Environ. Sci. Technol. 1998, 32, 2349-2353
Response to Comment on “An Alternative Minimum Level Definition for Analytical Quantification” SIR: We appreciate the opportunity to respond to the criticisms raised by Kahn et al. of our paper published in the July 1997 issue of this Journal and subsequent comments (1-3). Although we take considerable exception to the content and tone of their letter, we are pleased to respond point by point to their comments and hope the dialogue will help raise general awareness and understanding of the underlying scientific principles. Development of the AML. Kahn et al. provide a list of difficulties they had with the definitions and components of the AML. We apologize for any confusion caused by implicit definitions or sloppy terminology. None of these problems threaten the statistical integrity of the AML. Below are specific responses to concerns raised. Other concerns are addressed elsewhere in this response. (a) Equation 7 states
yˆi ) b0w + b1w,xi Following the standard convention, the value y is the predicted instrument response for any true concentration, xi, where variable y is used to indicate the instrument response and variable x is used to indicate the true concentration. There is nothing contradictory in the use of yˆ in eqs 7, 11, and 12; for consistency, we simply should have used the subscript, w, in eq 7, as we did in eqs 11 and 12:
yˆwi ) b0w + b1wxi (b) It was our oversight that we did not define b0w, the estimate of the intercept, computed using WLS. The mathematical definition can be found in any textbook covering WLS. It is
b0w ) yjw - b1wxj (c) Kahn et al. are apparently confused regarding the use of the index i. Rather than to tediously elaborate how the index is used in each case, we hope it will suffice to state that i refers to each measurement, yi, and that the ki are all the same for all replicates at a single spike concentration. With respect to the appropriateness of the t-distribution used in computing the AML, Kahn et al. are confused regarding the fundamental difference between the WLS estimated linear model used in our paper
ywi ) b0w + b1wxi + ewi and the maximum likelihood estimated (MLE) nonlinear model given by Rocke and Lorenzato (4), i.e.
yi ) b0 + b1xieη +ei In the Rocke and Lorenzato model, η represents proportional error at higher true concentrations, and ei represents additive error at low concentrations. Kahn et al. are correct that, for the Rocke-Lorenzato model described above, errors at larger concentrations are log-normally distributed; however, we are fitting a linear model and using an approximation to the Rocke and Lorenzato model, i.e. S0013-936X(98)02003-3 CCC: $15.00 Published on Web 06/20/1998
1998 American Chemical Society
σx ) xa0 + a1(x)2 to obtain the estimated weights used in WLS estimation of the prediction bounds. As such, we are correct in assuming that, at concentration x, the deviations from the fitted regression line are normally distributed with mean 0 and standard deviation σx as given above. Use of the t-distribution is therefore completely justified for our linear model whereas it is not for the nonlinear model of Rocke and Lorenzato. The reader should note that direct estimation of the parameters of the nonlinear Rocke-Lorenzato model is far more complex than the computations involved in computing the AML. Comparison of the two strategies, however, is an interesting topic for future research. We are puzzled that Kahn et al. find our two models for the weights (i.e., exponential or Rocke and Lorenzato) so objectionable yet they want us to use them to obtain estimated weights instead of using the actual observed values. There is, of course, a long tradition in the statistical literature of doing both, and each has its strength and weakness. It should be noted, however, that the difference in AML values between using observed versus estimated weights was 42 vs 43 ng/L for 114Cd and 8 vs 8 ng/L for 111Cd. Kahn et al. state that no motivation (in the sense of a real world application) has been provided for the one-sided tolerance limit (used to calculate the critical level, LC). We welcome the opportunity to provide such motivation. There are three types of statistical limits (also called “bounds” or, if two-sided, called “intervals”) (5): confidence limit, prediction