Skip to main content
SpringerLink
Log in

Parametric modelling of growth curve data: An overview

  • Published:
Test Aims and scope Submit manuscript

Abstract

In the past two decades a parametric multivariate regression modelling approach for analyzing growth curve data has achieved prominence. The approach, which has several advantages over classical analysis-of-variance and general multivariate approaches, consists of postulating, fitting, evaluating, and comparing parametric models for the data's mean structure and covariance structure. This article provides an overview of the approach, using unified terminology and notation. Well-established models and some developed more recently are described, with emphasis given to those models that allow for nonstationarity and for measurement times that differ across subjects and are unequally spaced. Graphical diagnostics that can assist with model postulation and evaluation are discussed, as are more formal methods for fitting and comparing models. Three examples serve to illustrate the methodology and to reveal the relative strengths and weaknesses of the various parametric models.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Altman, N.S. (1990). Kernel smoothing of data with correlated errors.Journal of the American Statistical Association,90, 508–515.

    Article  MathSciNet  Google Scholar 

  • Altman, N.S. and G. Casella (1995). Nonparametric empirical Bayes growth curve analysis.Journal of the American Statistical Association,85, 749–759.

    Article  MathSciNet  Google Scholar 

  • Boularan, J., L. Ferré and P. Vieu (1994). Growth curves: a two-stage nonparametric approach.Journal of Statistical Planning and Inference,38, 327–350.

    Article  MATH  MathSciNet  Google Scholar 

  • Boularan, J., L. Ferré and P. Vieu (1995). A nonparametric model for unbalanced longitudinal data with application to geophysical data.Computational Statistics,10, 285–298.

    MATH  MathSciNet  Google Scholar 

  • Byrne, P.J. and S.F. Arnold (1983). Inference about multivariate means for a nonstationary autoregressive model.Journal of the American Statistical Association,78, 850–855.

    Article  MathSciNet  Google Scholar 

  • Chen, H. (1995). Analysis of Irregularly Spaced Longitudinal Data Using a Kernel Smoothing Approach. Unpublished Ph.D. Dissertation, Department of Statistics and Actuarial Science, The University of Iowa.

  • Crowder, M.J. and D.J. Hand (1990).Analysis of Repeated Measures, Chapman & Hall, London.

    MATH  Google Scholar 

  • Cryer, J.D. (1986).Time Series Analysis, PWS-Kent, Boston.

    MATH  Google Scholar 

  • Diggle, P.J. and M.F. Hutchinson (1988). On spline smoothing with autocorrelated errors.Australian Journal of Statistics,31, 166–182.

    MathSciNet  Google Scholar 

  • Diggle, P.J., K.Y. Liang and S.L. Zeger (1994).Analysis of Longitudinal Data. Oxford University Press, New York.

    Google Scholar 

  • Diggle, P.J. and A.P. Verbyla (1998). Nonparametric estimation of covariance structure in longitudinal data.Biometrics,54, 401–415.

    Article  MATH  Google Scholar 

  • Ferreira, E., V. Núñez-Antón and J. Rodríguez-Poo (1997). Kernel regression estimates of growth curves using nonstationary correlated errors.Statistics and Probability Letters,34, 413–423.

    Article  MATH  MathSciNet  Google Scholar 

  • Fraiman, R. and J. Meloche (1994). Smoothing dependent observations.Statistics and Probability Letters,21, 203–214.

    Article  MATH  MathSciNet  Google Scholar 

  • Gantz, B.J., R.S. Tyler, J.F. Knutson, G.G. Woodworth, P. Abbas, B.F. McCabe, J. Hinrichs, N. Tye-Murray, C. Lansing, F. Kuk and C. Brown (1988). Evaluation of five different cochlear implant designs: Audiologic assessment and predictors of performance.Laryngoscope,98, 1100–1106.

    Article  Google Scholar 

  • Glasbey, C.A. (1988). Standard errors resilient to error variance misspecification.Biometrika,75, 201–206.

