Statistical Significance of Clustering for High-Dimension, Low-Sample Size Data

Clustering methods provide a powerful tool for the exploratory analysis of high-dimension, low-sample size (HDLSS) data sets, such as gene expression microarray data. A fundamental statistical issue in clustering is which clusters are ''really there'', as opposed to being artifacts of the natural sampling variation. We propose SigClust as a simple and natural approach to this fundamental statistical problem. In particular, we define a cluster as data coming from a single Gaussian distribution and formulate the problem of assessing statistical significance of clustering as a testing procedure. This Gaussian null assumption allows direct formulation of p values that effectively quantify the significance of a given clustering. HDLSS covariance estimation for SigClust is achieved by a combination of invariance principles, together with a factor analysis model. The properties of SigClust are studied. Simulated examples, as well as an application to a real cancer microarray data set, show that the proposed method works remarkably well for assessing significance of clustering. Some theoretical results also are obtained.