On asymptotic tests in three-factor factorial designs with no replications

We revisit the problems of testing three-factor classification models with a single observation per cell. A common approach in analyzing such nonreplicated data is to omit the highest order interaction and regard it as error. This paper discusses the use of a multiplicative model (See and Smith, 1996 and 1998) which is applied on residuals in order to separate the variability due to three-factor interaction from what is counted as random error. In particular, to test the significance of the interaction term, we derived an approximated distribution of the likelihood ratio test statistic based on the quadrilinear model known as Tucker’s three-mode principal component model. The derivation utilizes the distribution of the eigenvalues of the Wishart matrix.