This dissertation is concerned with the application of a method for decomposing multivariate data, three-mode principal component analysis, to a three-way table with one observation per cell. It is based on the class of multiplicative models for three-way tables (s x t x u) whose general form has expectation

E(y_ijk) = μ + α_i + β_j + γ_k + (αβ)_ij + (αγ)_ik + (βγ)_jk + s Σ _p=1 t Σ _q=1 u Σ _r=1 c_pqr g_ip h_jq e_kr.

The application is related to tests for nonadditivity in the two-way analysis of variance with noreplication.

Three-factor interaction can be assessed for three-way cross classified tables with only one observation per treatment combination. This is done by partitioning the three-factor interaction sum of squares into a portion related to the interaction and a portion associated with random error. In particular, the estimated interaction matrix is decomposed by three-mode principal component analysis to separate significant interaction from random error. Three test procedures are presented for assessing interaction: randomization tests, Monte Carlo methods, and likelihood ratio tests. Examples illustrating the use of these approaches are presented.

In addition to the above testing approaches, a graphical procedure, joint plots, is investigated for diagnosing the type of model to fit to three-way arrays of data. The plot is a multiway analogue of the biplot graphical analysis for two-way matrices. Each observation is represented by a linear combination of inner products of markers which are obtained from three-mode principal component analysis. The relationship between various models and the geometrical configurations of the plots on Euclidean spaces of such markers allows one to diagnose the type of model which fits the data. An example is given to illustrate the simplicity of the technique and the usefulness of this graphical approach in diagnosing models.