limit, and tolerance limit. The statistical tolerance limit is used when one wishes to make statistical inferences about a quantile of an arbitrarily large number of future measurements or values from a population distribution. A quantile is defined as a percentage point of a distribution. For example, the median is the 50% quantile (half of the distribution is below the median), and the 99% quantile is the level that exceeds 99% of a distribution. For example, for a standard normal distribution, the 99% quantile (or “critical value”) can be found in any elementary statistics textbook to be 2.33. Perfect standard normal deviates with mean known to be 0 and standard deviation known to be 1 will exceed the value of 2.33 only 1% of the time, and 99% of the time will fall below 2.33. Unfortunately, one does not typically know the mean nor the standard deviation of a distribution of interest. Both values must be estimated and are always done so with uncertainty. Suppose 20 measurements are available to estimate the mean, standard deviation, and a level (L), and one wants to choose L so that with 95% certainty L will exceed 99% of all future measurements. Then that level must be set to the statistical tolerance limit, L ) xj + 3.30S ) 3.30 for the standard normal distribution. A higher number is needed (3.30, as opposed to 2.33) due to the uncertainty in the data. L would have to be infinite to achieve 100% confidence of exceeding any percentage of the population or to have any confidence of exceeding 100% of the population. As far as motivation or appropriateness, Hahn and Meeker state in their definitive textbook on statistical intervals (5): “Practical applications often require the construction of one-sided tolerance bounds. For example, in response to a request by a regulatory agency, a manufacturer has to make a statement concerning the maximum noise that ... is met by ... 95% of a particular model of a jet engine. The statement is to be based upon measurements from a random sample of 10 engines, and is to be made with 90% confidence. In VOL. 32, NO. 15, 1998 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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this case, the manufacturer desires a one-sided upper tolerance bound that will be met (i.e., not exceeded) by at least 95% of the population of jet engines, based on the previous test results. A one-sided upper tolerance bound is appropriate here because the regulatory agency is concerned principally with how noisy, and not how quiet, the engines might be. A one-sided upper 95% tolerance bound is the same as an upper 95% confidence bound on the 99th percentile of the population distribution.” If we substitute “concentration” in place of “noise” and “process” in place of “jet engine”, the situations are nearly identical. Application of the MDL Procedure. Kahn et al. objected strongly to our discussion of the MDL procedure because (a) we used 114Cd rather than 111Cd in our original paper and (b) we did not reference any initial estimate of the MDL. (a) The purpose of our paper was to illustrate computation of the AML using low-level data. Our intentions were not to set national standards for the analysis of Cd. As such, use of 114Cd may have been unfortunate for regulatory purposes, but it was not a critical flaw as an illustration in a paper largely devoted to presenting a statistical technique. Nevertheless, in subsequent discussion of the paper (3), we provided a parallel analysis of 111Cd and found quite similar results. The confusion regarding the blank correction is unfortunate and we apologize for any role we may have played in creating this misunderstanding. The careful review and thoughtful comments of Kimbrough (2) helped make this an instructive error. Please note, however, that contrary to Kahn et al.’s understanding, the AML (6) and DETECT (7) computer programs can analyze either raw instrument responses or measured concentrations. (b) In terms of initial estimates of the MDL, the U.S. EPA suggests four general alternatives (8): (1) the concentration that provides a signal to noise ratio between 2.5 and 5, (2) three times the standard deviation of a blank, (3) the point at which the slope changes in the calibration curve, or (4) instrument limitations. The MDL is then obtained by spiking samples at