An Alternative Minimum Level Definition for Analytical Quantification

Environ. Sci. Technol. 1997, 31, 2071-2077

An Alternative Minimum Level Definition for Analytical Quantification ROBERT D. GIBBONS* University of Illinois at Chicago, 912 South Wood Street, Chicago, Illinois 60614 DAVID E. COLEMAN Alcoa Technical Center, Applied Math and Computer Technology, AMCT (483)-D-10, 100 Technical Drive, Alcoa Center, Pennsylvania 15069 RAYMOND F. MADDALONE TRW, Building 01, Room 2040, 1 Space Park, Redondo Beach, California 90278

This paper presents a new approach to quantitative determination in analytical chemistry. Various historical approaches to determining the quantification limit are reviewed. The Alternative Minimum Level (AML) directly models the relationship between variability and concentration and can incorporate variability from multiple analysts, instruments, and laboratories. In addition, the AML incorporates uncertainty in the calibration function that relates instrument response to true concentration and uncertainty in the standard deviation at the AML. The net result is that the AML is robust to selection of spiking concentration and only requires that we have a reasonably accurate model of the relationship between concentration and standard deviation of instrument response or measured concentration. Computation of the AML and comparative approaches is illustrated in detail using research data provided by the U.S. EPA on ICPMS Method 1638 for cadmium in reagent water.

Introduction Quantitative determination in analytical chemistry is a twostage process. First, we must make the binary decision of whether or not the compound is present in the sample. Second, in the event the compound is present, we must determine if the estimated concentration supports quantitative determination. The first determination is governed by the detection limit (1-9). These investigators have defined the limit of detection to include control of both false positive and false negative rates, varying levels of confidence, uncertainty in the concentration mean and variance and in the calibration function that relates instrument response to concentration, variance components related to multiple instruments and analysts, application to multiple future detection decisions, nonconstant variance, and baseline instrument response. In contrast, a much smaller literature exists on the second stage of the decision process, i.e., quantification. Currie (1) originally described the “determination limit” (LQ) as the concentration (x) at which the relative standard * Corresponding author telephone: 312-413-7755; fax: 312-9962113; e-mail: [email protected].

S0013-936X(96)00899-1 CCC: $14.00

 1997 American Chemical Society

deviation (RSD) is 10% (a signal to noise ratio of 10:1):

LQ ) 10σLQ

(1)

In many ways, this definition remains the most general and useful in that it requires the signal to noise ratio at or above the quantification limit to be small. Whether small corresponds to 10%, 15%, or 20% is somewhat arbitrary; however, the basic idea can be extended to any %RSD, and the choice of RSD can be based on data quality objectives (DQOs). A major limitation of the method is that in Currie’s original formulation, he provided no method for accommodating nonconstant variance, and he assumed that the true population standard deviation σ at LQ was known, which is rarely, if ever, the case. To actually solve for LQ in eq 1 would require (a) a model for the relationship between σ and x, (b) a model for the relationship between true and measured concentration, and (c) an iterative solution of eq 1. Note, however, that in practice it is not possible to solve eq 1 if the rate of change in σ as a function of x is greater than 0.1. A major contribution of the Alternative Minimum Level (AML) described here is generalization of Currie’s LQ to the case of nonconstant variance and providing a solution that will typically yield an %RSD of approximately 10% in most practical applications. Keith (10) defined the limit of quantitation (LOQ) as the concentration that is 10 standard deviation units above the average blank response:

LOQ ) xj 0 + 10s0

(2)

