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.