Tests and relevancies for the hypotheses of an orthogonal family in a model with orthogonal block structure

A model has an orthogonal block structure if it has, as covariance matrix, a linear combination of pairwise orthogonal projection matrices, that add up to the identity matrix. The range space of these matrices are associated to hypotheses of an orthogonal family. In this paper we show how to obtain tests for these hypotheses when normality is assumed and how to consider their relevance when normality is discarded. Besides the notion of relevance, we formulate hypotheses in a general way that may be applied to models with orthogonal block structure, whose factors may have fixed and/or random effects. The results are applied to prime basis factorial models and an example is presented.