where xj 0 is the average blank measurement and s0 is the standard deviation of the blank measurements. As written, the model makes no assumptions regarding the relationship between variability and concentration, since both xj and s are based on blank samples. In many cases, however, instruments censor measured concentrations less than zero (and even instrument responses such as peak areas below a peak area rejection level), therefore, it may not even be possible to obtain a valid estimate of s0. Unlike Currie’s LQ, the RSD at the LOQ will typically be much higher than 10%, making quantitative determination doubtful at best. The U.S. EPA (11) defined the practical quantitation limit (PQL) as “the lowest level achievable by good laboratories within specified limits during routine laboratory operating conditions.” This definition has been operationally defined by the U.S. EPA (11) as 5 or 10 times the method detection limit, or the concentration at which 75% of the laboratories in an interlaboratory study report concentrations (20% of the true value, or the concentration at which 75% of the laboratories report concentrations within (40% of the true value. The first operational definition is arbitrary and depends completely on the validity of the corresponding method detection limit, about which serious questions have been raised (5, 9). The second and third operational definitions are somewhat better, however, the interlaboratory studies are often done at a single concentration (maximum contaminant level, MCL) in experienced government laboratories that “knew they were being tested with standard samples in distilled water without matrix interferences” (11). Furthermore, it is unclear whether all measurements made by a single laboratory must be within (20% or if this criterion can be satisfied by just one or two measurements. In practice, the PQLs published by the U.S. EPA have taken on fixed values that were arrived at by consensus (e.g., the PQLs for most volatile organic priority pollutant compounds are either 5 or 10 µg/L). Gibbons and co-workers (12) have provided a more statistically rigorous estimator of LQ that accommodates the

VOL. 31, NO. 7, 1997 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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effects of nonconstant variance, uncertainty in the parameters of the recovery curve and uncertainty in the sample-based estimator of the true population variance. The method is based on a variance stabilizing square root transformation of both true and measured concentration (5). The method performs well when the transformation brings about homoscedasticity; however, it can yield biased results when the data remain heteroscedastic following transformation. More recently, the U.S. EPA (13) has defined the Interim Minimum Level (ML) as

ML ) 3.18(MDL) ) 3.18(3.14)sx ) 10sx

(3)

where sx is the standard deviation of seven replicate measurements spiked at true concentration x. The constant 3.14 is Student’s t for R ) 0.01 on six degrees of freedom. In deriving the Method Detection Limit (MDL), Glaser and coworkers (4) assume that variability is a linear function of concentration, such that for a limited number of analyses

sx ∼ a0 + a1x

(4)

where sx is the standard deviation of n replicate analyses at true concentration x and a0 and a1 are the intercept and the slope of the linear regression of standard deviation on concentration. Equation 4 reveals that the result of eq 3 (the MDL and hence the ML) is a direct function of the choice of spiking concentration x. Anticipating this problem to some degree, the U.S. EPA provides guidance requiring the ratio of spiking concentration to estimated MDL to be no greater than 5:1. If the ratio is greater than 5:1, the spiking concentration is iteratively reduced until the 5:1 criterion is achieved. While this guidance certainly reduces gross discrepancies between the MDL and the actual spike level, it can still lead to considerable uncertainty in the ML as illustrated in the example presented in a following section. In addition, in order to meet the 5:1 criterion, several concentrations will generally need to be tested, leading to the same data required to compute the AML. In addition, as described by the U.S. EPA the ML is not based on a permittee’s MDL but rather on a single MDL developed by a single EPA laboratory under ideal conditions. This further reduces generalizability of the ML to application under routine conditions in which analysts are blind to the true concentration, and any one sample can be analyzed by different analysts, instruments, and even laboratories.

σx ) xa0 + a1(x)2

The exponential model can be fitted by substituting sx for σx and by using nonlinear least-squares (e.g., Gauss-Newton) or using unweighted least squares regression of natural log transformed observed standard deviation on actual concentration (15). The Rocke and Lorenzato model can be fitted either by maximum likelihood estimation (MLE), nonlinear least-squares (NLS), or weighted least-squares (WLS) regression (i.e., fit a model to s2 using the inverse variance at concentration x as a weight). The MLE solution described by Rocke and Lorenzato is complicated in that it involves numerical integration; however, both WLS and NLS are straightforward and have been successfully applied by the authors to numerous problems. The theoretical basis for the Rocke and Lorenzato (R&L) model is that it is based on a combination of additive and multiplicative response variation, such that variance is nearly constant at low levels (a0) but thereafter exhibits a linear relation with concentration (i.e., (a1)1/2x) providing a function that looks like a “hockey stick”. The exponential model provides a smoothed approximation to the R&L model that can also accommodate nonlinearity in the function at higher concentrations. An added advantage of the WLS estimation procedure is that it gives greater weight to the often more precise results at lower concentrations; therefore, the inclusion of higher concentrations even orders of magnitude above the true AML will have little or no effect on the estimated AML. Details of the WLS regression of s on x are identical to those for the WLS regression of measured concentration on true concentration (i.e., y ) s2x and x ) x2) described in the following step. 2. Compute the WLS regression of measured concentration or instrument response (y ) on true concentration (x ) for the linear model (16, 17):

