Classic and Kruskal-Wallis ANOVA methods test the explanatory power of a partitioning by an independent categorical variable on the associated response variables, either using their actual values or ranks. Nonparametric ANOVA extends the classic methodology to a case where a distance metric between data points can be defined, rather than response values themselves. Even though considerably widening the applicability of the ANOVA, it still does not provide a principled framework for the case where the responses do not follow a metric, such as BLAST similarities and distances in non-ultrametric trees. Here we consider ANOVA models for non-metric spaces. We consider cases where metric properties (identity, symmetry, and subadditivity) are each relaxed in turn to derive the resulting nonparametric ANOVA, and then focus on the special case where ranks of the similarity between responses are used. We show that nonparametric ANOVA follows as a special case if the response variable follows a proper metric. As a practical example, the methodology is applied to assess the confidence of bipartitions in a consensus species tree of 149 land plants estimated from pairwise BLAST scores of plastid genes.