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Response to Comment on “An Alternate Minimum Level Definition for Analytical Quantification” SIR: We appreciate the opportunity to respond to the comments made by Rigo on our paper published in the July 1997 issue of this Journal and subsequent comments (1-5). We are quite pleased that our paper has raised such a high level of interest and scientific debate. Such interactions can only lead to scientific advancement in this interesting area of applied statistics. Rigo provides a different perspective on the problem of detection and quantification than those previously raised in our original article and subsequent correspondence. His area of interest and application is in the comparison of stack emissions data (i.e., air monitoring) to regulatory standards. In such applications interest is typically in comparing an upper confidence limit for the true mean concentration in a series of m monitoring samples (in his example m ) 3) to a regulatory standard or criterion. This is, in fact, a much different problem than the usual detection and quantification problem in which we are attempting to make the binary presence or absence decision (i.e., detection) and, subsequently, whether a quantitative determination can be made. For stack emissions, we are attempting to characterize the concentration distribution at the stack and compare a statistic of that distribution to an external standard. By contrast, the detection and quantification problem addressed in our original paper concerned determinations made for a single new sample without reference to any external standard or criterion (i.e., is the analyte present in the sample and is the potential disparity between the measured concentration and the true concentration at an acceptable level). While these are clearly both important yet quite different questions, leading to different statistical applications, the connection between the two is that in both cases we must incorporate the effects of heteroscedasticity into our statistical estimators. It is gratifying to learn that the nonconstant relationship between concentration and variability is found in other media as well. To model this process in air quality data, Rigo has essentially applied the Rocke and Lorenzato (6) (R-L) model to data that have been previously normalized via Box-Cox (7) type transformations. While this is an interesting extension of the original model described by Rocke and Lorenzato, it remains unclear how useful it is even in the air monitoring application presented by Rigo. For example, he points out that the R-L model is invalid because the “residuals are not randomly distributed”. However, the fit of the R-L model or his transformed version of it to his stack emission data are never presented. In Figure lb, the transformation has resulted in an inverse relationship between variance and concentration with no apparent area in which there is a “low concentration region where uncertainty is essentially constant, a transition zone and then a region where the uncertainty increases with concentration”. As such, it is hard to appreciate Rigo’s conclusion that “Figure lb shows that the residuals will be well behaved and eq 1 describes the standard deviation-concentration relationship”. 10.1021/es992002t CCC: $18.00 Published on Web 02/20/1999
1999 American Chemical Society
A related concern has to do with the use of power transformations on the subsequent construction of a tolerance limit used in comparing a “three-run average” to a regulatory standard. Since a regulatory standard typically represents an average exposure concentration in the original concentration metric, the tolerance bound must be related back to that original metric. A standard inference problem occurs when data are transformed to meet distributional requirements of a spherical normal linear model and a confidence or tolerance interval is required for the mean of a variate in the original, untransformed scale (8). It is unclear how this would be achieved in the current setting, and the simple naive approach of reverse transformation of the resulting tolerance limit estimate can, in fact, be considerably biased (8). There are several points in Rigo’s correspondence that bear further clarification. First, the R-L model is implied when b ) 0, and c and d are arbitrary in this context. Similarly, the R-L model is implied when a, b > 0 not just a, b + 0. It is also unclear to us why c ) 0 / implies a “logarithmic model” and why d ) 0 implies an “exponential model.” We would encourage Rigo to publish his work on this problem in a more expanded form so that the foundation for his claims could be more fully explicated. Finally, we would be concerned about the potential for overfitting data using Rigo’s eq 1. In Rigo’s Figure, the relationship between sample standard deviation and sample mean was shown. However, it is preferable to collect data in a study using known standards (albeit difficult to obtain such standards for certain combinations of analyte and medium) and relate sample standard deviation to the known “true” concentration. In such a calibration or validation study, it is not uncommon to have only five levels of known standards (9), leaving a paltry single degree of freedom for error, if the four-parameter eq 1 is used. This is inadequate: either a simpler function must be used, or a possibly unrealistic number of analyte levels must be included in the study. In summary, regression models that incorporate heterogeneity of variance have numerous applications in the environmental sciences. We have previously illustrated their application in deriving detection and quantification limit estimators that can be applied to analysis of individual sample measurements to determine the presence or absence of an analyte and relative uncertainty in a concentration estimate. Rigo has illustrated a similar application for estimating upper tolerance bounds for stack emission data for the purpose of comparison to regulatory standards. His conclusion that the only important question is “just whether or not a measurement is likely to indicate compliance with a regulation (contract requirement) or not” reflects his own background and field of expertise and is in our opinion an unfortunately narrow perspective. In fact, even within the environmental regulatory framework, water discharge requirements are often set for individual measurements at the quantification limit. Having a precise estimate of the quantification limit is therefore important in the regulatory context as well.
Literature Cited (1) Gibbons, R. D.; Coleman, D. E.; Maddalone, R. F. Environ. Sci. Technol. 1997, 31, 2071-2077. VOL. 33, NO. 8, 1999 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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(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. F. Environ. Sci. Technol. 1998 32, 2349-2353. (6) Rocke, D. M.; Lorenzato, S. Technometrics 1995, 37, 176-184. (7) Box, G. E. P.; Cox, D. R. J. Royal Statistical Soc. Ser. B 1964, 26, 176-184. (8) Land, C. E. Annals Mathematical Statistics 1971, 42, 11871205.
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(9) ASTM Standard Practice for 99(IDE) for Analytical Methods with Negligible Calibration Error; ASTM D6091-97, Section 6.
Robert D. Gibbons* University of Illinois at Chicago
David E. Coleman Alcoa Technical Center
Ray Maddalone TRW ES992002T
