HYPOTHESIS-TESTING FOR COMPONENTS OF THE SHIFTED MULTIPLICATIVE MODEL FOR A NONADDITIVE 2-WAY TABLE

To determine the number of multiplicative components in a shifted multiplicative model (SHMM) for a r x c table (c less-than-or-equal-to r) given by y(ij) = beta + SIGMA(k=1)t lambda(k)delta(ik)gamma(jk) + e(ij), likelihood ratio tests of H-0: lambda(t) = 0 versus H(a): lambda(t) not-equal 0, lambda(t+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 greater-than-or-equal-to 2, a method analogous to one developed by Marasinghe (1985) and Schott (1986) for the additive main effect and multiplicative interaction (AMMI) model y(ij) = mu + tau(i) + eta(j) + SIGMA(k=1)t theta(k)u(ik)v(jk) + e(ij) is used. namely, that the t(th) term is tested as thought it is the first term for a (r - t + 1) x (c - t + 1) table. Type I error rates for tests of lambda2 and lambda3 agreed closely with nominal significant levels 0.01, 0.05 and 0.10 if previous lambda values were sufficiently large, but tended to be conservative otherwise. The procedure is illustrated with two examples from the literature.