## The Annals of Statistics

### Interactions and outliers in the two-way analysis of variance

#### 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 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.

#### Article information

Source
Ann. Statist., Volume 26, Number 4 (1998), 1279-1305.

Dates
First available in Project Euclid: 21 June 2002

https://projecteuclid.org/euclid.aos/1024691243

Digital Object Identifier
doi:10.1214/aos/1024691243

Mathematical Reviews number (MathSciNet)
MR1647657

Zentralblatt MATH identifier
0930.62070

Subjects
Primary: 62J10: Analysis of variance and covariance
Secondary: 62F35: Robustness and adaptive procedures

#### Citation

Terbeck, Wolfgang; Davies, P. Laurie. Interactions and outliers in the two-way analysis of variance. Ann. Statist. 26 (1998), no. 4, 1279--1305. doi:10.1214/aos/1024691243. https://projecteuclid.org/euclid.aos/1024691243

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