regulation – from an industry perspective or: relationships between detection limits, quantitation...

Ž .Chemometrics and Intelligent Laboratory Systems 37 1997 71–80

Regulation – From an industry perspective orRelationships between detection limits, quantitation limits, and

significant digits

David Coleman ), Jay Auses, Nancy GramsAMCT-D-ATC Alcoa Lab., Alcoa Center, PA 15069, USA

Abstract

This article develops the concepts of detection and quantitation limits, as advanced by Currie. It interprets and extendsthe calibration-based Hubaux–Vos detection limit, DL , in the context of trace-level measurements. RecommendationsH –V

are made for statistically sound data reporting and decision-making under uncertainty. It is shown that relative measurementerror derived from calibration error bands can be related to significant digits, DL , and a quantitation limit that ensures atH –V

least one significant digit in a reported measurement. An example is given of a calibrationrDL study for measuring PCBconcentrations in water by GC.

Keywords: Method detection limit; Practical quantitation limit; Calibration; Measurement error; Detection; False positive; False negative;Signal-to-noise ratio; Least squares

1. Introduction

Many definitions and interpretations of criticallevels, detection limits and quantitation limits fortrace-level measurement have been developed over

w xthe years 1–3 . The basic concepts are simple.The ‘critical level,’ LC, is the lowest measured

concentration above which detection can be confi-w xdently asserted to have occurred 2 . It is the lowest

measurement which is unlikely to have been ob-tained from a blank sample. Because of this, LC issometimes referred to as the detection threshold. Notethat there is no assurance that samples with concen-

) Corresponding author.

trations at or below LC will be detected. For that, oneneeds to consider a higher concentration — at thedetection limit.

Ž .A detection limit DL is the lowest concentrationat or above which an analyte can be confidently de-

Ž w xtected i.e., distinguished from zero; Currie 2 used.the designation L , or ‘limit of detection’ . Thus, theD

detection limit defines the lowest concentration atwhich the measurement signal consistently emergesfrom the noise. That is, the reported measurementwill, with high confidence, exceed LC.

ŽQuantitation limits QL; L in Currie’s lexiconQw x.2 are defined in diverse ways. In concept, a QL isthe lowest concentration at which there is some con-fidence in the accuracy of the reported measurement.This is what is commonly meant by ‘quantitate’ or

0169-7439r97r$17.00 Copyright q 1997 Elsevier Science B.V. All rights reserved.Ž .PII S0169-7439 97 00026-9

( )D. Coleman et al.rChemometrics and Intelligent Laboratory Systems 37 1997 71–8072

‘quantify’. In practice, some of the definitions usedare as follows:

Ž .The quantitation limit QL is the lowest concen-tration at or above which...

Ž .a ...one can quantitate. This definition is too am-biguous.

Ž .b ...one can have ‘assurance’ of detection. This isinconsistent with the common usage of the relatedwords: quantify, quantity, quantitative, and is incon-sistent with the historical use of the term, introduced

w xby Currie 2 .Ž .c ...measurements have a low, prescribed stan-

dard deviation, say, 5 ppb. This is a reasonable defi-nition for ensuring a certain number of significantdigits of ppb measurement for a known range ofmeasurement values, say 100–999 ppb.

Ž .d ...measurements have limited relative standardŽ .deviation RSD , e.g., RSD-10%, where RSD is the

estimated measurement standard deviation divided bythe estimated level. This is a reasonable definition ifs is constant or is estimated at the QL. It is used bythe American Chemical Society to define the limit of

Ž . w xquantitation LOQ 4 , and Gibbons requires RSD-

w x10% 5 .Ž .e ...measurements have limited relative measure-

Ž . Žment error RME , e.g., RME 5%, or some otherprescribed proportion at some level of confidencesuch as 95%. Relative measurement error is definedas the estimated measurement uncertainty divided by

Žthe estimated level such as twice the measurement.standard deviation . This is also a reasonable defini-

tion, and is directly pertinent to the user of measure-ments.

Ž .In this paper, we develop e , which will be shownto be linked to the number of significant digits.

