Keywords
random coefficients models
means models
data analysis strategies
Modelos de regresión lineal multinivel
modelos de coeficientes aleatorios
modelos de medias
estrategias de analisis de datos
How to Cite
Abstract
We propose a mixed-effects linear model for analyzing growth curves data obtained using a two-way classification experiment. The model combines an unconstrained means model and a regression model on the time, in which the coefficients are considered random. The model allows for experimental unit covariates so as to study the trend and the variability of the individual growth curves. Comments on data analysis strategies are provided. An application of the model is illustrated using a data-set comes from a chrysanthemum growth experiment.
References
Bryk, A.S.; Raudenbush, S. W. (1992) Hierarchical Linear Models: Applications and Data Analysis Methods. Sage Publications, Thousand Oaks CA.
Chatfield, C. (1995) Problem Solving; A Statistician’s Guide, Second Edition. Chapman and Hall, London.
Cox, D.R.; Snell, E.J. (1981) Applied Statistics. Chapman and Hall, London.
Crowder, M.J.; Hand, D.J. (1990) Analysis of Repeated Measures. Chapman and Hall, London.
Daniel, C.; Wood, F. S. (1980) Fitting Equations to Data. John Wiley, New York.
Dempster, A.P.; Laird, M.N.; Rubin, D.B. (1977) “Maximum likelihood from incomplete data via the EM algorithm”, Journal of The Royal Statistical Society, Series B 39: 1–38.
Dempster, A.P.; Rubin, D.B.; Tsutakawa, R.K. (1981) “Estimation in covariance components models”, Journal of the American Statistical Association 76: 341–353.
Diggle, P.J.; Liang, K.Y.; Zeger, S.L. (1994) Analysis of Longitudinal Data. Oxford University Press, New York.
Fussillier, L. (1996) Un Modelo para Analizar Curvas de Crecimiento. Tesina, Especialización en Métodos Estadísticos, Universidad Veracruzana, Veracruz, México.
Geisser, S. (1980) “Growth curve analysis”, in: A.M. Kshirsagar (Ed.) Handbook of Statistics, Vol. 1, North-Holland, Amsterdam: 89–115.
Gibbons, R.D. (2000) “Mixed-Effects models for mental health services research”, Health Services 6 Outcomes Research Methodology 1(2): 91–129.
Gibbons, R.D.; Bhaumik. D.K. (2001) “Weighted random-effects regression models with application to interlaboratory calibration”, Technometrics 43(2): 192–198.
Goldstein, H. (1986) “Multilevel mixed linear model analysis using iterative generalized least squares”, Biometrika 73: 43–56.
Goldstein, H. (1987) Multilevel Models in Educational and Social Research. Grimn,
London.
Goldstein, H. (1989a) “Models for multilevel response variables with an application to growth curves”, in: D.R. Bock (Ed.) Multilevel Analysis of Educational Data, Academic Press, New York.
Goldstein, H. (1989b) “Restricted unbiased iterative generalized least squares estimation”, Biometrika 76: 622–623.
Goldstein, H. (1995) Multilevel Statistical Models, Second Edition. Halsted Press, New York.
Hand, D.J.; Crowder, M.J. (1996) Practical Longitudinal Data Analysis. Chapman and Hall, London.
Hartiey, H.O.; Rao, J.N.K. (1967) “Maximum-likelihood estimation for the mixed analysis of variance model”, Biometrika 54: 93–108.
Henderson, C.R. (1986) “Recent developments in variance and covariance estimation”, Journal of Animal Sciences 63: 208–216.
Hocking, R.R. (1985) The Analysis of Linear Models. Brooks/Cole, Monterey, CA.
Hui, S.L. (1984) “Curve fitting for repeated measurements made at irregular time-points”, Biometrics 40: 691–697.
Jenrich, R.I.; Schluchter, M.D. (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) “Review of five multilevel analysis programs: BMDP-5V, GENMOD, HLM, ML3, VARCL”, The American Statistician 48: 324–335.
