Hypothesis Testing With Rank Conditions in Phylogenetics

A phylogenetic model of sequence evolution for a set of n taxa is a collection of probability distributions on the 4(n) possible site patterns that may be observed in their aligned DNA sequences. For a four-taxon model, one can arrange the entries of these probability distributions into three flattening matrices that correspond to the three different unrooted leaf-labeled four-leaf trees, or quartet trees. The flattening matrix corresponding to the tree parameter of the model is known to satisfy certain rank conditions. Methods such as ErikSVD and SVDQuartets take advantage of this observation by applying singular value decomposition to flattening matrices consisting of empirical data. Each possible quartet is assigned an “SVD score” based on how close the flattening is to the set of matrices of the predicted rank. When choosing among possible quartets, the one with the lowest score is inferred to be the phylogeny of the four taxa under consideration. Since an n-leaf phylogenetic tree is determined by its quartets, this approach can be generalized to infer larger phylogenies. In this article, we explore using the SVD score as a test statistic to test whether phylogenetic data were generated by a particular quartet tree. To do so, we use several results to approximate the distribution of the SVD score and to give upper bounds on the p-value of the associated hypothesis tests. We also apply these hypothesis tests to simulated phylogenetic data and discuss the implications for interpreting SVD scores in rank-based inference methods.