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.

翻译：在多元回归模型中，我们证明了针对感兴趣变量的系数t检验是一致最大功效无偏检验，其中其他参数被视为冗余参数。该证明基于具有Neyman结构的检验理论，且不要求检验统计量的无偏性或线性假设。我们进一步证明，设计矩阵的Gram-Schmidt分解导出了一族回归模型，该模型可能为相应的变换回归量提供功效更强的检验。最后，我们讨论了Gram-Schmidt回归相较于标准多元回归模型的解释性与性能标准，并展示了功效差异如何对研究设计产生重要影响。