Langford, I.H.; Lewis, T. (1998) “Outliers in multilevel data”, Journal of The Royal Statistical Society Series A 161: 121–160.
Laird, N.M.; Ware. J.H. (1982) “Random-effects models for longitudinal data”, Biometrics 38: 963–974.
Lindsey, J. K.(1993) Models for Repeated Measurements. Oxford University Press, New York.
Longford, N. (1987) “A fast scoring algorithm for maximum likelihood estimation in unbalanced mixed-models with nested random effects”, Biometrika 74: 817–827.
Longford, N. T. (1993) Random Coefficient Models. Oxford University Press, New York.
Longford, N.T. (1995) “Random coefficient models”, in: G. Arminger, C.C. Clogg & M.E. Sobel (Eds.) Handbook of Statistical Models for the Social and Behavioral Sciences. Plenum Press, New York.
Mason, W.M.; Wong, G.Y.; Entwisle, B. (1983) “Contextual analysis through the multilevel linear model”, in: S. Leinhardt (Ed.) Sociological Methodology, Jossey-Bass, San Francisco.
McClean, R.A.; Sanders, W.L.; Stroup, W.W. (1991) “A unified approach to mixed linear models”, The American Statistician 45: 54–64.
Ojeda, M. M.; Juárez C. S. (1996) “Biplot display for diagnostics in a two-level regression model for growth curves analysis”, Computational Statistics and Data Analysis 22: 583–597.
Pan, H.Q.; Goldstein. H.; Guo, D. (1989) “A two-level cross-sectional model using grafted polynomials”, Annals of Human Biology 19: 337–346.
Potthoff, P.F.; Rao, J.N.K. (1964) “A generalized multivariate analysis of variance model useful especially for growth curve problems”, Biometrika 51: 313–326.
Prosser, R., Rassbash, J.; Goldstein, H. (1990) ML3: Software for Tree Level Analysis. Institute of Education, University of London, London.
Prosser, R., Rassbash, J.; Goldstein, H. (1991) Data Analysis with ML3. Institute of Education, University of London, London.
Ramsay, J.O.; Silverman, B.W. (1997) The Analysis of Functional Data. Springer–Verlag, New York.
Rao, C.R. (1965) “The Theory of least squares when the parameters are stochastic and its application to growth curves”, Biometrics 52: 447–458.
Rawlings, J.O., Pantula; S.G.; Dicky, D.A. (1998) Applied Regression Analysis: A Research Tool. Second Edition. Springer–Verlag, New York.
Rencher, A. C. (2000). Linear Models in Statistics. John Wiley & Sons, New York.
Searle, S.R.; Casella, G.; McCunoch, C.E. (1992) Variance Components. John Wiley & Sons, New York.
Seber, G.A.F. (1984) Multivariate Observations. John Wiley & Sons, New York.
Singer. J. D. (1998). Using SAS PROC MIXED to fit multilevel models, hierarchical models and individual growth models. Journal of Educational and Behavioral Statistics. 24, No. 4, 323-355.
Shi, L.; Ojeda, M.M. (2004) “Local influence in multilevel regression for growth curves”, Journal of Multivariate Analisys 91(2): 282–304.
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.
Verbeke, G.; Molenberghs (eds.) (1997) Linear Mixed Models in Practice: A SAS-Oriented Approach. Springer Verlag, New York.
Vonesh, E.F.; Chichilli, V.M. (1997) Linear and Nonlinear Models for the Analisys of Repeat Measurements. Marcel Dekker, New York.
Weissfeid, L. A.; Kshirsagar, A. M. (1992) “A modified growth curve model and its application to clinical studies”, Australian Journal of Statistics 34: 161–168.
Yang, M., Goldstein, H.; Rasbash. J. (1996) MLn Macros for Advanced Multilevel Modelling. Institute of Education, University of London, London.
Ware, J. H. (1985) “Linear models for the analysis of longitudinal studies”, The American Statistician 39: 95–101.