18 Spectrostopy 21 (9) September 2006 www.spectroscopyonline.com

Chemometrics In Specfroscopy

Limitations in AnalyticalAccuracy, Part I: Horwitz'sTrumpetTwo technical papers recognized as significant early contributions in the discussion ofthe limitations of analytical accuracy and uncertainty include those by Horwitz of theU.S. FDA (1,2). For this next series of articles, we will be discussing both the topic andthe approaches to this topic taken by those classic papers.

Jerome Workman, Jr. and Howard Mark

Two technical papers recognized as significant earlycontributions in the discussion of the limitations ofanalytical accuracy and uncertainty include those by

Horwitz of the U.S. FDA (1,2). For this next series of arti-cles, we will be discussing both the topic and theapproaches to this topic taken by the classic papers just ref-erenced. The determination and understanding of analyti-cal error is often approached using interlaboratory collabo-rative studies. We have previously delved into that subjectin "Chemometrics in Spectroscopy" witb a multipart col-umn series (3-8).

Horwitz points out in his Analytical Chemistry A-pagespaper (1), inserting the statement made by John Mandel,"the basic objective of conducting interlaboratory tests isnot to detect tbe known statistically significant differencesamong laboratories: 'The real aim is to acbieve the practicalintercbangeability of test results.' Interlaboratory tests areconducted to determine how much allowance must bemade for variability among laboratories in order to makethe values interchangeable."

Horwitz also points out the universal recognition of irre-producible differences in supposedly identical methodresults between laboratories. It has even been determinedtbat when the same analyst is moved between laboratories,the variability of results obtained by that analyst increases.

One government laboratory study concluded that variabilityin results could be minimized only if one was to "conductall analyses in a single laboratory . . . by the same analyst."So if we must always have interlaboratory variability, bowmuch allowance in results should be regarded as valid — orlegally permissible — as indicating "identical" results? Whatare the practical limits of acceptable variability betweenmetbods of analysis, especially for regulatory purposes?

We will address aspects of reproducibility, which has beendefined previously as "tbe precision between laboratories." Italso has been defined as "total between-laboratory precision."Tbis is a measure of tbe ability of different laboratories toevaluate eacb otber. Reproducibility includes all tbe measure-ment errors or variances, including the with in-laboratoryerror. Otber terms include precision, defined as "the closenessof agreement between independent test results obtainedunder stipulated conditions" (9); and repeatability, or "theprecision for the same analyst within tbe same laboratory, orwitbin-laboratory precision." Note that for none of these defi-nitions do we require the "true value for an analytical sam-ple." In practice, we do not know the true analyte value unlesswe have created the sample, and then it is only known to agiven certainty (that is, within a determined uncertainty).

Systematic error is also known as bias. The bias is tbeconstant value difference between a measured value (or set

20 Spectroscopy 21(9) September 2OO6 www.spectroscopyonline.coni

of values) and a consensus value (ortrue value if known). Specificity is theanalytical property of a method ortechnique to be insensitive to interfer-ences and to yield a signal relative tothe analyte of interest only. Limit ofreliable measurement predates the useof minimum detection limit (MDL).The MDL is the minitnum amount ofanalyte present that can be detectedwith known certainty. Standard errorof the laboratory (SEL) represents theprecision of a laboratory method. Astatistical definition is given in the fol-lowing paragraph. The SEL can bedetermined by using one or moresamples properly aliquoted and ana-lyzed in replicate by one or more lab-oratories. The average analytical valuefor the replicates on a single sample isdetermined as

Table I: Characteristics and allowable uncertainty for different analyticalmeth"*-

X::[1]

SEL is given by

SEL =ZZ1=1 j= l

[2]

where the i index represents differ-ent samples and the; index differentmeasurements on the same sample.

