## Interactions and Outliers in the Two-Way Analysis of Variance (1998)

Venue: | Annals of Statistics |

Citations: | 6 - 3 self |

### BibTeX

@ARTICLE{Terbeck98interactionsand,

author = {W. Terbeck and P. L. Davies},

title = {Interactions and Outliers in the Two-Way Analysis of Variance},

journal = {Annals of Statistics},

year = {1998},

volume = {26},

pages = {1279--1305}

}

### OpenURL

### Abstract

. 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 hown that the standard definition of interactions is counter intuitive and obfuscates rather than clarifies. A different definition of interaction is given which amongst 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 behaviour of any equivariant location functional from which it follows that the L 1 -functional has optimal breakdown behaviour. 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 ...

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Citation Context ...ith respect to AR, AC and IRC. Terbeck [33] contains more information on this topic. 1.3 Previous work The definition of interaction we give in Section 2:1 below is not new and may be found in Daniel =-=[8]-=-. Daniel shows how the least squares residuals may be used to detect certain patterns of outliers. In many cases Tukey's median polish correctly identifies outliers in the two-way table but it cannot ... |

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Citation Context ...reakdown point. A discussion of the problem of outliers in the analysis of variance is to be found in Hampel et al [18]. He, Jureckov'a, Koenker and Portnoy [19], Bradu [4] and Ellis and Morgenthaler =-=[14]-=- consider the breakdown behaviour of the L 1 -functional for fixed regressors. Their work leads to necessary and sufficient conditions for a subset of the regressors to be safe for the L 1 -functional... |

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Citation Context ...-way table. The results we give may be applied to the multiplicative model by taking logarithms. Many other structures can be constructed such as the row- and column-linear models developed by Mandel =-=[25]-=-, [26]. They face the same problems and as yet have not been robustified. The linear model has the great advantage of simplicity and even if it is not an adequate model the residuals from a robust fit... |

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Citation Context ....M., Rousseeuw, P.J., Stahel, W.A. [18] Section 1.1d. We note that in the context of this paper there is no difference between an interaction and an outlier and we shall use both terminologies. Tukey =-=[35]-=- calls such observations exotic. Failure to detecting interactions is equivalent to failure to detect outliers and in the context of outliers such a failure is referred to as breakdown. When we theref... |

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Citation Context ...p operations. Most but not all methods suggested in the literature are equivariant. An exception is Tukey's median polish (Tukey [34]) which is not equivariant with respect to AR, AC and IRC. Terbeck =-=[33]-=- contains more information on this topic. 1.3 Previous work The definition of interaction we give in Section 2:1 below is not new and may be found in Daniel [8]. Daniel shows how the least squares res... |

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Citation Context ...y identifiable interaction pattern P then C is a solution of the least absolute deviation problem. In generally solutions to the L 1 -problem in the two-way-layout are not unique (Armstrong and Frome =-=[1]-=-). The following theorem is therefore stronger than Lemma 2.10: Theorem 2.11 Let X be a matrix and C and C 0 be two residual matrices in C(X) such that C has an unconditionally identifiable interactio... |

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Citation Context ...ction pattern and C 0 minimizes the least absolute deviation. Then C = C 0 . An adaptation of general simplex methods to the special case of a two-way table has been given by Armstrong and Frome [1], =-=[2]-=-. Theorem 2.11 is no longer valid for a matrix C whose interaction pattern is not unconditionally identifiable as the following example shows; 0 @ 0 0 1 2 3 0 0 0 0 0 0 0 0 0 0 1 A : whose unique L 1 ... |

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Citation Context ...tisfy either Corollary 2.7 or Corollary 2.8. They do not however find all unconditionally identifiable interaction patterns as defined below. More detailed information is given in Terbeck [33]. Bradu =-=[3]-=-, Bradu and Hawkins [5], Gentleman and Wilk [15] and [16] have also considered the problem of identifying multiple outliers in the twoway analysis of variance. Hubert [23] has treated the correspondin... |

