Tests for two trees using likelihood methods.

This article considers two similar likelihood-based test statistics for comparing two fixed trees, the Kishino-Hasegawa (KH) test statistic and the likelihood ratio (LR) statistic, as well as a number of different methods for determining thresholds to declare a significant result. An explanation is given for why the KH test, which uses the KH test statistic and normal theory thresholds, need not give correct type I error probabilities under the appropriate null hypothesis. Simulations show that the KH test tends to give much smaller type I error probabilities than expected. The article presents a computationally efficient normal-theory parametric bootstrap method for determining better KH test statistic thresholds. For the LR statistic, existing mixture of chi-squares results for determining thresholds are extended to cases in which a tree with two or three zero edge-lengths exhibits the two trees being compared. The resulting chi-bar test and use of the KH test statistic with normal bootstrap are shown through simulation to give good performance but are more difficult to implement than the KH test. Two conservative approaches are presented which require only log likelihoods and simple chi-square thresholds. While they did not perform as well as chi-bar and normal bootstrap methods in the simulations considered, they gave better performance than the KH test and have just as simple an implementation. As a by-product of parametric bootstrap considerations, an adjustment to the Swofford-Olsen-Waddell-Hillis (SOWH) test is proposed.