Keith R. Eberhardt

Calibration and simultaneous tolerance intervals for regression

Simultaneous calibration (or discrimination) intervals in regression were proposed by Lieberman, Miller, and Hamilton (1967) and by Scheffe (1973). Those procedures enable one to construct confidence intervals for the unobserved values of the independent variable corresponding to an unlimited sequence of observations of the dependent variable in a regression model. These calibration intervals are conservative in that they are obtained from simultaneous tolerance intervals for which the actual confidence level exceeds the nominal level. Furthermore, all other existing simultaneous tolerance intervals in regression are likewise conservative. In this article, we propose simultaneous tolerance intervals that are narrower than previous intervals. Given the tables of factors included in this article, they are also simple to construct and use in straightline calibration applications.

A minimax approach to combining means, with practical examples☆

Abstract Eberhardt, K.R., Reeve, C.P. and Spiegelman, C.H., 1989. A minimax approach to combining means, with practical examples. Chemometrics and Intelligent Laboratory Systems , 5: 129–148. This paper describes a method for combining sample means that accounts for bias in those means. It compares the unweighted mean, the weighted mean using reciprocal estimated variances for weights, and a minimax weighted mean. When the individual means are subject to nontrivial biases we show that the minimax estimator can lead to important decreases in mean squared error and confidence interval width. Our recommendations are based on statistical theory and on simulations based on three Standard Reference Material data sets.

Computing factors for exact two-sided tolerance limits for a normal distribution

A self-contained FORTRAN subroutine is provided which computes factors for Wald-Wolfowitz type tolerance limits allowing arbitrary combinations of sample size n and degrees of freedom ν. The exact calculations from our program reveal inadequacies of two existing approximations, especially when ν ≫ n. Numerous applications where ν ≠ n − 1 are cited; two of these are discussed and illustrated.

A Careful Consideration of the Calibration Concept

This paper presents a detailed discussion of the technical aspects of the calibration process with emphasis on the definition of the measurand, the conditions under which the calibration results are valid, and the subsequent use of the calibration results in measurement uncertainty statements. The concepts of measurement uncertainty, error, systematic error, and reproducibility are also addressed as they pertain to the calibration process.

A comparison of uncertainty criteria for calibration

Calibration consists of using a fitted regression line to estimate the value of an unobserved independent variable x corresponding to an observed dependent variable y. To construct a confidence interval for a single x, Eisenhart introduced a procedure that consists of inverting prediction intervals around the regression line. Numerous other inference procedures have been proposed for multiple-use calibration, in which a single fitted regression line is used repeatedly to estimate many xs, We provide a synthesis of this literature and offer some numerical comparisons. We also attempt to motivate the use of various criteria based on the particular points of view of the various parties involved in determining the calibration or using the results. In addition, we derive probability expressions for computing exact simultaneous prediction intervals that enable the construction of tighter limits than are currently available based on that criterion.

Constant-Width Calibration Intervals for Linear Regression

Calibration, in the sense of inverse regression, is widely used in measurement science and other applications. For univariate regression models, simultaneous calibration intervals enable one to construct confidence intervals for the unobserved values of..

Fabrics for testing the ignition propensity of cigarettes

This paper reports an analysis of data from a study conducted by the cigarette industry to determine whether the fabrics used in a measurement method for cigarette ignition propensity reasonably represent the ignition behaviour of actual upholstery fabrics. A ‘consistency score’ is defined to evaluate objectively the relative agreement of ignition test results on various test fabrics compared with the cotton duck fabrics used in the measurement method. Particular attention is paid to those cases where the cigarettes show statistically significant di⁄erences by the chi-squared test. This analysis finds that the aggregated set of 79 industry fabrics ranks the four test cigarettes in the same order as do the three cotton duck fabrics in the measurement method. Thus, to the extent that the industry set is representative of those fabrics used in upholstery, it would be proper to use the three test fabrics as surrogates for the purpose of determining the relative ignition propensity of a cigarette. The analysis does identify six to ten fabrics that would be expected to show persistent reversals compared to the aggregate ordering; however, three-fourths of the fabrics rank the cigarettes consistently with the cotton duck fabrics. ( 1997 John Wiley & Sons, Ltd.