inormal | Brainder.

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).

References

Van der Waerden BL. Order tests for the two-sample problem and their power. Proc Koninklijke Nederlandse Akademie van Wetenschappen. Ser A. 1952; 55:453–458.
Blom G. Statistical estimates and transformed beta-variables. Wiley, New York, 1958 (page 71).
Tukey JW. The future of data analysis. Ann Math Stat. 1962; 33:1–67.
Bliss CI. Statistics in biology. McGraw-Hill, New York, 1967 (pages 117–120).
Beasley TM, Erickson S, Allison DB. Rank-based inverse normal transformations are increasingly used, but are they merited? Behav Genet. 2009; 39(5):580-95.