Chemometrics In Spectroscopy Limitations in Analytical Accuracy, Part I: Horwitz s Trumpet

Transcription

1 18 Spectroscopy 1(9) September 006 Chemometrcs In Spectroscopy Lmtatons n Analytcal Accuracy, Part I: Horwtz s Trumpet Two techncal papers recognzed as sgnfcant early contrbutons n the dscusson of the lmtatons of analytcal accuracy and uncertanty nclude those by Horwtz of the U.S. FDA (1,). For ths next seres of artcles, we wll be dscussng both the topc and the approaches to ths topc taken by those classc papers. Jerome Workman, Jr. and Howard Mark Two techncal papers recognzed as sgnfcant early contrbutons n the dscusson of the lmtatons of analytcal accuracy and uncertanty nclude those by Horwtz of the U.S. FDA (1,). For ths next seres of artcles, we wll be dscussng both the topc and the approaches to ths topc taken by the classc papers just referenced. The determnaton and understandng of analytcal error s often approached usng nterlaboratory collaboratve studes. We have prevously delved nto that subject n Chemometrcs n Spectroscopy wth a multpart column seres (3 8). Horwtz ponts out n hs Analytcal Chemstry A-pages paper (1), nsertng the statement made by John Mandel, the basc objectve of conductng nterlaboratory tests s not to detect the known statstcally sgnfcant dfferences among laboratores: The real am s to acheve the practcal nterchangeablty of test results. Interlaboratory tests are conducted to determne how much allowance must be made for varablty among laboratores n order to make the values nterchangeable. Horwtz also ponts out the unversal recognton of rreproducble dfferences n supposedly dentcal method results between laboratores. It has even been determned that when the same analyst s moved between laboratores, the varablty of results obtaned by that analyst ncreases. One government laboratory study concluded that varablty n results could be mnmzed only f one was to conduct all analyses n a sngle laboratory... by the same analyst. So f we must always have nterlaboratory varablty, how much allowance n results should be regarded as vald or legally permssble as ndcatng dentcal results? What are the practcal lmts of acceptable varablty between methods of analyss, especally for regulatory purposes? We wll address aspects of reproducblty, whch has been defned prevously as the precson between laboratores. It also has been defned as total between-laboratory precson. Ths s a measure of the ablty of dfferent laboratores to evaluate each other. Reproducblty ncludes all the measurement errors or varances, ncludng the wthn-laboratory error. Other terms nclude precson, defned as the closeness of agreement between ndependent test results obtaned under stpulated condtons (9); and repeatablty, or the precson for the same analyst wthn the same laboratory, or wthn-laboratory precson. Note that for none of these defntons do we requre the true value for an analytcal sample. In practce, we do not know the true analyte value unless we have created the sample, and then t s only known to a gven certanty (that s, wthn a determned uncertanty). Systematc error s also known as bas. The bas s the constant value dfference between a measured value (or set

2 0 Spectroscopy 1(9) September 006 f the SEL values show a statstcally sgnfcant varaton as a functon of x. (Note: a useful descrpton of the F-statstc and ts uses for comparng data sets s found n reference 11.) Any analytcal method nherently carres wth t lmtatons n terms of speed, allowable uncertanty (as MDL), and specfcty. These characterstcs of a method (or analytcal technque) determne where and how the method can be used. Table I shows a template to relate purpose of analytof values) and a consensus value (or true value f known). Specfcty s the analytcal property of a method or technque to be nsenstve to nterferences and to yeld a sgnal relatve to the analyte of nterest only. Lmt of relable measurement predates the use of mnmum detecton lmt (MDL). The MDL s the mnmum amount of analyte present that can be detected wth known certanty. Standard error of the laboratory (SEL) represents the precson of a laboratory method. A statstcal defnton s gven n the followng paragraph. The SEL can be determned by usng one or more samples properly alquoted and analyzed n replcate by one or more laboratores. The average analytcal value for the replcates on a sngle sample s determned as SEL s gven by SEL = r x = = j 1 n r = 1 j= 1 where the ndex represents dfferent samples and the j ndex dfferent measurements on the same sample. Ths can apply whether the replcates were performed n a sngle laboratory or whether a collaboratve study was undertaken at multple laboratores. Addtonal technques for plannng collaboratve tests can be found n reference 10. Some care must be taken n applyng equaton. If all of the analytcal results are from a sngle analyst n a sngle lab, then the repeatablty of the analyss s defned as t(n(r 1),95%) SEL, where t(n(r 1), 95%) s the Student s t value for the 95% confdence level and n(r 1) degrees of freedom. If the analytcal results are from multple analysts and laboratores, the same calculaton yelds the reproducblty of the analyss. For many analytcal tests, SEL can vary wth the magntude of x. SEL values calculated for samples havng dfferent x can be compared by an F-test to determne x (x j x n(r 1) j ) [1] [] Table I: Characterstcs and allowable uncertanty for dfferent analytcal methods Purpose Speed Bas Allowed Comments Detecton Rapd YES No false postves or negatves Survey Rapd YES Specfc but not accurate Montor Medum Constant bas s Specfc and tracks well wth allowed nterferences Complance Slow OK NO Specfc and Accurate CV% exponent ( exp ) Analyte concentraton exponent (10 exp ) Fgure 1: Relatonshp of laboratory CV (as %) wth analyte concentraton as powers of 10 exp. (For example, 6 on the abscssa represents a concentraton of 10 6, or 1 ppm.) Note: The shape of the curves has been referred to as Horwtz s trumpet. cal methods to the speed of analyss and error types permtted. Bas s allowed between laboratores when constant and determnstc. For any method optmzaton, we must consder the requrements for precson and bas, specfcty, and MDL. Horwtz clams that rrespectve of the complexty found wthn varous analytcal methods, the lmts of analytcal varablty can be expressed or summarzed by plottng the calculated mean coeffcent of varaton Table II: Relatonshp of laboratory CV (%) (as powers of ) wth analyte concentraton (as powers of 10) CV (%) Analyte Conc. Absolute Conc. Conc. n ppm Near 100% % % % ppm ppb ppt 10 6

