Evaluation of Approximate Methods for Calculating the Limit of Detection and Limit of Quantification

Transcription

1 Environ. Sci. Technol. 1999, 33, Evlution of Approximte Methods for Clculting the Limit of Detection nd Limit of Quntifiction MICHAEL E. ZORN, ROBERT D. GIBBONS, AND WILLIAM C. SONZOGNI*, Wter Chemistry Progrm, University of WisconsinsMdison, Mdison, Wisconsin 53706, nd Deprtment of Biosttistics, University of IllinoissChicgo, Chicgo, Illinois In previous pper, computtionl method ws presented for determining sttisticlly rigorous limits of detection nd quntifiction. The min purpose of this study is to evlute similr but less computtionlly complex methods. These pproximte methods use dt t multiple spiking concentrtions, re itertive, cn be derived from either prediction intervls or sttisticl tolernce intervls, nd require t minimum ordinry lest-squres regression for clculting the intercept nd slope. Approximte detection nd quntifiction limits clculted for vrious PCB congeners were similr to those clculted using the computtionlly exct method. Although the exct methods should be employed whenever possible (they include uncertinty in the clibrtion function nd provide greter weight to less vrible dt), pproximte methods cn provide detection nd quntifiction limits tht re sufficiently ccurte for most pplictions. Prcticl ppliction of multiconcentrtion-bsed methods my involve the use of routinely generted qulity control dt in the sttisticl clcultions. Introduction Limits of detection nd quntifiction re used routinely by nlyticl chemists to evlute dt qulity. The limit of detection is used to decide whether n nlyte is present, while the limit of quntifiction is used to decide whether the concentrtion of n nlyte cn be relibly determined. Both limits re most commonly clculted bsed on vribility in nlyte response t single, nd often rbitrry, spiked concentrtion [e.g., U.S. EPA s MDL (1) nd ML (2)]. Although these single concentrtion designs re computtionlly simple, they cn yield quite vrible results depending on the choice of spiking concentrtion (3, 4). More rigorous methods use replictes spiked t series of concentrtions. These clibrtion designs provide more ccurte estimtes of the limits of detection nd quntifiction becuse they ccount for vribility in nlyte response t multiple concentrtions. However, these methods re computtionlly nd experimentlly more complex thn single concentrtion designs. In recent pper (5), clibrtion-bsed detection nd quntifiction limits were clculted for 16 polychlorinted * Corresponding uthor phone: (608) ; fx: (608) ; e-mil: sonzogni@fcstff.wisc.edu. University of WisconsinsMdison. University of IllinoissChicgo. biphenyl (PCB) congeners using weighted lest-squres prediction intervls nd sttisticl tolernce intervls (subsequently referred to simply s tolernce intervls); weights t the limits were estimted by modeling response vribility s function of concentrtion. Although this full model provides very ccurte estimtes of the detection nd quntifiction limits, the sttisticl clcultions re quite complex. The min purpose of this study is to evlute less computtionlly complex (pproximte) methods for clculting detection nd quntifiction limits tht reduce the sttisticl computtions required. These pproximte methods re lso clibrtion-bsed; however, they do not incorporte uncertinty in the clibrtion curve (typiclly smll component of the overll vribility). Limits clculted using the pproximte methods re compred to results obtined using the full model provided in ref 5. Prcticl ppliction of clibrtion-bsed methods is lso briefly discussed. Lest-Squres Regression Anlysis. Clibrtion designs require mesurement of replicte spikes t series of nlyte concentrtions spnning the estimted detection nd quntifiction limits. An ordinry lest-squres (OLS) regression nlysis is performed of response (Y) on nlyte concentrtion (X) expressed s liner, first-order model of the form Y ) b 0 + b 1 X + ɛ (1) where b 0 is the intercept, b 1 is the slope, nd ɛ represents error in the response mesurement or devition from the fitted regression line. Errors re ssumed to be independent nd normlly distributed with men zero nd constnt vrince. Often, the errors do not exhibit constnt vrince; nonconstnt vrince hs been previously documented for vrious chemicl nlyses nd nlytes (5-17). Severl uthors (5, 11, 13, 14) hve suggested using weighted lestsqures (WLS) regression for clculting limits of detection nd quntifiction in situtions of nonconstnt vrince. WLS regression is modifiction of OLS tht gives greter emphsis (or weight) to lower vribility, more relible dt. The WLS model is Y ) b 0w + b 1w X + ɛ (2) where b 0w is the weighted intercept nd b 1w is the weighted slope [see ref 5 or Drper nd Smith (18) for more detiled discussion of OLS nd WLS regression]. Clibrtion design detection nd quntifiction limit estimtors re bsed on one-sided sttisticl intervls. Prediction intervls provide (1 - R)100% confidence of including the next single instrument response t the true concentrtion (X j), wheres tolernce intervls provide (1 - R)100% confidence of including (P)100% of the entire popultion of instrument responses t the true concentrtion. For exmple, tolernce intervl with R)0.01 nd P ) 0.95 would provide 99% confidence of including 95% of future instrument responses t X j. As such, tolernce intervls re better suited to the typicl routine production lbortory in which re estimted once nd pplied to lrge nd potentilly unknown number of future detection decisions. Weighted prediction nd tolernce intervl equtions re provided in ref 5. Approximte Prediction nd Tolernce Intervls. Onesided prediction intervls round predicted response (Ŷ j) /es981133b CCC: $ Americn Chemicl Society VOL. 33, NO. 13, 1999 / ENVIRONMENTAL SCIENCE & TECHNOLOGY Published on Web 05/22/1999

2 t concentrtion X j cn be pproximted s where t (1-R,n-p-2) is the upper (1 -R)100 percentge point of Student s t-distribution on n - p - 2 df (where p is the number of prmeters used to model the stndrd devition, see below) nd s j is the stndrd devition t X j. The pproximtion ignores uncertinty in the clibrtion function tht reltes instrument response (or mesured concentrtion) to true concentrtionsthis component of vribility is typiclly smll in prctice. Tolernce intervls re wider nd will provide lrger estimtes of detection nd quntifiction limits thn corresponding prediction intervls. However, inference to lrge nd potentilly unknown number of future detection decisions is possible with high degree of confidence, mking tolernce intervls ttrctive for routine ppliction in commercil lbortories. A one-sided tolernce limit for predicted response (Ŷ j) t concentrtion X j cn be pproximted by where K (P,1-R;n) is single smple tolernce limit fctor for confidence (R) nd coverge (P) bsed on n vilble mesurementsssee Gibbons (15), Guttmn (19), or Hhn nd Meeker (20) for tbulted vlues, or see Link (21) for n eqution tht does not require the use of tbles. The method developed by Link is not recommended for clculting K (P,1-R;n) with smll vlues of n. For exmple, using K (0.99,0.99;n) with fewer thn 10 mesurements results in n error of greter thn 10%; however, t n ) 40, the error is decresed to bout 1%. In situtions of nonconstnt vrince, estimtion of the stndrd devition s j t X j is required to clculte prediction nd tolernce limits using eqs 3 nd 4. This cn be chieved by modeling the stndrd devition s function of concentrtion. The following models of stndrd devition s x hve been previously proposed: proportionl model (13, 15), s x ) 1X; liner model (13, 15), s x ) 0 + 1X; qudrtic model (5), s x ) 0 + 1X + 2X 2 ; n exponentil model (3, 5), s x ) 0e 1 x ; nd two-component model (16), pproximted by s x ) ( 0 + 1X 2 ) 1/2. Limit of Detection. As described by Currie (6), the criticl level (L C) provides specificlly defined flse-positive (type I) error rte nd is concerned with the signl or mesured concentrtion tht is significntly greter thn bckground instrumentl noise. However, the