Multivariate stimulus-response designs can be described by a three-way array of stimuli by responses by individuals. Its underlying structure can be represented by a network based on the Tucker2 component model in which stimulus components are connected with response components by means of the links that differ between individuals. For each individual such links are represented in a slice of the extended core array. For a proper understanding of these links, it is desirable that [1] the individual core slices as well as the component matrices have simple structures and [2] the differences of core slices between individuals are as few as possible. For attaining [1] and [2] we propose a method in which both the component matrices and the core slices of a Tucker2 solution are transformed simultaneously in order that the component matrices match simple target matrices and the core slices are summarized by a simple target slice. The proposed method is evaluated in a simulation study and illustrated with a three-way data array of semantic differential ratings.

The rank of a three-way array refers to the smallest number of rank-one arrays (outer products of three vectors) that generate the array as their sum. It is also the number of components required for a full decomposition of a three-way array by CANDECOMP/PARAFAC. The typical rank of a three-way array refers to the rank a three-way array has almost surely. The present paper deals with typical rank, and generalizes existing results on the typical rank of I × J × K arrays with K = 2 to a particular class of arrays with K ≥ 2. It is shown that the typical rank is I when the array is tall in the sense that JK − J < I < JK. In addition, typical rank results are given for the case where I equals JK − J.