( )1.1. Practical implications of the critical leÕel LC ,( ) ( )detection limit DL , and quantitation limit QL

Conceptually, the real number line can be parti-tioned into four intervals into which a trace levelmeasurement, M, can fall. Fig. 1 represents this par-titioning of the number line, graphically. The section

Ž . Ž .of the number line corresponding to a – d above arenoted, as are the LC, DL, and QL concepts reviewedprior.

Ž .a M is indistinguishable from zero concentra-tion and hence should be considered a non-detect.

Fig. 1. Partitioning of the real number line.

Ž .b M is treated as a detection, but in this inter-val, there is low confidence of detection.

Ž .c M is treated as a detection, and any concentra-tion in this interval is likely to be detected. M is avery noisy measurement value which should only bereported with an error interval, and only used withextreme caution in comparison or computation.

Ž . Žd M can be reported as a measurement prefer-.ably with an error interval , and can generally be used

for comparison and computation.

2. Measurement error and its uses

The error associated with measurements at tracelevels is of great statistical and practical importance.

ŽBy the very definition of ‘trace level’ i.e., measure-ments in the range from zero to slightly above the

.QL , the error associated with M is large relative toa magnitude of measurement. In many contexts, un-evaluated or underestimated measurement errors cancorrupt decisions of large consequence which are

Žbased on measurements or functions of measure-.ments , such as:

Ž .a Comparison to compliance limits — resultingin unwarranted fines, undetected excursions, and un-necessary validation re-samples.

Ž .b Comparison to other measurements to decide:v the need for environmental classification or

cleanup — resulting in costly and unnecessary wasteprocessing or remediation, or contamination,

v research and development direction — result-ing in missed opportunities, and dead-end develop-ments,

v life-and-death medical choices — resulting indeath, ill health, and useless treatments.

Ž .c Comparison to product specification limits —resulting in poor choices, possibly with costly

Ž .‘cascading’ consequences as in product recalls .Ž .d Comparison to process specification limits —

resulting in out-of-spec products and less consistentproducts due to over-control or under-control of pro-cesses.

( )D. Coleman et al.rChemometrics and Intelligent Laboratory Systems 37 1997 71–80 73

2.1. Types of measurement error

w xMeasurement error has many components 6 , butit consists of two classes of deviation from the true

Žvalue: static error i.e., fixed ‘bias,’ sometimes known. Žas ‘inaccuracy’ , and stochastic error i.e., random,

ch an g in g in tim e , co m m o n ly k n o w n as.‘imprecision’ . In this paper, we will focus on

stochastic error, which will subsequently be referredto as ‘measurement error.’

The measurement error can be quantified in sev-eral ways. It can be computed theoretically, based onan underlying measurement principle, such as Vs IR.Such formulas can be used to develop worst-case or

Žtypical-case estimates perhaps with linearizing or.other approximations . They can also be used in sim-

ulation if hypotheses about the distributions of con-tributing sources of variation can be made. Most of-ten, an empirical approach is followed. One can makerepeated measurements of a ‘known’ analyte concen-

Ž .tration. The sample standard deviation s , derivedfrom the measurements, is an estimate of the preci-sion of the measurement system at that known con-centration.

2.2. Not reporting measurement error

Measurement error can be reported in several ways— but commonly, it is not reported at all. Unfortu-nately, this contributes ignorance to the decisions oflarge consequence described above. The risks of be-ing wrong in a simple comparison of numbers areunknown when the measurement error is unknown.This reflects an incomplete appreciation of the cost ofinformation — which enters into a step-by-step deci-sion-making process. For example, assume a deci-

Žsion-maker e.g., engineer, scientist, regulator, or. Žmanager wants to choose between two products or

.two processes , A or B, and proceeds as follows:Ž .1 estimate the cost of obtaining a measurement

Ž .or a set of measurements , M, of the difference be-Žtween A and B which will indicate with unknown

.error which is better;Ž .2 compare that cost to the relative benefit of

Ž .making the right versus the wrong , weighted by theperceived improved likelihood of making the rightdecision due to M;

Ž . Ž3 buy M if supported by the costrbenefit anal-.ysis ;

Ž . Ž .4 choose B instead of A or vice versa , basedon M.

