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Uncertainty functions, a compact way of summarising or specifying the behaviour of analytical systems
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Trends in Analytical Chemistry, Vol. 30, No. 7, 2011
Uncertainty functions, a compact way of summarising or specifying the behaviour of analytical systems Michael Thompson Analytical chemists can advantageously use an uncertainty function to describe the performance of an analytical system in terms of the standard uncertainty or standard deviation as a function of the concentration of the analyte. This ‘‘characteristic function’’ is useful for estimating uncertainty at a new concentration. A similar function can be used to prescribe the uncertainty that is regarded as fit for purpose for a particular application. This ‘‘fitness function’’ is useful in setting standards of accuracy in proficiency tests and similar exercises without revealing the concentration of the analyte. In combination, these two functions provide a rational basis for method selection. ª 2011 Elsevier Ltd. All rights reserved. Keywords: Analytical system; Characteristic function; Concentration; Fitness for purpose; Fitness function; Method selection; Precision; Uncertainty; Uncertainty function; Variation with concentration of analyte Note on terminology: The term ‘‘concentration’’ is used in this review in a general sense for any mixture unit and mass fraction, not exclusively for ‘‘per-volume’’ units. The term ‘‘analytical system’’ refers to the use of a defined analytical method applied to a specific type of test material under constant conditions, so that the precision and uncertainty can be considered time invariant.
1. Uncertainty functions Michael Thompson School of Science, Birkbeck University of London, Malet Street, London WC1E 7HX, UK
E-mail: [email protected]
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In an ‘‘analytical system’’, as defined above, the uncertainty of the value of a measurement result varies with the concentration of the analyte. In some instances this variation can be ignored. For instance, if the test material is the product of a controlled industrial process, the concentration range of the analyte may be very small. The uncertainty associated with the result can then be taken as invariant. In other analytical systems, such as the analysis of environmental materials, the concentration of the analyte in successive samples might range over several orders of magnitude. In that case an assumption of constant uncertainty is almost certainly invalid and potentially misleading. To meet the latter circumstance we would need a relationship that could realistically accommodate the whole concentration range likely to be encountered. In such systems it would be a reasonable assumption that (a) the uncertainty would vary smoothly with concentration and (b) the relationship
could be expressed by a fairly simple mathematical function. Functions referring to the uncertainty or precision behaviour of analytical systems have been called ‘‘characteristic functions’’. Such a function could bring together in a compact form the information that is latent in the patchwork of overlapping concepts and misleading limits that is traditionally accepted as comprising validation [1]. All of the relevant information could be built into the uncertainty function. The function would comprise a useful and compact way of summarising the behaviour of the analytical system. This would enable the analyst to ascribe an uncertainty (or standard deviation) to any particular concentration determined under the given conditions. In other words it would show the analyst how to interpolate correctly. As an example of such interpolation, a characteristic function can be used to normalise differences between duplicated results used for internal quality control. Within an analytical system, differences between duplicated results from a
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Trends in Analytical Chemistry, Vol. 30, No. 7, 2011
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the name of the analyte and the type of test material to which the method is to be applied; an expression F c ¼ f ðcÞ showing the relationship between the uncertainty of measurement F c and the concentration c of the analyte; and a statement of the concentration range over which the function is applicable. A characteristic function can be determined by validation of the analytical ‘‘system’’. The uncertainty or standard deviation is estimated by the analyst, in an appropriate fashion, at several concentrations of the analyte and the values obtained used to estimate empirically the parameters of the selected mathematical function. The uncertainty of a result at any other valid concentration (that is, within the validated range) is inferred via the function. Uncertainty is, by definition, a property of the value of a measurement result rather than of a measurement method. As a consequence, a characteristic function would be applicable only to analyses carried out in specific circumstances, for example, in a particular laboratory, by trained analysts, in other words, in an analytical system. Off-the-shelf analytical methods therefore have to be at least partly re-validated before first use in a laboratory [2], and consistent performance subsequently ensured by internal quality control methods [3].
population with a repeatability standard deviation rr would be a random variable with standard deviation pffiffiffi 2rr . Under the assumptions of normality and statistical control, pffiffiffi observed absolute differences should exceed 3 2rr 4:2rr with a probability of about 0.003. (This probability corresponds to the action limits of a Shewhart chart.) However, the concentration of the analyte, and therefore the value of rr , will vary from one test material to another. The analyst could use a characteristic function of suitable form to estimate rr at any observed concentration. An uncertainty function could describe equally well a conceptually quite distinct aspect of uncertainty, that is, the uncertainty that is fit for purpose in a specific application. Such a function has been called the ‘‘fitness function’’ [2]. The fitness function would allow the specification of a maximum allowable or optimal uncertainty without reference to a specific method, test material, analyte or concentration. This would have obvious applications in activities such as proficiency testing and internal quality control. In another important application of the function, the selection of a suitable analytical method would comprise the comparison of one or more characteristic functions with the prescribed fitness function over the whole of the relevant concentration range. Such a comparison could be executed graphically (Fig. 1). Where the characteristic function falls below the fitness function, the method is fit for purpose or better. (If the characteristic function falls considerably below the fitness function, the analytical method is probably unnecessarily accurate and therefore unnecessarily expensive. This follows from the optimality definition of fitness for purpose.)
3. Finding the functional form of a characteristic function experimentally The first priority is to explore the shape of some characteristic functions. If a plausible functional form could be shown experimentally to be widely applicable, we could then apply it to the validation of specific instances. However, it is important to realise that the experimental requirements for exploring the shape of characteristic functions are quite different from those required for validation.
2. The characteristic function To summarise the performance of an analytical system by a characteristic function we would need only the following:
Standard uncertainty
0.25 0.20 0.15 0.10 0.05 0.00 0
1
2
3
4
5
Concentration qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Figure 1. Characteristic function F c ¼ 0:032 þ ð0:04cÞ2 (solid line) and fitness function F f ¼ 0:05c (dashed line). The method is fit for purpose (or even more accurate) where F c < F f [i.e. in the range 1–5 (arbitrary units)].
