Data from: Low-parameter phylogenetic inference under the general Markov model

In their 2008 and 2009 papers, Sumner and colleagues introduced the
"squangles" -- a small set of Markov invariants for phylogenetic
quartets. The squangles are consistent with the general Markov model (GM)
and can be used to infer quartets without the need to explicitly estimate
all parameters. As GM is inhomogeneous and hence non-stationary, the
squangles are expected to perform well compared to standard approaches
when there are changes in base-composition amongst species. However, the
GM model assumes constant rates across sites, so the squangles should be
confounded by data generated with invariant sites or other forms of
rate-variation across sites. Here we implement the squangles in a
least-squares setting that returns quartets weighted by either confidence
or internal edge lengths, and we show how these weighted quartets can be
used as input into a variety of supertree and supernetwork methods. For
the first time, we quantitatively investigate the robustness of the
squangles to breaking of the constant rates-across-sites assumption on
both simulated and real data sets; and we suggest a modification that
improves the performance of the squangles in the presence of invariant
sites. Our conclusion is that the squangles provide a novel tool for
phylogenetic estimation that is complementary to methods that explicitly
account for rate-variation across sites, but rely on homogeneous -- and
hence stationary -- models.