A wiser process is:Ž .1 Estimate the cost of obtaining M, plus an esti-

mate, s, of uncertainty in M;Ž .2 Compare that cost to the relative benefit of

Ž .making the right versus the wrong , weighted by theperceived improved likelihood of making the rightdecision due to M and s;

Ž .3 Buy the measurements needed to produce MŽ .and s if supported by the costrbenefit analysis ;

Ž .4 Use M and s to compute a probability distri-bution for M and an expected benefit of A versus B.Additionally, M and s are used to estimate the costsand benefits of more measurements — to supple-ment M or refine s;

Ž . Ž .5 Choose B instead of A or vice versa , basedon M and s — or decide to obtain more measure-

Ž .ments, and go to step 3 .

2.3. Inadequate reporting

Often, measurement error is implicitly communi-Ž .cated by the number of displayed or reported digits.

Displayed digits are supposedly significant digits. Forexample, 3.918 supposedly has four significant dig-its, while 4 has only one — where a significant digitis commonly understood to be a digit with some in-formation content. However, the number of dis-played digits often has no bearing on the number ofdigits with information content. For example, if mea-surement error for both values is not known, it is notnecessarily wise to choose water source B with 3.918ppb contaminant over water source A with 4 ppb, andeven if 3.918 has four significant digits, it is not nec-essarily true that B is below the standard of 3.92. Noris it known if A exceeds the standard.

These reporting practices are not intellectuallyhonest, and can damage the credibility of measure-

Žment-oriented professions such as analytical chem-.istry .

Another common measurement error reporting ap-proach is of the form, Ms4.3"0.1, where the ab-

Žsolute error, represented by ‘0.1,’ should be but.generally is not identified as one of the following

forms in the next section.


2.4. Possible absolute error identification

Absolute error is one of the following:Ž .a the true standard deviation, s;Ž . Žb a sample standard deviation, s or standard er-

.ror, if appropriate ;Ž .c two or three times a sample standard deviation

Ž .or standard error, if appropriate ; orŽ . Žd half of a statistical interval e.g., a confidence

.interval at, say, 95% or 99% confidence.Any of these explicit representations of measure-

ment error is more appropriate than none at all, butŽ .each is limited for these respective reasons: a the

Ž . Ž .true standard deviation is rarely known; b and cŽ .the number of degrees of freedom dof for the sam-

ple standard deviation must also be provided, so thatthe correct factor for a statistical interval can be de-

Ž .termined; and d a particular statistical interval isperhaps most helpful for a specific use. However, theuser of the measurement may need a different inter-val or a different confidence level, yet cannot gener-ate it without knowing the degrees of freedom asso-ciated with the error term.

Similar to the absolute error format, Ms4.3"

0.1, is the relative error version, M s 4.3 " 2%,which has the same limitations as cited above.

2.5. Recommended data reporting format

Superior to all of the above is the following for-Ž . Žmat: Ms4.3 ss0.1; dofs11 , or Ms4.3 ss.2%; dofs11 . The number of degrees of freedom

applies to s, not M. If M is a function of multiplemeasurements rather than a single measurement, thenthe number of degrees of freedom for M should alsobe noted. This format allows the measurement user toconstruct any statistical interval desired, and thusjudge the risks associated with the decisions beingmade based on these data.

3. Importance, concept, and definition of signifi-cant digits

3.1. Importance of significant digits

Precise trace-level measurement is important be-cause binary decisions must be made based on sim-ple A-to-B comparisons. For example, a company or

regulatory agency may check for compliance bycomparing the measured contaminant concentration

Žin today’s 9.00 a.m. hourly sample or monthly aver-.age , 11 ppb, to the permit limit, 10 ppb. Typically,

neither the company nor the agency have the re-sources to conduct a complex statistical analysis. Asimple comparison should be made — provided thatthe measurement error is so small that the wrong de-cision is unlikely. As briefly described above, theconsequences of wrong decisions are so onerous thatthe risk of a wrong decision must be kept low.

