ranknormalization

Rank normalization, also known as the rank-based inverse normal transformation or normal scores transformation, is a data preprocessing technique that aims to transform a numeric variable so its distribution resembles a standard normal distribution. It is a nonparametric method that relies on the relative ordering of observations rather than their absolute values and preserves the monotonic relationship with the original data.

Method: for each variable with n observations, compute the ranks r_i (using the average rank for ties). Map each rank to a uniform probability p_i = (r_i − 0.5)/n. Transform using x_i' = Φ^{-1}(p_i), where Φ^{-1} is the inverse cumulative distribution function of the standard normal distribution. The resulting transformed values are on a standard-normal scale and preserve the rank order of the original data. Variants differ mainly in how ties are handled or how p_i is defined, with the van der Waerden transformation and the rank-based inverse normal transformation (INT) being common examples.

Properties and notes: the transform is monotone and nonparametric, reducing the impact of skewness and outliers while producing data that are closer to normal. It does not preserve original distances or units, and interpretation of transformed values is in terms of standard normal units rather than the original scale. It can alter relationships between variables, particularly if nonlinear associations exist.

Applications and limitations: rank normalization is used to satisfy normality or homoscedasticity assumptions in parametric tests and regression, improve model performance in some machine learning pipelines, and is widely employed in fields such as statistics and genomics (for example, to normalize phenotypes in GWAS). It may be less appropriate when the exact scale or interpretable units of the original data are essential, or when many tied values reduce discrimination after transformation.