    Article  MATH  Google Scholar 

  • Gumpertz, M.L. and S.G. Pantula (1992). Nonlinear regression with variance components.Journal of the American Statistical Association,87, 201–209.

    Article  MATH  MathSciNet  Google Scholar 

  • Hall, P. and P. Patil (1994). Properties of nonparametric estimators of autoco-variance for stationary random fields.Probability Theory and Related Fields,99, 399–424.

    Article  MATH  MathSciNet  Google Scholar 

  • Hall, P., N.I. Fisher and B. Hoffmann (1994). On the nonparametric estimation of covariance functions.Annals of Statistics,22, 2115–2134.

    MATH  MathSciNet  Google Scholar 

  • Hart, J.D. and T.E. Wehrly (1986). Kernel regression estimation using repeated measurements data.Journal of the American Statistical Association,81, 1080–1088.

    Article  MATH  MathSciNet  Google Scholar 

  • Harville, D.A. (1977). Maximum likelihood approaches to variance component estimation and to related problems.Journal of the American Statistical Association,72, 320–338.

    Article  MATH  MathSciNet  Google Scholar 

  • Hastie, T. and R. Tibshirani (1990).Generalized Additive Models. Chapman & Hall, London.

    MATH  Google Scholar 

  • IMSL, Inc. (1991a).Fortran Subroutines for Mathematical Applications.MATH LIBRARY Version 2.0 IMSL, Inc., Houston.

    Google Scholar 

  • IMSL, Inc., (1991b).Fortran Subroutines for Mathematical Applications, STAT LIBRARY Version 2.0. IMSL, Inc., Houston.

    Google Scholar 

  • Jennrich, R.L. and M.D. Schluchter (1986). Unbalanced repeated-measures models with structured covariance matrices.Biometrics,42, 805–820.

    Article  MATH  MathSciNet  Google Scholar 

  • Johnson, K.L. (1989). Higher-order antedependence models. Unpublished Ph.D. Dissertation, Department of Statistics, Pennsylvania State University.

  • Kenward, M.C. (1987). A method for comparing profiles of repeated measurments.Applied Statistics,36, 296–308.

    Article  MathSciNet  Google Scholar 

  • Laird, N.M. and J.H. Ware (1982). Random effects models for longitudinal data.Biometrics,38, 963–974.

    Article  MATH  Google Scholar 

  • Lindsey, J.K. (1993).Models for Repeated Measurements, Oxford University Press, Oxford.

    Google Scholar 

  • Lindstrom, M.J. and D.M. Bates (1990). Nonlinear mixed effects models for repeated measures data.Biometrics,46, 673–687.

    Article  MathSciNet  Google Scholar 

  • Macchiavelli, R.E. and S.F. Arnold (1994). Variable order antedependence models.Communications in Statistics.Theory and Methods,23, 2683–2699.

    MATH  MathSciNet  Google Scholar 

  • Moyeed, R.A. and P.J. Diggle (1994). Rates of convergence in semi-parametric modelling of longitudinal data.Australian Journal of Statistics,36, 75–93.

    MATH  MathSciNet  Google Scholar 

  • Müller, H.G. (1988).Nonparametric Regression Analysis of Longitudinal Data. Lecture Notes in Statistics. 41. Springer-Verlag, Berlin.

    MATH  Google Scholar 

  • Núñez-Antón, V. (1997). Longitudinal data analysis: non-stationary error structures and antedependent models.Applied Stochastic Models and Data Analysis,13, 279–287.

    Article  MATH  Google Scholar 

  • Núñez-Antón, V. and G.G. Woodworth (1994). Analysis of longitudinal data with unequally spaced observations and time-dependent correlated errors.Biometrics,50, 445–456.

    Article  MATH  Google Scholar 

  • Núñez-Antón, V., J.M. Rodríguez-Poo and P. Vieu (1999). Longitudinal data with nonstationary errors: a nonparametric three-stage approach.Test,8, 201–231.

    MATH  MathSciNet  Google Scholar 

  • Núñez-Antón, V. and D.L. Zimmerman (2000). Modelling nonstationary longitudinal data.Biometrics,56 699–705.