detection limit (L D) provides specific flse-positive nd flse-negtive (type II) error rtes nd represents the true concentrtion tht is significntly greter thn zerossee refs 6, 15, or22 for more detiled discussion. Previous uthors hve clculted the criticl level nd limit of detection using WLS prediction (3, 5, 11, 13, 15, 22) nd WLS tolernce intervls (5, 15, 22). The prediction intervl-bsed criticl level in response units nd detection limit in concentrtion units cn be pproximted s nd Y j ) Ŷ j ( t (1-R,n-p-2) s j n respectively, where the intercept (b 0*) nd slope () re clculted using either OLS or WLS regression. Vlues of s 0 (3) Y j ) Ŷ j ( K (1-R,P;n) s j (4) Y C ) b 0* + t (1-R,n-p-2) s n t (1-β,n-p-2) s LD L D ) L C n (5) (6) nd s LD (the stndrd devition t zero nd t the detection limit, respectively) cn be estimted by modeling the stndrd devition s function of concentrtion. Similrly, the tolernce intervl-bsed criticl level in response units nd detection limit in concentrtion units cn be pproximted by nd Y C ) b 0* + K (1-R,P;n) s 0 (7) K (1-β,P;n) s LD L D ) L C + (8) respectively. The criticl level in concentrtion units is clculted s L C ) (Y C - b 0*)/ using either prediction or tolernce intervls. Equtions 7 nd 8 re sttisticlly equivlent to those used by the Americn Society for Testing nd Mterils to clculte the interlbortory detection estimte, IDE (23). Limit of Quntifiction. The limit of quntifiction is used to decide whether the concentrtion of n nlyte cn be relibly determined. Gibbons et l. (3) hve suggested n lterntive minimum level, or AML, procedure for clculting the limit of quntifiction. This pproch defines Y Q (the determintion limit in response units) s 10 times the stndrd devition t the lowest detectble signl (L C) plus the weighted intercept (to ccommodte bis). The AML is the concentrtion tht provides n upper bound for the opertionlly defined level L Q.Inref5, WLS prediction nd tolernce intervls were used to clculte L C, s LC, Y Q, nd AML. In this study, Y Q is clculted s Y Q ) 10s LC + b 0* (9) where s LC is the stndrd devition t the criticl level. The determintion limit in concentrtion units is clculted s L Q ) (Y Q - b 0*)/. The prediction intervl-bsed AML cn be pproximted s t (1-β,n-p-2) s LQ AML ) L Q n (10) using eq 5 to obtin estimtes for L C, s LQ, nd Y Q nd the tolernce intervl-bsed AML cn be pproximted by K (1-β,P;n) s LQ AML ) L Q + (11) using eq 7 to obtin estimtes for L C, s LQ, nd Y Q. [Note tht L Q is bsed on vribility t L C (see eq 9), wheres the AML is bsed on vribility t L Q (see eqs 10 nd 11).] Experimentl Section Gs chromtogrphic pek res of 16 PCB congeners [numbered ccording to Bllschmiter nd Zell (24)] were obtined from ref 5ssee the originl reference for informtion on PCB stndrds nd gs chromtogrphic prmeters. Sttisticl nlyses were performed using Minitb progrm softwre (Relese 9.2). Vrious mcros were written to fcilitte regression nlyses nd clcultion of detection nd quntifiction limits using the methods developed bove. Due to significnt lck of fit (p < 0.05) of the liner model for five of the PCB congeners (61, 77, 101, 128, nd 180) reported in ref 5, these congeners re not included in this study. In prctice, nlysis of these congeners would typiclly ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 33, NO. 13, 1999

3 TABLE 1. Ordinry nd Weighted Lest-Squres Prmeters Determined from Regression Anlyses of Response (Y) s Function of Concentrtion (X) intercept slope PCB n OLS WLS OLS WLS Numbered ccording to Bllschmiter nd Zell (24). TABLE 2. Detection Limits nd Alterntive Minimum Levels (ng/ml) Clculted Using WLS Regression Anlyses of Response (Y) s Function of Concentrtion (X) with Weighted Prediction Intervls (WPI) nd Weighted Tolernce Intervls (WTI) PCB WPI b WTI c WPI b WTI c Numbered ccording to Bllschmiter nd Zell (24). b 99% confidence (i.e., R)β ) 0.01). c 99% confidence nd 99% coverge (i.e., R)β ) 0.01 nd P ) 0.99). be restricted to those concentrtions for which the liner model