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A Multivariate Reduced-Rank Growth Curve Model with Unbalanced Data

Published online by Cambridge University Press:  01 January 2025

Heungsun Hwang  and
Yoshio Takane
Heungsun Hwang*
Affiliation:
HEC Montreal
Yoshio Takane
Affiliation:
Mcgill University
*
Requests for reprints should be sent to Heungsun Hwang, HEC Montreal, Department of Marketing, 3000 Chemin de la Cote Ste-Catherine, Montreal, Quebec, H3T 2A7, CANADA. Email: heungsun.hwang@hec.ca
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Abstract

A multivariate reduced-rank growth curve model is proposed that extends the univariate reducedrank growth curve model to the multivariate case, in which several response variables are measured over multiple time points. The proposed model allows us to investigate the relationships among a number of response variables in a more parsimonious way than the traditional growth curve model. In addition, the method is more flexible than the traditional growth curve model. For example, response variables do not have to be measured at the same time points, nor the same number of time points. It is also possible to apply various kinds of basis function matrices with different ranks across response variables. It is not necessary to specify an entire set of basis functions in advance. Examples are given for illustration.

Keywords

Multivariate growth curve modelsreduced-rank restrictionlongitudinal data
Type
Theory And Methods
Information
Psychometrika , Volume 69 , Issue 1 , March 2004 , pp. 65 - 79
Copyright
Copyright © 2004 The Psychometric Society

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Footnotes

The work reported in this paper was supported by Grant A6394 from the Natural Sciences and Engineering Research Council of Canada to the second author. We thank Jennifer Stephan for her helpful comments on an earlier version of this paper. We also thank Patrick Curran and Terry Duncan for kindly letting us use the NLSY and substance use data, respectively. The substance use data were provided by Grant DA09548 from the National Institute on Drug Abuse.