This can apply whether the replicateswere performed in a single laboratoryor whether a collaborative study wasundertaken at multiple laboratories.Additional techniques for planning col-laborative tests can be found in refer-ence 10. Some care must be taken inapplying equation 2. If all of the analyt-ical results are from a single analyst in asingle lab, then the repeatability oi \Yieanalysis is defined as \/2 X r(n(r —1),95%) X SEL, where f(n(r- 1),95%) is the Student's t value for the95% confidence level and n{r — 1)degrees of freedom. If the analyticalresults are from multiple analysts andlaboratories, the same calculation yieldsthe reproducibility of the analysis. Formany analytical tests, SEL can vary withthe magnitude of x SEL values calcu-lated for samples having different x, canbe compared by an F-test to determine

DetectionSurveyMonitor

Compliance

RapidRapidMedium

Slow OK

YESYESConstant bias isallowedNO

No false positives or negativesSpecific but not accurateSpecific and tracks well withinterferencesSpecific and Accurate

aX

fN

+-•CC

oQ.X

8

6-

4 -

2 -

0^

-2 -

-4-

-8Analyte concentration exponent

Figure l : Relationship of laboratory CV (as %) with analyte concentration as powers of

(For example, 6 on the abscissa represents a concentration of 10^ or 1 ppm.) Note: The shape of

the curves has been referred to as Horwitz's trumpet.

if the SEL values show a statistically sig-nificant variation as a function of X;.(Note: a useful description of the F-sta-tistic and its uses for comparing datasets is found in reference 11.)

Any analytical method inherentlycarries with it limitations in terms ofspeed, allowable uncertainty (asMDL), and specificity. These charac-teristics of a method (or analyticaltechnique) determine where and howthe method can be used. Table I showsa template to relate purpose of analyt-

ical methods to the speed of analysisand error types permitted.

Bias is allowed between laboratorieswhen constant and deterministic. Forany method optimization, we mustconsider the requirements for preci-sion and bias, specificity, and MDL.

Horwitz claims that irrespective ofthe complexity found within variousanalytical methods, the limits of ana-lytical variability can be expressed orsummarized by "plotting the calcu-lated mean coefficient of variation

Table II: Relationship of laboratory CV (%} (as powers of 2} with analyteconcentration (as powers of 10y

CV (%)

2°2122232"2526

Anaiyte Cone. *

10010-110-210-310-610-910-12

Near 100%10%1.0%0.1%1 ppmIppbIppt

10̂10^10''103110-310-6

22 SpectresCQpy 2I(») September 2006 www.spectroscopyonline.coin

Table Ml: Relationship of laboratory CV (asconcentration (as powers of 10) ^̂ .̂i

CV% (as ZE'̂ P)

012

456

CV%

1248163264

Cone, (as 10 ^^P)

0-1-2- 3-6- 9-12

Absolute Cone.

Near 100%10%1.0%0.1%1 ppmIppbIppt

Cone. Tn ppm

10610510"103110-310-6

(CV), expressed as powers of two[ordinate], against the analyte leveltneasured, expressed as powers of 10[abscissa]." In an analysis of 150 inde-pendent Association of OfficialAnalytical Chemists (AOAC) interlab-oratory collaborative studies coveringnumerous methods such as chro-matography, atomic absorption,molecular absorption spectroscopy,spectrophotometry, and bioassay, itappears that the relationship describ-ing the CV of an analytical methodand the absolute analyte concentrationis independent of the analyte type orthe method used for detection.

Moving ahead to describe the

details of this claim, we need todevelop a few basic review concepts.The standard deviation of measure-ments is determined by first calculat-ing the mean, then taking the differ-ence of each control result from themean, squaring that difference, divid-ing by M — 1, then taking the squareroot. All of these operations areimplied in the following equation:

S =(n-l) [3

where 5 represents the standard

deviation, L means summation of allthe {Xj - x)2 values, x, is an individualanalysis result, x is the mean of theanalysis results, and n is the totalnumber of measurement resultsincluded in the group.

Percent CV refers to the "coeffi-cient of variation," which describesthe standard deviation as a percent-age of the mean, as shown in the fol-lowing equation:

CV{%) - (s/x) X 100

where s is the standard deviation,X is the mean, and the multiplier of100 is used to convert the (s/x) ratioto a percentage.

