Formal test for one-way versus two-way clustering

Develop and validate a formal statistical procedure to test one-way clustering against two-way clustering in linear regression models with clustered disturbances, and assess whether an adaptation of score-variance tests provides a correct and reliable solution.

Background

Determining whether dependence occurs along one or two clustering dimensions is crucial for valid inference. While two-way clustering methods exist, a formal test to choose between one-way and two-way clustering has not been established. The literature conjectures that score-variance tests might be adapted, but this remains to be shown rigorously.

References

Unfortunately, there is at present no formal way to test one-way clustering against two-way clustering, although \citet*[Remark~9]{MNW-testing} conjectures that a version of the score-variance test can be used.

— When Can We Trust Cluster-Robust Inference?  (2604.02000 - MacKinnon, 2 Apr 2026) in Subsection 5.2 (Two-Way Clustering)

We did not apply a permutation or Mantel test to account for this dependence, so the reported significance levels should be interpreted with this caveat in mind. A permutation test that permutes language identities, or a regression with random effects for source and target language, would account for this dependence; we leave this to future work.

— Improving Cross-Lingual Transfer for Sequential Sentence Classification in Research Papers via Structural Similarity  (2609.19650 - Yamauchi et al., 17 Sep 2026) in Section 'Limitations', subsection 'Lack of independence between language pairs'

These checks support the result, although the small-sample approximation remains uncertain.

— StudentBench: AI and human tutoring yield equivalent GRE learning gains  (2609.28470 - Northcutt et al., 23 Sep 2026) in Appendix B.2, Pooled AI-human equivalence