A Selective Review of Statistical Significance in Cluster Analysis

Abstract

Cluster analysis can return apparent subgroups even when the observations arise from a population without scientifically meaningful cluster structure. Formal tests of clustering significance therefore require a definition of homogeneity, a statistic measuring the observed structure, and a calibration procedure that accounts for the data-dependent construction of the partition. We organize the literature around three inferential questions: whether cluster structure is present, how much structure is supported by the data, and what can be inferred after clustering. We review global tests for cluster structure, methods for assessing cluster number, stability, and hierarchical subdivisions, and procedures for valid inference after data-dependent clustering. A central theme is that these approaches target different notions of cluster structure and therefore lead to different statistical interpretations. Particular attention is given to genomic and single-cell applications, where high dimensionality, count-valued measurements, data-dependent preprocessing, clustering resolution, and reuse of the same measurements for downstream molecular characterization interact.