yˆi ) b0w + b1wxi

n

∑[(x - xj )y /k ] i

b1w )

w

i

i

i)1

(8)

n

∑[(x - xj ) /k ] 2

i

w

i

i)1

n

The fundamental problem with the ML is not its definition (10sx ) but the fact that the sample standard deviation sx is a moving target and strongly depends on the choice of spiking concentration x. Similarly, the iterative solution of equation 1 is problematic because a 10% RSD may not be achievable. A reasonable alternative to both approaches is to compute the standard deviation at the lowest concentration that is differentiable from zero (i.e., Currie’s “Critical Level” LC) and use that standard deviation in computing the AML. In this way, the spiking concentration is no longer arbitrary, and the effect of nonconstant variance is incorporated up to the lowest nonzero concentration. Of course, this will not guarantee a 10% RSD at the AML; however, we can compute the actual %RSD at the AML by modeling association between s and x. The algorithm is as follows: 1. Using calibration data model the relationship between σ and x. Two excellent choices are the exponential model

σx ) a0ea1(x)

(5)

or the model of Rocke and Lorenzato (14)

9

(7)

where

∑[y /k ]

The AML

2072

(6)

ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 31, NO. 7, 1997

i

yjw )

i

i)1

(9)

n

∑[1/k ] i

i)1 n

∑[x /k ] i

xj w )

i

i)1

(10)

n

∑[1/k ] i

i)1

and the weight ki ) s2xi is the sample variance for sample i. The weighted residual variance is n

s2w )

∑[(y - yˆ i

wi)

2

/ki]/(n - 2)

i)1

and the estimated variance for a predicted value yˆwj is

(11)

TABLE 1. Method 1638 ICPMS Data from the U.S. EPA for Cadmium at Mass 114 (in ppt) replicate spike

1

2

3

4

5

6

7

0 10 20 50 100 200 500 1000 2000 5000 10000 25000

-53.900 -38.030 -26.000 -7.310 61.390 156.070 461.580 943.050 2021.740 4856.450 10023.110 25458.900

-48.990 -36.970 -36.820 7.580 61.030 155.110 459.970 963.690 1914.770 5048.990 9766.380 25063.110

-47.990 -37.750 -29.200 3.710 50.030 157.460 442.210 955.140 1914.770 4931.330 9905.440 25726.330

-49.500 -35.800 -26.220 1.170 51.760 145.370 456.660 938.130 1989.650 4963.420 9830.560 25384.020

-51.290 -36.130 -18.460 1.720 51.870 157.670 479.550 962.620 1957.550 4760.170 9937.530 25469.600

-41.520 -40.620 -35.800 -1.840 49.120 95.160 447.350 968.940 1978.950 5006.200 9980.320 24998.930

-47.820 -36.680 -29.200 8.150 52.340 156.710 433.440 986.590 1946.860 4760.170 9809.170 24731.500

[

]

FIGURE 1. Recovery curve with 99% WLS prediction interval for cadmium in ppt using ICPMS Method 1638 in reagent water.

V(yˆwj) ) s2w kj +

(xj - xj w)2

1 + n

n

∑(1/k ) ∑(x - xj ) /k 2

i

i)1

i

w

i)1

be obtained by iteratively computing

(12)

i

yD) yC + K0.95,0.99sLD ) yC + K0.95,0.99a0e(a1(yD-b0w)/b1w) (16) and LD is the corresponding true concentration

3. Compute yC as the upper 95% confidence 99% coverage tolerance limit for measured concentrations when the true concentration is zero

LD ) (yD - b0w)/b1w

(17)

which was originally suggested by Scott and co-workers (18). In either case, LC is the corresponding critical level, i.e.,