3 Spectroscopy 1(9) September 006 Table III: Relatonshp of laboratory CV (as powers of ) wth analyte concentraton (as powers of 10) CV% (as Exp ) CV% Conc. (as 10 Exp ) Absolute Conc. Conc. n ppm Near 100% % % % ppm ppb ppt 10 6 (CV), expressed as powers of two [ordnate], aganst the analyte level measured, expressed as powers of 10 [abscssa]. In an analyss of 150 ndependent Assocaton of Offcal Analytcal Chemsts (AOAC) nterlaboratory collaboratve studes coverng numerous methods such as chromatography, atomc absorpton, molecular absorpton spectroscopy, spectrophotometry, and boassay, t appears that the relatonshp descrbng the CV of an analytcal method and the absolute analyte concentraton s ndependent of the analyte type or the method used for detecton. Movng ahead to descrbe the detals of ths clam, we need to develop a few basc revew concepts. The standard devaton of measurements s determned by frst calculatng the mean, then takng the dfference of each control result from the mean, squarng that dfference, dvdng by n 1, then takng the square root. All of these operatons are mpled n the followng equaton: n = s = 1 ( x x) ( n 1) where s represents the standard [3] devaton, Σ means summaton of all the (x x ) values, x s an ndvdual analyss result, x s the mean of the analyss results, and n s the total number of measurement results ncluded n the group. Percent CV refers to the coeffcent of varaton, whch descrbes the standard devaton as a percentage of the mean, as shown n the followng equaton: CV(%) = (s/x ) 100 where s s the standard devaton, x s the mean, and the multpler of 100 s used to convert the (s/x ) rato to a percentage. The data for Fgures 1 and are shown n Tables II and III, respectvely. In revewng the data from the 150 studes, t was found that about 7% of all data reported could be consdered outler data as ndcated by a Dxon test. Some nternatonal refereed methods performed by experts had to accept up to 10% outlers resultng from best efforts n ther analytcal laboratores. Horwtz throws down the gauntlet Crcle 16 Crcle 17

4 September 006 1(9) Spectroscopy 3 CV (as percent error) Analyte concentraton exponent (10 exp ) Fgure : Relatonshp of laboratory CV (as powers of ) wth analyte concentraton as powers of 10 exp. (For example, 6 on the abscssa represents a concentraton of 10 6, or 1 ppm wth a CV (%) of 4.) to analytcal scentsts, statng that a general equaton can be formulated for the representaton of analytcal precson. He states ths as follows: CV(%) = (1 0.5log C) where C s the mass fracton of analyte as concentraton expressed n powers of 10 (for example, 0.1% s equal to C = 10 3 ). At hgh (macro) concentratons, CV doubles for every order of magntude that concentraton decreases; for low (mcro) concentratons, CV doubles for every three orders of magntude decrease n concentraton. Note that ths represents the between-labo- Crcle 18

5 4 Spectroscopy 1(9) September 006 ratory varaton. The wthn-laboratory varaton should be 50 66% of the between-laboratory varaton. Reflectng on Fgures 1 and, some have called ths Horwtz s trumpet. How nterestng that he plays such a tune for analytcal scentsts. Another form of expresson also can be derved, because CV (%) s another term for percent relatve standard devaton (% RSD) as follows (1): %RSD = (1 0.5log C) There are many tests for uncertanty n analytcal results and we wll contnue to present and dscuss these wthn ths seres. References (1) W. Horwtz, Anal. Chem. 54(1), 67A 76A (198). () W. Horwtz, R.K. Laverne, W.K. Boyer, J. Assoc. Off. Anal. Chem. 63(6), 1344 (1980). (3) J. Workman and H. Mark, Spectroscopy 15(1), 16 5 (000). (4) J. Workman and H. Mark, Spectroscopy 15(), 8 9 (000). (5) J. Workman and H. Mark, Spectroscopy 15(5), 8 9 (000). (6) J. Workman and H. Mark, Spectroscopy 15(6), 6 7 (000). (7) J. Workman and H. Mark, Spectroscopy 15(7), (000). (8) J. Workman and H. Mark, Spectroscopy 15(9), 7 (000). (9) ASTM E Form and Style for ASTM Standards, ASTM Internatonal, West Conshohocken, PA. ASTM E Standard Practce for Use of the Terms Precson and Bas n ASTM Test Methods. (10) S. Helland, Scand. J. Statst. 17, 97 (1990). (11) H. Mark and J. Workman. Statstcs n Spectroscopy (second edton) (Elsever, Amsterdam, 003), pp ; 13. (1) Personal communcaton wth G. Clark Dehne, Captal Unversty (Columbus, Oho) (004). Erratum A sharp-eyed (and careful) reader found a typographcal error n our prevous column (1). The A on the left-hand sde of equaton 1b should have the subscrpt, rather than 1. Our thanks to Kaho Kwok for pontng ths out to us. (1) Spectroscopy 1(5), (006). Jerome Workman, Jr. serves on the Edtoral Advsory Board of Spectroscopy and s drector of research, technology, and applcatons development for the Molecular Spectroscopy & Mcroanalyss dvson of Thermo Electron Corp. He can be reached by e-mal at: jerry.workman@thermo.com. Howard Mark serves on the Edtoral Advsory Board of Spectroscopy and runs a consultng servce, Mark Electroncs (Suffern, NY). He can be reached va e-mal: hlmark@prodgy.net. Crcle 19