Some regulatory and engineeringrscientific usesdo involve computations with typically noisy, trace-level measurements — such as a daily mean or max-imum, or a historical ‘average’ background concen-tration. The idea of a QL is to have negligible errorfor most comparisons and computations. This isequivalent to having assurance of significance in the

Ž .reported measurement digit s .

3.2. Concept of significant digits

ŽIn decimal numbers, it is commonly accepted but.often violated that a number should be displayed

with all of its significant digits and only its signifi-cant digits. This is the scientist’s or technician’s oathbefore taking the witness stand to report a measure-ment, as in: ‘‘The truth, the whole truth, and nothingbut the truth.’’ For example, a number representedhere as MsÕ.xyzP10 k would be assumed to havefour significant digits. They are each displayedŽwhere ‘Õ’, ‘x’, ‘y’, and ‘z’ each represent a digit,

.not necessarily unique . There is information contentin the fourth digit, z, but the digit may be uncertain.For example, the uncertainty might be "0.002P10 k.There are between three and four significant digits insuch a number. There would be greater than threesignificant digits in M if the measurement error in-terval — did not have some high probability of con-taining all ten decimal digits, zero through nine. Oth-erwise, z should not be presented as a significantdigit.

Note that the different possible amounts of infor-mation content in z can correspond to different lev-

Ž .els of relative measurement error RME — actually,to a wide range of relative errors. There are twoproblems with significant digits which are responsi-ble for this wide range. They are outlined below. Thefirst problem can be addressed by allowing fractions


of significant digits. The second problem precludes adirect relationship between relative error and signifi-

Žcant digits, but a bounding relationship i.e., an in-.equality is provided.

3.3. Problems with the traditional definition of sig-nificant digits

Causing a range of relative errors among mea-surement values with the same number of significantdigits.

Ž .A The information content in the last significantdigit may vary, by as much as an order of magni-tude. Using our example, MsÕ.xyzP10 k, the infor-mation content in z can be low: M may have an er-

k Žror of up to "0.005P10 i.e., its total error intervalk .width could be as much as 0.01P10 . Or, the infor-

mation content of z may be high: M may have errork Žas small as "0.0005 P 10 i.e., its error interval

k .width could be as little as 0.001P10 .Ž . Ž .B The value e.g., Õ.xyz for measurement M

can range over an order of magnitude. M could beany number from 1.000P10 k to 9.999P10 k, and stillhave four significant digits.

Ž .Problem A can be eliminated by introducingfractions of a significant digit, consistent with thewhole number definition of significant digits.

3.4. Definition of significant digits, including frac-tions

A number expressed in scientific notation is saidto have w significant digits if the value has an errorinterval of width 1r2P10ywq1. For example, if w isa whole number, then in correct scientific notation,there are wy1 significant digits to the right of thedecimal point and a single digit to the left of the dec-

Ž kimal point i.e., Õ.xyzP10 , with value Õ.xyz, and ex-.ponent k .

This implies that a number has w significant dig-its when the value has a total error interval width of10ywq1. Inversely, when the value has a total error

Ž .interval width of q, the number has ylog q q110

significant digits.Some examples are:

Õ. xyz"0.0005 P10 k has ylog 2P0.0005 q1Ž . Ž .10

s4 significant digits


s3 significant digits

Finally, since there is no reason to confine w towhole numbers,


s3.4 significant digitsThus, the obvious result is achieved for any givenmeasurement value with an error interval: as thewidth of the error interval decreases, the number ofsignificant digits increases, and the relative error de-creases, proportionally. The width of the error inter-val can decrease due to more precise measurementsystems, a greater number of individual measure-ments being averaged for a reported measurement, orthe selection of a reduced level of statistical confi-

Ž .dence e.g., 80% instead of 95% for the error inter-val.