    Article  MATH  Google Scholar 

  • Núñez-Antón, V. and D.L. Zimmerman (2001). Modelización de datos longitudinales con estructuras de covarianza no estacionarias: modelos de coeficientes aleatorios frente a modelos alternativos.Qüestiió, in press.

  • Pourahmadi, M. (1999). Joint mean-covariance models with applications to longitudinal data: unconstrained parameterisation.Biometrika,86, 677–690.

    Article  MATH  MathSciNet  Google Scholar 

  • Rice, J.A. and B.W. Silverman (1991). Estimating the mean and covariance structure nonparametrically when the data are curves.Journal of the Royal Statistical Society, B,53, 233–243.

    MATH  MathSciNet  Google Scholar 

  • Rosenbaum, P.R. and D.B. Rubin (1983). The central, role of the propensity score in observation studies for causal effects.Biometrika,70, 41–55.

    Article  MATH  MathSciNet  Google Scholar 

  • Sampson, P.D. and P. Guttorp (1992). Nonparametric representation of nonstationary spatial covariance structure.Journal of the American Statistical Association,87, 108–119.

    Article  Google Scholar 

  • Sandland, R.L. and C.A. McGilchrist (1979). Stochastic growth curve analysis.Biometrics,35, 255–271.

    Article  MATH  Google Scholar 

  • SAS Institute (1998).SAS/STAT Software: Changes and Enhancements through Release 6.12. SAS Institute, North Carolina.

    Google Scholar 

  • Shapiro, A. and J.D. Botha (1991). Variogram fitting with a general class of conditionally nonnegative definte functions.Computational Statistics and Data Analysis,11, 87–96.

    Article  Google Scholar 

  • Staniswalis, J.G. and J.J. Lee (1998). Nonparametric regression analysis of longitudinal data.Journal of the American Statistical Association,93, 1403–1418.

    Article  MATH  MathSciNet  Google Scholar 

  • Verbeke, G. and G. Molenberghs (2000).Linear Mixed Models for Longitudinal Data. Springer Verlag, New York.

    MATH  Google Scholar 

  • Wang, Y. and J.M.G. Taylor (1995). Inference for smooth curves in longitudinal data with application to an AIDS clinical trial.Statistics in Medicine,14, 1205–1218.

    Google Scholar 

  • Wolfinger, R.D. (1996). Heterogeneous variance-covariance structures for repeated measures.Journal of Agricultural, Biological, and Environmental Statistics,1, 205–230.

    Article  MathSciNet  Google Scholar 

  • Zeger, S.L. and P.J. Diggle (1994). Semiparametric models for longitudinal data with application to CD4 cell numbers in HIV seroconverters.Biometrics,50, 689–699.

    Article  MATH  Google Scholar 

  • Zimmerman, D.L., and V. Núñez-Antón (1997). Structured antedependence models for longitudinal data. InModelling Longitudinal and Spatially Correlated Data. Methods, Applications, and Future Directions, 63–76 (T.G. Gregoire, D.R. Brillinger, P.J. Diggle, E. Russek-Cohen, W.G. Warren, and R. Wolfinger, eds.) Springer-Verlag, New York.

    Google Scholar 

  • Zimmerman, D.L., V. Núñez-Antón, and H. El Barmi, (1988). Computational aspects of likelihood-based estimation of first-order antedependence models.Journal of Statistical Computation and Simulation,60, 67–84.

    Google Scholar 

  • Zimmerman, D.L. (2000). Viewing the correlation structure of longitudinal data through a PRISM.The American Statistician,54, 310–318.

    Article  MathSciNet  Google Scholar 

References

  • Gregoire, T.G. and O. Schabenberger (1996). A nonlinear mixed-effects model to predict cumulative bole volume of standing trees.Journal of Applied Statistics,23, 257–271.

    Article  Google Scholar 

  • Jones, R.H. (1990). Serial correlation or random subject effects.Communications in Statistics—Simulation and Computation,19, 1105–1123.