one to five times this initial estimate. Use of these four very different approaches can yield a wide variety of initial estimates of the MDL. As such, the purpose of our illustration was to point out how differences in starting point can lead to large differences in the computed MDL. Of course for many analytes, instruments automatically censor negative concentrations that would be found in approximately onehalf of all blank samples. In this case, the estimate of the standard deviation of blank measurements is meaningless and would yield a dramatic underestimate of the initial MDL. Our example also clearly illustrated that a wide variety of different MDLs satisfied the U.S. EPA’s final acceptance criterion for an MDL, i.e., a spike to MDL ratio of 5 or less. To make it clear that this problem is not restricted to 114Cd, Table 1 presents results for individual MDL and ML calculations for 107Ag, 111Cd, 63Cu, 60Ni, 123Sb, 82Se, 205Tl, and 66Zn. Clearly, there are enormous ranges of candidate MDLs and MLs that meet the EPA’s acceptance criterion of a spike to MDL ratio of 5:1 or less. As noted by Kahn et al., this does not mean that these candidate MDLs would have necessarily been obtained based on one of the four methods of selecting the initial spiking concentration. What Table 1 illustrates is what range of MDLs would be acceptable if the initial estimate of the MDL produced a computed MDL in that range. Perhaps most instructive are the results for 82Se, which produce acceptable candidate MDLs in the range of 0-2000 ng/L. With respect to 82Se, if we had estimated the initial MDL based on three times the standard deviation of a blank, we would have obtained an initial estimate of 151 ng/L (see Table 1). This would lead to acceptable spiking concentrations in the range of 151-755 ng/L. Resulting MDLs for these spiking concentrations range from 258 to 300 ng/L and 2350
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TABLE 1. Estimates of Candidate U.S. EPA MDLs and MLs for Each Compound and Spiking Concentration (in ng/L)a constituent
spike
SD
MDL
ML
spike/MDL
Ag-107
0 10 20 50 100 200 500 1000 2000 0 10 20 50 100 200 500 1000 2000 0 10 20 50 100 200 500 1000 2000 0 10 20 50 100 200 500 1000 2000 0 10 20 50 100 200 500 1000 2000 0 10 20 50 100 200 500 1000 2000 0 10 20 50 100 200 500 1000 2000 0 10 20 50 100 200 500 1000 2000
0.47 0.67 0.66 1.08 2.23 15.32 9.05 21.43 25.73 0.49 0.58 2.25 2.50 3.35 16.19 10.20 15.99 25.36 5.69 1.89 2.03 2.68 3.72 3.63 10.50 13.43 46.27 4.17 1.55 4.40 5.93 4.40 8.39 12.96 16.71 68.14 0.37 1.59 1.02 1.37 2.20 13.73 6.30 12.70 21.01 50.12 75.24 58.10 31.87 64.11 81.99 95.30 75.67 146.14 1.80 0.89 0.97 1.01 4.19 6.21 11.33 19.52 46.91 29.36 3.15 6.54 14.69 4.70 9.02 12.82 17.75 31.75
1.47 2.09 2.06 3.40 7.00 48.10 28.43 67.29 80.81 1.53 1.81 7.07 7.86 10.52 50.84 32.01 50.20 79.63 17.87 5.93 6.39 8.43 11.67 11.38 32.98 42.18 145.27 13.08 4.87 13.83 18.61 13.83 26.34 40.70 52.46 213.97 1.15 5.00 3.22 4.29 6.91 43.10 19.77 39.87 65.96 157.37 236.25 182.45 100.06 201.30 257.45 299.25 237.61 458.88 5.64 2.79 3.04 3.16 13.17 19.49 35.58 61.28 147.29 92.18 9.91 20.54 46.11 14.76 28.33 40.26 55.73 99.69
4.69 6.66 6.55 10.81 22.29 153.19 90.54 214.30 257.35 4.87 5.75 22.51 25.05 33.51 161.90 101.95 159.87 253.58 56.91 18.90 20.34 26.83 37.17 36.26 105.04 134.32 462.65 41.67 15.50 44.05 59.27 44.05 83.89 129.63 167.08 681.43 3.65 15.93 10.24 13.66 22.00 137.27 62.97 126.99 210.08 501.19 752.39 581.04 318.65 641.09 819.89 953.04 756.73 1461.42 17.95 8.89 9.67 10.05 41.94 62.08 113.32 195.17 469.08 293.58 31.55 65.43 146.86 46.99 90.23 128.21 177.48 317.49
0.00 4.78 9.72 14.73 14.28 4.16 17.59 14.86 24.75 0.00 5.54 2.83 6.36 9.50 3.93 15.62 19.92 25.12 0.00 1.69 3.13 5.93 8.57 17.57 15.16 23.71 13.77 0.00 2.05 1.45 2.69 7.23 7.59 12.28 19.06 9.35 0.00 2.00 6.22 11.66 14.48 4.64 25.29 25.08 30.32 0.00 0.04 0.11 0.50 0.50 0.78 1.67 4.21 4.36 0.00 3.58 6.59 15.84 7.59 10.26 14.05 16.32 13.58 0.00 1.01 0.97 1.08 6.78 7.06 12.42 17.94 20.06
Cd-111
Cu-63
Ni-60
Sb-123
Se-82
Tl-205
Zn-66
a Entry in boldface is the highest spiking concentration at which EPA criterion of a 5:1 ratio of spiking concentration to MDL is met. This does not mean that it would necessarily be the initial spiking concentration selected using one of the four methods described by EPA.
