  Cluster analysis has proved to be an invaluable tool for the exploratory and
unsupervised analysis of high dimensional datasets. Among methods for
clustering, hierarchical approaches have enjoyed substantial popularity in
genomics and other fields for their ability to simultaneously uncover multiple
layers of clustering structure. A critical and challenging question in cluster
analysis is whether the identified clusters represent important underlying
structure or are artifacts of natural sampling variation. Few approaches have
been proposed for addressing this problem in the context of hierarchical
clustering, for which the problem is further complicated by the natural tree
structure of the partition, and the multiplicity of tests required to parse the
layers of nested clusters. In this paper, we propose a Monte Carlo based
approach for testing statistical significance in hierarchical clustering which
addresses these issues. The approach is implemented as a sequential testing
procedure guaranteeing control of the family-wise error rate. Theoretical
justification is provided for our approach, and its power to detect true
clustering structure is illustrated through several simulation studies and
applications to two cancer gene expression datasets.
