Three-Way and Higher-Order Nested Classifications

  • Hardeo Sahai
  • Mario Miguel Ojeda


In the preceding chapter, we considered a random effects model involving a two-way nested classification. Examples of three and higher-order nested classifications occur frequently in many industrial experiments where raw material is first broken up into batches and then into subbatches, subsubbatches, and so forth. For example, in an experiment designed to identify various sources of variability in tensile strength measurements, one may randomly select a lots of raw material, b boxes are taken from each lot, c sample preparations are made from the material in each box, and finally n tensile strength tests are performed for each preparation. These factors often present themselves in a hierarchical manner and are appropriately specified as random effects. In this chapter, we consider a random effects model involving a three-way nested classification and indicate its generalization to higher-order nested classifications.


Variance Component Random Effect Model Computing Software Negative Estimate Exact Confidence Interval 
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  1. R. Ahmad and S. M. Mostafa (1987), Empirical Bayes estimation of variance components in balanced random models, J. Statist. Comput. Simul, 27, 143–153.MATHCrossRefGoogle Scholar
  2. C. A. Bennett (1954), Effect of measurement error on chemical process control, Indust. Quality Contol, 11-6, 17–20.Google Scholar
  3. L. D. Broemeling (1969a), Confidence intervals for measures of heritability, Biometrics, 25, 424–427.CrossRefGoogle Scholar
  4. L. D. Broemeling (1969b), Confidence intervals for variance ratios of random model, J. Amer. Statist. Assoc., 64, 660–664.CrossRefGoogle Scholar
  5. I. Bross (1950), Fiducial intervals for variance components, Biometrics, 6, 136–144.CrossRefGoogle Scholar
  6. K. A. Brownlee (1953), Industrial Experimentation, Chemical Publishing Company, New York.Google Scholar
  7. M. G. Bulmer (1957), Approximate confidence limits for components of variance, Biometrika, 44, 159–167.MathSciNetMATHGoogle Scholar
  8. R. K. Burdick and F. A. Graybill (1992), Confidence Intervals on Variance Components, Marcel Dekker, New York.MATHGoogle Scholar
  9. F. A. Graybill (1976), Theory and Application of the Linear Model, Duxbury, North Scituate, MA.MATHGoogle Scholar
  10. F. A. Graybill and C.-M. Wang (1979), Confidence intervals for proportions of variability in two-factor nested variance components models, J. Amer. Statist. Assoc., 74, 368–374.MathSciNetMATHGoogle Scholar
  11. F. A. Graybill and C.-M. Wang (1980), Confidence intervals on nonnegative linear combinations of variances, J. Amer. Statist. Assoc., 75, 869–873.MathSciNetCrossRefGoogle Scholar
  12. L. H. Herbach (1959), Properties of Model II type analysis of variance tests A: Optimum nature of the F-test for Model II in the balanced case, Ann. Math. Statist, 30, 939–959.MathSciNetMATHCrossRefGoogle Scholar
  13. N. L. Johnson and F. C. Leone (1964), Statistics and Experimental Design in Engineering and the Physical Sciences, Vol. 2., Wiley, New York.MATHGoogle Scholar
  14. R. Khattree (1989), On robustness of tests for random effects in balanced nested designs, Statistics, 20, 41–46.MathSciNetMATHCrossRefGoogle Scholar
  15. A. W. Kimball (1951), On dependent tests of significance in the analysis of variance, Ann. Math. Statist, 22, 600–602.MathSciNetMATHCrossRefGoogle Scholar
  16. F. C. Leone and L. S. Nelson (1966), Sampling distribution of variance components I: Empirical studies of balanced nested designs, Technometrics, 8, 457–468.MathSciNetCrossRefGoogle Scholar
  17. T.-F. C. Lu, F. A. Graybill, and R. K. Burdick (1987), Confidence intervals on the ratio of expected mean squares (θ1 + dθ2)/θ3, Biometrics, 43, 535–543.MathSciNetCrossRefGoogle Scholar
  18. S. Moriguti (1954), Confidence limits for a variance component, Rep. Statist Appl Res. (JUSE), 3, 7–19.Google Scholar
  19. H. Sahai (1975), Bayes equivariant estimators in high order hierarchical random effects models, J. Roy. Statist Soc Ser. B, 37, 193–197.MathSciNetMATHGoogle Scholar
  20. H. Sahai and R. L. Anderson (1973), Confidence regions for variance ratios of random models for balanced data, J. Amer. Statist. Assoc., 68, 951–952.MathSciNetMATHCrossRefGoogle Scholar
  21. R. D. Snee (1983), Graphical analysis of process variation studies, J. Qual. Tech., 15, 76–88.Google Scholar
  22. G. C. Tiao and G. E. P. Box (1967), Bayesian analysis of a three-component hierarchical design model, Biometrika, 54, 109–125.MathSciNetMATHGoogle Scholar
  23. N. Ting, R. K. Burdick, and F. A. Graybill (1991), Confidence intervals on ratios of positive linear combinations of variance components, Statist. Probab. Lett., 11, 523–528.MathSciNetMATHCrossRefGoogle Scholar
  24. N. Ting, R. K. Burdick, F. A. Graybill, S. Jeyaratnam, and T.-F. C. Lu (1990), Confidence intervals on linear combinations of variance components that are unrestricted in sign, J. Statist. Comput. Simul, 35, 135–143.MathSciNetCrossRefGoogle Scholar
  25. J. R. Trout (1985), Design and analysis of experiments to estimate components of variation: Two case studies, in R. D. Snee, L. B. Hare, and J. R. Trout, eds., Experiments in Industry, American Society for Quality Control, Milwaukee, 75–88.Google Scholar
  26. J. S. Williams (1962), A confidence interval for variance components, Biometrika, 49, 278–281.MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Hardeo Sahai
    • 1
  • Mario Miguel Ojeda
    • 2
  1. 1.Center for Addiction Studies School of MedicineUniversidad Central del CaribeBayamon, Puerto RicoUSA
  2. 2.Económico AdministrativaUniversidad VeracruzanaKalapa, VeracruzMéxico

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