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vations per point, the confidence regions for each point are at first sight surprisingly large. It is clear in this instance that a good fit has been obtained and that the scope for alternative well-fitting functions is limited: neither a constant standard deviation nor a constant relative standard deviation could provide an adequate fit. Understandably then there have been few determined efforts to study the shape of characteristic functions experimentally. What studies there are have exploited various situations where large collections of data have been collected for other purposes. Strictly speaking, most of such studies have addressed only standard deviation under various conditions of precision, which is, of course, not identical with uncertainty (except with restricted definitions of the measurand). The most useful of these are obtained where authors study reproducibility standard deviation, which has been found to be a useful benchmark for uncertainty in many instances. In that instance suitable data can be obtained from proficiency tests in which all of the participants use the same method. It is reiterated that this is an expedient for studying the shape of characteristic functions – not for determining characteristic functions for validating specific analytical systems. Experimenters in this field quickly come up against a fundamental aspect of science, not as a philosophical nicety but as a practical reality: we are unable to identify a true characteristic function by experiment. In practice we demonstrate that there is no significant lack of fit between the available data and a particular function. Even then there may be other functions that fit equally well over the relevant range. A practicable approach, therefore, is to identify a functional form that had broad
To study the shape of a curve we need data of high precision closely spaced over the whole of the relevant range. Validation data per se are clearly unsuitable. Economic realities in validation restrict the experimentation to only one or at most a few concentrations of the analyte. Moreover the uncertainty of each uncertainty estimate would be relatively large at each experimental point, because the estimate is usually based on a small number of degrees of freedom. The unpalatable truth is that precisions and uncertainties estimated in typical validations are themselves disturbingly uncertain. Validation data could therefore be consistent with a number of divergent uncertainty functions. Even collaborative trials (interlaboratory method performance studies), widely regarded as the most comprehensive method of validation and costing of the order of €50,000 or more, rarely supply sufficient information for present purposes. This statement may occasion surprise in those unfamiliar with the subject but is evident in Fig. 2, which shows results from a typical collaborative trial comprising nine participating laboratories and six different test materials [4]. The wide error bars show the 95% confidence regions for each estimated standard deviation. The data are accordingly consistent with a variety of functional forms: the three shown have all been suggested as appropriate in normative or other authoritative documents. This problem has to be overcome by the use of very large datasets. For example, Fig. 3 shows a characteristic function fitted to numerous closely-spaced points with reasonable precision in the dependent variable, the robust repeatability standard deviation based on 100 observations [5]. Even with this large number of obser-
Standard deviation
2
1
0 0
1
2
3
4
5
Concentration, mg/kg Figure 2. Results from a collaborative trial of a method for the determination of total fumonisins in corn, showing estimated reproducibility standard deviations at various concentrations (points) with their 95% confidence limits (vertical bars). The lines show a variety of good fits (i.e. with no significant lack of fit) to the data including: Equation (2) (solid line); constant relative standard deviation (dashed line); and, line of the form rR ¼ bc c (analogue of the Horwitz function with b–0:02) (dotted line).
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1.0
0.1 1
10
100
Concentration, mg/kg qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Figure 3. Example fit of repeatability data (nickel determined in soils) to the characteristic function F c ¼ 0:0312 þ ð0:0249cÞ2 (line). The vertical bars show 95% confidence limits around individual estimates of standard deviation at each concentration.
relevance in analytical measurement and then apply it to real-life validation. The ideal form would have the following properties: (a) parsimony, that is, few empirically-determined parameters; (b) technical plausibility based on standard theory; and (c) experimentallydemonstrated goodness of fit under stringent testing. Functions that have been suggested and meet these criteria to a greater or lesser extent are summarised in Table 1 [6–17].
4. Some unsuitable functions Several authors have suggested functions that fail to meet one or more of these criteria. Most of these can be summarised as examples of the general form r ¼ a þ bcc
orative trials in the food sector, at mass fractions between about 100 parts per billion and 10%. Collaborative trials, however, tend to avoid test materials with concentrations near detection limits and thus circumvent the need to consider a value for a. Moreover, where low-concentration test materials are included, common data recording practices of results near natural limits tend to render the required statistical analysis invalid or impossible. The Horwitz function thus effectively describes a trend in the value of b alone]. More generally, any form of Equation (1) is inherently suspect for describing a complete analytical system, even one with a > 0, because it implies the addition of terms that are independent standard deviations, which is technically incorrect – it is the variances that should be added.
ð1Þ
for standard deviation r at concentration c with parameters a; b; c. Specific examples can be seen in Table 1 Examples of Equation (1) with a ¼ 0 can be dismissed out of hand for general purposes as a characteristic function, because they cannot be correct at low concentrations. They all imply zero standard deviation at zero concentration, which is never true in measured results (otherwise an analytical system could have an exactly zero detection limit, a feature that is not observed in practice). Examples of Equation (1) with a ¼ 0; c ¼ 1 describe the familiar assumption of a constant relative standard deviation, which is serviceable in many instances but not generally applicable. [The Horwitz function (Table 1) is an example of Equation (1) and deserves special mention because of its widespread use. It describes very closely the observed trend of reproducibility standard deviation from collab-
5. Adding variances A suitable uncertainty function can be envisaged by analogy with the use of bathroom scales. Before getting on the scales we adjust the scale by setting its zero mark to the reference line. There is clearly some uncertainty in the resulting zero point that will affect all subsequent measurements. There is a further, independent uncertainty in the sensitivity, the distance moved by the scale in response to an applied weight. The uncertainty in this second term would depend on the strength of the spring and thus be proportional to the applied weight. The uncertainty in a reading would be determined by combining these two independent uncertainties in the proper way. Many, perhaps most, analytical systems are likely to behave like bathroom scales. There will be two independent uncertainty terms, a constant baseline
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Table 1. Functions suggested for accounting for variation of precision or uncertainty with the concentration of the analyte. SDr indicates standard deviation of repeatability; SDR indicates standard deviation of reproducibility. Units could be any appropriate mixture units except where indicated Equation qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ a þ bc, r ¼ a2 þ ðbcÞ2 r ¼ a þ bc r ¼ a þ bc, r ¼ bc c r ¼ 0:02c 0:8594
Application
pffiffiffi r¼b c qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ a þ bc, r ¼ a2 þ ðbcÞ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ a2 þ ðbc c Þ2 8 < 0:22c; c > 106:9 6:9 0:8495 r ¼ 0:02c < c < 0:14 pffiffiffi ; 10 : 0:01 c ; c > 0:14 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ a2 þ ðbcÞ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ a2 þ ðbcÞ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ a2 þ ðbcÞ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ a2 þ ðbcÞ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ a2 þ ðbcÞ2 , r ¼ a þ bc qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ a2 þ ðbcÞ2
Ref.