    MATH  Google Scholar 

  • Laird, N. and J.H. Ware (1982). Random effects models for longitudinal data.Biometrics,38, 963–974.

    Article  MATH  Google Scholar 

  • Schabenberger, O. and F.J. Pierce (2001).Contemporary Statistical Models for the Plant and Soil Sciences. CRC-Press.

References

  • Altman, N.S. and G. Casella (1995). Nonparametric empirical Bayes growth curve analysis.Journal of the American Statistical Society,90, 508–515.

    MATH  MathSciNet  Google Scholar 

  • Berkey, C.S. (1982). Bayesian approach for a nonlinear growth model.Biometrics,38, 953–961.

    Article  Google Scholar 

  • Castro, P.E., W.H. Lawton and E.A. Sylvestre (1986). Principal modes of variation for processes with continuous sample curves.Technometrics,28, 329–337.

    Article  MATH  Google Scholar 

  • Gasser, T., H.-G. Müller, W. Köhler, L. Molinari and A. Prader (1984). Nonparametric regression analysis of growth curves.Annals of Statistics,12, 210–229.

    MATH  MathSciNet  Google Scholar 

  • Graybill, F. A. (1976).Theory and Application of the Linear Model. Duxbury, Massachusetts.

  • Härdle, W. and J.S. Marron (1990). Semiparametric comparison of regression curves.Annals of Statistics,18, 63–89.

    MATH  MathSciNet  Google Scholar 

  • Hui, S.I. and J.O. Berger (1983). Empirical Bayes estimation of rates in longitudinal studies.Journal of the American Statistical Society,78, 753–760.

    MATH  Google Scholar 

  • Núñez-Antón, V., J.M. Rodríguez-Poo and P. Vieu (1999). Longitudinal data with nonstationary errors: a nonparametric three-stage approach.Test,8, 201–231.

    MATH  MathSciNet  Google Scholar 

  • Pham, D.T., J. Möcks, W. Köhler and T. Gasser (1987). Variable latencies of noisy signals: Estimation and testing in brain potential data.Biometrika,74, 525–533.

    Article  MATH  Google Scholar 

  • Ramsay, J.O. and C.J. Dalzell (1991). Some tools for functional data analysis (with discussion).Journal of the Royal Statistical Society, B,53, 539–561.

    MATH  MathSciNet  Google Scholar 

  • Ramsay, J.O. and B.W. Silverman (1997).Functional Data Analysis, Springer-Verlag.

  • Rao, C.R. (1987). Predictions of future observations in growth curve models (with discussion).Statistical Science,2, 434–471.

    MATH  MathSciNet  Google Scholar 

  • Ritov, Y. (1989). Estimating a signal with noisy nuisance parameters.Biometrika,76, 31–37.

    Article  MATH  MathSciNet  Google Scholar 

  • Silverman, B.W. (1995). Incorporating parametric effects into functional principal components analysis.Journal of the Royal Statistical Society, B,57, 673–689.

    MATH  Google Scholar 

  • Strenio, J.F., H.I. Weisberg and A.S. Bryk (1983). Empirical Bayes estimation of individual growth-curve parameters and their relationship to covariates.Biometrics,39, 71–86.

    Article  MATH  MathSciNet  Google Scholar 

  • Wang, K. and T. Gasser (1997). Alignment of curves by dynamic time warping.Annals of Statistics,25, 1251–1276.

    Article  MATH  MathSciNet  Google Scholar 

References

  • Kenward, M.G. and J.H. Roger (1997). Small sample inference for fixed effects estimators from restricted maximum likelihood.Biometrics,53, 983–997.

    Article  MATH  Google Scholar 

  • Satterthwaite, F.F. (1941). Synthesis of variance.Psychometrika,6, 309–316.

    Article  MATH  MathSciNet  Google Scholar 

  • Speed, T.P. (1991). Comment on: That BLUP is a good thing: the estimation of random effects, by G.K. Robinson.Statistical Science,6, 44.