FIGURE 1. (a) Kahn’s diagnostic for exponential model for Cd111, supplemented with pointwise 99% confidence intervals, plus candidate straight line. (b) Kahn’s diagnostic for R&L model for Cd111, supplemented with pointwise 99% confidence intervals, plus candidate straight line. MLs of 820 to 954 ng/L (see Table 1). By contrast, had we used the published MDL for 82Se for Method 1638 as our initial estimate (i.e., MDL ) 450 ng/L) (9), valid spiking concentrations would be in the range of 450-2250 ng/L,which based on the data in Table 1 would have yielded valid MDLs in the range of 300-459 ng/L and corresponding MLs from 954 to 1462 ng/L. These results are clearly dependent on the choice of initial estimate of the MDL and the corresponding concentration at which the samples are spiked. By contrast, using concentrations in the range of 0-2000 ng/L, the AML based on the Rock-Lorenzato model was 507 ng/L, which provides a relative standard deviation of 9%. Had we restricted the spiking concentration range from 0 to 1000 ng/L, the AML would have been 513 ng/L. Had we expanded the spiking concentration range from 0 to 25000 ng/L, the AML would have been 582 ng/L. Clearly, the AML is far more robust to the effects of varying spiking concentrations than the ML. Finally, in this example, the AML is in fact smaller than the corresponding MLs obtained following the approach advocated by Kahn et al. In light of these results, we fail to see how use of the AML would raise regulatory compliance levels and be inconsistent with EPA’s mandate to protect human health and the environment. As Kahn et al. point out in their criticism of our use of observed rather than estimated weights (i.e., “The weights suggested by Gibbons et al. are not appropriate”), the observed variances can vary wildly. Instead, they site Carroll and Rupert (10) as “Instead, they prefer the use of weights estimated based on modeling the relationship between measurement and the associated variation.” First, if the variances can vary wildly, then the MDL that is 3.14 times the square root of the variance can also vary wildly. Their stated preference is to model the relationship between variability and concentration, which inevitably leads to a calibration-based estimator such as the AML and leads us away from single concentration estimators such as the MDL which can vary wildly when the number of replicate measurements is small (e.g., the n ) 7 measurements used in computing the MDL). In criticizing our choice of weights, Kahn et al. have actually raised an issue of far greater concern for the MDL than the AML. Application of the Limit of Quantitation (LOQ). Kahn et al. correctly point out that since we missed the fact that the data were already adjusted for background, the correct equation is LOQ ) 10s0 and does not involve addition of the background mean as performed in our paper. We note, however, that it is still quite possible to get a negative LOQ
since with true concentration of zero the average measured concentration should be negative 50% of the time, and if s is small enough the LOQ will be negative. Of course, the real points made in our paper were that (a) the standard deviation in blanks may not be estimable due to instrument censoring and (b) quantitative determination has little to do with variability of background and much more to do with the ratio of real signals to noise (i.e., analyte present). Application of the AML. Kahn et al. state that “114Cd data and diagnostics provided here indicate that these models [exponential and Rocke-Lorenzato] do not fit these data.” This categorical statement is unwarranted and is not convincingly demonstrated. Kahn et al. developed “diagnostic plots” that transform the data differently for each model so that ideally, for a correct model, the data would “show points that indicate a straight line, with some allowance for random variation”. We applaud the concept of the transformation, but unfortunately, Kahn et al. do not provide any guidelines for the allowance for typical random variation. The judgment cannot be made by simply looking at the diagnostic plots and using a “calibrated eyeball”sfor two reasons: (a) The distributions of the log sample standard deviations and of the sample variances are not symmetric; they each have relatively long upper tails, especially the sample variance. Therefore, it is harder to discriminate high outliers from ordinary variation. (b) A true underlying relationship with measurement variation that increases with increasing spike concentration will result in noisier sample standard deviations at higher concentrations. This makes it more difficult to judge whether sample standard deviations at the higher concentrations are unusual. We believe that there may be other diagnostic plots that are more useful for judging the quality of fit for these two models. We also have a suggestion on how to supplement the proposed diagnostic plots so that they do not rely on subjective assessmentsfraught with the problems listed above. We have generated supplemented versions of the proposed diagnostic plots (see Figures 1 and 2). Since Kimbrough and Kahn et al. have pointed out that using 111Cd is preferable to using 114Cd for quantitative assessment, Figure l shows the same transformations as suggested by Kahn et al. but using 111Cd. We also restricted ourselves to a smaller range since we are most interested in low concentrations, and we are not attempting to demonstrate that the models necessarily hold for several orders of VOL. 32, NO. 15, 1998 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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FIGURE 