SDr
1971
[6]
SDr in geochemical analysis SDR, SDr in individual collaborative trials General trend of SDR in all collaborative trials in the food sector (Horwitz function) (r and c are mass fractions) SDr, iron and steel, iron ore SDr from geochemical analysis
1978 1979 1980
[7] [8] [9] Many follow-up references
1987 1988
[10] [11]
SDR, clinical biochemistry
1997
[12]
General trend of SDR in all collaborative trials in the food sector (r and c are mass fractions)
2000
[13]
Uncertainty, in general
2000
[14]
Uncertainty, in food analysis
2006
[2]
SDR from proficiency tests, many different types of analysis SD (‘‘instrumental precision’’) Ba isotopes by ICP-MS
2008
[15]
2009
[16]
Uncertainty, SDR, trace elements in water
2009
[17]
SDr from duplication from in-run quality control. Many different analytes by ICP-AES
2010
[5]
An example with a ¼ 0:03; b ¼ 0:04 (% mass fraction) is shown in Fig. 1. A function of this form was first proposed by Zitter and God [4] in respect of precision. This function accounts also for the commonly observed properties of relative uncertainty, as rearranging Equation (2) gives
uncertainty of magnitude a and a term bc that is proportional to the concentration. A proper combination of these two terms provides the required characteristic function, namely qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Fc ¼ a2 þ ðbcÞ ð2Þ
Relative standard uncertainty
Date
0.2
0.1
β =0.04
0.0 0
1
2
3
4
5
Concentration Figure 4. Relative standard uncertainty as a function of concentration according to Equation (2). The parameters are the same as those in Fig. 1.
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Fc ¼ c
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 þ b2 c2
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ð3Þ
which shows an asymptotic tendency to a constant relative uncertainty b at high concentrations and rapidly escalating values as concentrations fall towards zero (Fig. 4). Equation (2) is recommended for characterising uncertainty in the Eurachem Guide to Quantifying Uncertainty in Chemical Measurement [12], although lack of large datasets makes the function difficult to test adequately by experiment for uncertainty per se.
6. Recent experimental work Equation (2) has been carefully tested in relation to precision by making use of large blocks of data collected for other purposes. It was tested under ÔinstrumentalÕ conditions, by considering measurements of concentrations of barium isotopes determined by ICP-MS. The study involved 14 different concentrations of barium each measured 100 times. By virtue of the varied abundances of the isotopes, it was possible to cover a wide concentration range overall. The function provided an excellent fit to the data in all instances [14]. Equation (2) was further tested under randomised repeatability conditions by using a block of more than 3000 pairs of duplicated results accumulated over an extended period during routine internal quality control of a multielement analysis of soils and sediments by ICPAES. In nearly all instances there was no significant lack of fit observed. In the same study Equation (2) was separately tested by using more than 30000 duplicate determinations of ethanol in breath. In that very large study there was a small lack of fit detected in some of the data points, but these were evenly distributed over the concentration range and did not signify that the curve had an incorrect shape [16]. Precision under reproducibility conditions has been studied by using the results from proficiency tests. Most proficiency tests distribute the same type of test material over extended periods. That allows the accumulation of data from a large group of participants and round by round at many different levels of concentration. The studies exploited the fact that a large number of participants tend to use a single analytical method. In nearly all of the cases studied lack of fit to Equation (2) was nonsignificant or trivial in magnitude. In these circumstances the reproducibility standard deviation could be taken as a reasonable surrogate for uncertainty [13]. ´ lvarez-Prieto have reported Jime´nez-Chaco´n and A their studies of the variation of uncertainty with concentration in a substantial range of analytical systems and confirmed a good fit to Equation (2) for many of them [15]. However, they had access to a relatively
small number of uncertainty estimates over the concentration range. 7. Validation and the characteristic function Validation is carried out in practice in the most economic manner that will satisfy the requirements for accreditation, providing an estimate of uncertainty – usually not a very precise one – at one or at most a few concentrations of the analyte. Validation per se provides no information about the shape of the characteristic function, which makes the estimation of uncertainty at other concentrations problematic. The way forward seems to be, in the absence of contra-indications, to assume that Equation (2) holds and use the validation data to estimate its parameters, a and b, and to record the maximum concentration at which the equation was tested. This could be carried out rather simply. The value of a could be estimated as the standard uncertainty estimated in the usual manner at (or close to) zero concentration, with due precaution to avoid any censoring of the data. The value of b would be the relative standard uncertainty at 2 a high concentration, that is, where ðbcÞ a2 , typically when the concentration is more than 100 times the detection limit. The analyst would have to be convinced that Equation (2) was indeed the correct functional form and there are circumstances when it is not. It is known to be invalid in instances where the reported analyte is not the measured one. For example, we might report a result as Ôresidue on dryingÕ but actually measure loss on drying. Equation (2) could fit the latter but not the former [13]. However, the studies described above suggest that it would work well in many types of analysis. In particular, the equation has been shown to provide a good fit to values of reproducibility (interlaboratory) standard ^R encompassing methods with a variety of deviation r analytes, matrices, and physical measurement principles. This is relevant to characterising uncertainty. Of ^R cannot be taken as a reasonable estimate of course, r standard uncertainty without due consideration, but it has been shown to be a useful benchmark in many instances [18]. [Having estimated a and b, the analyst might consider using Equation (1), the simpler linear function r ¼ a þ bc for the characteristic function. Although Equations (1) and (2) with the same parameters predict identical or close uncertainties towards the extremities of the concentration range, they deviate at mid-range concentrations, Equation (1) predicting aprelative uncertainty that is too high by ffiffiffi factor of 2 at maximum, at a concentration cmax ¼ a=b. While this may be serviceable in some circumstances, it is not accurate].
