A Powerful Bayesian Test for Equality of Means in High Dimensions

ABSTRACT

We develop a Bayes factor-based testing procedure for comparing two population means in high-dimensional settings. In ‘large-p-small-n” settings, Bayes factors based on proper priors require eliciting a large and complex p × p covariance matrix, whereas Bayes factors based on Jeffrey’s prior suffer the same impediment as the classical Hotelling T² test statistic as they involve inversion of ill-formed sample covariance matrices. To circumvent this limitation, we propose that the Bayes factor be based on lower dimensional random projections of the high-dimensional data vectors. We choose the prior under the alternative to maximize the power of the test for a fixed threshold level, yielding a restricted most powerful Bayesian test (RMPBT). The final test statistic is based on the ensemble of Bayes factors corresponding to multiple replications of randomly projected data. We show that the test is unbiased and, under mild conditions, is also locally consistent. We demonstrate the efficacy of the approach through simulated and real data examples. Supplementary materials for this article are available online.

KEYWORDS:

• Bayes factor
• Random projection
• Restricted most powerful Bayesian tests
• Testing of hypotheses

Supplementary Material

The Appendix referenced in the article can be found online in the Appendix file. Supplementary material presents additional tables from the simulation study and a section presenting the derivation of a Bayes factor based on proper joint Normal-Inv-Wishart prior for the nuisance parameters. We have provided a table comparing power estimates between a deterministic projection-based test and a random projection-based test. We have also included some additional tables containing the results of a simulation comparing the power of RMPBT to that of a test obtained from a proper prior. A Julia code, implementing our approach, is also available as part of the supplementary material.

Acknowledgments

The authors are grateful to Dr. Robert Chapkin for sharing the organoids data with them. the authors also thank the associate editor and the anonymous referees for their comments and suggestions that helped greatly improve the article’s presentation.

Additional information

Funding

Carroll and Mallick’s research was supported by grant U01-CA057030 and by grant R01-CA194391, both from the National Cancer Institute.