Fig. 2

Pairwise comparisons introduction. a Star tree, all isolates equidistantly related. In this scenario, each isolate has a random and independently distributed probability of exhibiting each state and Fisher’s exact test is appropriate. b In non-star trees, the probability of exhibiting each state is confounded by the population structure, in this case meaning the evolutionary history of the sample. An appropriate way of handling this is shifting focus towards evolutionary transitions, as in the pairwise comparisons algorithm. This figure shows the basic idea of a contrasting pair. This tree has a maximum number of 1 non-intersecting, contrasting pairs, a 1–1|0–0 pair. c An illegit pairing. While the two middle isolates and the top and the bottom isolates are both able to form a contrasting pair, a single picking cannot pick both pairs as they would intersect (shared branch shown stapled in purple). Thus, the maximum number of contrasting pairs in this tree is 1. The “best” picking is the red pair (1–1|0–0), which supports gene = 1 - > trait = 1 and the “worst” picking is the blue pair (1–0|0–1), which supports gene = 0 - > trait = 1. The associated p value is equal to 1.0 in either case