If the true concentration is equal to LC then using the critical level LC as a decision rule will yield a false negative rate of 50% (i.e., in 50% of the cases we will fail to detect the constituent when its true concentration is equal to the critical level). In contrast, at the detection limit (LD), the probability of not detecting the constituent when using the critical level is only 1%. As an analogy, the critical level for the federal speed limit is 55 mph; however, police will rarely identify an exceedance until a driver exceeds 60 mph, and their confidence that the true speed (not just the measured speed) has exceeded the limit is high. For the critical level, emphasis is on measured concentration whereas for the detection limit emphasis is on true concentration. 4. Compute the standard deviation at the LC from Step 1 (i.e., sLC), for example, using the exponential model:

LC ) (yC - b0w)/b1w

sLC ) a0ea1(LC)

yC ) K0.95,0.99s0 + b0w ) K0.95,0.99a0 + b0w

(13)

where K0.95,0.99 is the 95% confidence 99% coverage one-sided normal tolerance limit factor for n observations, where in this case, n is the total number of measurements (8). In many cases, however, data at x ) 0 are unavailable (e.g., there is no measurable instrument response). As a computational alternative, we can iteratively compute yC as

yC ) K0.95,0.99sLC + b0w ) K0.95,0.99a0e(a1(yC-b0w)/b1w) + b0w (14)

(15)

Note that although it is not required for computing the AML, a Currie (1) type estimate of the detection limit (LD) can also

(18)

5. Compute the measured concentration at 10 times the standard deviation at the LC :


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TABLE 2. Estimates of the U.S. EPA MDL and ML for Cd at Mass 114 and Spiking Concentration (in ppt)a spike

SD

MDL

ML

spike/MDL

0 10 20 50 100 200 500 1000 2000 5000 10000 25000

3.81 1.62 6.26 5.38 5.10 22.92 15.04 16.31 39.41 115.02 94.56 342.19

11.98 5.09 19.65 16.91 16.01 71.96 47.24 51.21 123.74 361.17 296.92 1074.47

38.14 16.21 62.59 53.85 50.97 229.18 150.43 163.10 394.07 1150.21 945.60 3421.87

0.00 1.97 1.02 2.96 6.25 2.78 10.59 19.53 16.16 13.84 33.68 23.27

a The entry in boldface is the first spiking concentration at which the EPA criterion of a 5:1 ratio of spiking concentration to MDL is met. Data provided by U.S. EPA as reported by the U.S. EPA on April 26, 1996.

yQ ) 10sLC + b0w

(19)

Note that b0w is added to 10sLC to convert yQ into a measured concentration instead of a response variation. 6. The AML is then computed as the upper 99% prediction limit

xQ + (t/b1w)xV(yQ)

(20)

where xQ ) (yQ - b0w)/b1w. The value t is the upper 99th percentile of Student’s t -distribution on n - 2 - p degrees of freedom, where p is the number of unknown parameters in the standard deviation model. Alternatively when n is large (n > 25), the AML can be approximated by

AML ∼ xQ + tsyQ/b1w

(21)

since the two rightmost terms under the radical in eq 12 go to zero and the residual variance is generally close to one.

Illustration To illustrate application of the AML and comparison to single concentration-based estimates, Method 1638 ICPMS data from the U.S. EPA for cadmium at mass 114 were analyzed. The spiking concentrations ranged from 0 to 25 000 ppt. These data were obtained under the highly controlled nonblind research conditions in reagent-grade water and may therefore not be representative of routine practice; however, they illustrate the remarkable low level capabilities of state of the art analytical technology. All raw data are displayed in Table 1. Note that the data in Table 1 exhibit a considerable downward bias (i.e., measured concentrations are less than true spiking concentrations, particularly at the lower concentrations). Spiking concentrations, observed standard deviations, MDLs, MLs, and spike to MDL ratios are displayed in Table 2. Inspection of Table 2 reveals that using the U.S. EPA’s criterion of a spiking concentration to MDL ratio of 5:1 or less, we could have selected a spiking concentration of 10, 20, 50, or 200 ppt and obtained ML estimates ranging anywhere from 16 to 229 ppt. Alternatively, we could have guessed low and obtained an ML of 16 ppt. Interestingly, the 100 ppt spike did not meet the U.S. EPA’s 5:1 criterion but the 200 ppt spike did. Since, in general, following the U.S. EPA’s guidance we would begin at higher concentrations and iteratively decrease the spiking concentration until the 5:1 criterion was met, we would have obtained an ML of 229 ppt. As a comparison, the LOQ can also be computed from these data since uncensored measured concentrations in blanks (i.e., x ) 0) are available. The LOQ is computed as

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FIGURE 2. Relationship between concentration and variability for cadmium data (a) exponential model for all data; (b) exponential model with outlier removed; (c) Rocke and Lorenzato model with outlier removed.

