A Test of Incomplete Additivity in the Multiplicative Interaction Model

Abstract

Consider the multiplicative interaction model defined by y ij = μ + τ i + β j + λα i γ j + ∈ ij , i = 1, 2, …, t, j = 1, 2, …, b , where it is assumed that Σ i τ i = Σ j β j = Σ i α i = σ j γ j = 0 and γ i α i 2 = γ j γ j 2 = 1. It is also assumed that the ∈ ij are distributed NID (0, σ 2 ). This article derives the likelihood ratio test of H 0 : Hα = 0 and Gγ = 0 vs. H a : Hα ≈ 0 or Gγ ≈ 0 , where H is a q × t matrix of row contrasts of rank q and G is a r × b matrix of row constrasts of rank r. An approximation to the critical points of the test statistic is given and tables are given for a few selected values of b, t, q , and r. An improved estimator of σ 2 is derived, and all results are illustrated with an example.