Identification of dispersion effects in replicated two-level fractional factorial experiments.

*(English)*Zbl 1423.62097Summary: Tests for dispersion effects in replicated two-level factorial experiments assuming a location-dispersion model are presented. The tests use individual measures of dispersion that remove the location effects and also provide an estimate of pure error. Empirical critical values for two such tests are given for two-level full or regular fractional factorial designs with 8, 16, 32, and 64 runs. The powers of the tests are examined under normal, exponential, and Cauchy distributed errors. Our recommended test uses dispersion measures calculated as deviations of the data values from their cell medians, and this test is illustrated via an example.

##### MSC:

62K15 | Factorial statistical designs |

62P30 | Applications of statistics in engineering and industry; control charts |

##### Software:

GLIM
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\textit{C. Dingus} et al., J. Stat. Theory Pract. 7, No. 4, 687--702 (2013; Zbl 1423.62097)

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##### References:

[1] | Aitkin, M., Modelling variance heterogeneity in normal regression using GLIM, Appl. Stat., 36, 332-339, (1987) |

[2] | Ankenman, B. E.; Dean, A. M.; Rao, C. R (ed.); Khattree, R. (ed.), Quality improvement and robustness via design of experiments, 263-317, (2003), Amsterdam, The Netherlands |

[3] | Bartlett, M. S.; Kendall, D. G., The statistical analysis of variance-heterogeneity and the logarithmic transformation, Suppl. J. R. Stat. Soc., 8, 128-138, (1946) · Zbl 0063.00230 |

[4] | Bergman, B.; Hynén, A., Dispersion effects from unreplicated designs in the 2k−\(p\) series, Technometrics, 39, 191-198, (1997) · Zbl 0889.62068 |

[5] | Box, G. E.; Meyer, R. D., Dispersion effects from fractional designs, Technometrics, 28, 19-27, (1986) · Zbl 0586.62167 |

[6] | Box, G. E P., Signal-to-noise ratios, performance criteria, and transformations, Technometrics, 30, 1-17, (1988) |

[7] | Box, G. E. P., W. G. Hunter. and J. S. Hunter. 1978. Statistics for experimenters: An introduction to design, data analysis, and model building. New York, NY, John Wiley & Sons. · Zbl 0394.62003 |

[8] | Brenneman, W. A.; Nair, V. N., Methods for identifying dispersion effects in unreplicated factorial experiments, Technometrics, 43, 388-405, (2001) |

[9] | Brown, M.; Forsythe, A., Robust tests for the equality of variances, J. Am. Stat. Assoc., 69, 364-367, (1974) · Zbl 0291.62063 |

[10] | Bursztyn, D.; Steinberg, D. M.; Dean, A. M (ed.); Lewis, S. M (ed.), Screening experiments for dispersion effects, 21-47, (2006), New York, NY |

[11] | Cook, R. D.; Weisberg, S., Diagnostics for heteroscedasticity in regression, Biometrika, 70, 1-10, (1983) · Zbl 0502.62063 |

[12] | Dean, A., and D. Voss. 1999. Design and analysis of experiments. New York, NY, Springer. · Zbl 0910.62066 |

[13] | Dingus, C. A. V. 2005. Designs and methods for the identification of active location and dispersion effects. Ph.D. thesis, The Ohio State University, Columbus, OH. |

[14] | Haaland, P.; O’Connell, M., Inference for effect-saturated fractional fatorials, Technometrics, 37, 82-93, (1995) · Zbl 0825.62656 |

[15] | Harvey, A. C., Estimating regression models with multiplicative heteroscedasticity, Econometrica, 44, 461-465, (1976) · Zbl 0333.62040 |

[16] | Lenth, R. V., Quick and easy analysis of unreplicated factorials, Technometrics, 31, 469-473, (1989) |

[17] | Levene, H.; Olkin, I. (ed.); Ghurye, S. G (ed.); Hoeffding, W. (ed.); Madow, W. G (ed.); Mann, H. B (ed.), Robust tests for equality of variances, (1960), Stanford, CA |

[18] | Mackertich, N. A., J. C. Benneyan, and P. D. Kraus. 2003. Alternate dispersion measures in replicated factorial experiments. Unpublished manuscript. http://www1.coe.neu.edu/∼benneyan/papers/doe_dispersion.pdf |

[19] | McGrath, R. N.; Lin, D. K J., Confounding of location and dispersion effects in unreplicated fractional factorials, J. Quality Technol, 33, 129-139, (2001) |

[20] | McGrath, R. N.; Lin, D. K J., Testing multiple dispersion effects in unreplicated two-level fractional factorial designs, Technometrics, 43, 406-414, (2001) |

[21] | Miller, A., The analysis of unreplicated factorial experiments using all possible comparisons, Technometrics, 47, 51-63, (2005) |

[22] | Montgomery, D. C. 2009. Design and analysis of experiments. 7th ed. New York, NY, John Wiley & Sons. |

[23] | Nair, V. J.; Pregibon, D., Analyzing dispersion effects from replicated factorial experiments, Technometrics, 30, 247-257, (1988) · Zbl 0655.62097 |

[24] | Pan, G., The impact of unidentified location effects on dispersion—effects identification from unreplicated designs, Technometrics, 41, 313-326, (1999) · Zbl 0998.62064 |

[25] | Pignatiello, J. J.; Ramberg, J. S., Discussion of “Off-line Quality Control, Parameter Design, and the Taguschi Method” by R.N. Kackar, J. Quality Technol., 17, 198-206, (1985) |

[26] | Pignatiello, J. J.; Ramberg, J. S., Comments on “Performance Measures Independent of Adjustment: An Explanation and Extension of Taguchi’s Signal-to-noise Ratios.”, Technometrics, 29, 274-277, (1987) |

[27] | Verbyla, A. P., Modelling variance heterogeneity: residual maximum likelihood and diagnostics, J. R. Stat. Soc. Ser. B, 55, 493-508, (1993) · Zbl 0783.62051 |

[28] | Wang, P. C., Tests for dispersion effects from orthogonal arrays, Comput. Stat. Data Anal., 8, 109-117, (1989) |

[29] | Wolfe, D. A.; Dean, A. M.; Wiers, M. D.; Hartlaub, B. A., Nonparametric rank based main effects test procedures for the two-way layout in the presence of interaction, J. Nonparametric Stat., 1, 241-252, (1992) · Zbl 1263.62086 |

[30] | Wolfinger, R. D.; Tobias, R. D., Joint estimation of location, dispersion, and random effects in robust design, Technometrics, 40, 62-71, (1998) · Zbl 0896.62082 |

[31] | Wu, C. F. J., and M. Hamada. 2000. Experiments: Planning, analysis, and parameter design optimization. New York, NY, John Wiley & Sons. · Zbl 0964.62065 |

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