2. (a) Kahn’s diagnostic for exponential model for Cd114, supplemented with pointwise 99% confidence intervals, plus candidate straight line. (b) Kahn’s diagnostic for R&L model for Cd114, supplemented with pointwise 99% confidence intervals, plus candidate straight line. magnitude. Note that this is completely consistent with the basic idea of WLS regression (i.e., assigning greatest weight to results with smallest absolute standard deviation). Figure la shows loge(s) vs spike concentration (ng/L) supplemented with two piecewise linear curves that provide the (pointwise) 99% confidence interval for the true standard deviation at each spike concentration, based on the χ2 distribution. Thus, for each spike concentration, it is plausible that the true standard deviation would lie anywhere in between the two curves. The diagnostic question then becomes, “Is it plausible that a straightline relationship holds between loge(s) and spike concentration?” The credibility of the model then rests upon finding a straight line (if one exists) within the piecewise linear curves. Such a straight line CAN be found, and one is shown in Figure la (although this line should not be used to obtain coefficients for the exponential model). This does not prove that the exponential model is CORRECT or even the best of several candidate models. In fact, we think that the Rocke-Lorenzato model may be preferable to the exponential model. The message coming from Figure la is merely that the exponential model is plausible. Figure 1b is analogous to Figure la, except that the sample variance is plotted vs the square of spike concentration. The plot is supplemented by 99% confidence intervalssin the form of piecewise linear curvessand we attempt to find a straight line that will fit within the curves. Once again, such a straight line CAN be found, and the Rocke-Lorenzato model is plausible (although this line should not be used to obtain coefficients for the Rocke-Lorenzato model). This proposed diagnostic plotseven with the supplemental curvessis particularly inadvisable because the scales of the axes compress nearly all the plotted points into the lower left corner, obscuring the region in which we are most interested: trace concentration. We would never use this kind of plot, in practice. For 114Cd, Kahn and et al. erroneously claim that the first diagnostic plot “clearly does not indicate a straight line” (Figure 2a is our supplemented version). As with 114Cd, a plausible straight line CAN be found, and one is shown in the plot. Once again, this does not prove that the exponential model is correct or even the best of several candidate models, and the Rocke-Lorenzato model may be preferable. The message coming from Figure 2a is merely that the exponential model is plausible. Thus, Kahn et al. do not have a solid basis for their assertion that, “the exponential model is not appropriate for describing the 114Cd data.” 2352
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Similarly, Figure 2b is the diagnostic plot for the RockeLorenzato model, supplemented with confidence intervals. Kahn et al. erroneously claim, “If data fit this relationship, then a plot of the variance ... vs the square of the spike concentration would show points that indicate a straight line. This is not the case ...” Fitting a line within these confidence intervals is a trivial exercise, as can be seen in Figure 2b (similarly, Figure 1b). Thus, Kahn et al. do not have a solid basis for their assertion that, “the Rocke and Lorenzato model is not appropriate for describing the 114Cd data”. Finally, we challenge the claim made by Kahn et al. that “the conclusion regarding the robustness of the AML procedures to departures from the assumed relationship between true concentration and measurement is not justified”. That claim is wholly unsubstantiated. The attempt to demonstrate that the candidate standard deviation models are not appropriate could not logically discredit the robustness claim, even if the demonstration was successfulswhich it was not. If anything, such a finding could possibly be used to reinforce the claim, not discredit it. Kahn et al. provide no logic or data supporting their statement that “robustness of the AML procedure, to departures from the assumed relationship between true concentration and measurement variation, is unwarranted.” As an example, Figure 3a,b shows the observed and estimated standard deviations in the regions of interest for 114Cd and 111Cd, respectively. The fit of the exponential model for 114Cd and the fit of the RockeLorenzato model for 111Cd is quite reasonable at the low end of the concentration range. As previously noted (1,3), using all of the data from 0 to 25 000 ng/L (which Kahn et al. conclude “do not fit these data”) produced virtually no change in the estimated AML (i.e., 42 vs 46 ng/L for 114Cd and 8 vs 10 ng/L for 111Cd). How can that not be an endorsement for robustness? Clearly, the AML is not “based on the strong assumption that the relationship between true concentration and measurement variation has been appropriately modeled,” in that the “lack of fit” noted by Kahn et al. does not in any way alter the results. Inconsistencies with Quantitation Levels in Commonly Used Analytical Methods. In Table 1 of their paper, Kahn et al. note that the computed interlaboratory AMLs for the EPRI GFAAS data in reagent water, river water and ash pond overflow reported in Table 3 of our original paper were too high. They came to this conclusion on the basis of comparing the computed interlaboratory AMLs to various published standards (see Table 1 of their paper). For example, Kahn
