On hyperbolic transformations to normality

- A family of hyperbolic transformations towards normality is proposed/constructed.
- The family is effective in transforming skewed/platykurtic distributions to normal.
- The matching quantile approach is used for initial parameter estimates.
- The new family outperforms other well-known transformations in a simulation.
- Data examples on mathematics test scores and DNA microarrays are illustrated.

In biological and social sciences, it is essential to consider data transformations to normality for detecting structural effects and for better data representation and interpretation. An array of transformations to normality has been derived for data exhibiting skewed, leptokurtic and unimodal shapes, but is less amenable to data exhibiting platykurtic shapes, such as a nearly bimodal distribution. This study proposes and constructs a new family of hyperbolic power transformations for improving normality of raw data with varying degrees of skewness and kurtosis. An advantage this new family has is its effectiveness in transforming platykurtic or bimodal data distributions to normal. A simulation study and a real data example on mathematics achievement test scores are used to illustrate the wide-ranging applications of the proposed family of transformations. As a cautionary note, usefulness and limitations of the proposed method will be discussed for stabilizing the variance of DNA microarray data and for symmetrizing the data distribution towards normality. The empirical applications also illustrate an example of conservative t- and ANOVA F-tests when the assumption of normality is violated.