  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, GM includes the IID assumption, so the squangles
should be confounded by data generated with invariant sites or with
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 use these as input into a variety of quartet-based supertree
methods. For the first time, we quantitatively investigate the robustness of
the squangles to the breaking of IID assumptions 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.
