Accreditation and Quality Assurance

, Volume 13, Issue 4, pp 193–216

A comparison of location estimators for interlaboratory data contaminated with value and uncertainty outliers

General Paper

DOI: 10.1007/s00769-008-0360-3

Cite this article as:
Duewer, D.L. Accred Qual Assur (2008) 13: 193. doi:10.1007/s00769-008-0360-3

Abstract

While estimation of measurement uncertainty (MU) is increasingly acknowledged as an essential component of the chemical measurement process, there is little agreement on how best to use even nominally well-estimated MU. There are philosophical and practical issues involved in defining what is “best” for a given data set; however, there is remarkably little guidance on how well different MU-using estimators perform with imperfect data. This report characterizes the bias, efficiency, and robustness properties for several commonly used or recently proposed estimators of true location, μ, using “Monte Carlo” (MC) evaluation of “measurement” data sets drawn from well-defined distributions. These synthetic models address a number of issues pertinent to interlaboratory comparisons studies. While the MC results do not provide specific guidance on “which estimator is best” for any given set of real data, they do provide broad insight into the expected relative performance within broadly defined scenarios. Perhaps the broadest and most emphatic guidance from the present study is that (1) well-estimated measurement uncertainties can be used to improve the reliability of location determination and (2) some approaches to using measurement uncertainties are better than others. The traditional inverse squared uncertainty-weighted estimators perform well only in the absence of unrepresentative values (value outliers) or underestimated uncertainties (uncertainty outliers); even modest contamination by such outliers may result in relatively inaccurate estimates. In contrast, some inverse total variance-weighted-estimators and probability density function area-based estimators perform well for all scenarios evaluated, including underestimated uncertainties, extreme value outliers, and asymmetric contamination.

Keywords

Consensus valueInterlaboratory comparisonsMeasurement uncertaintyMixture modelsMonte Carlo evaluationProbability density functionRobustnessWeighting function

Abbreviations

CCQM

Comité Consultatif pour la Quantité de Matière

ITV

Inverse total variance

IU2

Inverse squared uncertainty

KC

Key comparison

Lp

Least power

MADe

Median absolute deviation from the median, expressed as a standard deviation

MC

Monte Carlo

MM

Mixture model

MU

Measurement uncertainty

n

Number of measurements in a data set

nBS

Number of bootstrap pseudo-data sets

nMC

Number of Monte Carlo simulation data sets

N(μσ)

Normal (Gaussian) distribution having mean μ and standard deviation σ

NMI

National Metrology Institute

P

Level of confidence

PC

Principal component

PDF

Probability density function

UI[u]

Uniform (rectangular) distribution of integers having lower limit and upper limit u

UR[u]

Uniform (rectangular) distribution of real numbers having lower limit and upper limit u

s

Estimate of dispersion, expressed as a standard deviation

\( s{\left( {\hat{\mu }} \right)} \)

Estimate of the variability of an estimator on replicate sampling of a population, expressed as a standard deviation

u(xi)

Uncertainty component of the ith measurement, expressed as a standard deviation

UP(xi)

Uncertainty component of the ith measurement, expressed as a coverage interval providing approximately P% level of confidence

wi

Weight in a given calculation given to the ith measurement

\( \ifmmode\expandafter\bar\else\expandafter\=\fi{x} \)

Arithmetic mean

xi

Value component of the ith measurement

μ

True location of a population

\( \hat{\mu } \)

Estimate of location

σ

True dispersion of a population, expressed as a standard deviation

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Analytical Chemistry Division, Stop 8390, Chemical Science and Technology LaboratoryNational Institute of Standards and TechnologyGaithersburgUSA