    Google Scholar 

  • Verbyla, A.P., B.R. Cullis, M.G. Kenward and S.J. Welham (1999). The analysis of designed experiments and longitudinal data using smoothing splines (with discussion).Applied. Statistics,48, 269–312.

    MATH  Google Scholar 

References

  • Brown, H. and R. Prescott (1999).Applied Mixed Model in Medicine. John Wiley.

  • Carlin, B.P. and T.A. Louis (1996).Bayes and Empirical Bayes Methods for Data Analysis. Chapman and Hall.

  • Davidian, M. and D.M. Giltinan (1995).Nonlinear Models For Repeated Measurement Data. Chapman and Hall.

  • Diggle, P.J. (1988). An approach to the analysis of repeated measures.Biometrics,44, 959–971.

    Article  MATH  MathSciNet  Google Scholar 

  • Goldstein, H. (1979).The Design and Analysis of Longitudinal Studies. Academic Press.

  • Goldstein, H. (1995).Multilevel Statistical Models. Edward Arnold.

  • Hand, D.J., and M.J., Crowder (1995).Practical Longitudinal Data Analysis. Chapman and Hall.

  • Hedeker, D. and R.D. Gibbons (1994). A random-effects ordinal regression model for multilevel analysis.Biometrics,50, 933–944.

    Article  MATH  Google Scholar 

  • Hedeker, D. and R.D. Gibbons (1996). MIXOR: A computer program for mixed-effects ordinal regression analysis.Computer Methods and Programs in Biomedicine,49, 157–176.

    Article  Google Scholar 

  • Kshirsagar, A.M., and W.B. Smith (1995).Growth Curves. Marcel Dekker.

  • Lesaffre, E. and G. Verbeke (1998). Local influence in linear mixed models.Biometrics,54, 570–582.

    Article  MATH  Google Scholar 

  • Longford, N.T. (1993)Random Coefficient Models. Oxford University Press.

  • Searle, S.R., G. Casella and C.E. McCulloch (1992).Variance Components. John Wiley.

  • Verbeke, G., E. Lesaffre and L.J. Brant (1998). The detection of residual serial correlation in linear mixed models.Statistics in Medicine,17, 1391–1402.

    Article  Google Scholar 

  • Verbeke, G. and G. Molenberghs (1997).Linear Mixed Models in Practice: A SAS Oriented Approach. Springer-Verlag.

  • Vonesh, E.F., and V.M. Chinchilli (1997).Linear and Nonlinear Models for the Analysis of Repeated Measurements. Marcel Dekker.

References

  • Davison, A.C. and S. Sardy (2000). The partial scatterplot matrix.Journal of Computational and Graphical Statistics,9, 750–758.

    Article  MathSciNet  Google Scholar 

  • Johnson, R.A. and D.W. Wichern (1998).Applied Multivariate Statistical Analysis, fourth edition. Prentice Hall.

  • Pourahmadi, M. (2001).Foundations of Time Series Analysis and Prediction Theory. John Wiley.

  • Searle, S.R., G. Casella and C.E. McCulloch (1992).Variance Components. John Wiley.

References

  • Besse, P., H. Cardot and F. Ferraty (1997). Simultaneous nonparametric regressions of unbalanced longitudinal data.Computational Statistics and Data Analysis,24, 255–270.

    Article  MATH  MathSciNet  Google Scholar 

  • Bosq, D. (2000).Linear Processes in Function Spaces. Theory and Applications, Lecture Notes in Statistics,149, Springer Verlag.

  • Kneip, A. and T. Gasser (1992). Statistical tools to analyse data representing a sample of curves.Annals of Statistics,20, 1266–1305.

    MATH  MathSciNet  Google Scholar 

  • Ramsay, J. and X. Li (1998). Curve registration.Journal of the Royal Statistical Society, B.,60, 351–363

    Article  MATH  MathSciNet  Google Scholar 

  • Ramsay, J. and B. Silverman (1996).Functional Data Analysis, Springer-Verlag.

