, Volume 16, Issue 3, pp 598–612 | Cite as

Bayesian tolerance intervals for the balanced two-factor nested random effects model

  • Abrie J. Van der Merwe
  • Johan HugoEmail author
Original Paper


Statistical intervals, properly calculated from sample data, are likely to be substantially more informative to decision makers than obtaining a point estimate alone and are often of paramount interest to practitioners and thus management (and are usually a great deal more meaningful than statistical significance or hypothesis tests). Wolfinger (1998, J Qual Technol 36:162–170) presented a simulation-based approach for determining Bayesian tolerance intervals in a balanced one-way random effects model. In this note the theory and results of Wolfinger are extended to the balanced two-factor nested random effects model. The example illustrates the flexibility and unique features of the Bayesian simulation method for the construction of tolerance intervals.


Tolerance intervals Random effects Bayesian procedure Monte Carlo simulation Predictive density 

Mathematics Subject Classification (2000)

62F15 62F25 62P30 65C05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bagui S, Bhaumik DK, Parnes M (1996) One-sided tolerance limit for unbalanced m-way random effects ANOVA models. J Appl Stat Sci 3:135–148 zbMATHMathSciNetGoogle Scholar
  2. Berger JO, Bernardo JM (1992) On the development of reference priors (with discussion). In: Bernardo JM, Berger JO, Dawid AP, Smith AFM (eds) Bayesian statistics, vol 4. Oxford University Press, Oxford, pp 35–60 Google Scholar
  3. Box GEP, Tiao GC (1973) Bayesian inference in statistical analysis. Addison-Wesley, Reading zbMATHGoogle Scholar
  4. Gelfand AE, Smith AFM (1991) Gibbs sampling for marginal posterior expectations. Commun Stat Theory Methods 20(5,6):1747–1766 MathSciNetGoogle Scholar
  5. Guttman I (1970) Statistical regions: classical and Bayesian. Charles Griffin and Co, London zbMATHGoogle Scholar
  6. Guttman I, Menzefricke U (2003) Posterior distributions for functions of variance components. Test 12(1):115–123 zbMATHCrossRefMathSciNetGoogle Scholar
  7. Hahn GJ, Meeker WQ (1991) Statistical intervals. A guide for practitioners. Wiley, New York zbMATHGoogle Scholar
  8. Krishnamoorthy K, Mathew T (2004) One-sided tolerance limits in balanced and unbalanced one-way random models based on generalized confidence intervals. Technometrics 46(1):44–52 CrossRefMathSciNetGoogle Scholar
  9. Laubscher NF (1996) A variance components model for statistical process control. South Afr Stat J 30:27–47 zbMATHGoogle Scholar
  10. Shewhart WA (1931) Economic control of quality of manufactured product. Republished by the American Society for Quality Control, Milwaukee, 1980 Google Scholar
  11. Tierney L (1994) Markov chains for exploring posterior distributions (with discussion). Ann Stat 22:1701–1762 zbMATHMathSciNetGoogle Scholar
  12. Van Der Merwe AJ, Hugo J (2004) Tolerance intervals for the balanced two-factor nested random effects model. Tech Rep 335, Department of Mathematical Statistics, University of the Free State, Bloemfontein Google Scholar
  13. Vangel MG (1992) New methods for one-sided tolerance limits for a one-way balanced random-effects ANOVA model. Technometrics 34:176–185 zbMATHCrossRefMathSciNetGoogle Scholar
  14. Vangel MG (1994) One-sided β-content tolerance intervals for mixed models. In: Spring research conference on statistics in industry and technology. Chapel Hill Google Scholar
  15. Wang CM, Iyer HK (1994) Tolerance intervals for the distribution of true values in the presence of measurement error. Technometrics 36:162–170 zbMATHCrossRefGoogle Scholar
  16. Wolfinger RD (1998) Tolerance intervals for variance component models using Bayesian simulation. J Qual Technol 30(1):18–32 Google Scholar
  17. Wolfinger RD, Kass RE (2000) Nonconjugate Bayesian analysis of variance component models. Biometrics 56:768–774 zbMATHCrossRefGoogle Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 2007

Authors and Affiliations

  1. 1.Department of Mathematical StatisticsUniversity of the Free StateBloemfonteinRepublic of South Africa
  2. 2.Department of StatisticsNelson Mandela Metropolitan UniversityPort ElizabethRepublic of South Africa

Personalised recommendations