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 them 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 L1 functional. The unconditionally identifiable interaction patterns describe the optimal breakdown behavior of any equivariant location functional from which it follows that the L1 functional has optimal breakdown behavior. Possible lack of uniqueness of the L1 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.