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.

KEY WORDS:

• Interaction
• Two-way models
• Multiplicative interaction models
• Nonadditivity