Analysis of Linear and Non-Linear Growth Models with Random Parameters

  • N. M. Laird
Part of the Advanced Series in Agricultural Sciences book series (AGRICULTURAL, volume 18)


This paper discusses the analysis of growth data using mixed models. Both Bayes and frequentist approaches are outlined for the linear model. The use of the EM algorithm is shown to offer a flexible and straightforward computational approach to the analysis. Extensions to non-linear models are described.


Stat Assoc Posterior Mode REML Estimate Linear Growth Model Individual Growth Curve 
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  1. Berkey CS (1982) Bayesian approach for a non-linear growth model. Biometrics 38:953–961PubMedCrossRefGoogle Scholar
  2. Berkey CS, Laird NM (1986) Non-linear growth curve analyses: estimating the population parameters. Ann Hum Biol 13:111–128PubMedCrossRefGoogle Scholar
  3. Bock RD, Thissen D (1980) Statistical problems of fitting individual growth curves. In: Johnstone FE, Roche AF, Susanne C (eds) Human physical growth and maturation. Plenum Press, New YorkGoogle Scholar
  4. Credeur KR (1980) Estimation from incomplete multinomial data. PhD Thesis, Harvard UnivGoogle Scholar
  5. Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the EM algorithm. J R Stat Soc Ser B 39:1–38Google Scholar
  6. Feam T (1975) A Bayesian approach to growth curves. Biometrika 62:89–100CrossRefGoogle Scholar
  7. Harville DA (1976) Extension of the Gauss-Markov theorem to include the estimation of random effects. Ann Stat 4:384–395CrossRefGoogle Scholar
  8. Harville DA (1977) Maximum likelihood approaches to variance component estimation and to related problems. J Am Stat Assoc 72:320–340CrossRefGoogle Scholar
  9. Harville DA and Mee RW (1984) A mixed-model procedure for analyzing ordered categorical data. Biometrics 40:393–408CrossRefGoogle Scholar
  10. Kackar RN, Harville DA (1984) Approximations for standard errors of estimates of fixed and random effects in mixed linear models. J Am Stat Assoc 79:853–862CrossRefGoogle Scholar
  11. Laird NM (1978a) Bayesian estimation methods for two-way contingency tables. Biometrika 65:581–590CrossRefGoogle Scholar
  12. Laird NM (1978b) Non-parametric maximum-likelihood estimation of a mixing distribution. J Am Stat Assoc 73:805–813CrossRefGoogle Scholar
  13. Laird NM (1982a) Computation of variance components using the E-M algorithm. J Stat Comput Simul 14:295–303CrossRefGoogle Scholar
  14. Laird NM (1982b) Empirical Bayes estimates using the non-parametric maximum likelihood estimate of the prior. J Stat Comput Simul 1:211–220CrossRefGoogle Scholar
  15. Laird NM, Louis TA (1986) Empirical Bayes confidence intervals based on bootstrap samples. Tech Rep, Dep Biostatistics, Harvard School of Public Health, BostonGoogle Scholar
  16. Lange N, Laird NM (1986) Random-effects and growth curve modeling for balanced and complete longitudinal data. Tech Rep, Dep Biostatistics, Harvard School of Public Health, BostonGoogle Scholar
  17. Lindley DV, Smith AFM (1972) Bayes estimates for the linear model. J R Stat Soc Ser B 34:1–41Google Scholar
  18. Lindsay BG (1983) The geometry of mixture likelihoods: a general theory. Ann Stat 11:86– 94CrossRefGoogle Scholar
  19. Mallet A (1986) A maximum likelihood estimation method for random coefficient regression models. Biometrika 73:645–656CrossRefGoogle Scholar
  20. Morris CN (1983a) Parametric empirical Bayes inference: theory and applications. J Am Stat Assoc 78:47–59CrossRefGoogle Scholar
  21. Morris CN (1983b) Parametric empirical bayes confidence intervals. In: Box GEP, Leonard T, Wu CF (eds) Scientific inference, data analysis, and robustness. Academic Press, New York pp 25–50Google Scholar
  22. O’Hagan A (1976) On posterior joint and marginal models. Biometrika 63:329–333CrossRefGoogle Scholar
  23. Patterson HD, Thompson R (1971) Recovery of interblock information when block sizes are unequal. Biometrika 58:545–554CrossRefGoogle Scholar
  24. Racine-Poon A (1985) A Bayesian approach to non-linear random effects models. Biometrics 41:1015–1024PubMedCrossRefGoogle Scholar
  25. Rao CR (1975) Simultaneous estimation of parameters in different linear models and applications to biometric problems. Biometrics 31:545–554PubMedCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  • N. M. Laird
    • 1
  1. 1.Department of BiostatisticsHarvard School of Public HealthBostonUSA

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