Multiple Comparisons of Log-Likelihoods with Applications to Phylogenetic Inference

The maximum-likelihood method for inferring mo-lecular phylogeny (Felsenstein 1981) is being widelyused. The probabilistic model for generating the molec-ular sequences is specified by the substitution processand the tree topology. The parameters for the substitu-tion process and the branch lengths are estimated bymaximizing the likelihood, and then the tree topology isestimated by maximizing the maximized likelihood. Toobtain the confidence limit of the topology, the test ofKishino and Hasegawa (1989), referred to as the KHtest, is often used in practice. The same idea that is thebasis for the KH test is also found in the statistical lit-erature (Linhart 1988; Vuong 1989). The KH test wasdesigned for comparing two topologies but is often usedfor comparing many topologies. This use of the KH testleads to overconfidence for a wrong tree, because thesampling error due to the selection of the topology isoverlooked in it. In this note, we present a modificationof the KH test to take into account a multiplicity oftestings.Let a index the topologies and L