4. Relationship between relative measurement er-( )ror RME , and significant digits

Ž .We now turn back to problem B — the range ofRME due to the range of possible values for the value,Õ.xyz. Different measurement values with the same

Žnumber of significant digits even the same frac-.tional significant digits will have different levels of

relative error. Some examples with 3.4 significantŽ .digits but different levels of RME follow:

Ž . Ž . k Ža 4. xyz"0.002 P10 has 3.4 sig. digits 3.4sŽ . .ylog 2 P 0.002 q 1, as seen above ; RME s10

0.002r4. xyzf0.05%Ž . Ž . kb 1.000"0.002 P10 also has 3.4 sig. digits;

RMEs0.002r1.000f0.2%Ž . Ž . kc 9.999"0.002 P10 also has 3.4 sig. digits;

RMEs0.002r9.999f0.02%These examples show that measurement values

within an order of magnitude of one another, holdingŽthe error interval constant i.e., the same number of

.significant digits have relatiÕe measurement errorsspanning an order of magnitude.

4.1. A relationship between significant digits andRME

Ž . Ž .Generalizing b and c , it can be shown that anumber with w significant digits satisfies the follow-ing inequality:1 1yw ywq1P10 FRME- P10 . 1Ž .2 2

Thus, RME and significant digits have a ‘boundingŽrelationship’ with one another i.e., they are related


.through an inequality . Note that these are extremebounds. Consider a number with one significant digitŽ .ws1 . It satisfies the relationship: 5%(RME-

50%. If the RME exceeds 50%, there cannot be asmuch as one significant digit, and to ensure that avalue has at least one significant digit, RME must beno more than 5%. If the relative error is less than 5%,the number will have more than one significant dig-

Ž .its possibly two .

4.2. A statistical definition of signal-to-noise

For convenience, we define the signal-to-noise-Ž .ratio SNR as:

signal measurementSNRs log s log X10 10ž / ž /noise total meas t error

sylog 2PRME ,Ž .10

where we let the signal be the reported measurementvalue, and the noise be the total measurement error

Ž .estimate i.e., uncertainty .Note that many definitions of the signal-to-noise

ratio in the literature are similar, but do not involvetaking the logarithm. It is done here for algebraicconvenience.

Ž .If we take Eq. 1 , multiply by 2, take base-tenlogarithms, and invert the signs, we get:

wGSNR)wy1. 2Ž .Ž .What remains to make Eq. 1 useful is a way to

obtain the total error of a measurement. This is pur-sued in the next section.

( )5. The Hubaux–Vos detection limit DL andH – V

relative measurement error

A DL procedure was proposed by Hubaux and Vosw x w x7 , and later modified by Clayton 8 . A version ofthis procedure is also used by the U.S. Army Toxic

w xand Hazardous Materials Agency 9 .Fig. 2 shows the H–V approach, including results

of a five-point straight-line calibration of raw re-Ž . Ž .sponse y versus concentration x by ordinary least

squares. Replicates are not shown here, for simplic-ity, but are generally recommended. Two additionalcurves are shown: each a one-sided prediction limit

for individual observations. The curves are functionsw x Žof the concentration 10 . The values of a for the

. Ž .upper curve and b for the lower curve can be cho-sen to be at any desired level. The curves need notbe symmetric about the calibration line. The uppercurve is the 1ya prediction limit — hence, for anysingle future sample at any concentration value, theprobability of obtaining a response less than the or-dinate on the upper curve at that concentration is 1ya . Similarly, for any single future sample at anyconcentration value, the probability of obtaining a re-sponse greater than the ordinate on the lower curveat that concentration is 1yb.

