Analysis of Variance for Random Models pp 333-381 | Cite as

# Three-Way and Higher-Order Nested Classifications

## Abstract

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.

## Keywords

Variance Component Random Effect Model Computing Software Negative Estimate Exact Confidence Interval## Preview

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## Bibliography

- 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 - C. A. Bennett (1954), Effect of measurement error on chemical process control,
*Indust. Quality Contol*,**11-6**, 17–20.Google Scholar - L. D. Broemeling (1969a), Confidence intervals for measures of heritability,
*Biometrics*,**25**, 424–427.CrossRefGoogle Scholar - L. D. Broemeling (1969b), Confidence intervals for variance ratios of random model,
*J. Amer. Statist. Assoc.*,**64**, 660–664.CrossRefGoogle Scholar - I. Bross (1950), Fiducial intervals for variance components,
*Biometrics*,**6**, 136–144.CrossRefGoogle Scholar - K. A. Brownlee (1953),
*Industrial Experimentation*, Chemical Publishing Company, New York.Google Scholar - M. G. Bulmer (1957), Approximate confidence limits for components of variance,
*Biometrika*,**44**, 159–167.MathSciNetMATHGoogle Scholar - R. K. Burdick and F. A. Graybill (1992),
*Confidence Intervals on Variance Components*, Marcel Dekker, New York.MATHGoogle Scholar - F. A. Graybill (1976),
*Theory and Application of the Linear Model*, Duxbury, North Scituate, MA.MATHGoogle Scholar - 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 - 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 - 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 - 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 - R. Khattree (1989), On robustness of tests for random effects in balanced nested designs,
*Statistics*,**20**, 41–46.MathSciNetMATHCrossRefGoogle Scholar - A. W. Kimball (1951), On dependent tests of significance in the analysis of variance,
*Ann. Math. Statist*,**22**, 600–602.MathSciNetMATHCrossRefGoogle Scholar - 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 - T.-F. C. Lu, F. A. Graybill, and R. K. Burdick (1987), Confidence intervals on the ratio of expected mean squares (θ
_{1}+*d*θ_{2})/θ_{3},*Biometrics*,**43**, 535–543.MathSciNetCrossRefGoogle Scholar - S. Moriguti (1954), Confidence limits for a variance component,
*Rep. Statist Appl Res. (JUSE)*,**3**, 7–19.Google Scholar - H. Sahai (1975), Bayes equivariant estimators in high order hierarchical random effects models,
*J. Roy. Statist Soc Ser. B*,**37**, 193–197.MathSciNetMATHGoogle Scholar - 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 - R. D. Snee (1983), Graphical analysis of process variation studies,
*J. Qual. Tech.*,**15**, 76–88.Google Scholar - G. C. Tiao and G. E. P. Box (1967), Bayesian analysis of a three-component hierarchical design model,
*Biometrika*,**54**, 109–125.MathSciNetMATHGoogle Scholar - 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 - 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 - 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 - J. S. Williams (1962), A confidence interval for variance components,
*Biometrika*,**49**, 278–281.MathSciNetMATHGoogle Scholar