Regression‐based heterogeneity analysis to identify overlapping subgroup structure in high‐dimensional data

Heterogeneity is a hallmark of complex diseases. Regression‐based heterogeneity analysis, which is directly concerned with outcome–feature relationships, has led to a deeper understanding of disease biology. Such an analysis identifies the underlying subgroup structure and estimates the subgroup‐specific regression coefficients. However, most of the existing regression‐based heterogeneity analyses can only address disjoint subgroups; that is, each sample is assigned to only one subgroup. In reality, some samples have multiple labels, for example, many genes have several biological functions, and some cells of pure cell types transition into other types over time, which suggest that their outcome–feature relationships (regression coefficients) can be a mixture of relationships in more than one subgroups, and as a result, the disjoint subgrouping results can be unsatisfactory. To this end, we develop a novel approach to regression‐based heterogeneity analysis, which takes into account possible overlaps between subgroups and high data dimensions. A subgroup membership vector is introduced for each sample, which is combined with a loss function. Considering the lack of information arising from small sample sizes, an l2$l_2$ norm penalty is developed for each membership vector to encourage similarity in its elements. A sparse penalization is also applied for regularized estimation and feature selection. Extensive simulations demonstrate its superiority over direct competitors. The analysis of Cancer Cell Line Encyclopedia data and lung cancer data from The Cancer Genome Atlas show that the proposed approach can identify an overlapping subgroup structure with favorable performance in prediction and stability.