Inverse prediction is used to convert future re-sponses into estimated concentrations. Likewise, theprediction interval defined by the curves — for anygiven response — provides a measurement error in-terval at prescribed levels of confidence. That is, forany given response, the concentrations where that re-sponse intersects the 1ya upper prediction limit andthe 1yb lower prediction limit provide the lower

Ž .and upper limits of an overall 100 1 y a y b %confidence measurement error interval, respectively.In Fig. 2, T is selected to ensure that, with probabil-ity a , the response from measuring a blank sampleŽ .xs0 will exceed T. If the response is T , the mea-

Ž .surement is LC, with 100 1yayb % confidenceŽ .interval, 0, DL .H – V

Hubaux and Vos’s article presents a way to com-Žpute a DL actually, a pair of values: DL and re-H – V

.sponse threshold, T which provides specified levels

Ž .Fig. 2. Sample Hubaux–Vos DL DL : Based on trace-levelH – V

calibration and measurement error.


of a s false positive and b s false non-detectionprobabilities, on average, over many calibrations. The

Ž .regression theory formula for the upper 100 1ya %w xindividual prediction limit is 12,13 :

1r22xyxŽ .aqbxq t s 1q1rnq1ya ,ny2 Sx x

saqbxq t PsPR x 3Ž . Ž .1ya ,ny2

where, as the intercept from ordinary least squaresregression, bs the slope from the regression, ss theestimate of the response error standard deviation,usually taken to be the root mean squared error fromthe calibration, ns the number of observations usedin the calibration, e.g., 5, t is Student’s t1ya ,ny2

Ž .100 1 y a % critical value for n y 2 degrees offreedom, xs the mean concentration in the calibra-

n Ž .2tion design, S sÝ x yx is the corrected sumx x is1 i

of squares of the design concentrations, and1r221 zyxŽ .

R z s 1q qŽ .n Sx x

is the factor which accounts for individual responseerror and calibration error.

Ž .The formula for the lower 100 1yb % individ-ual prediction limit is:

aqbxy t PsPR x . 4Ž . Ž .1yb ,ny2

These formulas are combined to produce a recursive

Ždefinition of DL oddly, this is not provided inH – V.the Hubaux–Vos article :

DL s srb P t PR 0 qtŽ . Ž .ŽH – V 1ya ,ny2 1yb ,ny2

PR DL . 5Ž . Ž ..H – V

w xAn analytical form of the solution exists 5 , but itcan be easier to find the solution by starting with an

Ž1. Ž .estimate, DL s5 srb , then substituting this es-H – VŽ .timate in the right-hand side of Eq. 5 to get a new

estimate, DLŽ2. , which in turn is substituted into theH – V

right-hand side to get DLŽ3. , and so on. The itera-H – V

tion is stopped when the estimates change relativelyŽlittle e.g., 1y DL iq1 r DL iŽ . Ž .Ž . Ž .Ž .H – V H – V.-1% , usually after four or five iterations.

6. Relationship between DL and significantH – V

digits

Fig. 3 is an example of the relationship betweencalibration, prediction limits, DL , and RME. ItH – V

shows the calibration line determined from thirty-fourŽactual GC measurements of Aroclor 1248 a PCB

.mixture spiked in river water samples. Iterative ap-Ž . Ž1.plication of Eq. 5 results in: DL s278 ppt; fol-H – V

lowed by DLŽ2. fDLŽ3. s . . . s236 ppt. Exam-H – V H – VŽ .ination of Eq. 5 and Fig. 3 reveals that DL isH – V

Ž .the concentration line segment A which approxi-Žmately equals the width of its prediction interval line

Ž .Fig. 3. Calibration, prediction limits, and DL for Aroclor 1248 measured in river water by GC ns34; asbs3% .H – V


.segment B when the prediction interval is made ap-proximately symmetric by setting asb. That is, the

Ž .100 1yayb % confidence interval for a measure-ment with concentration reported at DL is ap-H – V

ŽŽ . .proximately: DL " DL r2 , or DLH – V H -V H – VŽ ŽŽ . Ž . ."50% since RME s srb R x t rx, from1ya

.prediction theory . Note that at DL , RMEf50%,H – VŽ .therefore SNRsylog 2PRME f0. Returning to10

Žthe discussion of significant digits and RME Section. Ž .4 , and referring to Eq. 1 , DL is thus approxi-H – V

mately the lowest concentration at which measure-ments have at least zero significant digits.

This suggests the following.