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8. Fitness functions Logically an uncertainty that is fit for purpose has to be established before a method can be identified as suitable for an application. The resulting fitness function F f is conceptually different from the uncertainty associated with a particular analytical system. Before selecting an analytical method, the analyst needs to be aware of the level of uncertainty that is optimal for the customer (optimal in the sense of balancing the cost of analysis with the average penalty arising from an incorrect decision based on the result). This optimal uncertainty can be expressed as a function of concentration. While there is a mathematical approach to fitness for purpose based on decision theoretic principles [19], a fitness function based on professional experience in the application area could simply be agreed between the analyst and the customer. Alternatively, the function might be defined by an authority regulating a whole application sector of chemical analysis, or by the provider of a proficiency test in that sector. As a simple example the following fitness function Ff ¼ 0:05c; 0:1 < c < 5%
ð4Þ
specifies that the relative standard uncertainty should be 5% of the concentration (or less) over the concentration range 0.1-5% mass fraction (Fig. 1). When such a fitness function has been agreed upon, it can be used to determine whether a particular analytical method is suitable by comparison with its reported characteristic function F c . Subject to confirmatory validation, a method is suitable if its characteristic function falls below the fitness function over the whole relevant range (as, for example, in Fig. 1). Fitness functions have other uses. For example they can be used to specify in advance the degree of accuracy prescribed and used for scoring in a proficiency test, without reference to the actual concentration of the analyte in the test material. Fitness functions can follow any convenient mathematical form [20]. For example, in FAPAS, a large proficiency test operating in the food analysis sector, the Horwitz function is used to define fitness for purpose in most instances. Related criteria are used in GeoPT, a proficiency test in the geochemical sector. In many applications a function of the form of Equation (4), that is, constant relative uncertainty, would be too stringent at low concentrations. It could be modified by including an intercept term and taking the form corresponding to Equation (2).
9. Do we really need detection limits? The detection limit cL (or any other limit such as the limit of determination or the limit of quantification)
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artificially dichotomises the concentration axis in an analytical system. We are all subject to the human tendency to categorize things that in reality fall on a continuum, the propensity that Richard Dawkins calls (in a quite different context) the Ôtyranny of the discontinuous mindÕ [21]. This tendency encourages analysts and their customers to believe – incorrectly – that a result just below the detection limit (say at 0:9cL ) has a qualitatively different status to one just above the limit at 1:1cL . Analysts have consequently become unwilling to report a result that falls below such a limit, replacing the numerical result, originally on a rational or interval scale, with an ordinal form such as ‘‘not detected’’ or ‘‘less than cL ’’. Such reporting is detrimental because it renders much more difficult an unbiased statistical treatment of the resulting dataset. All that end-users of analytical data really need to know is the result and its correct uncertainty [22]. In a particular analytical system, the parameter a in Equation (2) is clearly related to the traditional detection limit cL . We would expect cL ka, where k is a coverage factor, probably between 2 and 3 depending of the preferred definition of detection limit. However, the widespread use of Equation (2) would dispense with the need for arbitrary limits.
10. Conclusions A rationale has been presented for the use of a characteristic function to describe the performance of analytical methods as a function of the concentration of the analyte. Experiments have shown that, in a large proportion in instances, a compact function of the form of Equation (2) will accurately fulfil the need. Where Equation (2) is applicable, its parameters for a specific analytical system can be quickly estimated. The characteristic function would bring together the multifarious aspects of validating an analytical method and to some extent overcome the practical limitations of validation. It could also eliminate the need for arbitrary limits such as the detection limit at the low end of the concentration range. A fitness function could likewise be used to specify the optimal level of uncertainty for a particular application or for a proficiency test and, in combination with the characteristic function, provide a comprehensive basis for the selection of appropriate analytical methods.
Note added in proof After the finalisation of this paper, the author became aware of a parallel development in the filed of clinical immunoassay. Interested readers can find a review of that work in [23].
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References [1] M. Thompson, S.L.R. Ellison, R. Wood, Pure Appl. Chem. 74 (2002) 835. [2] M. Thompson, R. Wood, Accred. Qual. Assur. 10 (2006) 471. [3] M. Thompson, R. Wood, Pure Appl. Chem. 67 (1995) 649. [4] C.B. Bird, B. Malone, R.G. Rice, P.F. Ross, R. Eppley, M.M. Abouzied, J. AOAC Int. 85 (2002) 404. [5] M. Thompson, B.J. Coles, Accred. Qual. Assur. 16 (2010) 13. [6] H. Zitter, C. God, Fresenius Z. Anal. Chem. 255 (1971) 1. [7] M. Thompson, R.J. Howarth, J. Geochem. Explor. 9 (1978) 23. [8] International Standards Organization (ISO), ISO 5725, ISO, Geneva, Switzerland, 1979. [9] W. Horwitz, L.R. Kamps, K.W. Boyer, J. Assoc. Off. Anal. Chem. 63 (1980) 1344. [10] H. Hughes, P.W. Hurley, Analyst (Cambridge, UK) 112 (1987) 1445. [11] M. Thompson, Analyst (Cambridge, UK) 113 (1988) 1587. [12] D.L. Duewer, J.B. Thomas, M.C. Kline, W.A. MacCrehan, R. Schaffer, K.E. Sharpless, W.E. May, J.A. Crowell, Anal. Chem. 69 (1997) 1406.
Trends [13] M. Thompson, Analyst (Cambridge, UK) 125 (2000) 385. [14] S.L.R. Ellison, M. Rosslein, A. Williams, Eurachem CITAC Guide CG4, Quantifying uncertainty in chemical measurement, Second Edition, 2000 (http://www.citac.cc/QUAM2000-1.pdf). [15] M. Thompson, K. Mathieson, A.P. Damant, R. Wood, Accred. Qual. Assur. 13 (2008) 223. [16] M. Thompson, B.J. Coles, Accred. Qual. Assur. 14 (2009) 147. ´ lvarez-Prieto, Accred. Qual. Assur. 14 [17] J. Jime´nez-Chaco´n, M. A (2009) 15. [18] S.L.R. Ellison, K. Mathieson, Accred. Qual. Assur. 13 (2008) 231. [19] T. Fearn, S. Fisher, M. Thompson, S.L.R. Ellison, Analyst (Cambridge, UK) 127 (2002) 818. [20] AMC Technical Briefs No. 19A (www.rsc.org/images/brief%2019A_tcm18-134927.pdf). [21] R. Dawkins, The AncestorÕs Tale, Weidenfeld and Nicholson, London, UK, 2004. [22] Analytical Methods Committee, Accred. Qual. Assur. 13 (2008) 29. [23] W.A. Sadler, Clin. Biochem. Rev. 29 (Suppl.1) (2008) S33.
