A generalised Box–Cox transformation for the parametric estimation of clinical reference intervals

Abstract

Parametric methods for the calculation of reference intervals in clinical studies often rely on the identification of a suitable transformation so that the transformed data can be assumed to be drawn from a Gaussian distribution. In this paper, the two-stage transformation recommended by the International Federation for Clinical Chemistry is compared with a novel generalised Box–Cox family of transformations. Investigation is also made of sample sizes needed to achieve certain criteria of reliability in the calculated reference interval. Simulations are used to show that the generalised Box–Cox family achieves a lower bias than the two-stage transformation. It was found that there is a possibility that the two-stage transformation will result in percentile estimates that cannot be back-transformed to obtain the required reference intervals, a difficulty not observed when using the generalised Box–Cox family introduced in this paper.

Keywords:

• Box–Cox family
• transformations
• reference intervals

Acknowledgements

The author is grateful for the advice and help received from Terence Iles and Barry Nix (both retired members of staff from Cardiff University). He would also like to acknowledge useful discussions with Dafydd Evans (Cardiff School of Mathematics, Cardiff University).