Testing Normality of Data Transformed by Maximum Likelihood Box Cox

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.