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Uncertainty functions, a compact way of summarising or specifying the behaviour of analytical systems Michael Thompson Analytical chemists can advantageously use an uncertainty function to describe the performance of an analytical system in terms of the standard uncertainty or standard deviation as a function of the concentration of the analyte. This ‘‘characteristic function’’ is useful for estimating uncertainty at a new concentration. A similar function can be used to prescribe the uncertainty that is regarded as fit for purpose for a particular application. This ‘‘fitness function’’ is useful in setting standards of accuracy in proficiency tests and similar exercises without revealing the concentration of the analyte. In combination, these two functions provide a rational basis for method selection. ª 2011 Elsevier Ltd. All rights reserved. Keywords: Analytical system; Characteristic function; Concentration; Fitness for purpose; Fitness function; Method selection; Precision; Uncertainty; Uncertainty function; Variation with concentration of analyte Note on terminology: The term ‘‘concentration’’ is used in this review in a general sense for any mixture unit and mass fraction, not exclusively for ‘‘per-volume’’ units. The term ‘‘analytical system’’ refers to the use of a defined analytical method applied to a specific type of test material under constant conditions, so that the precision and uncertainty can be considered time invariant.
1. Uncertainty functions Michael Thompson School of Science, Birkbeck University of London, Malet Street, London WC1E 7HX, UK
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In an ‘‘analytical system’’, as defined above, the uncertainty of the value of a measurement result varies with the concentration of the analyte. In some instances this variation can be ignored. For instance, if the test material is the product of a controlled industrial process, the concentration range of the analyte may be very small. The uncertainty associated with the result can then be taken as invariant. In other analytical systems, such as the analysis of environmental materials, the concentration of the analyte in successive samples might range over several orders of magnitude. In that case an assumption of constant uncertainty is almost certainly invalid and potentially misleading. To meet the latter circumstance we would need a relationship that could realistically accommodate the whole concentration range likely to be encountered. In such systems it would be a reasonable assumption that (a) the uncertainty would vary smoothly with concentration and (b) the relationship
could be expressed by a fairly simple mathematical function. Functions referring to the uncertainty or precision behaviour of analytical systems have been called ‘‘characteristic functions’’. Such a function could bring together in a compact form the information that is latent in the patchwork of overlapping concepts and misleading limits that is traditionally accepted as comprising validation [1]. All of the relevant information could be built into the uncertainty function. The function would comprise a useful and compact way of summarising the behaviour of the analytical system. This would enable the analyst to ascribe an uncertainty (or standard deviation) to any particular concentration determined under the given conditions. In other words it would show the analyst how to interpolate correctly. As an example of such interpolation, a characteristic function can be used to normalise differences between duplicated results used for internal quality control. Within an analytical system, differences between duplicated results from a
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the name of the analyte and the type of test material to which the method is to be applied; an expression F c ¼ f ðcÞ showing the relationship between the uncertainty of measurement F c and the concentration c of the analyte; and a statement of the concentration range over which the function is applicable. A characteristic function can be determined by validation of the analytical ‘‘system’’. The uncertainty or standard deviation is estimated by the analyst, in an appropriate fashion, at several concentrations of the analyte and the values obtained used to estimate empirically the parameters of the selected mathematical function. The uncertainty of a result at any other valid concentration (that is, within the validated range) is inferred via the function. Uncertainty is, by definition, a property of the value of a measurement result rather than of a measurement method. As a consequence, a characteristic function would be applicable only to analyses carried out in specific circumstances, for example, in a particular laboratory, by trained analysts, in other words, in an analytical system. Off-the-shelf analytical methods therefore have to be at least partly re-validated before first use in a laboratory [2], and consistent performance subsequently ensured by internal quality control methods [3].
population with a repeatability standard deviation rr would be a random variable with standard deviation pffiffiffi 2rr . Under the assumptions of normality and statistical control, pffiffiffi observed absolute differences should exceed 3 2rr 4:2rr with a probability of about 0.003. (This probability corresponds to the action limits of a Shewhart chart.) However, the concentration of the analyte, and therefore the value of rr , will vary from one test material to another. The analyst could use a characteristic function of suitable form to estimate rr at any observed concentration. An uncertainty function could describe equally well a conceptually quite distinct aspect of uncertainty, that is, the uncertainty that is fit for purpose in a specific application. Such a function has been called the ‘‘fitness function’’ [2]. The fitness function would allow the specification of a maximum allowable or optimal uncertainty without reference to a specific method, test material, analyte or concentration. This would have obvious applications in activities such as proficiency testing and internal quality control. In another important application of the function, the selection of a suitable analytical method would comprise the comparison of one or more characteristic functions with the prescribed fitness function over the whole of the relevant concentration range. Such a comparison could be executed graphically (Fig. 1). Where the characteristic function falls below the fitness function, the method is fit for purpose or better. (If the characteristic function falls considerably below the fitness function, the analytical method is probably unnecessarily accurate and therefore unnecessarily expensive. This follows from the optimality definition of fitness for purpose.)
3. Finding the functional form of a characteristic function experimentally The first priority is to explore the shape of some characteristic functions. If a plausible functional form could be shown experimentally to be widely applicable, we could then apply it to the validation of specific instances. However, it is important to realise that the experimental requirements for exploring the shape of characteristic functions are quite different from those required for validation.
2. The characteristic function To summarise the performance of an analytical system by a characteristic function we would need only the following:
Standard uncertainty
0.25 0.20 0.15 0.10 0.05 0.00 0
1
2
3
4
5
Concentration qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Figure 1. Characteristic function F c ¼ 0:032 þ ð0:04cÞ2 (solid line) and fitness function F f ¼ 0:05c (dashed line). The method is fit for purpose (or even more accurate) where F c < F f [i.e. in the range 1–5 (arbitrary units)].
