Abstract
Hierarchically structured data are common in many areas of scientific research. Such data are characterized by nested membership relations among the units of observation. Multilevel analysis is a class of methods that explicitly takes the hierarchical structure into account. Repeated measures data can be considered as having a hierarchical structure as well: measurements are nested within, for instance, individuals. In this paper, an overview is given of the multilevel analysis approach to repeated measures data. A simple application to growth curves is provided as an illustration. It is argued that multilevel analysis of repeated measures data is a powerful and attractive approach for several reasons, such as flexibility, and the emphasis on individual development.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bock, R. D. (1989). Multilevel Analysis of Educational Data, San Diego, CA: Academic Press.
Box, G. E. P. (1950). Problems in the analysis of growth and wear curves, Biometrics 6: 362–389.
Bryk, A. S. & Raudenbush, S. W. (1987). Application of hierarchical linear models to assessing change, Psychological Bulletin 101: 147–158.
Bryk, A. S. & Raudenbush, S. W. (1992). Hierachical Linear Models: Applications and Data Analysis Methods, Newbury Park, CA: Sage Publications.
Bryk, A. S., Raudenbush, S. W., Seltzer, M. & Congdon, R. T. (1988). An Introduction to HLM: Computer Program and User's Guide, University of Chicago.
Burstein, L., Linn, R. L. & Capell, F. J. (1978). Analyzing multilevel data in the presence of heterogeneous within-class regressions, Journal of Educational Statistics 3: 347–383.
De Leeuw, J. & Kreft, I. G. G. (1986). Random coefficient models for multilevel analysis, Journal of Educational Statistics 11: 57–85.
De Lury, D. B. (1950). Values and integrals of the orthogonal polynomials up to n = 50, Toronto: University of Toronto Press.
Dempster, A. P., Rubin, D. B. & Tsutakawa, R. K. (1981). Estimation in covariance components models, Journal of the American Statistical Association 76: 341–353.
Dixon, W. J. (1988). BMDP Statistical Software Manual Vol. 2. Berkeley, CA: University of California Press.
Francis, B., Green, M. & Payne, C. (1993). The GLIM System, Release 4 Manual, Oxford, GB: Clarendon Press.
Goldstein, H. (1986a). Efficient statistical modelling of longitudinal data, Annals of Human Biology 13: 129–141.
Goldstein, H. (1986b). Multilevel mixed linear model analysis using iterative generalized least squares, Biometrika 73: 43–56.
Goldstein, H. (1987). Multilevel Models in Educational and Social Research, London, GB: Oxford University Press.
Goldstein, H. (1989). Models for multilevel response variables with an application to growth curves, in R. D. Bock (Ed.), Multilevel analysis of educational data, San Diego, CA: Academic Press, 107–125.
Goldstein, H. (1995). Multilevel Statistical Models, (2nd edn), London, GB: Edward Arnold.
Harville, D. A. (1977). Maximum likelihood approaches to variance component estimation and to related problems, Journal of the American Statistical Association 72: 320–340.
Hills, M. (1974). Statistics for Comparative Studies, New York, NY: John Wiley.
Jennrich, R. & Schluchter, M. (1986). Unbalanced repeated measures models with structured covariance matrices, Biometrics 42: 805–820.
Kreft, I. G. G., De Leeuw, J. & Van der Leeden, R. (1994). A review of five multilevel analysis programs: BMDP-5V, GENMOD, HLM, ML3, VARCL, The American Statistician 48: 324–335.
Laird, N. M. & Ware, J. H. (1982). Random effects models for longitudinal data, Biometrics 38: 963–974.
Lindley, D. V. & Smith, A. F. M. (1972). Bayes estimates for the linear model, Journal of the Royal Statistical Society B34: 1–41.
Longford, N. T. (1987). A fast scoring algorithm for maximum likelihood estimation in unbalanced mixed models with nested random effects, Biometrika 74: 817–827.
Longford, N. T. (1990). VARCL. Software for Variance Component Analysis of Data with Nested Random Effects (Maximum Likelihood), Princeton, NJ: Educational Testing Service.
Longford, N. T. (1993). Random Coefficient Models, Oxford, GB: Clarendon Press.
McCullagh, P. & Nelder, J. A. (1989). Generalized linear models, 2nd edn. London, GB: Chapman & Hall.
Morris, C. N. (1983). Parametric empirical Bayes inference: Theory and applications, Journal of the American Statistical Association 78: 47–65.
Potthoff, R. F. & Roy, S. N. (1964). A generalized multivariate analysis of variance model useful especially for growth curve problems, Biometrika 51: 313–326.
Prosser, R., Rasbash, J. & Goldstein, H. (1991). ML3. Software for Three-Level Analysis. User's Guide for V. 2, London, GB: Institute of Education, University of London.
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: 447–458.
Rasbash, J., Yang, M., Woodhouse, G. & Goldstein, H. (1995). MLn: command reference guide, London, GB: Institute of Education, University of London.
Raudenbush, S. W. (1988). Educational applications of hierarchical linear models: A review, Journal of Educational Statistics 13: 85–116.
Raudenbush, S. W. (1989). The analysis of longitudinal, multilevel data, International Journal of Educational Research 13: 721–740.
Strenio, J. F. (1981). Empirical Bayes estimation for a hierarchical linear model, Unpublished doctoral dissertation, Harvard University.
Strenio, J. F., Weisberg, H. I. & Bryk, A. S. (1983). Empirical Bayes estimation of individual growth curve parameters and their relationship to covariates, Biometrics 39: 71–86.
Tate, R. L. & wongbundit, Y. (1983). Random versus non-random coefficient models for multilevel analysis, Journal of Educational Statistics 8: 103–120.
Van der Leeden, R., Vrijburg, K. & De Leeuw, J. (1996). A review of two different approaches for the analysis of growth data using longitudinal mixed linear models: comparing hierarchical linear regression (ML3, HLM) and repeated measures design with structured covariance matrices (BMDP5V), Computational Statistics and Data Analysis 21: 581–605.
Visser, R. A. (1985). Analysis of longitudinal data in behavioural and social research, Leiden, The Netherlands: DSWO Press
Ware, J. H. (1985). Linear models for the analysis of longitudinal studies, Journal of the American Statistical Association 80: 95–101.
Wilks, S. (1962). Mathematical Statistics, New York, NY: John Wiley.
Willett, J. B. (1988). Questions and answers in the measurement of change, in E. Rithkopf (ed.), Review of Research in Education (1988–89), Washington, DC: American Educational Research Association, 345–422.
Wishart, J. (1938). Growth rate determinations in nutrition studies with the bacon pig, and their analysis, Biometrika 30: 16–28.
Zeger, S. L., Liang, K.-Y. & Albert, P. S. (1988). Models for longitudinal data: a generalized estimating equation approach, Biometrics 44: 1049–1060.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Van Der Leeden, R. Multilevel Analysis of Repeated Measures Data. Quality & Quantity 32, 15–29 (1998). https://doi.org/10.1023/A:1004233225855
Issue Date:
DOI: https://doi.org/10.1023/A:1004233225855