A Characterization of Most(More) Powerful Test Statistics with Simple Nonparametric Applications

Abstract

Data-driven most powerful tests are statistical hypothesis decision-making tools that deliver the greatest power against a fixed null hypothesis among all corresponding data-based tests of a given size. When the underlying data distributions are known, the likelihood ratio principle can be applied to conduct most powerful tests. Reversing this notion, we consider the following questions. (a) Assuming a test statistic, say T, is given, how can we transform T to improve the power of the test? (b) Can T be used to generate the most powerful test? (c) How does one compare test statistics with respect to an attribute of the desired most powerful decision-making procedure? To examine these questions, we propose one-to-one mapping of the term “most powerful” to the distribution properties of a given test statistic via matching characterization. This form of characterization has practical applicability and aligns well with the general principle of sufficiency. Findings indicate that to improve a given test, we can employ relevant ancillary statistics that do not have changes in their distributions with respect to tested hypotheses. As an example, the present method is illustrated by modifying the usual t-test under nonparametric settings. Numerical studies based on generated data and a real-data set confirm that the proposed approach can be useful in practice.

Keywords:

• Ancillary statistic
• Likelihood ratio
• Most powerful test
• Nonparametric test
• Sufficiency
• t-test
• Test for median

Supplementary Materials

The supplementary materials contain: the proofs of the theoretical results presented in the article; and Table S1 that displays the analysis of variance related to the linear regression fitted to the observed log-transformed HDL-cholesterol measurements using the log-transformed TBARS measurements as a factor, in Section 5.

Acknowledgments

The authors are grateful to the Editor, Associate Editor, and an anonymous referee for suggestions that led to a substantial extension and improvement of the presented results.

Additional information

Funding

This work was supported by a National Cancer Institute (NCI) Cancer Center Support Grant (CCSG) to Roswell Park Comprehensive Cancer Center (grant no. P30CA016056). The second author’s research is supported by the following three NCI grants: NRG Oncology Statistical and Data Management Center grant (grant no. U10CA180822); Immuno-Oncology Translational Network (IOTN) Moonshot grant (grant no. U24CA232979-01); Acquired Resistance to Therapy Network (ARTN) grant (grant no.U24CA274159-01).