  We propose a statistical framework to investigate whether a given
subpopulation lies between two other subpopulations in a multivariate feature
space. This methodology is motivated by a biological question from a
collaborator: Is a newly discovered cell type between two known types in
several given features? We propose two in-betweenness indices (IBI) to quantify
the in-betweenness exhibited by a random triangle formed by the summary
statistics of the three subpopulations. Statistical inference methods are
provided for triangle shape and IBI metrics. The application of our methods is
demonstrated in three examples: the classic Iris data set, a study of risk of
relapse across three breast cancer subtypes, and the motivating neuronal cell
data with measured electrophysiological features.
