  We extend the work of Robinson and Turner to use hypothesis testing with
persistence homology to test for measurable differences in shape between point
clouds from three or more groups. Using samples of point clouds from three
distinct groups, we conduct a large-scale simulation study to validate our
proposed extension. We consider various combinations of groups, samples sizes
and measurement errors in the simulation study, providing for each combination
the percentage of $p$-values below an alpha-level of 0.05. Additionally, we
apply our method to a Cardiotocography data set and find statistically
significant evidence of measurable differences in shape between normal, suspect
and pathologic health status groups.

Methodology;Algebraic Topology