Approximate prediction intervals in the original units when fitting models using data transformations

Abstract

AbstractIn this paper, we consider prediction interval estimation in the original units of observation after fitting a linear model to an appropriately transformed response variable. We assume that the residuals obtained from fitting the linear model in the transformed space are iid zero‐mean normal random variables, at least approximately. We discuss the bias in the retransformed mean and derive a reduced‐bias estimator for the kth moment of the original response, given settings of the design variables. This is then used to compute reduced‐bias estimates for the mean and variance of the untransformed response at various locations in design space. We then exploit a well‐known probability inequality, along with our proposed moment estimator, to construct an approximate 100(1−α)% prediction interval on the original response, given settings of the design factors. We used Monte Carlo simulation to evaluate the performance of the proposed prediction interval estimator relative to 2 commonly used alternatives. Our results suggest the proposed method is often the better alternative when the sample size is small and/or when the underlying model is misspecified. We illustrate the application of our new method by applying it to a real experimental data set obtained from the literature, where machine tool life was studied as a function of various machining parameters.