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Interactions and outliers in the two-way analysis of variance. (English) Zbl 0930.62070
Summary: The two-way analysis of variance with interactions is a well established and integral part of statistics. In spite of its long standing, it is shown that the standard definition of interactions is counterintuitive and obfuscates rather than clarifies. A different definition of interaction is given which among other advantages allows the detection of interactions even in the case of one observation per cell.
A characterization of unconditionally identifiable interaction patterns is given and it is proved that such patterns can be identified by the \(L^1\) functional. The unconditionally identifiable interaction patterns describe the optimal breakdown behavior of any equivariant location functional from which it follows that the \(L^1\) functional has optimal breakdown behavior. Possible lack of uniqueness of the \(L^1\) functional can be overcome using an \(M\) functional with an external scale derived independently from the observations. The resulting procedures are applied to some data sets including one describing the results of an interlaboratory test.

MSC:
62J10 Analysis of variance and covariance (ANOVA)
62F35 Robustness and adaptive procedures (parametric inference)
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[1] Armstrong, R. D. and Frome, E. L. (1976). The calculation of least absolute value estimators for two-way-tables. In Proceedings of the Statistical Computing Section 101-106. Amer. Statist. Assoc., Washington DC. · Zbl 0344.62055
[2] Armstrong, R. D. and Frome, E. L. (1979). Least-absolute-values-estimators for one-way and two-way tables. Naval Res. Logist. 26 79-96. · Zbl 0406.62052
[3] Bradu, D. (1975). E.D.V. in Biologie und Medizin 6, Heft 4. Fischer, Stuttgart.
[4] Bradu, D. (1997). Identification of outliers by means of L1 regression: Safe and unsafe configurations. Comput. Statist. Data Anal. 24 271-281. · Zbl 0900.62390
[5] Bradu, D. and Hawkins, D. M. (1982). Location of multiple outliers in two-way tables, using tetrads. Technometrics 24 103-108.
[6] Cochran, W. G. and Cox, F. M. (1957). Experimental Designs, 2nd ed. Wiley, New York. · Zbl 0077.13205
[7] Daniel, C. (1960). Locating outliers in factorial experiments. Technometrics 2 149-156. · Zbl 0091.14901
[8] Daniel, C. (1976). Applications of Statistics to Industrial Experimentation. Wiley, New York. · Zbl 0345.62058
[9] Daniel, C. (1978). Patterns in residuals in the two-way-lay out. Technometrics 20 385-395. · Zbl 0401.62056
[10] Davies, P. L. (1993). Aspects of robust linear regression. Ann. Statist. 21 1843-1899. · Zbl 0797.62026
[11] Davies, P. L. (1995). Data features. Statist. Neerlandica 49 185-245. · Zbl 0831.62001
[12] Donoho, D. L. and Huber, P. J. (1983). The notion of breakdown point. In A Festschrift for Erich L. Lehmann (P. J. Bickel, K. A. Doksum and J. L. Hodges, Jr., eds.) 157-184. Wadsworth, Belmont, CA. · Zbl 0523.62032
[13] El-Attar, R. A., Vidy asagar, M. and Dutta, S. R. K. (1979). An algorithm for l1-norm minimization with application to nonlinear l1-approximation. SIAM J. Numer. Anal. 16 70-86. JSTOR: · Zbl 0401.90089
[14] Ellis, S. P. and Morgenthaler, S. (1992). Leverage and breakdown in L1 regression. J. Amer. Statist. Assoc. 87 143-148. JSTOR: · Zbl 0781.62101
[15] Gentleman, J. F. and Wilk, M. B. (1975a). Detecting outliers in a two-way table. I. Statistical behavior of residuals. Technometrics 17 1-14. · Zbl 0322.62084
[16] Gentleman, J. F. and Wilk, M. B. (1975b). Detecting outliers in a two-way table. II. Supplementing the direct analysis of residuals. Biometrics 31 387-410. · Zbl 0322.62084
[17] Hampel, F. R. (1975). Bey ond location parameters: robust concepts and methods. In Proceedings of the 40th Session of the ISI 46 375-391. · Zbl 0349.62029
[18] Hampel, F. R., Ronchetti, E. M., Rousseeuw, P. J. and Stahel, W. A. (1986). Robust Statistics. Wiley, New York. · Zbl 0593.62027
[19] He, X., Jureckova, R., Koenker, R. and Portnoy, S. (1990). Tail behavior of regression estimators and their breakdown points. Econometrica 58 1195-1214. JSTOR: · Zbl 0745.62030
[20] Hoaglin, D. D., Mosteller F. and Tukey, J. W. (1983). Understanding Robust and Exploratory Data Analy sis. Wiley, New York. · Zbl 0599.62007
[21] Hoaglin, D. C., Mosteller, F. and Tukey J. W. (1985). Exploring Data Tables, Trends, and Shapes. Wiley, New York. · Zbl 0659.62002
[22] Huber, P. J. (1981). Robust Statistics. Wiley, New York. · Zbl 0536.62025
[23] Huber, P. J. (1995). Robustness: Where are we now? Student 1 75-86.
[24] Hubert, M. (1996). The breakdown value of the L1 estimator in contingency tables. Statist. Probab. Lett. 33 419-425. · Zbl 0899.62073
[25] Lischer, P. (1993). Ringversuche zur bestimmung des qualitätsstandards von laboratorien. In Seminar der Region Oesterreich-Schweiz der Internationalen Biometrischen Gesellschaft, Innsbruck.
[26] Mandel, J. (1991). Evaluation and Control of Measurements. Dekker, New York. · Zbl 0813.62090
[27] Mandel, J. (1995). Analy sis of Two-Way Lay outs. Chapman and Hall, New York.
[28] Nelder, J. A., Mead, R. (1965). A simplex method for function minimization. Computer Journal 7 308-313. · Zbl 0229.65053
[29] Roberts, J. and Cohrssen, J. (1968). Hearing levels of adults. U.S. National Center for Health Statistics Publications, Series 11, No. 31, p. 36, Table 4, Rockland, MD.
[30] Rousseeuw, P. J. (1984). Least median of squares regression. J. Amer. Statist. Assoc. 79 871-880. JSTOR: · Zbl 0551.62049
[31] Rousseeuw, P. J. (1985). Multivariate estimation with high breakdown point. In Mathematical Statistics and Applications B. Proceedings of the 4th Pannonian Sy mp. Math. Statist. (W. Grossmann, C. G. Pflug, I. Vincze and W. Wertz, eds.) Akadémiai Kiadó, Budapest. · Zbl 0609.62054
[32] Rousseeuw, P. J. and Croux, C. (1993). Alternatives to the median absolute deviation. J. Amer. Statist. Assoc. 88 1273-1283. JSTOR: · Zbl 0792.62025
[33] Rousseeuw, P. J. and Leroy, A. M. (1987). Robust Regression and Outlier Detection. Wiley, New York. · Zbl 0711.62030
[34] Terbeck, W. (1996). Interaktionen in der zwei-faktoren-varianzanalyse, Ph.D. dissertation, Univ. Essen. · Zbl 0879.62059
[35] Tukey, J. W. (1977). Exploratory Data Analy sis. Addison-Wesley, Reading, MA. · Zbl 0409.62003
[36] Tukey, J. W. (1993). Exploratory analysis of variance as providing examples of strategic choices. In New Directions in Statistical Data Analy sis and Robustness (S. Morgenthaler, E. Ronchetti and W. A. Stahel, eds.). Birkhäuser, Basel. · Zbl 0846.62003
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