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vations per point, the confidence regions for each point are at first sight surprisingly large. It is clear in this instance that a good fit has been obtained and that the scope for alternative well-fitting functions is limited: neither a constant standard deviation nor a constant relative standard deviation could provide an adequate fit. Understandably then there have been few determined efforts to study the shape of characteristic functions experimentally. What studies there are have exploited various situations where large collections of data have been collected for other purposes. Strictly speaking, most of such studies have addressed only standard deviation under various conditions of precision, which is, of course, not identical with uncertainty (except with restricted definitions of the measurand). The most useful of these are obtained where authors study reproducibility standard deviation, which has been found to be a useful benchmark for uncertainty in many instances. In that instance suitable data can be obtained from proficiency tests in which all of the participants use the same method. It is reiterated that this is an expedient for studying the shape of characteristic functions – not for determining characteristic functions for validating specific analytical systems. Experimenters in this field quickly come up against a fundamental aspect of science, not as a philosophical nicety but as a practical reality: we are unable to identify a true characteristic function by experiment. In practice we demonstrate that there is no significant lack of fit between the available data and a particular function. Even then there may be other functions that fit equally well over the relevant range. A practicable approach, therefore, is to identify a functional form that had broad
To study the shape of a curve we need data of high precision closely spaced over the whole of the relevant range. Validation data per se are clearly unsuitable. Economic realities in validation restrict the experimentation to only one or at most a few concentrations of the analyte. Moreover the uncertainty of each uncertainty estimate would be relatively large at each experimental point, because the estimate is usually based on a small number of degrees of freedom. The unpalatable truth is that precisions and uncertainties estimated in typical validations are themselves disturbingly uncertain. Validation data could therefore be consistent with a number of divergent uncertainty functions. Even collaborative trials (interlaboratory method performance studies), widely regarded as the most comprehensive method of validation and costing of the order of €50,000 or more, rarely supply sufficient information for present purposes. This statement may occasion surprise in those unfamiliar with the subject but is evident in Fig. 2, which shows results from a typical collaborative trial comprising nine participating laboratories and six different test materials [4]. The wide error bars show the 95% confidence regions for each estimated standard deviation. The data are accordingly consistent with a variety of functional forms: the three shown have all been suggested as appropriate in normative or other authoritative documents. This problem has to be overcome by the use of very large datasets. For example, Fig. 3 shows a characteristic function fitted to numerous closely-spaced points with reasonable precision in the dependent variable, the robust repeatability standard deviation based on 100 observations [5]. Even with this large number of obser-
Standard deviation
2
1
0 0
1
2
3
4
5
Concentration, mg/kg Figure 2. Results from a collaborative trial of a method for the determination of total fumonisins in corn, showing estimated reproducibility standard deviations at various concentrations (points) with their 95% confidence limits (vertical bars). The lines show a variety of good fits (i.e. with no significant lack of fit) to the data including: Equation (2) (solid line); constant relative standard deviation (dashed line); and, line of the form rR ¼ bc c (analogue of the Horwitz function with b–0:02) (dotted line).
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1.0
0.1 1
10
100
Concentration, mg/kg qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Figure 3. Example fit of repeatability data (nickel determined in soils) to the characteristic function F c ¼ 0:0312 þ ð0:0249cÞ2 (line). The vertical bars show 95% confidence limits around individual estimates of standard deviation at each concentration.
relevance in analytical measurement and then apply it to real-life validation. The ideal form would have the following properties: (a) parsimony, that is, few empirically-determined parameters; (b) technical plausibility based on standard theory; and (c) experimentallydemonstrated goodness of fit under stringent testing. Functions that have been suggested and meet these criteria to a greater or lesser extent are summarised in Table 1 [6–17].
4. Some unsuitable functions Several authors have suggested functions that fail to meet one or more of these criteria. Most of these can be summarised as examples of the general form r ¼ a þ bcc
orative trials in the food sector, at mass fractions between about 100 parts per billion and 10%. Collaborative trials, however, tend to avoid test materials with concentrations near detection limits and thus circumvent the need to consider a value for a. Moreover, where low-concentration test materials are included, common data recording practices of results near natural limits tend to render the required statistical analysis invalid or impossible. The Horwitz function thus effectively describes a trend in the value of b alone]. More generally, any form of Equation (1) is inherently suspect for describing a complete analytical system, even one with a > 0, because it implies the addition of terms that are independent standard deviations, which is technically incorrect – it is the variances that should be added.
ð1Þ
for standard deviation r at concentration c with parameters a; b; c. Specific examples can be seen in Table 1 Examples of Equation (1) with a ¼ 0 can be dismissed out of hand for general purposes as a characteristic function, because they cannot be correct at low concentrations. They all imply zero standard deviation at zero concentration, which is never true in measured results (otherwise an analytical system could have an exactly zero detection limit, a feature that is not observed in practice). Examples of Equation (1) with a ¼ 0; c ¼ 1 describe the familiar assumption of a constant relative standard deviation, which is serviceable in many instances but not generally applicable. [The Horwitz function (Table 1) is an example of Equation (1) and deserves special mention because of its widespread use. It describes very closely the observed trend of reproducibility standard deviation from collab-
5. Adding variances A suitable uncertainty function can be envisaged by analogy with the use of bathroom scales. Before getting on the scales we adjust the scale by setting its zero mark to the reference line. There is clearly some uncertainty in the resulting zero point that will affect all subsequent measurements. There is a further, independent uncertainty in the sensitivity, the distance moved by the scale in response to an applied weight. The uncertainty in this second term would depend on the strength of the spring and thus be proportional to the applied weight. The uncertainty in a reading would be determined by combining these two independent uncertainties in the proper way. Many, perhaps most, analytical systems are likely to behave like bathroom scales. There will be two independent uncertainty terms, a constant baseline
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Table 1. Functions suggested for accounting for variation of precision or uncertainty with the concentration of the analyte. SDr indicates standard deviation of repeatability; SDR indicates standard deviation of reproducibility. Units could be any appropriate mixture units except where indicated Equation qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ a þ bc, r ¼ a2 þ ðbcÞ2 r ¼ a þ bc r ¼ a þ bc, r ¼ bc c r ¼ 0:02c 0:8594
Application
pffiffiffi r¼b c qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ a þ bc, r ¼ a2 þ ðbcÞ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ a2 þ ðbc c Þ2 8 < 0:22c; c > 106:9 6:9 0:8495 r ¼ 0:02c < c < 0:14 pffiffiffi ; 10 : 0:01 c ; c > 0:14 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ a2 þ ðbcÞ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ a2 þ ðbcÞ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ a2 þ ðbcÞ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ a2 þ ðbcÞ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ a2 þ ðbcÞ2 , r ¼ a þ bc qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ a2 þ ðbcÞ2
Ref.