could be pplied. Results nd Discussion OLS nd WLS (w i ) 1/s i2 ) regression nlyses were performed of response (Y) s function of nlyte concentrtion (X), nd the intercepts nd slopes re listed in Tble 1. Becuse the dt exhibit incresing vribility with concentrtion (5), the intercept nd slope re more ccurtely estimted using WLS; nevertheless, there is resonble similrity in estimtes of the intercept nd slope for OLS nd WLS regression models. Tble 2 lists the detection limit nd lterntive minimum level (AML) clculted using the rigorous methods presented in ref 5. Vlues were clculted using WLS regression nlyses of response (Y) s function of nlyte concentrtion (X) with weighted prediction intervls, WPIs (R)β ) 0.01), nd weighted tolernce intervls, WTIs (R ) β ) 0.01, P ) 0.99). A qudrtic model of the stndrd devition s function of concentrtion ws used to estimte the weights. In ref 5, qudrtic model (where p ) 3) ws shown to provide better fit to these dt thn n exponentil model or twocomponent model (16). As shown in Figure 1, the width of the weighted prediction nd tolernce intervls chnge with concentrtion, ccurtely representing the ctul error. Approximte Detection nd Quntifiction Limits. Tble 3 lists nd clculted using WLS regression nlyses (prmeters: b 0w nd b 1w) of response FIGURE 1. Weighted lest-squres (WLS) regression nlysis of response (Y) s function of concentrtion (X) for PCB congener 14. Includes the clibrtion line (s), weighted prediction intervls (- - -), nd weighted tolernce intervls ( ). TABLE 3. Detection Limits nd Alterntive Minimum Levels (ng/ml) Clculted Using WLS Regression Anlyses of Response (Y) s Function of Concentrtion (X) with Approximte Prediction Intervls (API w ) nd Approximte Tolernce Intervls (ATI w ) PCB API b w ATI c w API b w ATI c w Numbered ccording to Bllschmiter nd Zell (24). b 99% confidence (i.e., R)β ) 0.01). c 99% confidence nd 99% coverge (i.e., R)β ) 0.01 nd P ) 0.99). (Y) s function of nlyte concentrtion (X) with pproximte prediction intervls (API w: R)β ) 0.01) nd pproximte tolernce intervls, (ATI w: R)β ) 0.01, P ) 0.99). A qudrtic model for the stndrd devition ws used. Vlues of K were estimted s in Link (21), where K (0.99,0.99;54) ) 3.113, K (0.99,0.99;55) ) 3.104, nd K (0.99,0.99;56) ) Approximte limits re very similr to limits clculted using WLS prediction nd tolernce intervls. All vlues in Tble 3 re within 20% of the corresponding vlues in Tble 2 except the tolernce intervl-bsed detection limit for congener 65 (22%). As shown in Figure 2, pproximte tolernce intervls (with WLS estimtes of the regression prmeters) re indistinguishble from WLS tolernce intervls t very low concentrtion. The required clcultions would be further simplified if ordinry estimtes of the intercept nd slope could be substituted for the weighted estimtes, thereby eliminting the need to perform the more difficult WLS regression. As discussed bove, the intercept nd slope re not gretly ffected by weighting. Clcultions were repeted using OLS regression nlyses (prmeters: b 0 nd b 1) of response (Y) s function of nlyte concentrtion (X) with pproximte prediction intervls (API o: R)β ) 0.01) nd pproximte tolernce intervls (ATI o: R)β ) 0.01, P ) 0.99) nd VOL. 33, NO. 13, 1999 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

4 FIGURE 2. Weighted lest-squres (WLS) regression nlysis of response (Y) s function of concentrtion (X) for PCB congener 14 (four lowest concentrtions shown). Includes the clibrtion line (s), weighted tolernce intervls ( ), nd pproximte tolernce intervls (- - -). The pproximte tolernce intervls re virtully indistinguishble from the weighted tolernce intervls, yielding very good estimtes of detection nd quntifiction limits. TABLE 4. Detection Limits nd Alterntive Minimum Levels (ng/ml) Clculted Using OLS Regression Anlyses of Response (Y) s Function of Concentrtion (X) with Approximte Prediction Intervls (API o ) nd Approximte Tolernce Intervls (ATI o ) PCB API b o ATI c o API b o ATI c o Numbered ccording