FIGURE 3. (a) Standard deviation vs concentration for Cd114 in ppt. (b) Standard deviation vs concentration for D111 in ppt. et al. cite quantification levels from Methods for Chemical Analysis of Water and Wastes of 5 µg/L for chromium, nickel, and lead in reagent water; whereas, we computed AMLs of 29, 33, and 63, µg/L, respectively. They suggest that the source of these “high” AMLs is our use of data that were spiked at too high a concentration to meaningfully estimate quantification limits. The lowest spiking concentration used to estimate the AMLs for these compounds in the interlaboratory example was 2 µg/L, which is at or below all but one of the published quantification limits cited by Kahn et al. The more likely explanation for the discrepancy is that the estimated AML from the EPRI data include the real world effects of deliberately running the study “blind” (i.e., analysts were unaware of the actual concentrations in the test samples) and multiple laboratories. Together, including these effects provides a more realistic estimate of the true measurement variability in routine applications. These sources of variability are not included in the MDL or ML. Conclusions. In their conclusions, Kahn et al. voice the concern that the AML will become a regulatory standard that will preclude use of data from laboratories that can demonstrate attainment of low detection and quantification limits. They also conclude that the AML is based on collection of data from many laboratories. They also appear to conclude that the AML is in some way uniquely matrix and method specific as if this somehow limits its usefulness. From a purely statistical perspective, there is nothing about the AML that requires the use of interlaboratory data. Indeed the illustration is based on data from a single lab. The fact that the AML is matrix, analyte, and method specific should come as no surprise since all detection and quantification limit estimators are specific to the relevant matrix and method to which they are applied. We fail to understand how the ML is any less matrix, method, or analyte specific, and it is unclear to us how the AML would raise regulatory compliance levels relative to the ML.
The question of whether data from multiple laboratories should be used is an empirical one, and the answer is very much dependent on the particular question being asked. We strongly disagree with the notion that the AML should be computed in some selected laboratory and the results should become a national standard. The AML is a statistical estimate and can vary from lab to lab, analyst to analyst, instrument to instrument, and day to day. Any laboratory can compute its own AML, and data from multiple laboratories should be used to set limits on what is or is not an acceptable AML. If national standards are to be used, then applications that involve the use of multiple laboratories should incorporate interlaboratory variability into the AML. Where this is not the case (e.g., for use in quality control or addressing a unique matrix), laboratory-specific AMLs could be used. In summary, we hope that we have made it very clear that our paper (1) is not riddled with errors and omissions as Kahn et al. suggest. The AML presents a statistically valid approach to quantitative determination that allows one to estimate the concentration at which the signal to noise ratio is typically on the order of 10:1. In light of this, measurements above the AML are quantifiable in the sense that the true and measured concentrations will be sufficiently similar for most practical applications. As clearly illustrated here and in our original paper, the AML will not lead to elevated estimates of the quantification limit that would “raise regulatory compliance levels” and “would be inconsistent with EPA’s mandate to protect human health and the environment”. As pointed out by Kahn et al., the observed variances at any single concentration can vary wildly, and the preferred approach involves “use of weights estimated based on modeling the relationship between measurement and the associated variation”. We could not agree more.
Literature Cited (1) Gibbons R. D.; Coleman, D. E.; Maddalone, R. F. Environ. Sci. Technol. 1997, 31, 2071-2077. (2) Kimbrough, D. E. Environ. Sci. Technol. 1997, 31, 3727-3728. (3) Gibbons, R. D.; Coleman, D. E.; Maddalone, R. F. Environ. Sci. Technol. 1997, 31, 3729-3731. (4) Rocke, D. M.; Lorenzato, S. Technometrics 1995, 37, 176-184. (5) Hahn, G. J.; Meeker, W. Q. Statistical Intervals: A Guide for Practitioners; Wiley: New York, 1991. (6) Gibbons, R. D. AML: A computer program for computing the Alternative Minimum Level; Scientific Software International: Chicago, 1996. (7) Gibbons, R. D. DETECT: A computer program for computing detection and quantification limits. Scientific Software International: Chicago, 1996. (8) U.S. EPA. Appendix B to Part 136; Definition and procedure for determination of the method detection limit, revision 1.11. Fed. Regist. 1984, 49 (209), 43430. (9) U.S. EPA. Method 1638: Determination of trace elements in ambient waters by inductively coupled plasma-mass spectrometry; EPA 821-R96-005; U.S. EPA: Washington, DC, January 1996. (10) Carroll, R. J.; Rupert, D. Transformation and Weighting in Regression; Chapman and Hall: New York, 1988.
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 (Y83)-D-10100 Technical Drive Alcoa Center, Pennsylvania 15069
Ray Maddalone TRW Building 01, Room 2040 1 Space Park Redondo Beach, California 90278 ES9820036 VOL. 32, NO. 15, 1998 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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