  • STAPH (2000).Actes du groupe de travail en Statistique Fonctionnelle recueillis par H. Cardot, F. Ferraty, Y. Romain, P. Sarda et P. Vieu. Laboratoire de Statistiqué et Probalité, Toulouse III, France.

    Google Scholar 

References

  • Crowder, M.I. and D.J. Hand (1990).Analysis of Repeated Measures. Chapman and Hall, London.

    MATH  Google Scholar 

  • Delgado, M.A. and W. González-Manteiga (2001). Significance testing in nonparametric regression based on the bootstrap.Annals of Statistics, in press.

  • Diggle, P.J., K.Y. Liang and S.L. Zeger (1994).Analysis of Longitudinal Data. Oxford University Press, New York.

    Google Scholar 

  • Fan, J. and Y. Li (1996). Consistent model specification tests: omitted variables and semiparametric functional forms.Econometrica,64, 865–890.

    Article  MATH  MathSciNet  Google Scholar 

  • Ferraty, F. and P. Vieu (2001). Functional nonparametric model for scalar response. Preprint.

  • Frank, I.E. and J.H. Friedman (1993). A statistical view of some chemometrics regression tools (with discussion).Technometrics,35, 109–148.

    Article  MATH  Google Scholar 

  • Green, P.J. (1985). Linear models for field trials, smoothing and cross-validation.Biometrika,72, 523–537.

    Article  Google Scholar 

  • Hand, D.J. and M.J. Crowder (1996).Practical Longitudinal Data Analysis. Chapman and Hall, London.

    MATH  Google Scholar 

  • Härdle, W. and E. Mammen (1993). Comparing nonparametric versus parametric regression fits.Annals of Statistics,21, 1926–1947.

    MATH  MathSciNet  Google Scholar 

  • Hastie, T., A. Buja and R. Tibshirani (1995). Penalized discriminant analysis.Annals of Statistics,23, 73–102.

    MATH  MathSciNet  Google Scholar 

  • Hastie, T. and C. Mallows (1993). Discussion of: A Statistical View of Some Chemometrics Regression Tools, by I.E. Frank and J.H. Friedman.Technometrics,35, 140–143.

  • Hastie, T.J. and R.J. Tibshirani (1990).Generalized Additive Models. Chapman and Hall, London.

    MATH  Google Scholar 

  • Hidalgo, J. (1999). Nonparametric tests for model selection with time series data.Test,8, 365–398.

    MATH  MathSciNet  Google Scholar 

  • Jones, R.H. (1990). Serial correlation or random subject effects?Communications in Statistics—Simulation and Computation,19, 1105–1123.

    MATH  Google Scholar 

  • Kenward, M.G. and J.H. Roger (1997). Small sample inference for fixed effects estimators from restricted maximum likelihood.Biometrics,53, 983–997.

    Article  MATH  Google Scholar 

  • Kuchibhata, M. and J.D. Hart (1996). Smoothing based lack of fit tests: variation on a theme.Nonparametric Statistics,7, 1–22.

    MathSciNet  Google Scholar 

  • Lee, A. (1994). Asymptotic distribution of the Ullah-type specification test against the nonparametric alternative.Journal of Qualitative Economics,10, 73–92.

    Google Scholar 

  • Núñez-Antón, V., J.M. Rodríguez-Póo and P. Vieu (1999). Longitudinal data with nonstationary errors: a nonparametric three-stage approach.Test,8, 201–231.

    MATH  MathSciNet  Google Scholar 

  • Núñez-Antón, V. and D.L. Zimmerman (2000). Modelling nonstationary longitudinal data.Biometrics,56, 699–705.

    Article  MATH  Google Scholar 

  • Núñez-Antón, V. and D.L. Zimmerman (2001). Modelización de datos longitudinales con estructuras de covarianza no estacionarias: modelos de coeficientes aleatorios frente a modelos alternativos.Qüestiió, in press.

