A variety of null hypotheses have been used to test for associations which might be construed as evidence of biologically interesting relationships between binary character states exhibited by taxa. These models assume that particular regions of a phylogenetic tree are independent with respect to their probabilities of character evolution, but they differ in the regions they specify. Early analyses specified terminal taxa (usually species); more recent developments have specified all or a subset of branches, or infinitesimally short sections of branches. Yet the central problem for comparative biologists is that branches throughout a phylogeny are not independent with respect to their evolutionary possibilities. Tests which assume that they are may indeed provide evidence for non-random association between characters according to the particular model of randomness used, but do not necessarily provide the basis for rationally inferring the existence of a biologically interesting link between the characters. In particular, they suffer from pseudoreplication of lineage-specific factors.
By way of contrast, we resurrect in this context a different model of randomness, the random assignment of treatments, which we argue provides a rationally acceptable basis for inference. States are assumed to be randomly assigned amongst sister taxa exhibiting different states of both variables. This model allows that the probabilities of character evolution vary throughout the tree, but does not require that these be specified, nor assumptions to be made about how evolution occurs. We illustrate this approach with reference to controversial associations between (i) warning coloration and larval gregariousness in butterflies, for which we find some support, and (ii) hybrid fitness and heterogamety (Haldane's Rule), for which we find no support, in contrast to Ridley's method which demonstrates the opposite Rule.