On the Efficiency of Generally Balanced Designs Analysed by Restricted Maximum Likelihood

  • H. Monod
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 51)


Restricted maximum likelihood (reml) is commonly used in the analysis of incomplete block designs. With this method, treatment contrasts are estimated by generalized least squares, using an estimated variance-covariance matrix of the observations as if it were known. This leads to under-estimation of the variance of treatment contrasts, because uncertainty on the variance components is not adequately taken into account. To correct for this bias, Kackar and Harville (1984) and Kenward and Roger (1997) propose adjusted estimators of the treatment variance-covariance matrix, based on Taylor series expansions.

We consider small experiments with an orthogonal block structure. The adjusted estimator of Kenward and Roger (1997) is calculated when the design is generally balanced. A small modification is proposed that leads to a simple expression for the adjustment, as a function of the efficiency factors of the design, the variance components and the dimensions of the block strata. The behaviour of the adjusted estimator is assessed through a simulation study based on a semi-Latin square for twelve treatments.


generally balanced design restricted maximum likelihood 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Azaïs, J.-M., Monod, H. and Bailey, R.A.B. (1998). The influence of design on validity and efficiency of neighbour methods. Biometrics 54, 1374–1387.zbMATHCrossRefGoogle Scholar
  2. Bailey, R.A.B. (1984). Discussion of the paper by T. Tjur “Analysis of variance models in orthogonal designs”. International Statistical Review 52, 65–76.CrossRefGoogle Scholar
  3. Bailey, R.A.B. (1991). Strata for randomized experiments. Journal of the Royal Statistical Society B 53, 27–66.zbMATHGoogle Scholar
  4. Bailey, R.A.B. (1992). Efficient semi-Latin squares. Statistica Sinica 2, 413–437.MathSciNetzbMATHGoogle Scholar
  5. Bailey, R.A.B. (1993). General balance: artificial theory or practical relevance? In Proceedings of the International Conference on Linear Statistical Inference LINSTAT’93 Eds T. Calinsky and R. Kala, pp.171–184. Dordrecht: Kluwer.Google Scholar
  6. Bardin, A. and Azaïs, J.-M. (1991). Une hypothèse minimale pour une théorie des plans d’expérience randomisés. Revue de Statistique Appliquée 38, 5–20.Google Scholar
  7. Houtman, A.M. and Speed, T.R (1985). Balance in designed experiments with orthogonal block structure. Annals of Statistics 11, 1069–1085.MathSciNetGoogle Scholar
  8. Kackar, A.N. and Harville, D.A. (1984). Approximations for standard errors of estimators of fixed and random effects in mixed linear models. Journal of the American Statistical Association 79, 853–862.MathSciNetzbMATHGoogle Scholar
  9. Kenward, M. and Roger, J.H. (1997). Small sample inference for fixed effects from restricted maximum likelihood. Biometrics 53, 983–997.zbMATHCrossRefGoogle Scholar
  10. Neider, J.A. (1965). The analysis of randomized experiments with orthogonal block structure: block structure and the null analysis of variance. Proceedingsof the Royal Society A 283, 147–162.CrossRefGoogle Scholar
  11. Searle, S.R., Casella, G. and McCulloch, C.E. (1992). Variance Components. New York: Wiley.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • H. Monod
    • 1
  1. 1.Unité de BiométrieINRA-VersaillesVersailles CedexFrance

Personalised recommendations