Data transformations facilitate regression analysis by addressing issues such as skewness and nonlinearity, yet many popular transformations make hypothesis testing misleading because results change with measurement units. Using data from a randomized experiment, we demonstrate that natural rescaling can drastically alter conclusions under widely used transformations. We formalize a minimal coherence requirement for empirical analysis: re-expressing the same data in different units should neither change _t_-statistics nor distort predicted values beyond mechanical rescaling. Measurement-unit invariance links OLS, GLM, and Tobit specifications to logarithmic and power forms. We discuss conditions for interpreting outputs from transformed-outcome regressions as structural, conditional-mean, and conditional-median semi-elasticities.