Watson, M. (1981). A one-factor multivariate time series model of metropolitan wage rates.Journal of the American Statistical Association, 76, 774–781.
Harshman, R.A. (1972). Determination and proof of minimum uniqueness conditions for PARAFAC1.UCLA Working Papers in Phonetics, 22, 111–117.
Harshman, R.A., & Lundy, M.E. (1984a). The PARAFAC model for three-way factor analysis and multidimensional scaling. In H.G. Law, C.W. Conrad, Jr., J.A. Hattie & R.P. McDonald (Eds.),Research methods for multimode data analysis (pp. 122–215). New York, NY: Praeger.
Harshman, R.A. & Lundy, M.E. (1984b). Data preprocessing and the extended PARAFAC model. In H.G. Law, C.W. Conrad, Jr., J.A. Hattie, & R.P. McDonald (Eds.),Research methods for multimode data analysis (pp. 216–284). New York: Praeger.
Hastie, T, Tibshirani, R., & Friedman, J. (2001).The elements of statistical learning: Data mining, inference, and prediction. New York, NY: Springer.
Immink, W. (1986).Parameter estimation in Markov models and dynamic factor analysis. Unpublished doctoral thesis, University of Utrecht, Utrecht, Netherlands.
Jones, C.J., & Nesselroade, J.R. (1990). Multivariate, replicated, single-subject, repeated measures designs and P-technique factor analysis: A review of intraindividual change studies.Experimental Aging Research, 16, 171–183.
Kaiser, H.F. (1958). The varimax criterion for analytic rotation in factor analysis.Psychometrika, 23, 187–200.
Kiers, H.A.L., & ten Berge, J.M.F. (1994). Hierarchical relations between methods for simultaneous component analysis and a technique for rotation to a simple simultaneous structure.British Journal of Mathematical and Statistical Psychology, 47, 109–126.
Kiers, H.A.L., ten Berge, J.M.F., & Bro, R. (1999). PARAFAC2-Part 1. A direct fitting algorithm for the PARAFAC2 model.Journal of Chemometrics, 13, 275–294.
Little, R.J.A., & Rubin, D.B. (1987).Statistical analysis with missing data. New York, NY: Wiley.
Louwerse, D.J., Smilde, A.K., & Kiers, H.A.L. (1999). Cross-validation of multiway component models.Journal of Chemometrics, 13, 491–510.
Molenaar, P.M. (1985). A dynamic factor model for the analysis of multivariate time series. Psychometrika,50, 181–202.
Nesselroade, J.R., & Molenaar, P.C.M. (1999). Pooling lagged covariances structures based on short, multivariate time series for dynamic factor analysis. In R.H. Hoyle (Ed.),Statistical strategies for small sample research (pp. 223–250). Thousand Oaks, CA: Sage Publications.
Niesing, J. (1997).Simultaneous component and factor analysis methods for two or more groups: A comparative study. Leiden, Netherlands: DSWO Press.
Shifren, K., Hooker, K., Wood, P., & Nesselroade, J.R. (1997). Structure and variation of mood in individuals with Parkinson's disease: A dynamic factor analysis.Psychology and aging, 12, 328–339.
Timmerman, M.E. (2001).Component analysis of multisubject multivariate longitudinal data. Unpublished doctoral thesis, University of Groningen, Groningen, Netherlands.
Timmerman, M.E., & Kiers, H.A.L. (2002). Three-way component analysis with smoothness constraints.Computational Statistics and Data Analysis, 3, 447–470.
Tucker, L.R. (1951).A method for synthesis of factor analysis studies (Personnel Research sections Rep. No. 984). Washington, DC: Department of the Army.
van Buuren, S. (1990).Optimal scaling of time series. Leiden, Netherlands: DSWO Press.
Watson, D. (1988). Intraindividual and interindividual analyses of positive and negative affect: Their relation to health complaints, perceived stress, and daily activities.Journal of personality and social psychology, 54, 1020–1030.
Watson, D., Clark, L.A., & Tellegen, A. (1988). Development and validation of brief measures of positive and negative affect: The PANAS scales.Journal of Personality and Social Psychology, 54, 1063–1070.
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This research has been made possible by funding from the Netherlands Organization of Scientific Research (NWO) to the first author. The authors are obliged to Tom A.B. Snijders, Jos M.F. ten Berge and three anonymous reviewers for comments on an earlier version of this paper, and to Kim Shifren for providing us with her data set, which was collected at Syracuse University.
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Timmerman, M.E., Kiers, H.A.L. Four simultaneous component models for the analysis of multivariate time series from more than one subject to model intraindividual and interindividual differences. Psychometrika 68, 105–121 (2003). https://doi.org/10.1007/BF02296656
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DOI: https://doi.org/10.1007/BF02296656