ISO/TS 17503:2015
(Main)Statistical methods of uncertainty evaluation — Guidance on evaluation of uncertainty using two-factor crossed designs
Statistical methods of uncertainty evaluation — Guidance on evaluation of uncertainty using two-factor crossed designs
ISO/TS 17503:2015 describes the estimation of uncertainties on the mean value in experiments conducted as crossed designs, and the use of variances extracted from such experiments and applied to the results of other measurements (for example, single observations). ISO/TS 17503:2015 covers balanced two-factor designs with any number of levels. The basic designs covered include the two-way design without replication and the two-way design with replication, with one or both factors considered as random. Calculations of variance components from ANOVA tables and their use in uncertainty estimation are given. In addition, brief guidance is given on the use of restricted maximum likelihood estimates from software, and on the treatment of experiments with small numbers of missing data points. Methods for review of the data for outliers and approximate normality are provided. The use of data obtained from the treatment of relative observations (for example, apparent recovery in analytical chemistry) is included.
Méthodes statistiques d'évaluation de l'incertitude — Lignes directrices pour l'évaluation de l'incertitude des modèles à deux facteurs croisés
General Information
Standards Content (Sample)
TECHNICAL ISO/TS
SPECIFICATION 17503
First edition
2015-11-01
Statistical methods of uncertainty
evaluation — Guidance on evaluation
of uncertainty using two-factor
crossed designs
Méthodes statistiques d’évaluation de l’incertitude — Lignes
directrices pour l’évaluation de l’incertitude des modèles à deux
facteurs croisés
Reference number
©
ISO 2015
© ISO 2015, Published in Switzerland
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ii © ISO 2015 – All rights reserved
Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Symbols . 2
5 Conduct of experiments . 4
6 Preliminary review of data — Overview. 4
7 Variance components and uncertainty estimation . 4
7.1 General considerations for variance components and uncertainty estimation . 4
7.2 Two-way layout without replication . 5
7.2.1 Design . 5
7.2.2 Preliminary inspection . 5
7.2.3 Variance component estimation. 5
7.2.4 Standard uncertainty for the mean of all observations . 6
7.2.5 Degrees of freedom for the standard uncertainty. 6
7.3 Two-way balanced experiment with replication (both factors random) . 7
7.3.1 Design . 7
7.3.2 Preliminary inspection . 7
7.3.3 Variance component extraction . 7
7.3.4 Standard uncertainty for the mean of all observations . 8
7.3.5 Degrees of freedom for the standard uncertainty. 9
7.4 Two-way balanced experiment with replication (one factor fixed, one factor random) .10
7.4.1 Design .10
7.4.2 Preliminary inspection .10
7.4.3 Variance component extraction .11
7.4.4 Standard uncertainty for the mean of all observations .11
7.4.5 Degrees of freedom for the standard uncertainty.12
8 Application to observations on a relative scale .12
9 Use of variance components in subsequent measurements .12
10 Alternative treatments .13
10.1 Restricted (or residual) maximum likelihood estimates .13
10.2 Alternative methods for model reduction .13
11 Treatment with missing values .13
Annex A (informative) Examples .14
Bibliography .19
Foreword
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Subcommittee SC 6, Measurement methods and results.
iv © ISO 2015 – All rights reserved
Introduction
Uncertainty estimation usually requires the estimation and subsequent combination of uncertainties
arising from random variation. Such random variation may arise within a particular experiment under
repeatability conditions, or over a wider range of conditions. Variation under repeatability conditions
is usually characterized as repeatability standard deviation or coefficient of variation; precision under
wider changes in conditions is generally termed intermediate precision or reproducibility.
The most common experimental design for estimating the long- and short-term components of variance
is the classical balanced nested design of the kind used by ISO 5725-2. In this design, a (constant)
number of observations are collected under repeatability conditions for each level of some other factor.
Where this additional factor is ‘Laboratory’, the experiment is a balanced inter-laboratory study, and
can be analysed to yield estimates of within-laboratory variance, σ , the between-laboratory
r
2 22 2
component of variance, σ , and hence the reproducibility variance, σσ=+σ . Estimation of
L RL r
uncertainties based on such a study is considered by ISO 21748. Where the additional grouping factor is
another condition of measurement, however, the between-group term can usefully be taken as the
uncertainty contribution arising from random variation in that factor. For example, if several different
extracts are prepared from a homogeneous material and each is measured several times, analysis of
variance can provide an estimate of the effect of variations in the extraction process. Further
elaboration is also possible by adding successive levels of grouping. For example, in an inter-laboratory
study the repeatability variance, between-day variance and between-laboratory variance can be
estimated in a single experiment by requiring each laboratory to undertake an equal number of
replicated measurements on each of two days.
While nested designs are among the most common designs for estimation of random variation, they
are not the only useful class of design. Consider, for example, an experiment intended to characterize
a reference material, conducted by measuring three separate units of the material in three separate
instrument runs, with (say) two observations per unit per run. In this experiment, unit and run are
said to be ‘crossed’; all units are measured in all runs. This design is often used to investigate variation
in ‘fixed’ effects, by testing for changes which are larger than expected from the within-group or
‘residual’ term. This particular experiment, for example, could easily test whether there is evidence
of significant differences between units or between runs. However, the units are likely to have been
selected randomly from a much larger (if ostensibly homogeneous) batch, and the run effects are also
most appropriately treated as random. If the mean of all the observations is taken as the estimate of
the reference material value, it becomes necessary to consider the uncertainties arising from both run-
to-run and unit-to-unit variation. This can be done in much the same way as for the nested designs
described previously, by extracting the variances of interest using two-way analysis of variance. In the
statistical literature, this is generally described as the use of a random-effects or (if one factor is a fixed
effect) mixed-effects model.
Variance component extraction can be achieved by several methods. For balanced designs, equating
expected mean squares from classical analysis of variance is straightforward. Restricted (sometimes
also called residual) maximum likelihood estimation (REML) is also widely recommended for estimation
of variance components, and is applicable to both balanced and unbalanced designs. This Technical
Specification describes the classical ANOVA calculations in detail and permits the use of REML.
Note that random effects rarely include all of the uncertainties affecting a particular measurement
result. If using the mean from a crossed design as a measurement result, it is generally necessary
to consider uncertainties arising from possible systematic effects, including between-laboratory
effects, as well as the random variation visible within the experiment, and these other effects can be
considerably larger than the variation visible within a single experiment.
This present Technical Specification describes the estimation and use of uncertainty contributions
using factorial designs.
TECHNICAL SPECIFICATION ISO/TS 17503:2015(E)
Statistical methods of uncertainty evaluation — Guidance on
evaluation of uncertainty using two-factor crossed designs
1 Scope
This Technical Specification describes the estimation of uncertainties on the mean value in experiments
conducted as crossed designs, and the use of variances extracted from such experiments and applied to
the results of other measurements (for example, single observations).
This Technical Specification covers balanced two-factor designs with any number of levels. The
basic designs covered include the two-way design without replication and the two-way design with
replication, with one or both factors cons
...
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