A Prediction Interval Estimator for the Original Response When Using Box-Cox Transformations

• Activity Number: 133
• Type: Contributed
• Date/Time: Monday, August 4, 2014 : 8:30 AM to 10:20 AM
• Sponsor: Quality and Productivity Section
• Author(s): Michael Walker*+ and Marcus Perry
• Companies: University of Alabama and University of Alabama
• Keywords: Box-Cox transformations ; transformation bias ; designed experiments ; linear models ; prediction intervals

Abstract

Motivated by electron microscopy experiments, we develop an approximate prediction interval on the response variable Y, assuming a normal-theory linear model is fit using a transformed version of Y, contained in the Box-Cox family. We derive a closed-form approximation to the kth moment of Y, which is then used to estimate the mean and variance of Y, given parameter estimates obtained from fitting the model in the transformed domain. Chebychev's inequality is then used to construct a prediction interval estimator on Y. Using Monte Carlo simulation, we assess the performance of our estimators for the mean and variance of Y, as well as the width performance of our proposed Chebychev interval, relative to that obtained by a more common interval construction approach. General results suggest that, for a given level of expected coverage, the proposed interval estimator will achieve a smaller mean and variance of the interval width estimates, especially as the degrees of freedom beyond that required to estimate model terms is small. We apply our method to two real experimental data sets, one involving a standard design, and the other involving a design with a split-plot error structure.