6.1. Alternate definition of the Hubaux–Vos detec-tion limit, DLH – V

The Hubaux–Vos detection Limit, DL , is ap-H – V

proximately equal to the lowest concentration withŽ .RME s 50%, SNR s 0, and equivalently at least

zero significant digits in the measurement.Zero significant digits may initially seem pathetic,

or technically dubious, but it actually is the only log-ical definition of detection: a concentration so lowthat there is no information content in the reportedmeasurement, other than a confident determination ofthe presence or absence of the analyte. ‘Confidence’is defined here as acceptably low average risks offalse detection and false non-detection.

Application of this definition can be seen in Fig.4, where both RME and SNR are plotted as functionsof concentration. Recall from the discussion of RME

Fig. 4. Relationships between RME, SNR and DL for mea-H – VŽsurement of Aroclor 1248 in river water by GC ns34, a s b s

.3% .

that with decreasing concentration, RME increases— slowly at first, and then very rapidly. TheHubaux–Vos detection limit is approximately found

Žat the point where RME reaches 0.5 50%; line C in.Fig. 4 . Simultaneous with the increase in RME, SNR

Ž .necessarily decreases as M decreases. The amountof information in M declines — finally to the pointwhere there is no more information than a confidentdetection — at DL , where SNR equals zero.H – V

There is no assurance of even a fraction of a signifi-Ž .cant digit line D in Fig. 4 .

7. A quantitation limit based on Hubaux–Vos( )QL and relative measurement errorH – V

As discussed in Section 1, there are diverse names,definitions, and procedures to estimate quantitation

Ž .limits QL’s . Concepts and formulas have been pre-sented in the previous sections to allow the practicalapplication of the following definition, a slight modi-fication of what was presented in Section 1:

Ž .The quantitation limit QL is the lowest concen-tration at or above which...

Ž .e ...measurements have at least 1.0 significantŽ .digit at high confidence , and, equivalently, have

limited relative measurement error, RME F 5%.Equivalently, SNRG1, i.e., there is a lower bound onSNR at the QL.

Ž .Recall Eq. 1 :1 1yw ywq1P10 FRME- P102 2

As seen above, this inequality asserts that a measure-Ž .ment with one significant digit ws1 satisfies the

inequality, 0.05FRME-0.5, and to ensure at leastone significant digit, one must have RMEF0.05, soone defines the QL as the lowest concentration, x, forwhich this holds. That is, using prediction limits inthe same manner as Hubaux–Vos, the QL is the so-

Ž . ŽŽ . Ž . .lution x to RME s srb R x t rx s 5%1y a

Ž .taking a s b , if such a solution exists. Equiva-Ž .lently, recall Eq. 2 to get:

wq1GSNRq1)wGSNR)wy1 6Ž .A measurement with one significant digit satisfies theinequality, 1GSNR)0, and to ensure at least onesignificant digit, one must have SNRG1. This sug-gests the formal definition in the next section.


7.1. Definition of the quantitation limit based on( )Hubaux–Vos QLH – V

The Quantitation Limit based on Hubaux–VosŽ .QL is the lowest concentration with RMEsH – V

5%, or equivalently, SNRs1. At this concentration,at least one significant digit can be ensured in themeasurement at a specified level of confidence,

Ž .100 1yayb %.Fig. 5 shows the results from actual PCB mea-

surements, using the same calibration study as inprevious examples. We see that QL f4100 pptH – V

f4 ppb. We would use this estimate to say that at orabove 4 ppb, the first digit of a reported measure-

Žment is correct with 94% confidence after rounding.to one digit . As with any confidence interval, the ac-

tual coverage is unknown, and will vary from cali-bration to calibration, but on average, 94% or moreof the reported measurements will have the correctfirst digit, after rounding.

In this case, the trace-level calibration and predic-tion intervals were extrapolated to a higher concen-tration range. In practice, if a QL is to be calculated,it is advisable to design the calibration to span a widerrange, bracketing the QL. If measurement error wereconstant, we would expect to have QL f 10 PH – V

DL , since RMEs50% at DL , and RMEsH – V H – V

50%r10s5% at QL . This does not quite holdH – V

true due to the ‘flaring’ of the prediction intervalsaway from the calibration line as the concentrationmoves further from the value of x in the calibrationdesign.

