Correspondence Comment on “An Alternate Minimum Level Definition for Analytical Quantification” SIR: I have been following the Alternate Minimum Level [AML] discussion (1-4) triggered by Gibbons et al. (5) and believe that both sides have considerable merit while missing the real problem. The relevant question is not the concentration that an analytical laboratory can measure with (30% precision in a given matrix (6) but the precision associated with a measured concentration. Kahn, Telliard, and White (4) raise arguments that apply to homoscedastic (constant) variance data sets. Unfortunately for many measurements, especially those conducted to determine stack emissions or characteristics of potentially hazardous solids, the variance is not constant with concentration. Similarly, the Gibbons et. al. approach is correct when a heteroscedastic variance is described by the RockeLorenzato equation. Unfortunately, the simultaneous stack emissions data that I have analyzed do not support this assumption. The observations underlying the need for something like the AML, coupled with those of Carroll et al. (7), lead me to conclude that there is, for complete sampling, analysis and data reduction systems, a low concentration region where uncertainty is essentially constant, a transition zone, and then a region where the uncertainty increases with concentration, albeit in a nonlinear but monotonic manner. As a result, I apply the following generalized equation to model measurement method uncertainty
s ) xa + bxj c d
(1)
where s is the standard deviation determined by simultaneous measurements; xj is the pair-wise average concentration; and a, b, c and d are constants selected to describe the data and create homogeneous residuals. The homoscedastic case assumed by current MDL (8) methodology is modeled when d is 1 and b is 0. The RockeLorenzato (9) model occurs when d and c are 2 and a and b are nonzero. If only a is 0, eq 1 describes a power curve (log-log transformation). When only c is 0, it is a logarithmic model. For d equal to 0, it is an exponential equation. Solving eq 1 is not difficult today but requires somewhat sophisticated software. The same coefficients are determined using either (A) nonlinear regression or (B) Cook and Weisberg’s (10) visual regression techniques or Box-Cox transformations (11) to establish c and d followed by applying ordinary least squares regression to the transformed data to determine a and b. Figure la shows the simultaneous Method 29 mercury data I have collected. Notice that the standard deviation increases with concentration so the data are not homoscedastic; the Spearman rank-correlation coefficient is significant. Even fitting the Rock-Lorenzato equation to this data is invalid since the residuals are not randomly distributed. Once the data are transformed using the exponents identified on the axes of (b); however, Figure lb shows that the residuals are well behaved, and eq 1 describes the standard deviation-concentration relationship. To determine measurement precision at a desired concentration, the upper confidence limit for the regression line 10.1021/es9811086 CCC: $18.00 Published on Web 02/24/1999
1999 American Chemical Society
FIGURE 1. Scatter plots of raw and normally transformed between pair standard deviations and averages for Method 29 mercury measurements. is determined and the S estimate returned to the data plane. This value is then multiplied by the upper tolerance limit factor as explained by Gibbons (5) and Hahn and Meeker (12) to establish the imprecision associated with a specified concentration. Traditionally, the 95% statistical confidence level (13) and 99% (two-sided) probability of inclusion (14) are used for air pollution measurements. Our experience indicates that overall measurement imprecision is much greater than the analytic laboratory’s contribution. For example, if one is interested in determining compliance with a 28 µg/dsm3 @ 7% O2 mercury standard, available data indicates that for three-run averages, the way compliance is determined under federal stack emissions regulations; concentrations less than 37.5 µg/dsm3 @ 7% O2 are statistically indistinguishable from the standard. But, for the published detection limit, one-third of the MQL is 0.7 µg/dsm3 (15). Along similar lines, when the precision of ITEQ dioxin measurements is determined, a three-run test average of 0.36 ng/dnm3 @ 11% O2 cannot be distinguished from the 0.1 ng/dnm3 @ 11% O2 European standard. The DL, however, is about 0.04 ng/dsm3 @ 11% O2. Avoiding false positivessfinding sources in violation when they are really in compliancesrequires that precision be known at levels of concern. It really does not matter what the quantitation limit is, just whether or not a measurement is likely to indicate compliance with a regulation (contract requirement) or not.
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) Kahn, H. D.; Telliard, W. A.; White, C. D. Environ. Sci. Technol. 1998, 32, 2346-2348. (5) Gibbons, R. D.; Coleman, D. E.; Maddalone, R. Environ. Sci. Technol. 1998, 32, 2349-2353. (6) 40 CFR 63, Appendix A, Method 301, Field Validation of Pollutant Measurement Methods from Various Waste Media. (7) Carroll, R. J.; Ruppert, D. Transformation and Weighting in Regression; Monographs on Statistics and Applied Probability 30, Chapman & Hall, 1988; 147. (8) Glaser, J. A.; Forest, D. L.; McKee, G. D.; Quave, S. A.; Budde, W. L. Environ. Sci. Technol. 1981, 15, 1426-1435. (9) Rocke, D. M.; Lorenzato, S. Technometrics 1995, 37, 176-184. (10) Cook, R.D.; Weisberg, S. An Introduction to Regression Graphics; Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics Section; Wiley: New York, 1994. VOL. 33, NO. 8, 1999 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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(11) Box, G. E. P.; Cox, D.R. J. Royal Statistical Soc. Ser. B 1964, 26, 211-246. (12) Hahn, G. J.; Meeker, W. Q. Statistical Intervals: A Guide for Practitioners; Wiley: New York, 1991. (13) 40 CFR 60, see for example, Appendix A, Method 19. (14) USEPA/OSW, Measurement Precision Overview of EPA/OSW Analysis; presented to ASME ReMAP Subcommittee, Washington, DC, February 10, 1998.
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(15) 40 CFR 60, Appendix A, Method 9, Table 29-1.
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