LOQ ) xj 0 + 10s0 ) -48.72 + 10(3.81) ) -10.62 ppt (22) The LOQ is, in fact, negative (i.e., -10.62 ppt) since there appears to be significant bias (i.e., the mean blank measured concentration is -48.72 ppt). The calibration-based methods will not produce this type of degenerate result since the bias is directly incorporated in the intercept of the recovery curve. Figure 1 displays the actual data for concentrations in the range of 0-200 ppt. As is clear both from Table 2 and Figure 1, variability is increasing with concentration even at these low levels. Figure 2a displays the relationship between variability and concentration for concentrations in the range of 0-1000 ppt for all available data. The relationship is reasonably described by an exponential model of the form

sx ) 4.804e0.0016x

(23)

(note that for the range of 0-200 ppt the model is sx )

FIGURE 3. Computation of the AML for the cadmium example data. 2.95e0.0095x). The exponential regression model provided better fit to these data than the R&L model in terms of minimizing mean square error (average squared deviation between observed and predicted values of s). Figure 3 displays full computational details of the analysis and reveals that the estimated AML is 42 ppt with %RSD of 10.47%. Note that if concentrations in the range of 0 to 1000 ppt are used in computing the AML, the estimated AML is 41 ppt. Even using the entire range of concentrations from 0-25000 ppt, the estimated AML is 46 ppt. Inspection of the data in Table 1 reveals an outlier at the 200 ppt concentration. Recomputing the AML without this value yielded a virtually identical AML of 45 ppt with %RSD of 8.6%; however, the fit of the exponential model to the observed standard deviations is improved (see Figure 2b). Figure 2c displays the fit of the R&L model to the same data. Comparison of Figure 2, panels b and c, reveals very similar functions; however, the exponential model is curvilinear throughout the entire concentration range. The estimated AML based on the R&L model fitted to these data was 41 ppt with all data and 39 ppt with the outlier removed. These data reveal that calibration-based methods do not necessarily lead to higher estimates of detection and quan-

TABLE 3. EPRI Interlaboratory GFAAS Data (in µg/L) for Three Compounds in Three Matrices LC

LD

no. of labs

total n

Chromium 11.89 28.80 13.35 14.02 30.94 22.46 9.39 22.63 12.37

21 23 19

86 92 76

AML

%RSD

reagent river ash pond

5.45 5.07 4.45


6.21 7.28 15.89

13.67 23.89 42.76

Nickel 33.13 46.33 96.55

14.02 25.88 30.80

21 23 21

91 95 87


10.45 14.52 10.33

27.47 36.71 33.05

Lead 62.74 85.70 64.52

28.82 25.59 25.39

21 22 16

88 91 90

tification limits. In addition, unlike single concentrationbased methods, calibration-based methods are robust to the inclusion of spiking concentrations that are even orders of magnitude higher than the true quantification limit as long as there is adequate coverage of the low end of the recovery curve. Whether the data in this example, which were obtained


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under ideal research laboratory conditions, generalize to routine laboratory practice is an open question.