SDr
1971
[6]
SDr in geochemical analysis SDR, SDr in individual collaborative trials General trend of SDR in all collaborative trials in the food sector (Horwitz function) (r and c are mass fractions) SDr, iron and steel, iron ore SDr from geochemical analysis
1978 1979 1980
[7] [8] [9] Many follow-up references
1987 1988
[10] [11]
SDR, clinical biochemistry
1997
[12]
General trend of SDR in all collaborative trials in the food sector (r and c are mass fractions)
2000
[13]
Uncertainty, in general
2000
[14]
Uncertainty, in food analysis
2006
[2]
SDR from proficiency tests, many different types of analysis SD (‘‘instrumental precision’’) Ba isotopes by ICP-MS
2008
[15]
2009
[16]
Uncertainty, SDR, trace elements in water
2009
[17]
SDr from duplication from in-run quality control. Many different analytes by ICP-AES
2010
[5]
An example with a ¼ 0:03; b ¼ 0:04 (% mass fraction) is shown in Fig. 1. A function of this form was first proposed by Zitter and God [4] in respect of precision. This function accounts also for the commonly observed properties of relative uncertainty, as rearranging Equation (2) gives
uncertainty of magnitude a and a term bc that is proportional to the concentration. A proper combination of these two terms provides the required characteristic function, namely qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Fc ¼ a2 þ ðbcÞ ð2Þ
Relative standard uncertainty
Date
0.2
0.1
β =0.04
0.0 0
1
2
3
4
5
Concentration Figure 4. Relative standard uncertainty as a function of concentration according to Equation (2). The parameters are the same as those in Fig. 1.
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Fc ¼ c
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 þ b2 c2
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ð3Þ
which shows an asymptotic tendency to a constant relative uncertainty b at high concentrations and rapidly escalating values as concentrations fall towards zero (Fig. 4). Equation (2) is recommended for characterising uncertainty in the Eurachem Guide to Quantifying Uncertainty in Chemical Measurement [12], although lack of large datasets makes the function difficult to test adequately by experiment for uncertainty per se.
6. Recent experimental work Equation (2) has been carefully tested in relation to precision by making use of large blocks of data collected for other purposes. It was tested under ÔinstrumentalÕ conditions, by considering measurements of concentrations of barium isotopes determined by ICP-MS. The study involved 14 different concentrations of barium each measured 100 times. By virtue of the varied abundances of the isotopes, it was possible to cover a wide concentration range overall. The function provided an excellent fit to the data in all instances [14]. Equation (2) was further tested under randomised repeatability conditions by using a block of more than 3000 pairs of duplicated results accumulated over an extended period during routine internal quality control of a multielement analysis of soils and sediments by ICPAES. In nearly all instances there was no significant lack of fit observed. In the same study Equation (2) was separately tested by using more than 30000 duplicate determinations of ethanol in breath. In that very large study there was a small lack of fit detected in some of the data points, but these were evenly distributed over the concentration range and did not signify that the curve had an incorrect shape [16]. Precision under reproducibility conditions has been studied by using the results from proficiency tests. Most proficiency tests distribute the same type of test material over extended periods. That allows the accumulation of data from a large group of participants and round by round at many different levels of concentration. The studies exploited the fact that a large number of participants tend to use a single analytical method. In nearly all of the cases studied lack of fit to Equation (2) was nonsignificant or trivial in magnitude. In these circumstances the reproducibility standard deviation could be taken as a reasonable surrogate for uncertainty [13]. ´ lvarez-Prieto have reported Jime´nez-Chaco´n and A their studies of the variation of uncertainty with concentration in a substantial range of analytical systems and confirmed a good fit to Equation (2) for many of them [15]. However, they had access to a relatively
small number of uncertainty estimates over the concentration range. 7. Validation and the characteristic function Validation is carried out in practice in the most economic manner that will satisfy the requirements for accreditation, providing an estimate of uncertainty – usually not a very precise one – at one or at most a few concentrations of the analyte. Validation per se provides no information about the shape of the characteristic function, which makes the estimation of uncertainty at other concentrations problematic. The way forward seems to be, in the absence of contra-indications, to assume that Equation (2) holds and use the validation data to estimate its parameters, a and b, and to record the maximum concentration at which the equation was tested. This could be carried out rather simply. The value of a could be estimated as the standard uncertainty estimated in the usual manner at (or close to) zero concentration, with due precaution to avoid any censoring of the data. The value of b would be the relative standard uncertainty at 2 a high concentration, that is, where ðbcÞ a2 , typically when the concentration is more than 100 times the detection limit. The analyst would have to be convinced that Equation (2) was indeed the correct functional form and there are circumstances when it is not. It is known to be invalid in instances where the reported analyte is not the measured one. For example, we might report a result as Ôresidue on dryingÕ but actually measure loss on drying. Equation (2) could fit the latter but not the former [13]. However, the studies described above suggest that it would work well in many types of analysis. In particular, the equation has been shown to provide a good fit to values of reproducibility (interlaboratory) standard ^R encompassing methods with a variety of deviation r analytes, matrices, and physical measurement principles. This is relevant to characterising uncertainty. Of ^R cannot be taken as a reasonable estimate of course, r standard uncertainty without due consideration, but it has been shown to be a useful benchmark in many instances [18]. [Having estimated a and b, the analyst might consider using Equation (1), the simpler linear function r ¼ a þ bc for the characteristic function. Although Equations (1) and (2) with the same parameters predict identical or close uncertainties towards the extremities of the concentration range, they deviate at mid-range concentrations, Equation (1) predicting aprelative uncertainty that is too high by ffiffiffi factor of 2 at maximum, at a concentration cmax ¼ a=b. While this may be serviceable in some circumstances, it is not accurate].