  • Verbyla, A.P., B.R. Cullis, M.G. Kenward and S.J. Welham (1999). The analysis of designed experiments and longitudinal data by using smoothing splines (with discussion).Applied Statistics,48, 269–311.

    MATH  Google Scholar 

  • Vieu, P. (2001).Personal Communication.

  • Zheng, J. (1996). A consistent test of functional form via nonparametric estimation techniques.Journal of Econometrics,75, 263–289.

    Article  MATH  MathSciNet  Google Scholar 

  • Zimmerman, D.L. and V. Núñez-Antón (1997). Structured antedependence models for longitudinal data. InModelling Longitudinal and Spatially Correlated Data. Methods, Applications, and Future Directions, 63–76 (T.G. Gregoire, D.R. Brillinger, P.J. Diggle, E. Russek-Cohen, W.G. Warren, and R. Wolfinger, eds.) Springer-Verlag, New York.

    Google Scholar 

  • Zimmerman, D.L. (2000). Viewing the correlation structure of longitudinal data through a PRISM.The American Statistician,54(4), 310–318.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics and Actuarial Science, The University of Iowa, USA

    Dale L. Zimmerman

  2. Departamento de Econometría y Estadística (E.A. III), Universidad del País Vasco, Avenida del Lehendakari Aguirre 83, 48015, Bilbao, Spain

    Vicente Núñez-Antón

  3. Yale University, USA

    Timothy G. Gregoire

  4. Virginia Polytechnic Institute and State University, USA

    Oliver Schabenberger

  5. Texas A&M Univesity, USA

    Jeffrey D. Hart

  6. London School of Hygiene and Tropical Medicine, UK

    Michael G. Kenward

  7. Limburgs Universitair Centrum, Belgium

    Geert Molenberghs

  8. Katholieke Universiteit Leuven, Belgium

    Geert Verbeke

  9. Northern Illinois University, USA

    Mohsen Pourahmadi

  10. Université Paul Sabatier, France

    Philippe Vieu

Authors
  1. Dale L. Zimmerman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vicente Núñez-Antón
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Timothy G. Gregoire
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Oliver Schabenberger
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Jeffrey D. Hart
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Michael G. Kenward
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Geert Molenberghs
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. Geert Verbeke
    View author publications

    You can also search for this author in PubMed Google Scholar

  9. Mohsen Pourahmadi
    View author publications

    You can also search for this author in PubMed Google Scholar

  10. Philippe Vieu
    View author publications

    You can also search for this author in PubMed Google Scholar

  11. Dela L. Zimmerman
    View author publications

    You can also search for this author in PubMed Google Scholar

  12. Vicente Núñez-Antón
    View author publications

    You can also search for this author in PubMed Google Scholar

  13. Timothy G. Gregoire
    View author publications

    You can also search for this author in PubMed Google Scholar

  14. Oliver Schabenberger
    View author publications

    You can also search for this author in PubMed Google Scholar

  15. Jeffrey D. Hart
    View author publications

    You can also search for this author in PubMed Google Scholar

  16. Michael G. Kenward
    View author publications

    You can also search for this author in PubMed Google Scholar

  17. Geert Molenberghs
    View author publications

    You can also search for this author in PubMed Google Scholar

  18. Geert Verbeke
    View author publications

    You can also search for this author in PubMed Google Scholar

  19. Mohsen Pourahmadi
    View author publications

    You can also search for this author in PubMed Google Scholar

  20. Philippe Vieu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vicente Núñez-Antón.

Additional information

Zimmerman's work was partially supported by NSF grant 9628612. Núñez-Autón's work was supported by DGESIC (Ministerio Español de Educación y Cultura). Universidad del País Vasco (UPV/EHU) and Gobierno Vasco, under research grants PB98-0149, UPV 038.321-HA129/99 and PI-1999-46.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zimmerman, D.L., Núñez-Antón, V., Gregoire, T.G. et al. Parametric modelling of growth curve data: An overview. Test 10, 1–73 (2001). https://doi.org/10.1007/BF02595823

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02595823

Key words

AMS subject classification

Search

Navigation