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Citation Context ...onal has the highest possible breakdown point. A discussion of the problem of outliers in the analysis of variance is to be found in Hampel et al [18]. He, Jureckov'a, Koenker and Portnoy [19], Bradu =-=[4]-=- and Ellis and Morgenthaler [14] consider the breakdown behaviour of the L 1 -functional for fixed regressors. Their work leads to necessary and sufficient conditions for a subset of the regressors to... |

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Citation Context ...2.7 or Corollary 2.8. They do not however find all unconditionally identifiable interaction patterns as defined below. More detailed information is given in Terbeck [33]. Bradu [3], Bradu and Hawkins =-=[5]-=-, Gentleman and Wilk [15] and [16] have also considered the problem of identifying multiple outliers in the twoway analysis of variance. Hubert [23] has treated the corresponding problem for two-way c... |

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Citation Context ...mple was also known to Daniel [7] and is of itself sufficient to discredit the usual definition of interaction. A practical example with the same structure is Table 5.6 on page 164 of Cochran and Cox =-=[6]-=-. It has been analysed by Daniel [7] Sections 4.3 and 4.7 and Hampel, F.R., Ronchetti, E.M., Rousseeuw, P.J., Stahel, W.A. [18] Section 1.1d. We note that in the context of this paper there is no diff... |

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Citation Context ...ey do not however find all unconditionally identifiable interaction patterns as defined below. More detailed information is given in Terbeck [33]. Bradu [3], Bradu and Hawkins [5], Gentleman and Wilk =-=[15]-=- and [16] have also considered the problem of identifying multiple outliers in the twoway analysis of variance. Hubert [23] has treated the corresponding problem for two-way contingency tables. She sh... |

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Citation Context ... however find all unconditionally identifiable interaction patterns as defined below. More detailed information is given in Terbeck [33]. Bradu [3], Bradu and Hawkins [5], Gentleman and Wilk [15] and =-=[16]-=- have also considered the problem of identifying multiple outliers in the twoway analysis of variance. Hubert [23] has treated the corresponding problem for two-way contingency tables. She shows that ... |

1 |
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Citation Context ...d. The linear model has the great advantage of simplicity and even if it is not an adequate model the residuals from a robust fit provide a good starting point for developing an improved model. Huber =-=[22]-=- states Embarrassingly, the robustification of the statistics of two-way tables is still wide open. We hope this paper reduces the embarrassment. 1.4 Contents In Section 2 we consider the no-noise mod... |

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Citation Context ... given in Terbeck [33]. Bradu [3], Bradu and Hawkins [5], Gentleman and Wilk [15] and [16] have also considered the problem of identifying multiple outliers in the twoway analysis of variance. Hubert =-=[23]-=- has treated the corresponding problem for two-way contingency tables. She shows that in this situation the L 1 -functional has the highest possible breakdown point. A discussion of the problem of out... |

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Ringversuche zur Bestimmung des Qualitatsstandards von Laboratorien, Seminar der Region Oesterreich-Schweiz der Internationalen Biometrischen Gesellschaft
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Citation Context ...h as error scale being dependent on the level of concentration. More work remains to be done on this topic and so we do not attempt to give a full analysis of the data below first analysed by Lischer =-=[24]-=-. It consists of ten samples of sewage sludge which were sent to 21 laboratories each of which had to report the lead concentration of each sample. The data are 0 B B B B B B B B B B B B B B B B B B B... |

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Hearing Levels of Adults, Table 4, p
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Citation Context ...1 is a bias effect caused by the outliers. 5.2 Daniel's example The second example is taken from Daniel [9]. The data are the results of ear tests and were originally published by Roberts and Corssen =-=[28]-=-. They have also been analysed by Bradu [3] and by Bradu and Hawkins [5]. The rows correspond to sound frequencies and the columns represent different occupational groups. The data are 0 B B B B B B B... |