  When the process underlying DNA substitutions varies across evolutionary
history, the standard Markov models underlying standard phylogenetic methods
are mathematically inconsistent. The most prominent example is the general time
reversible model (GTR) together with some, but not all, of its submodels. To
rectify this deficiency, Lie Markov models have been developed as the class of
models that are consistent in the face of a changing process of DNA
substitutions. Some well-known models in popular use are within this class, but
are either overly simplistic (e.g. the Kimura two-parameter model) or overly
complex (the general Markov model). On a diverse set of biological data sets,
we test a hierarchy of Lie Markov models spanning the full range of parameter
richness. Compared against the benchmark of the ever-popular GTR model, we find
that as a whole the Lie Markov models perform remarkably well, with the best
performing models having eight parameters and the ability to recognise the
distinction between purines and pyrimidines.