to Bllschmiter nd Zell (24). b 99% confidence (i.e., R)β ) 0.01). c 99% confidence nd 99% coverge (i.e., R)β ) 0.01 nd P ) 0.99). qudrtic model for the stndrd devition. The results re listed in Tble 4. These vlues re lso quite close to the vlues in Tble 2. Only the tolernce intervl-bsed detection limit for PCB congeners 31 (23%) nd 65 (27%) differ by more thn 20%. It should be noted tht OLS does not chnge the width of the pproximte intervls; however, the ordinry intercept nd slope move the fitted regression line (see Figure 3), thereby chnging the clculted limits. Prcticl Appliction of Multiconcentrtion-Bsed Methods. A considertion in using multiple spiking concentrtion methods for clculting limits of detection nd quntifiction, even with the less computtionlly complex pproximte methods described bove, is tht the overll lbortory effort cn be substntil. For exmple, mny lbortories routinely nlyze thousnds of nlytes using hundreds of nlyticl procedures. Also, regultory requirements re incresingly clling for frequent reclcultion of for qulity control purposes. Clcultion of detection nd quntifiction limits using multiconcentrtion pproch could dd substntil time nd expense to the nlyticl process reltive to the flwed single concentrtion methods now widely in use. FIGURE 3. Ordinry lest-squres (OLS) regression nlysis of response (Y) s function of concentrtion (X) for PCB congener 14 (four lowest concentrtions shown). Includes the clibrtion line (s) nd pproximte tolernce intervls (- - -). The width of the pproximte tolernce intervl is the sme for OLS nd WLS (t ny given concentrtion); however, the ordinry intercept nd slope move the fitted regression line, thus chnging the clculted limits. The weighted lest-squres (WLS) regression line ( ) is lso shown for reference. A potentil option for reducing the lbor involved in performing multiconcentrtion detection nd quntifiction limit determintions is to utilize routinely generted qulity control dt in the sttisticl clcultions. For exmple, results from lbortory-fortified mtrixes (LFMs) [or mtrix spikes, s defined in Stndrd Methods for the Exmintion of Wter nd Wstewter (25)] represent lrge pool of dt from which to pply multiconcentrtion pproch. It is recommended tht, s minimum, one LFM be included with ech smple set (btch) or on 5% bsis, whichever is more frequent (25). By vrying the fortifiction (i.e., spiking) level, lbortory could ccumulte mtrix-specific dt for multiconcentrtion clcultions with little extr effort. Also, for lbortories not currently performing routine LFMs, it might be possible to use instrument clibrtion dt in detection limit determintions. Perhps reltionship could be estblished between clibrtion smples nd rel world (i.e., mtrix specific) smples. An dded dvntge of using extnt, routinely generted dt (s opposed to using dt generted on single dy) is tht dy-to-dy vribility (inevitble in routine mesurement) is incorported. Even with reducing the effort to determine multiconcentrtion detection nd quntifiction limits, single concentrtion designs re still required by mny regultions. Consequently, single concentrtion techniques will likely continue in use for some time. Given this sitution nd the dependence of the clculted limits on the spiking concentrtion, the uthors recommend tht, when the single concentrtion technique must be used, the chosen spiking concentrtion lwys be reported long with the clculted limits. Without knowledge of the spiking concentrtion when using the single concentrtion designs, detection nd quntifiction limits produced by different methods or different lbortories should not be compred. Where possible, however, multiconcentrtion (clibrtion) methods should be used for clculting detection nd quntifiction limits. Supporting Informtion Avilble A computtionl exmple for PCB congener 65 is vilble s Supporting Informtion (4 pges with 2 tbles). This ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 33, NO. 13, 1999

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