Interlaboratory Examples In many applications (e.g., NPDES monitoring) split samples are obtained, and any one of a number of laboratories may be called upon to analyze the data. In this case, the betweenlaboratory component of variability must be included in the computation of the AML. The simplest solution is to include all of the data from all laboratories in computing the AML. This approximate approach is biased in that ignoring the variance components structure (i.e., measurements nested within laboratories) will underestimate the true total variance (8) which is correctly computed by estimating and then summing the individual variance components (e.g., between and within laboratory variance components). Unfortunately, in the nonlinear case, variance component models have not been well studied. Even in the linear case, if the number of measurements varies across laboratories, maximum likelihood methods are required, which lead to far more complex solutions than presented here. As such, the approach described here will lead to small underestimates of the true AMLs. The Electric Power Research Institute (EPRI) has funded extensive interlaboratory validations studies (19) of selected elements in aqueous discharge matrices for graphite furnace atomic absorption spectroscopy (As, Cd Cr, Cu, Ni, Pb, Se), flame AAS (Fe, Zn), cold vapor AAS (Hg), and inductively coupled plasma-atomic emission spectroscopy (Al, Ba, Be, B, Cd, Cr, Cu, Fe, Pb, Mn, Mo, Ni, V, Zn). The matrices included reagent-grade water, river water, ash pond overflow, seawater, and treated metal chemical cleaning wastes. Approximately 12-30 laboratories participated in each study, depending on the method and matrix. The data were processed for outliers following the ASTM D-2777 protocols, and the outlier-free data were used to compute an AML in reagent-grade water, river water, and ash pond overflow. Spiking concentrations ranged from 1 to 88 µg/L for chromium, from 1 to 63 µg/L for nickel, and 2 µg/L to 87 µg/L for lead. Results are reported in Table 3. Table 3 reveals that LC, LD, and the AML are generally lowest in reagent-grade water, higher in river water, and highest in ash pond overflow, reflecting increased interference in the two real world matrices. The RSDs (approximately 20%) are representative of interlaboratory precision at concentrations typically found in utility matrices.

Computing Currie’s Determination Limit An alternative that departs somewhat from the AML is to estimate the true concentration that leads to a measured %RSD of 10% or 20% or whatever is achievable. Requiring 10% will not always lead to a solution since the rate of change in standard deviation relative to concentration may be greater than 0.1. The solution for the case of nonconstant variance (16) involves iterative solution of

yQ )

1 V(yˆ ) rx Q

(25)

When applied to the cadmium data, application of eqs 24 and 25 yield LQ ) 44 ppt for an RSD of 10%. This result is quite similar to the AML of 42 ppt with %RSD of 10.47%. The LQ is slightly larger than the AML because its %RSD is slightly lower.

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Discussion A major limitation of single concentration-based quantification limit estimators is that very different results are obtained for different spiking concentrations. In contrast, the AML fixes the spiking concentration at the lowest nonzero concentration (the LC) and incorporates uncertainty in the calibration function that relates instrument response to true concentration and uncertainty in the standard deviation at the AML. The net result is that while both the ML and AML have the same definition (i.e., the concentration at which the instrument response is 10 times an estimate of the standard deviation s), the use of sx by the U.S. EPA can yield a variety of different results depending on x (i.e., the concentration at which the samples are spiked) whereas the AML is robust to the range of spiking concentration and only requires that we have a reasonably accurate model of the relationship between concentration and standard deviation of instrument response or measured concentration. Of course, if low-level concentrations are not included in the calibration study, the computed AML will generally yield an overestimate of the true AML. For example, if the true AML is 1 ppb and the lowest concentration in the calibration curve is 1 ppm, there will be no available information regarding variability at the true AML, and any extrapolated value will likely overestimate the true value. As illustrated here, however, including spiking concentrations 3 orders of magnitude larger than the AML (i.e., 0-25 000 ppt) produced virtually identical results to only including concentrations in the vicinity of the AML (i.e., 0-200 ppt). The reader should note that neither the ML or the AML are the ideal statistical estimates of the quantification level LQ because neither guarantees a specified signal to noise ratio (e.g., 10%). As shown in the previous section, direct solution for the LQ with specified RSD is possible; however, a 10% or even 20% RSD may not always be achievable for a given constituent and analytical method (8, 12, 19, 20). Although the AML and iterative solution for LQ are computationally more complex than single concentration-based methods such as the LOQ, PQL, and ML, the computer software used in preparing Figures 1-3 is now available to perform these computations directly from the measured concentrations and/or instrument responses (21, 22).

Acknowledgments (24)

where r represents the relative standard deviation (r ) 0.1 for a 10% RSD). The estimate, referred to as LQ based on Currie’s “determination limit” is then

LQ ) (yQ - b0w)/b1w

Note that this estimator is, in fact, more statistically and conceptually rigorous than the AML in that it uses an estimate of σQ to estimate LQ (i.e., the standard deviation at the determination limit) in contrast to the AML, which uses an estimate of σLC (i.e., the standard deviation at the critical level). The primary advantage of the AML over the determination limit is that, in many cases, some analytical methods in some environmental matrices cannot attain the ideal 10% RSD for quantification. The AML will provide a solution in all cases and provide an asymptotic limit on the minimum achievable RSD.


