# Inverse normal transformation in SOLAR

SOLAR software can, at the discretion of the user, apply a rank-based inverse-normal transformation to the data using the command inormal. This transformation is the one suggested by Van der Waerden (1952) and is given by:

$\tilde y_i = \Phi^{-1}\left\{\dfrac{r_i}{n+1}\right\}$

where $\tilde y_i$ is the transformed value for observation $i$, $\Phi^{-1}\left\{\cdot\right\}$ is the probit function, $r_i$ is the ordinary rank of the $i$-th case among $n$ observations.

This transformation is a particular case of the family of transformations discussed in the paper by Beasley et al. (2009). The family can be represented as:

$\tilde y_i = \Phi^{-1}\left\{\dfrac{r_i+c}{n-2c+1}\right\}$

where $c$ is a constant and the remaining variables are as above. The value of $c$ varies for different proposed methods. Blom (1958) suggests $c=3/8$, Tukey (1962) suggests $c=1/3$, Bliss (1967) suggests $c=1/2$ and, as just decribed, Van der Waerden suggests $c=0$.

Interesting enough, the Q-Q plots produced by Octave use the Bliss (1967) transformation.

An Octave/matlab function to perform these transformations in arbitrary data is here: inormal.m (note that this function does not require or use SOLAR).

## Version history

• 23.Jul.2011: First public release.
• 19.Jun.2014: Added ability to deal with ties, as well as NaNs in the data.