<p>Cross-classified random effects models are attractively used in recent social, medical, and behavioral research. These models gain their popularity due to the prevalence of cross-classified sources to which data are known to belong. Selection of a flexible modeling framework by testing the need for subset of the crossed effects thus constitutes a cornerstone in handling the complexity of the data structure in hand. Existing tests are restricted to linear models. For generalized linear models, test procedures for crossed random effects are either not well-developed or limited to normally distributed random effects, an assumption that is not easily verified in practice. While the impact of relevant tests has been noticed in the literature, preserving correct Type-I error rate when the error components are non-normally distributed is admitted to few tests that are only applied to linear models. Interestingly, distribution-free permutation tests are among the competing solutions. Nevertheless, such methods are not oriented to test for subsets of crossed random effects, which constitutes our proposal in this paper. By standardizing the linear model’s covariance matrix, the performance of a proposed analysis-of-variance test statistic is explored and is shown, via simulation studies, to possess favorable power compared to existing tests. An algorithm to obtain a correct size of test based on an approximate finite-sample distribution is provided via a simple permutation method. Importantly, a straightforward extension of the test procedure to generalized linear crossed random effects models is presented and assessed. Further illustration is provided using real datasets.</p>