We would like to acknowledge the very helpful comments and discussions that we have had with Steve Koorse, Jim Rice, Nancy Grams, Babu Nott, Kenneth Roller, Ron Filadelfo, Judy Scott, and James Stine, which have helped formulate these ideas. We would also like to thank Dr. Anna Liao and Dr. Deborah Hockman of Waste Management, Inc., and to thank Michael Zorn and Dr. Bill Sonzogni of the University of Wisconsin for providing several data sets and helpful comments through the course of our research.

Literature Cited (1) Currie, L. A. Anal. Chem. 1968, 40, 586-593. (2) Currie, L. A. Pure Appl. Chem. 1995, 67, 1699-1723.

(3) Hubaux, A.; Vos, G. Anal. Chem. 1970, 42, 849-855. (4) Glaser, J. A.; Foerst, D. L.; McKee, G. D.; Quane, S. A.; Budde, W. L. Environ. Sci. Technol. 1981, 15, 1426-1435. (5) Clayton, C. A.; Hines, J. W.; Elkins, P. D. Anal. Chem. 1987, 59, 2506-2514. (6) Lambert, D.; Peterson, B.; Terpenning, I. J.Am. Stat. Assoc. 1991, 86, 266-277. (7) Gibbons, R. D.; Jarke, F. H.; Stoub, K. P. In Waste Testing and Quality Assurance; ASTM STP 1075; Friedman, D., Ed.; American Society for Testing and Materials: Philadelphia, PA, 1991; pp 377-390. (8) Gibbons, R. D. Statistical Methods for Groundwater Monitoring; Wiley, New York, 1994. (9) Gibbons, R. D. J. Environ. Ecol. Stat. 1995, 2, 125-167. (10) Keith, L. H. Anal. Chem. 1983, 55, 2210-2218. (11) U.S. EPA. Fed. Regist. 1985, 50 (Nov 13), 46906. (12) Gibbons, R. D.; Grams N. E.; Jarke F. H.; Stoub K. P. Chemom. Intell. Lab. Syst. 1992, 12, 225-235. (13) U.S. EPA. Guidance on Evaluation, Resolution, and Documentation of Analytical Problems Associated with Compliance Monitoring; EPA 821-B-93-001; U.S. Government Printing Office: Washington, DC, June, 1993. (14) Rocke, D. M.; Lorenzato, S. Technometrics 1995, 37, 176-184. (15) Snedecor, G. W.; Cochran, W. G. Statistical Methods; Iowa State University Press: Ames, IA, 1980. (16) Oppenheimer, L.; Capizi, T. P.; Weppelman, R. M.; Mehta, H. Anal. Chem. 1983, 55, 638-643.

(17) Caulcutt, R.; Boddy, R. Statistics for Analytical Chemists; Chapman and Hall: New York, 1983. (18) Scott, J. W.; Whiddon, N. T.; Maddalone, R. F. Compliance Monitoring Detection and Quantitation Levels for Utility Aqueous Discharges; EPRI Final Report TR-103205; EPRI: Palo Alto, 1993. (19) Maddalone, R. F.; Scott, J. W.; Frank, J. Round-Robin Study of Methods for Trace Metal Analysis, Vols. 1-3; EPRI CS-5910; EPRI: Palo Alto, August, 1988. (20) Coleman, D. Proceedings of International Conference on Environmetrics and Chemometrics, Las Vegas, NV, 1995; to be published in Chemom. Intell. Lab. Syst. (21) Gibbons, R. D. AML: A computer program for computing the Alternative Minimum Level; Scientific Software International: Chicago, 1996. (22) Gibbons, R. D. DETECT: A computer program for computing detection and quantification limits; Scientific Software International; Chicago, 1996.

Received for review October 21, 1996. Revised manuscript received February 25, 1997. Accepted March 5, 1997.X ES960899D X

Abstract published in Advance ACS Abstracts, May 15, 1997.


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An Alternative Minimum Level Definition for Analytical Quantification

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