  Clustering methods have led to a number of important discoveries in
bioinformatics and beyond. A major challenge in their use is determining which
clusters represent important underlying structure, as opposed to spurious
sampling artifacts. This challenge is especially serious, and very few methods
are available, when the data are very high in dimension. Statistical
Significance of Clustering (SigClust) is a recently developed cluster
evaluation tool for high dimensional low sample size data. An important
component of the SigClust approach is the very definition of a single cluster
as a subset of data sampled from a multivariate Gaussian distribution. The
implementation of SigClust requires the estimation of the eigenvalues of the
covariance matrix for the null multivariate Gaussian distribution. We show that
the original eigenvalue estimation can lead to a test that suffers from severe
inflation of type-I error, in the important case where there are a few very
large eigenvalues. This paper addresses this critical challenge using a novel
likelihood based soft thresholding approach to estimate these eigenvalues,
which leads to a much improved SigClust. Major improvements in SigClust
performance are shown by both mathematical analysis, based on the new notion of
Theoretical Cluster Index, and extensive simulation studies. Applications to
some cancer genomic data further demonstrate the usefulness of these
improvements.
