Abstract

Under supervised heterogeneity analysis, samples within a population

form groups, and different groups have different regression models.

In most

of the existing analyses, a single level of heterogeneity structure is considered.

Partly motivated by multi-level unsupervised analysis such as hierarchical clustering, we consider multi-level supervised heterogeneity analysis. Consider for

example a two-level analysis. At the higher level, “coarse” information is used,

and samples form a smaller number of groups. At the lower level, “more subtle” information is used, and samples form a larger number of subgroups. To

achieve more lucid interpretations, we further consider the scenario where only

some variables are relevant at each level, different groups (subgroups) have the

same set of relevant variables, and the important variables at the higher level are

nested in those at the lower level. A penalized estimation and selection approach

is developed, and its theoretical and computational properties are established.

Simulation demonstrates competitive performance of the proposed approach. In

the analysis of TCGA breast cancer data, the proposed approach leads to sensible

grouping/subgrouping, identification, and estimation results. Overall, this study

expands the scope of heterogeneity analysis and delivers a practically useful tool.

Information

Preprint No.SS-2025-0147
Manuscript IDSS-2025-0147
Complete AuthorsRuiyue Wang, Sanguo Zhang, Shuangge Ma
Corresponding AuthorsShuangge Ma
Emailsshuangge.ma@yale.edu

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Acknowledgments

We thank the editors and reviewers for their careful review and insightful

comments. This work was supported by the National Natural Science Foundation of China No.12571298, Fundamental Research Funds for the Central

Universities, NIH CA204120, and NSF 220968.

Supplementary Materials

Online Supplementary Materials contains additional theoretical (referenced

in Sections 3), numerical (referenced in Section 5 and Section 6), and

methodological (referenced in Sections 7) developments, which is available

on the Statistica Sinica website.

Supplementary materials are available for download.