Uniformly most powerful tests in linear models

TL;DR

This paper proves the t-test's optimality in linear models and introduces a Gram-Schmidt-based approach for potentially more powerful tests, impacting study design and interpretation.

Contribution

It establishes the uniform most powerful unbiased property of the t-test in linear models without assuming unbiasedness or linearity, and proposes a Gram-Schmidt decomposition for improved testing.

Findings

t-test is uniformly most powerful unbiased in linear models

Gram-Schmidt decomposition can lead to more powerful tests

Implications for study design and test interpretation

Abstract

In the multiple regression model we prove that the coefficient t-test for a variable of interest is uniformly most powerful unbiased, with the other parameters considered nuisance. The proof is based on the theory of tests with Neyman-structure and does not assume unbiasedness or linearity of the test statistic. We further show that the Gram-Schmidt decomposition of the design matrix leads to a family of regression model with potentially more powerful tests for the corresponding transformed regressors. Finally, we discuss interpretation and performance criteria for the Gram-Schmidt regression compared to standard multiple regression, and show how the power differential has major implications for study design.