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Two-Way Nested Classification

  • Hardeo Sahai
  • Mario Miguel Ojeda

Abstract

In the preceding two chapters, we have considered experimental situations where the levels of two factors are crossed. In this and the following chapter we onsider experiments where the levels of one of the factors are nested within the levels of the other factor. The data for a two-way nested classification are similar hat of a single factor classification except that now replications are grouped into different sets arising from the levels of the nested factor for a given level of the main factor. Suppose the main factor A has a levels and the nested factor B has ab levels which are grouped into a sets of b levels each, and n observations are made at each level of the factor B giving a total of abn observations. The nested or hierarchical designs of this type are very important in many industrial and genetic investigations. For example, suppose an experiment is designed to investigate the variability of a certain material by randomly selecting a batches, b samples are made from each batch, and finally n analyses are performed on each sample. The purpose of the investigation may be to make inferences about the relative contribution of each source of variation to the total variance or to make inferences about the variance components individually. For another example, suppose in a breeding experiment a random sample of a sires is taken, each sire is mated to a sample of b dams, and finally n offspring are produced from each sire-dam mating. Again, the purpose of the investigation may be to study the relative magnitude of the variance components or to make inferences about them individually.

Keywords

Mean Square Error Variance Component Computing Software Marginal Posterior Distribution Exact Confidence Interval 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Bibliography

