Testing for interaction in two-way random and mixed effects models: the fully nonparametric approach

In a recent paper, Gaugler and Akritas (unpublished manuscript) considered testing for no main effect in a two-factor mixed effects design when the traditional assumptions do not hold. Here we extend the nonparametric modeling to the random effects design and consider the problem of testing for no interaction effect. The new models for these designs allow for dependence among the random effects, heteroscedasticity in the error and interaction terms, and do not require normality. At a more systemic level, these models differ from the classical ones in that they do not consider the random interaction term as an additional, extraneous source of variability. The proposed test procedure applies to settings where the random factor in the case of the mixed model or at least one of the random factors in the case of the random effects model has many levels. The number of replications can be small and possibly unbalanced. Moreover, the model and test procedure are general enough to accommodate data missing at random (MAR), provided the missingness mechanism is the same for each level of the random effect. The limiting distribution of the test statistic is normal. Extensive simulations indicate that our test procedure, with or without missing data, maintains the nominal Type I error rate in all simulation settings. On the contrary, the standard procedures (the F-test of PROC GLM in SAS, and the ML and REML methods of PROC MIXED in SAS), as well as the exact F-test of Khuri, Mathew, and Sinha (1998 in Statistical Tests for Mixed Linear Models), are extremely liberal in heteroscedastic settings, while under homoscedasticity and normality, the proposed test procedure is comparable to them. An analysis of a dataset from the Mussel Watch Project is presented.