The data for Figures 1 and 2 areshown in Tables II and III, respectively.

In reviewing the data from the 150studies, it was found that about 7% ofall data reported could be consideredoudier data as indicated by a Dixontest. Some international refereed meth-ods performed by experts had to acceptup to 10% outliers resulting from bestefiforts in tbeir analytical laboratories.

Horwitz throws down the gauntlet

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www.spectroscopyonline.com September 2006 21(9} 23

a. _

u

Analyte concentration exponent

Figure 2: Relationship of laboratory CV (as powers of 2) with analyte concentration as powers ofrepresents a concentration of 10~^ or 1 ppm with a CV (%) of 2''.)

to analytical scientists, stating that ageneral equation can be formulatedfor the representation of analyticalprecision. He states this as follows:

CV(%) =

where C is the mass fraction of ana-lyte as concentration expressed inpowers of 10 (for example, 0.1% isequal to C= 10-3).

At high (macro) concentrations,

P. (For example, 6 on the abscissa

CV doubles for every order of magni-tude that concentration decreases; forlow (micro) concentrations, CV dou-bles for every three orders of magni-tude decrease in concentration. Notethat this represents the between-labo-

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24 Spectnscopf 21(9) September 2006 www.spectroscopyonline.coin

ratory variation. The within-labora-tory variation should he 50-66% ofthe hetween-lahoratory variation.Reflecting on Figures 1 and 2, somehave called this Horwitz's trumpet.How interesting that he plays such atune for analytical scientists.

Another form of expression also canbe derived, because CV {%) is anotherterm for percent relative standarddeviation (% RSD) as follows (12):

There are many tests for uncer-tainty in analytical results and we willcontinue to present and discuss thesewithin this series.

References(!) W. Horwitz,/Ino/. Cbem. 54(1),

67A-76A(1982).

(2) W. Horwitz, R.K. Laverne, W.K. Boyer,

J. Assoc. Off. Anal. Chem. 63(6),

1344 (1980).

(3) J. Workman and H. Mark, Spec-

troscopy 15(1), T6-25 (2000).

(4) J. Workman and H. Mark, Spec-

troscopy 15(2), 28-29 (2000).

(5) J. Workman and H. Mark, Spec-

troscopy 15{5), 28-29 (2000).

(6) J. Workman and H. Mark, Spec-

troscopy 15(6), 26-27 (2000).

(7) J. Workman and H. Mark, Spec-

troscopy 15(7), 16-19 (2000).

(8) J. Workman and H. Mark, Spec-

troscopy 15(9), 27 (2000).

(9) ASTM E177 - 86. Form and Style for

ASTM Standards, ASTM International,

West Conshohocken, PA. ASTM El 77

- 86 "Standard Practice for Use of

the Terms Precision and Bias in ASTM

Test Methods."

(10) S. Helland, Scand. J. Statist. 17, 97

(1990).

(11) H. Mark and J. Workman. Statistics

in Spectroscopy (second edition) (El-

sevler, Amsterdam, 2003), pp.

205-211; 213-222.

(12) Personal communication with C.

Clark Dehne, Capital University

(Columbus, Ohio) (2004).

ErratumA sharp-eyed (and careful) readerfound a typographical error in ourprevious column (1). The "A" on theleft-hand side of equation !2bshould have the subscript 2, ratherthan 1. Our thanks to Kaho Kwokfor pointing this out to us.

(1) Spectroscopy 2} (5). 34-38 (2006).

Jerome Workman, Jr.serves on the EditorialAdvisory Board ofSpectroscopy and is directorof research, technology, andapplications development forthe Molecular Spectroscopy &

Microanalysis division of Thermo ElectronCorp. He can be reached by e-mail at:jerry.workman@thermo.com.

Howard Mark serveson the Editorial AdvisoryBoard of Spectroscopy andruns a consulting service,Mark Electronics (Suffern,NY). He can be reached viae-mail: hlmark@prodigy.net.

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