  1. R. Ahmad and S. M. Mostafa (1985), Empirical Bayes quadratic estimators of variance components in mixed normal models, Pak. J. Statist., 1, 59–65.MathSciNetMATHGoogle Scholar
  2. L. D. Broemeling (1969a), Confidence intervals for measures of heritability, Biometrics, 25, 424–427.CrossRefGoogle Scholar
  3. L. D. Broemeling (1969b), Confidence intervals for variance ratios of random model, J. Amer. Statist. Assoc., 64, 660–664.CrossRefGoogle Scholar
  4. K. A. Brownlee (1965), Statistical Theory and Methodology in Science and Engineering, 2nd ed., Wiley, New York.MATHGoogle Scholar
  5. R. K. Burdick and F. A. Graybill (1992), Confidence Intervals on Variance Components, Marcel Dekker, New York.MATHGoogle Scholar
  6. R. R. Corbeil and S. R. Searle (1976), A comparison of variance component estimators, Biometrics, 32, 779–791.MathSciNetCrossRefMATHGoogle Scholar
  7. O. L. Davies and P. L. Goldsmith, eds. (1972), Statistical Methods in Research and Production, 4th ed., Oliver and Boyd, Edinburgh.Google Scholar
  8. F. A. Graybill (1976), Theory and Application of the Linear Model, Duxbury, North Scituate, MA.MATHGoogle Scholar
  9. F. A. Graybill and W. H. Robertson (1957), Calculating confidence intervals for genetic heritability, Poultry Sci, 36, 261–265.CrossRefGoogle 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. J. Hartung and B. Voet (1987), An asymptotic X2-test for variance components, in W. Sendler, ed., Contributions to Stochastics, Physica-Verlag, Heidelberg, 153–163.CrossRefGoogle Scholar
  13. 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.MathSciNetCrossRefMATHGoogle Scholar
  14. J. L. Hodges, Jr. and E. L. Lehmann (1951), Some applications of the Cramér-Rao inequality, in L. Lecam and J. Neyman, Proceedings Second Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, CA, 13–22.Google Scholar
  15. W. G Howe (1974), Approximate confidence limits on the mean of X + Y where X and Y are two tabled independent random variables, J. Amen Statist. Assoc., 69, 789–794.MathSciNetMATHGoogle Scholar
  16. H. Jeffreys (1961), Theory of Probability, 3rd ed., Clarendon Press, Oxford, UK; 1st ed., 1939; 2nd ed., 1948.MATHGoogle Scholar
  17. 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
  18. R. Khattree (1989), On robustness of tests for random effects in balanced nested designs, Statistics, 20, 41–46.MathSciNetCrossRefMATHGoogle Scholar
  19. R. Khattree and D. N. Naik (1994), Optimal tests for nested designs with circular stationary dependence, J. Statist. Plann. Inference, 41, 231–240.MathSciNetCrossRefMATHGoogle Scholar
  20. A. I. Khuri (1981), Simultaneous confidence intervals for functions of variance components in random models, J. Amer. Statist. Assoc., 76, 878–885.MathSciNetCrossRefGoogle Scholar
  21. A. W. Kimball (1951), On dependent tests of significance in the analysis of variance, Ann. Math. Statist., 22, 600–602.MathSciNetCrossRefMATHGoogle Scholar
  22. K. R. Lee and C. H. Kapadia (1984), Variance component estimators for the balanced two-way mixed model, Biometrics, 40, 507–512.CrossRefGoogle Scholar
  23. E. L. Lehmann (1986), Testing Statistical Hypotheses, 2nd ed., Wiley, New York; reprint, 1997, Springer-Verlag, New York.CrossRefMATHGoogle Scholar
  24. S.-H. Li and J. H. Klotz (1978), Components of variance estimation for the split-plot design, J. Amer. Statist. Assoc., 73, 147–152.MathSciNetCrossRefMATHGoogle Scholar
  25. 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
  26. T.-F. C. Lu, F. A. Graybill, and R. K. Burdick (1989), Confidence intervals on the ratio of expected mean squares (θ1dθ2)/θ3, J. Statist. Plann. Inference, 21, 179–190.MathSciNetCrossRefMATHGoogle Scholar
  27. R. Osborne and W. S. B. Patterson (1952), On the sampling variance of heri-tability estimates derived from variance analysis, Proc Roy. Soc. Edinburgh Ser. B, 64, 456–461.Google Scholar
  28. S. Portnoy (1971), Formal Bayes estimation with application to a random effects model, Ann. Math. Statist., 42, 1379–1402.MathSciNetCrossRefMATHGoogle Scholar
  29. P. S. R. S. Rao and C. E. Heckler (1997), Estimators for the three-fold nested random effects model, J. Statist. Plann. Inference, 64, 341–352.MathSciNetCrossRefMATHGoogle Scholar
  30. H. Sahai (1974a), Non-negative maximum likelihood and restricted maximum likelihood estimators of variance components in two simple linear models, Util. Math., 5, 151–160.MathSciNetMATHGoogle Scholar
  31. H. Sahai (1974b), Some formal Bayes estimators of variance components in the balanced three-stage nested random effects models, Comm. Statist., 3, 233–242.MathSciNetMATHGoogle Scholar
  32. H. Sahai (1974c), Simultaneous confidence intervals for variance components in some balanced random effects models, Sankhyā Ser. B, 36, 278–287.MathSciNetMATHGoogle Scholar
  33. H. Sahai (1975), Bayes equivariant estimators in high order hierarchical random effects models, J. Roy. Statist. Soc. Ser. B, 37, 193–197.MathSciNetMATHGoogle Scholar
  34. H. Sahai (1976), A comparison of estimators of variance components in the balanced three-stage nested random effects model using mean squared error criterion, J. Amer. Statist. Assoc., 71, 435–444.MathSciNetCrossRefMATHGoogle Scholar
  35. 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.MathSciNetCrossRefMATHGoogle Scholar
  36. B. Seifert (1978), A note on the UMPU character of a test for the mean in balanced randomized nested classification, Statistics, 9, 185–189.MathSciNetMATHGoogle Scholar
  37. L. Shi (1997), UMPU test of outliers in random-effects model of two-way nested classification, Chinese J. Appl. Probab. Statist., 13, 125–132.MathSciNetMATHGoogle Scholar
  38. G. W. Snedecor and W. G. Cochran (1989), Statistical Method, 8th ed., Iowa State University Press, Ames, IA; 6th ed., 1967; 7th ed., 1980.Google Scholar
  39. R. D. Snee (1983), Graphical analysis of process variation studies, J. Qual. Tech., 15, 76–88.Google Scholar
  40. R. R. Sokal and F. J. Rohlf (1995), Biometry, 3rd ed., W. H. Freeman, New York; 1st ed., 1969; 2nd ed., 1981.Google Scholar
  41. C. Stein (1964), Inadmissibility of the usual estimator for the variance of a normal distribution with unknown mean, Ann. Inst. Statist. Math. (Japan), 16, 155–160.CrossRefMATHGoogle Scholar
  42. G. C. Tiao and G. E. P. Box (1967), Bayesian analysis of a three-component hierarchical design model, Biometrika, 54, 109–125.MathSciNetMATHGoogle Scholar
  43. 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.MathSciNetCrossRefMATHGoogle Scholar
  44. 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
  45. L. R. Verdooren (1980), On estimation of variance components, Statist. Neer-land., 34, 83–106.MathSciNetCrossRefMATHGoogle Scholar
  46. L. R. Verdooren (1988), Exact tests and confidence intervals for ratio of variance components in unbalanced two-and three-stage nested designs, Comm. Statist. A Theory Methods, 17, 1197–1230.MathSciNetCrossRefMATHGoogle Scholar
  47. C. M. Wang (1978), Confidence Intervals on Functions of Variance Components, Ph.D. dissertation, Colorado State University, Fort Collins, CO.Google Scholar
  48. C. M. Wang and F. A. Graybill (1981), Confidence intervals on a ratio of variances in the two-factor nested components of variance model, Comm. Statist. A Theory Methods, 10, 1357–1368.MathSciNetCrossRefGoogle 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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