References

Albert, J. M., & Kshirsagar, A. M. (1993). The reduced-rank growth curve model for discriminant analysis of longitudinal data. Australian Journal of Statistics, 35, 345–357CrossRefGoogle Scholar
Anderson, T. W. (1951). Estimating linear restrictions on regression coefficients for multivariate normal distributions. Annals of Mathematical Statistics, 22, 327–351CrossRefGoogle Scholar
Anderson, T. W. (1984). An Introduction to Multivariate Statistical Analysis. New York: John Wiley and SonsGoogle Scholar
Bijleveld, C. C. J. H., & de Leeuw, J. (1991). Fitting longitudinal reduced-rank regression models by alternating least squares. Psychometrika, 56, 433–447CrossRefGoogle Scholar
Böckenholt, U., & Takane, Y. (1994). Linear constraints in correspondence analysis. In Greenacre, M. J., & Blasius, J. (Eds.), Correspondence Analysis in Social Sciences (pp. 112–127). London: Academic PressGoogle Scholar
Carter, E. M., & Hubert, J. J. (1984). A growth-curve model approach to multivariate quantal bioassay. Biometrics, 40, 699–706CrossRefGoogle Scholar
Chinchilli, V. M., & Elswick, R. K. (1985). A mixture of the MANOVA and GMANOVA models. Communications in Statistics: Theory and Methods, 14, 3075–3089CrossRefGoogle Scholar
Curran, P. J. (1998). Introduction to hierarchical linear models of individual growth: An applied example using the SAS data system. Paper presented at the First International Institute on Developmental Science. Chapel Hill: University of North CarolinaGoogle Scholar
Curran, P. J., & Bollen, K. A. (1999). A hybrid latent trajectory model of stability and change: Applications in developmental psychology. Paper presented at the biennial meeting of the Society for Research on Child Development, Albuquerque, New Mexico.Google Scholar
Davies, P. T., & Tso, M. K.-S. (1982). Procedures for reduced-rank regression. Applied Statistics, 31, 244–255CrossRefGoogle Scholar
Duncan, T. E., Duncan, S. C., Alpert, A., Hops, H., Stoolmiller, M., & Muthén, B. (1997). Latent variable modeling of longitudinal and multilevel substance use data. Multivariate Behavioral Research, 32, 275–318CrossRefGoogle ScholarPubMed
de Leeuw, J. (1989). Fitting reduced rank regression models by alternating maximum likelihood. UCLA Statistics Series 35. Department of Statistics, University of California at Los Angeles.Google Scholar
Gabriel, K. R., & Zamir, S. (1979). Low rank approximation of matrices by least squares with any choice of weights. Technometrics, 21, 489–498CrossRefGoogle Scholar
Gifi, A. (1990). Nonlinear Multivariate Analysis. Chichester: John Wiley and SonsGoogle Scholar
Grizzle, J. E., & Allen, D. M. (1969). Analysis of growth and dose response curves. Biometrics, 25, 357–381CrossRefGoogle ScholarPubMed
Izenman, A. J. (1975). Reduced-rank regression for the multivariate linear model. Journal of Multivariate Analysis, 5, 248–264CrossRefGoogle Scholar
Khatri, C. G. (1966). A note on a MANOVA model applied to problems in growth curves. Annals of the Institute of Statistical Mathematics, 18, 75–86CrossRefGoogle Scholar
Kiers, H. A. L., & ten Berge, J. M. F. (1989). Alternating least squares algorithms for simultaneous components analysis with equal component weight matrices in two or more populations. Psychometrika, 54, 467–473CrossRefGoogle Scholar
Laird, N. M., & Ware, J. H. (1982). Random-effects models for longitudinal data. Biometrics, 38, 963–974CrossRefGoogle ScholarPubMed
Liang, K. Y., & Zeger, S. L. (1986). Longitudinal data analysis using generalized linear models. Biometrika, 73, 13–22CrossRefGoogle Scholar
Lundbye-Christensen, S. (1991). A multivariate growth curve model for pregnancy. Biometrics, 47, 637–657CrossRefGoogle ScholarPubMed
Nummi, T. (1997). Estimation in a random effects growth curve model. Journal of Applied Statistics, 24, 157–168CrossRefGoogle Scholar
Nummi, T., & Möttönen, J. (2000). On the analysis of multivariate growth curves. Metrika, 52, 77–89CrossRefGoogle Scholar
Potthoff, R. F., & Roy, S. N. (1964). A generalized multivariate analysis of variance model useful especially for growth curve problems. Biometrika, 51, 313–326CrossRefGoogle Scholar
Ramsay, J. O. (in press). Multilevel modeling of longitudinal and functional data. To appear in D. Moskowitz & S. Hershberger (Eds.), Modeling Intraindividual Variability with Repeated Measures Data: Methods and Applications. New York: Erlbaum Associates.Google Scholar
Ramsay, J. O., & Silverman, B. W. (1997). Functional Data Analysis. New York: Springer-VerlagCrossRefGoogle Scholar
Rao, C. R. (1964). The use and interpretation of principal component analysis in applied research. Sankhyã A, 26, 329–358Google Scholar
Rao, C. R. (1965). The theory of least squares when the parameters are stochastic and its application to the analysis of growth curves. Biometrika, 52, 447–458CrossRefGoogle Scholar
Rao, C. R. (1980). Matrix approximations and reduction of dimensionality in multivariate statistical analysis. In Krishnaiah, P. R. (Eds.), Multivariate Analysis (pp. 3–22). Amsterdam: North-HollandGoogle Scholar
Reinsel, G. C. (1982). Multivariate repeated-measurement or growth curve models with multivariate random-effects covariance structure. Journal of the American Statistical Association, 77, 190–195CrossRefGoogle Scholar
Reinsel, G. C., & Velu, R. P. (1998). Multivariate Reduced-Rank Regression: Theory and Applications. New York: Springer-VerlagCrossRefGoogle Scholar
Sakamoto, Y., Ishiguro, M., & Kitagawa, G. (1986). Akaike Information Criterion Statistics. Boston: D. ReidelGoogle Scholar
Seber, G. A. F. (1984). Multivariate Observations. New York: John Wiley and SonsCrossRefGoogle Scholar
Takane, Y., & Shibayama, T. (1991). Principal component analysis with external information on both subjects and variables. Psychometrika, 56, 97–120CrossRefGoogle Scholar
Takane, Y., Yanai, H., & Mayekawa, S. (1991). Relationships among several methods of linearly constrained correspondence analysis. Psychometrika, 56, 667–684CrossRefGoogle Scholar
van den Wollenberg, A. L. (1977). Redundancy analysis: An alternative for canonical analysis. Psychometrika, 42, 207–219CrossRefGoogle Scholar
van der Leeden, R. (1990). Reduced Rank Regression with Structured Residuals. Leiden: DSWO PressGoogle Scholar
von Rosen, D. (1991). The growth curve model: A review. Communications in Statistics: Theory and Methods, 9, 2791–2822CrossRefGoogle Scholar
Young, F. W. (1981). Quantitative analysis of qualitative data. Psychometrika, 46, 357–388CrossRefGoogle Scholar
Zeger, S. L., & Liang, K. Y. (1986). The analysis of discrete and continuous longitudinal data. Biometrics, 42, 121–130CrossRefGoogle Scholar