Title: When Are Estimates Independent of Measurement Units?

Abstract: Data transformations often facilitate regression analysis, yet many commonly used transformations make hypothesis testing misleading because the results depend on the measurement units of the data. This paper aims to address this issue by characterizing the set of transformations where measurement units do not affect conclusions in linear regressions. The equivalence theorem establishes that desirable properties—scale-equivariant coefficient estimates, scale-invariant t-statistics, and scale-invariant semi-elasticities—arise if and only if the transformation is a logarithmic or a power function. Power transformations thus offer a natural extension of logarithmic transformations that both preserves the essential feature of obtaining unit-independent estimates for unitless quantities of interest and can handle zero or negative values. On the other hand, popular alternatives that approximate the shape of the logarithmic function at large values, such as adding a small positive constant before applying a logarithmic transformation or the inverse hyperbolic sine transformation, result in similar inferences as in an untransformed linear regression when expressing outcomes in large measurement units and imply arbitrarily large effect sizes or arbitrarily large confidence intervals when expressing outcomes in small measurement units. We demonstrate using data from a randomized experiment that such transformations reverse the sign or significance of treatment effect estimates for up to 15 out of 49 outcomes variables when measurement units are changed to natural alternatives (e.g., from US dollars to local currency).