Hypothesis testing for components of the shifted multiplicative model for a nonadditive two-way table

To determine the number of multiplicative components in a shifted multiplicative model (SHMM) for a r × c table (c≤r) given by likelihood ratio tests of Ho: λt = 0 versus Ha:λt≠0, λt+1=0 are developed in a stepwise procedure with t successively set equal to 1,…,p − 1, where p=min(r−1,c). For t = 1, the empirical distribution and percentage points of the likelihood ratio test statistic are obtained through a Monte Carlo simulation study. Distribution of the likelihood ratio test statistic is also approximated by a Beta distribution. For t ≥ 2, a method analogous to one developed by Marasinghe (1985) and Schott (1986) for the additive main effect and multiplicative interaction (AMMI) model is used, namely, that the tth term is tested as thought it is the first term for a (r−t+1)×(c−t+1)table. Type I error rates for tests of λ2 and λ3 agreed closely with nominal significant levels 0.01,0.05 and 0.10 if previous λ values were sufficiently large, but tended to be conservative otherwise. The procedure is illustrated with two examples from the literature