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On Identifying Dispersion Effects in Unreplicated Fractional Factorial Experiments

  • Seiichi Yasui
  • Yoshikazu Ojima
  • Tomomichi Suzuki
Chapter

Abstract

The analysis of dispersion effects is as important as the location effect analysis in the quality improvement. Unreplicated fractional factorial experiments are useful to analyse not only location effects but also dispersion effects. The statistics introduced by Box and Meyer (1986) to identify dispersion effects are based on residuals subtracting the estimates for large location effects from observations. The statistic is a simple form, however, the property is not completely discovered. In this article, the distribution of the statistic under the null hypothesis is derived in unreplicated fractional factorial experiments using an orthogonal array. The distribution under the null hypothesis cannot be expressed uniquely. The statistic has different null distributions depending on the combination of columns allocating factors. We concluded that the distributions can be classified into three types, i.e. the F distribution, unknown distributions close to the F distribution and the constant (not stochastic variable) which is one. Finally, the power of the test for detection of a single active dispersion effect is evaluated.

Keywords

Orthogonal Array Design Matrix Dispersion Model Null Distribution Stochastic Variable 
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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References

  1. 1.
    Bartlett, M. S. and Kendall, D. G. 1946, The Statistical Analysis of variance-Heterogeneity and the Logarithmic Transformation, Journal of the Royal Statistical Society, Ser. B, 8, 128-138.MathSciNetGoogle Scholar
  2. 2.
    Bergman, B., and Hynen, A., 1997, Dispersion Effects From Unreplicated Designs in the 2pq Series, Technometrics, 39, 191-198.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Blomkvist O., Hynen A. and Bergman B., 1997, A Method to Identify Dispersion Effects from unreplicated multilevel experiments, Quality and Reliability Engineering International, 13, 127-138.CrossRefGoogle Scholar
  4. 4.
    Box, G. E. P., and Meyer, R. D., 1986, Dispersion Effects From Fractional Designs, Technometrics, 28, 19-27.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Ferrer, A. J. and Romero R., 1993, Small Samples Estimation of Dispersion Effects From Unreplicated Data, Communications in Statistics - Simulation and Computation, 22, 975-995.CrossRefGoogle Scholar
  6. 6.
    Harvey, A. C., 1976, Estimating Regression Models with Multiplicative Heteroscedasticity, Econometrica, 44, 461-465.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Holm S. and Wiklander K., 1999, Simultaneous Estimation of Location and Dispersion in Two-Level Fractional Factorial Designs, Journal of Applied Statistics, 26, 235-242.MATHCrossRefGoogle Scholar
  8. 8.
    Lee, H. S., 1994, Estimates For Mean and Dispersion Effects in Unreplicated Factorial Designs, Communications in Statistics - Theory and Methods, 23, 3593-3608.MATHCrossRefGoogle Scholar
  9. 9.
    Liao, C. T., 2000, Identification of dispersion effects from Unreplicated 2nk Fractional Factorial Designs, Computational Statistics & Data Analysis, 33, 291-298.MATHCrossRefGoogle Scholar
  10. 10.
    McGrath, R. N., 2003, Separateing Location and Dispersion Effects in Unreplicated Fractional Factorial Designs, Journal of Quality Technology, 35, 306-316.Google Scholar
  11. 11.
    McGrath, R. N. and Lin, D. K., 2001a, Testing Multiple Dispersion Effects in Unreplicated Fractional Factorial Designs, Technometrics, 43, 406-414.CrossRefMathSciNetGoogle Scholar
  12. 12.
    McGrath, R. N. and Lin, D. K., 2001b, Confounding of Location and Dispersion Effects in Unreplicated Fractional Factorials, Journal of Quality Technology, 33, 129-139.Google Scholar
  13. 13.
    Nair, V. N. and Pregibon, D., 1988, Analyzing Dispersion Effects from Replicated Factorial Experiments, Technometrics, 30, 247-257.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Ojima, Y., Suzuki T. and Yasui S., 2004 An Alternative Expression of the Fractional Factorial Designs for Two-level and Three-level Factors, Frontiers in Statistical Quality Control, Vol. 7, pp. 309-316.MathSciNetGoogle Scholar
  15. 15.
    Pan, G., 1999, The Impact of Unidentified Location Effects on Dispersion-Effects Identification From Unreplicated Fractional Factorial Designs, Technometrics, Vol. 41, pp. 313-326.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Schoen, E. D., 2004, Dispersion-effects Detection after Screening for Location Effects in Unreplicated Two-level Experiments, Journal of Statistical Planning and Inference, 126, 289-304.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Taguchi, G., and Wu, Y., 1980, An Introduction to Off-Line Quality Control, Nagoya, Japan: Central Japan Quality Control Association.Google Scholar
  18. 18.
    Wang, P. C., 1989, Tests for Dispersion Effects from Orthogonal Arrays, Computational Statistics & Data Analysis, 8, 109-117.CrossRefGoogle Scholar
  19. 19.
    Wang, P. C., 2001, Testing Dispersion Effects From General Unreplicated Fractional Factorial Designs, Qualtiy and Reliability Engineering International, 17, 243-248.CrossRefGoogle Scholar
  20. 20.
    Wiklander, K., 1998, A Comparison of Two Estimators of Dispersion Effects, Communications in Statistics - Theory and Methods, 27, 905-923.MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Wiklander, K. and Holm S., 2003, Dispersion Effects in Unreplicated Factorial Designs, Applied Stochastic Models in Business and Industry, 19, 13-30.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Physica-Verlag Heidelberg 2010

Authors and Affiliations

  • Seiichi Yasui
    • 1
  • Yoshikazu Ojima
    • 1
  • Tomomichi Suzuki
    • 1
  1. 1.Department of Industrial AdministrationTokyo University of ScienceChibaJAPAN

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