Testing Normality of Data Transformed by Maximum Likelihood Box Cox

TL;DR

This paper investigates the bias in normality tests after Box-Cox transformations and proposes a recalibration method to correct this bias, enhancing the reliability of parametric analyses in biomarker and environmental studies.

Contribution

It introduces a recalibration approach to correct bias in normality tests applied to Box-Cox transformed data, including the Anderson Darling and Shapiro-Wilk tests.

Findings

Bias in normality tests is severe after Box-Cox transformation.

Recalibration effectively reduces bias in multiple normality tests.

Improved normality testing supports more accurate parametric analysis.

Abstract

Transforming a random variable to improve its normality leads to a followup test for whether the transformed variable follows a normal distribution. Previous work has shown that the Anderson Darling test for normality suffers from resubstitution bias following Box-Cox transformation, and indicates normality much too often. The work reported here extends this by adding the Shapiro-Wilk statistic and the two-parameter Box Cox transformation, all of which show severe bias. We also develop a recalibration to correct the bias in all four settings. The methodology was motivated by finding reference ranges in biomarker studies where parametric analysis, possibly on a power-transformed measurand, can be much more informative than nonparametric. Setting environmental standards illustrates another potential application.