BSI Standards Publication Statistical methods of uncertainty evaluation — Guidance on evaluation of uncertainty using two-factor crossed designs PD ISO/TS 17503:2015National foreword This Published Document is the UK implementation of ISO/TS 17503:2015. The UK participation in its preparation was entrusted to Technical Committee SS/6, Precision of test methods. A list of organizations represented on this committee can be obtained on request to its secretary. This publication does not purport to include all the necessary provisions of a contract. Users are responsible for its correct application. © The British Standards Institution 2015. Published by BSI Standards Limited 2015 ISBN 978 0 580 89566 1 ICS 17.020 Compliance with a British Standard cannot confer immunity from legal obligations. This Published Document was published under the authority of the Standards Policy and Strategy Committee on 30 November 2015. Amendments/corrigenda issued since publication Date Text affected PUBLISHED DOCUMENT PD ISO/TS 17503:2015© ISO 2015 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 TECHNICAL SPECIFICATION ISO/TS 17503 Reference number ISO/TS 17503:2015(E) First edition 2015-11-01 PD ISO/TS 17503:2015 ISO/TS 17503:2015(E)ii © ISO 2015 – All rights reserved COPYRIGHT PROTECTED DOCUMENT © ISO 2015, Published in Switzerland All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized otherwise in any form or by any means, electronic or mechanical, including photocopying, or posting on the internet or an intranet, without prior written permission. Permission can be requested from either ISO at the address below or ISO’s member body in the country of the requester. ISO copyright office Ch. de Blandonnet 8 • CP 401 CH-1214 Vernier, Geneva, Switzerland Tel. +41 22 749 01 11 Fax +41 22 749 09 47

[email protected] www.iso.org PD ISO/TS 17503:2015 ISO/TS 17503:2015(E)Foreword iv Introduction v 1 Scope . 1 2 Normative references 1 3 T erms 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 uncertainty12 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 © ISO 2015 – All rights reserved iii Contents Page PD ISO/TS 17503:2015 ISO/TS 17503:2015(E) Foreword ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies (ISO member bodies). The work of preparing International Standards is normally carried out through ISO technical committees. Each member body interested in a subject for which a technical committee has been established has the right to be represented on that committee. International organizations, governmental and non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization. The procedures used to develop this document and those intended for its further maintenance are described in the ISO/IEC Directives, Part 1. In particular the different approval criteria needed for the different types of ISO documents should be noted. This document was drafted in accordance with the editorial rules of the ISO/IEC Directives, Part 2 (see www.iso.org/directives). Attention is drawn to the possibility that some of the elements of this document may be the subject of patent rights. ISO shall not be held responsible for identifying any or all such patent rights. Details of any patent rights identified during the development of the document will be in the Introduction and/or on the ISO list of patent declarations received (see www.iso.org/patents). Any trade name used in this document is information given for the convenience of users and does not constitute an endorsement. For an explanation on the meaning of ISO specific terms and expressions related to conformity assessment, as well as information about ISO’s adherence to the WTO principles in the Technical Barriers to Trade (TBT) see the following URL: Foreword - Supplementary information The committee responsible for this document is ISO/TC 69, Applications of statistical methods, Subcommittee SC 6, Measurement methods and results.iv © ISO 2015 – All rights reserved PD ISO/TS 17503:2015 ISO/TS 17503:2015(E) 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 component of variance, σ , and hence the reproducibility variance, σσ σ . Estimation of 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.© ISO 2015 – All rights reserved v PD ISO/TS 17503:2015 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 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. 2 Normative references The following documents, in whole or in part, are normatively referenced in this document and are indispensable for its application. For dated references, only the edition cited applies. For undated references, the latest edition of the referenced document (including any amendments) applies. ISO 3534-1, Statistics — Vocabulary and symbols — Part 1: General statistical terms and terms used in probability ISO 3534-3, Statistics — Vocabulary and symbols — Part 3: Design of experiments 3 T erms a nd definiti ons For the purposes of this document, the terms and definitions in ISO 3534-1, ISO 3534-3 and the following apply. 3.1 factor predictor variable that is varied with the intent of assessing its effect on the response variable Note 1 to entry: A factor may provide an assignable cause for the outcome of an experiment. Note 2 to entry: The use of factor here is more specific than its generic use as a synonym for predictor variable. Note 3 to entry: A factor may be associated with the creation of blocks. [SOURCE: ISO 3534-3:2013, 3.1.5, modified — cross-references within ISO 3534-3 omitted from Notes to entry] 3.2 level potential setting, value or assignment of a factor Note 1 to entry: A synonym is the value of a predictor variable. TECHNICAL SPECIFICATION ISO/TS 17503:2015(E) © ISO 2015 – All rights reserved 1 PD ISO/TS 17503:2015 ISO/TS 17503:2015(E) Note 2 to entry: The term “level” is normally associated with a quantitative characteristic. However, it also serves as the term describing the version or setting of qualitative characteristics. Note 3 to entry: Responses observed at the various levels of a factor provide information for determining the effect of the factor within the range of levels of the experiment. Extrapolation beyond the range of these levels is usually inappropriate without a firm basis for assuming model relationships. Interpolation within the range may depend on the number of levels and the spacing of these levels. It is usually reasonable to interpolate, although it is possible to have discontinuous or multi-modal relationships that cause abrupt changes within the range of the experiment. The levels may be limited to certain selected fixed values (whether these values are or are not known) or they may represent purely random selection over the range to be studied. EXAMPLE The ordinal-scale levels of a catalyst may be presence and absence. Four levels of a heat treatment may be 100 °C, 120 °C, 140 °C and 160 °C. The nominal-scale variable for a laboratory can have levels A, B and C, corresponding to three facilities. [SOURCE: ISO 3534-3:2013, 3.1.12] 3.3 f i x e d e f f e c t s a n a l y s i s o f v a r i a n c e analysis of variance in which the levels of each factor are pre-selected over the range of values of the factors Note 1 to entry: With fixed levels, it is inappropriate to compute components of variance. This model is sometimes referred to as a model 1 analysis of variance. [SOURCE: ISO 3534-3:2013, 3.3.9] 3.4 random effects analysis of variance analysis of variance in which each level of each factor is assumed to be sampled from the population of levels of each factor Note 1 to entry: With random levels, the primary interest is usually to obtain components of variance estimates. This model is commonly referred to as a model 2 analysis of variance. EXAMPLE Consider a situation in which an operation processes batches of raw material. “Batch” may be considered a random factor in an experiment when a few batc