Testing Subhypotheses in the Multiplicative Interaction Model

Abstract

The problem of analyzing a two-way cross-classified treatment structure with only one observation per treatment combination is considered. A test procedure is given that will enable the data analyst to determine subareas of the data in which the data are additive. The procedure is developed by assuming that a multiplicative interaction model adequately fits the data. Such a model is given by y ij , = μ + τ i + β j + λα i γ j + ∊ ij ; where i = 1, 2, …, t and j = 1, 2, …, b. It is assumed that the ∊ ij , are distributed independently and normally with mean zero and variance σ2. The other parameters are assumed to be unknown constants. In general, the problem may be stated as one of testing H 0,: H α = 0 versus H a ,: H α ≠ 0, where α = (α1, …, α t )′ and H is a q × t contrast matrix. A likelihood ratio statistic for this testing problem is derived and approximate critical points are given for the cases q = 1 and q = 2. The procedures are illustrated with an example.