Testing for Main Random Effects in Two-Way Random and Mixed Effects Models: Modifying the Statistic

A procedure for testing the significance of the main random effect is proposed under a model which does not require the traditional assumptions of symmetry, homoscedasticity, and normality for the error term and random effects. To accommodate this level of model generality, and also unbalanced designs, suitable adjustments to the F-test are made. The extensive simulations performed under the random effects model, and the unrestricted and restricted versions of the mixed effects model, indicate that the classical F procedure is extremely liberal under heteroscedasticity and unbalancedness. The proposed test procedure performs well in all settings and is comparable to the classical F-test when the classical assumptions are met. An analysis of a dataset from the Mussel Watch Project is presented. 1. Introduction Consider a two-factor mixed or random effects design, and let denote the th observation at level of the row factor and level of the column factor. In the case of a mixed effects design we assume that the row factor is fixed. The classical two-way model, compare [1–5], uses the decomposition For the random effects model, the ’s, ’s, ’s, and ’s are mutually independent, the ’s are iid , the ’s are iid , the ’s are iid , and the ’s are iid . For the mixed effects model, there are two common definitions of the effects. The unrestricted version of the model assumes and the are all independent. The restricted version of the model keeps the above assumptions but requires that implying that are correlated. Both versions assume the to be iid and independent from the other random effects. There is a dichotomy of opinion in the statistical literature as to which model should be used. Cornfield and Tukey [6], Scheffé [1], Winer [7], and Khuri et al. [5], among others, advocate the restricted version. Searle [2], Hocking [8], and Searle et al. [4] advocate the unrestricted version. While the statistic for testing for no main random effect differs in the two versions, both use what Scheffé [1, page 264] calls the symmetry assumption. Basically, this is the assumption of independence of the random main and interaction effects. This assumption was criticized in [9, 10] as unrealistic in most practical situations. More importantly, simulations demonstrated that the classical -test for testing the significance of the main fixed effect is very sensitive to departures from the symmetry assumption even in balanced designs with homoscedastic errors. Additional simulations showed the classical -test for testing the significance of the interaction effect to also be very