A profile log-likelihood function to compare inverse normal transformations in a linear model framework

Inverse normal transformations (INTs) are widely used to ameliorate non-normality, although it remains unclear whether the four main variants—van der Waerden, Rankit, Tukey, and Blom—produce meaningfully different results. This paper provides the first empirical assessment of the profile log-likelihood measure (PLM) proposed by McCullagh and Tresoldi as a principled diagnostic for comparing INTs. Across simulation studies spanning varied sample sizes, skewed distributions, and heavy-tailed errors, we find that INT choice matters primarily in small samples. Below N = 50, the four methods can diverge substantially, with the van der Waerden INT showing particular sensitivity to tail behavior. Above N = 100, all four become effectively interchangeable under the PLM, with relative percentage ranges below 1% across all conditions. These patterns are replicated in an empirical dataset characterized by strong skewness and heavy tails. The PLM thus offers a robust, distribution-free tool for evaluating INT performance and clarifies when transformation choice is consequential for applied research.