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8. Fitness functions Logically an uncertainty that is fit for purpose has to be established before a method can be identified as suitable for an application. The resulting fitness function F f is conceptually different from the uncertainty associated with a particular analytical system. Before selecting an analytical method, the analyst needs to be aware of the level of uncertainty that is optimal for the customer (optimal in the sense of balancing the cost of analysis with the average penalty arising from an incorrect decision based on the result). This optimal uncertainty can be expressed as a function of concentration. While there is a mathematical approach to fitness for purpose based on decision theoretic principles [19], a fitness function based on professional experience in the application area could simply be agreed between the analyst and the customer. Alternatively, the function might be defined by an authority regulating a whole application sector of chemical analysis, or by the provider of a proficiency test in that sector. As a simple example the following fitness function Ff ¼ 0:05c; 0:1 < c < 5%
ð4Þ
specifies that the relative standard uncertainty should be 5% of the concentration (or less) over the concentration range 0.1-5% mass fraction (Fig. 1). When such a fitness function has been agreed upon, it can be used to determine whether a particular analytical method is suitable by comparison with its reported characteristic function F c . Subject to confirmatory validation, a method is suitable if its characteristic function falls below the fitness function over the whole relevant range (as, for example, in Fig. 1). Fitness functions have other uses. For example they can be used to specify in advance the degree of accuracy prescribed and used for scoring in a proficiency test, without reference to the actual concentration of the analyte in the test material. Fitness functions can follow any convenient mathematical form [20]. For example, in FAPAS, a large proficiency test operating in the food analysis sector, the Horwitz function is used to define fitness for purpose in most instances. Related criteria are used in GeoPT, a proficiency test in the geochemical sector. In many applications a function of the form of Equation (4), that is, constant relative uncertainty, would be too stringent at low concentrations. It could be modified by including an intercept term and taking the form corresponding to Equation (2).
9. Do we really need detection limits? The detection limit cL (or any other limit such as the limit of determination or the limit of quantification)
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artificially dichotomises the concentration axis in an analytical system. We are all subject to the human tendency to categorize things that in reality fall on a continuum, the propensity that Richard Dawkins calls (in a quite different context) the Ôtyranny of the discontinuous mindÕ [21]. This tendency encourages analysts and their customers to believe – incorrectly – that a result just below the detection limit (say at 0:9cL ) has a qualitatively different status to one just above the limit at 1:1cL . Analysts have consequently become unwilling to report a result that falls below such a limit, replacing the numerical result, originally on a rational or interval scale, with an ordinal form such as ‘‘not detected’’ or ‘‘less than cL ’’. Such reporting is detrimental because it renders much more difficult an unbiased statistical treatment of the resulting dataset. All that end-users of analytical data really need to know is the result and its correct uncertainty [22]. In a particular analytical system, the parameter a in Equation (2) is clearly related to the traditional detection limit cL . We would expect cL ka, where k is a coverage factor, probably between 2 and 3 depending of the preferred definition of detection limit. However, the widespread use of Equation (2) would dispense with the need for arbitrary limits.
10. Conclusions A rationale has been presented for the use of a characteristic function to describe the performance of analytical methods as a function of the concentration of the analyte. Experiments have shown that, in a large proportion in instances, a compact function of the form of Equation (2) will accurately fulfil the need. Where Equation (2) is applicable, its parameters for a specific analytical system can be quickly estimated. The characteristic function would bring together the multifarious aspects of validating an analytical method and to some extent overcome the practical limitations of validation. It could also eliminate the need for arbitrary limits such as the detection limit at the low end of the concentration range. A fitness function could likewise be used to specify the optimal level of uncertainty for a particular application or for a proficiency test and, in combination with the characteristic function, provide a comprehensive basis for the selection of appropriate analytical methods.
Note added in proof After the finalisation of this paper, the author became aware of a parallel development in the filed of clinical immunoassay. Interested readers can find a review of that work in [23].
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References [1] M. Thompson, S.L.R. Ellison, R. Wood, Pure Appl. Chem. 74 (2002) 835. [2] M. Thompson, R. Wood, Accred. Qual. Assur. 10 (2006) 471. [3] M. Thompson, R. Wood, Pure Appl. Chem. 67 (1995) 649. [4] C.B. Bird, B. Malone, R.G. Rice, P.F. Ross, R. Eppley, M.M. Abouzied, J. AOAC Int. 85 (2002) 404. [5] M. Thompson, B.J. Coles, Accred. Qual. Assur. 16 (2010) 13. [6] H. Zitter, C. God, Fresenius Z. Anal. Chem. 255 (1971) 1. [7] M. Thompson, R.J. Howarth, J. Geochem. Explor. 9 (1978) 23. [8] International Standards Organization (ISO), ISO 5725, ISO, Geneva, Switzerland, 1979. [9] W. Horwitz, L.R. Kamps, K.W. Boyer, J. Assoc. Off. Anal. Chem. 63 (1980) 1344. [10] H. Hughes, P.W. Hurley, Analyst (Cambridge, UK) 112 (1987) 1445. [11] M. Thompson, Analyst (Cambridge, UK) 113 (1988) 1587. [12] D.L. Duewer, J.B. Thomas, M.C. Kline, W.A. MacCrehan, R. Schaffer, K.E. Sharpless, W.E. May, J.A. Crowell, Anal. Chem. 69 (1997) 1406.
Trends [13] M. Thompson, Analyst (Cambridge, UK) 125 (2000) 385. [14] S.L.R. Ellison, M. Rosslein, A. Williams, Eurachem CITAC Guide CG4, Quantifying uncertainty in chemical measurement, Second Edition, 2000 (http://www.citac.cc/QUAM2000-1.pdf). [15] M. Thompson, K. Mathieson, A.P. Damant, R. Wood, Accred. Qual. Assur. 13 (2008) 223. [16] M. Thompson, B.J. Coles, Accred. Qual. Assur. 14 (2009) 147. ´ lvarez-Prieto, Accred. Qual. Assur. 14 [17] J. Jime´nez-Chaco´n, M. A (2009) 15. [18] S.L.R. Ellison, K. Mathieson, Accred. Qual. Assur. 13 (2008) 231. [19] T. Fearn, S. Fisher, M. Thompson, S.L.R. Ellison, Analyst (Cambridge, UK) 127 (2002) 818. [20] AMC Technical Briefs No. 19A (www.rsc.org/images/brief%2019A_tcm18-134927.pdf). [21] R. Dawkins, The AncestorÕs Tale, Weidenfeld and Nicholson, London, UK, 2004. [22] Analytical Methods Committee, Accred. Qual. Assur. 13 (2008) 29. [23] W.A. Sadler, Clin. Biochem. Rev. 29 (Suppl.1) (2008) S33.
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