Upper Tolerance Intervals Adjusted for Multiple Nuisance Uncertainties

Abstract

In many product-quality situations, it is of interest to state with X% confidence that at least Y% of the product made during a given period has true (unknown) values of a property below a calculated value. Such a statistical statement can be made using an upper Y%-content tolerance interval with X% confidence, which is also referred to as an X%/Y% upper tolerance interval (X%/Y% UTI). In this article, we present the methodology and formulas for calculating X%/Y% UTIs on the true property values when there are sources of uncertainty from nuisance factors (nuisance uncertainties) in addition to the source of variation of interest. The nuisance uncertainties may include regression model uncertainty if a regression model is used to predict the product property of interest. The X%/Y% UTI methodology is developed and illustrated using an example in which there are sampling, analytical, and regression model nuisance uncertainties in addition to the source of variation for which a statistical tolerance statement is desired. The design and results of a simulation study conducted to assess the performance of the X%/Y% UTI method are described. The X%/Y% UTI method that adjusts for nuisance uncertainties is shown to generally achieve the nominal X% and Y% values and yields significantly smaller UTIs than not adjusting for the nuisance uncertainties.