Fig. 5. Relationships between RME, SNR and QL for mea-H – VŽsurement of Aroclor 1248 in river water by GC ns34, a s b s

.3% .

8. Conclusions, recommendations, and further ex-tensions

Ž .a The error associated with measurements attrace levels is of great statistical and practical impor-tance. All measurements between 0 and a statisticallysound QL are noisy. The values LC, DL, and QL —if defined and computed correctly — can be used todivide the measurement number line into four inter-vals — each with a practical interpretation and usefor environmental regulation.

Ž . Žb The best reporting format for trace-level i.e.,.noisy measurements is:

Ms4.3 ss0.1; dofs11 ,Ž .or its relative cousin,

Ms4.3 ss2%; dofs11Ž .With these formats, users of M can construct anyneeded intervals. Alternatively, if the type and confi-dence level of the interval can be anticipated, a dis-play such as the following can be used:

Ms4.3"0.1 at 95% confidencesP t s0.045P2.2s0.1 .Ž .11 ,Ž1y0.05r2.

Ž .c For environmental regulation situations wheredetection and quantitation limits are called for, or forreporting formats with known significant digits,DL and QL are recommended.H – V H – V

Ž .d DL is approximately the lowest concen-H – V

tration with:Ž . Ž .i a non-negative number fractions permitted of

significant digits;Ž . Ž .ii relative measurement error RME F50%; and

equivalently,Ž . Ž .iii a signal-to-noise-ratio SNR, defined herein

G0.Ž .DL with an LC threshold is appropriate toH – V

use as a regulatory level for reliably ensuring detec-tion of concentrations above zero.

Ž .e QL is approximately the lowest concentra-H – V

tion with:Ž .i 1.0 significant digit in the reported measure-

Ž .ment at a specified level of confidence ;Ž .ii RMEF5%; and equivalently;Ž .iii SNRG1.QL is appropriate to use as a minimum regu-H – V

latory concentration for direct numerical comparison,or for most computations.


Ž . Žf Where an acceptable value for QL orH – V.DL cannot be obtained, one may:H – V

Ž . Ž . Ži Relax i.e., reduce the confidence level, 100 1.y a y b %. This has the effect of decreasing the

Students’ t multiplier, reducing the limit.Ž .ii Replace single measurements with k replicate

measurements, and report the average. The new lim-its can be calculated mathematically without re-doingthe calibration study, and displayed as functionswhich decrease with increasing k. Alternatively, onecould use a scheme where successive measurementsare compared to a threshold, until a specified number

w xof them exceed that threshold 11 .Ž .iii Re-do the calibration study with a more ap-

propriate range of concentrations, with more concen-trations, or with more data points at the extremes ofthe calibration interval.

Note that the development of DL and QLH – V H – V

is based on straight line regression using ordinaryleast squares. This can be generalized to curvilinearregression and to non-constant response variation,

Žw x.where warranted 5,12–14 . Additionally, this arti-cle has been confined to DL’s and QL’s based onprediction limits about the calibration line, but theapproach can be generalized to statistical tolerance

Žw x.limits 5,10 . The confidence level is arbitrary, butthis is unavoidable, and at least it is controlled, un-like what is found in most assertions of significantdigits.

Acknowledgements

This work has depended on the contributions ofmany parties. P. Ramsey helped develop many of theinitial ideas. K. Eberhardt and R. Mee made substan-tial contributions and corrections. L. Blayden, P.Britton, D. Daniel, C. Davis, C. Dobbs, R. Gibbons,J. Goodman, L. LaFleur, D. Lewis, J. Longbottom, R.Maddalone, J. Marks, B. Nott, B. Novic, R. Poole, J.

Rice, K. Roller, J. Scott, W. Snee, J. Wellman andvarious members of ASTM D19.02 all provided in-put which strengthened the analysis and technical in-terpretation. T. Ceraso made the measurements usedin the example, and R. Bathelt and R. Green partici-pated in the study.

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