Letters
Proper Statistical Evaluation of Calibration Data Sir: The R E P O R T by G. L. Long and
J. D. Winefordner on "Limit of Detec tion" (Anal. Chem. 1983,55, 712 A) describes many of the pitfalls in estab lishing the limit of detection in analyt ical procedures. Unfortunately, many of the procedures discussed really ad dress the problem of detecting a single analytical signal, which differs from many replicate blank signals. This, of course, is the problem addressed in the IUPAC definition of a "detection limit." What is frequently wanted in practice is a method for distinguishing the average of many blank signals. Once this is achieved, the calibration curve may be used to obtain the corre sponding analyte concentration by in terpolation. This meets the ACS defi nition that "the limit of detection is the lowest concentration of an analyte that the analytical process can reliably detect" ("Guidelines for Data Acquisi tion and Data Quality Evaluation in Environmental Chemistry," Anal. Chem. 1980,52, 2242). A detection limit meeting the ACS definition (but not the IUPAC defini tion, which requires an approximation approach) can be obtained from confi dence interval estimates of sample concentrations interpolated from a calibration curve. When the calibra tion curve is linear and the measure ment errors can be considered to fol low a Gaussian distribution, well-es tablished statistical methods are avail able for obtaining confidence interval estimates for the analyte concentra tion in an unknown sample interpolat ed from replicate measurements using a calibration curve (see, e.g., Graybill, F. A. "An Introduction to Linear Sta tistical Models"; McGraw-Hill: New York, 1961; pp. 125-27; Natrella, M. G. "Experimental Statistics," Na tional Bureau of Standards Handbook 91, U.S. Department of Commerce, Washington, D.C., 1963; 5-4.1.4). Con fidence interval estimates from cali bration curves that are not straight lines can be obtained as described by L. M. Schwartz ("Nonlinear Calibra tion," Anal. Chem. 1977,49, 2062-68). Confidence interval estimates when measurement errors are not Gaussian can be obtained by replacing student's t in either of the above methods with k, where k is selected so that the req uisite probability is obtained from Tsehebyscheff's inequality, as illus trated by Long and Winefordner. The
concentration CL, such that its lower confidence limit is zero, would then be taken as the detection limit. The limit of detection defined in this way is a function not only of the assay precision σ (or its experimental ly measurable substitute, the standard deviation s) but also of the number of standards and blanks used to con struct the calibration curve, the num ber of replicate analyses of each un known, and the statistical design of the calibration experiment. The detec tion limit can be made arbitrarily small for any analytical procedure by simultaneously increasing the number of standards and blanks analyzed, in creasing the number of replicate de terminations of each unknown, and al tering the design of the calibration ex periment to include more standards at low concentrations and more blanks, provided that the analytical method generates an analog signal, and that the unknown sample is not limited in size. Two randomly distributed popu lations can always be distinguished if it is possible to obtain a sufficient number of analog measurements on each. If the unknown sample is of lim ited size, so that only one or a few rep licate determinations can be made, or if the analytical method generated a digital output signal, so that signal changes smaller than the least signifi cant digit are not measurable, then a finite detection limit will exist. In most cases, however, the limit of de tection is a reflection of the amount of effort the analyst is willing to make in performing replicate analyses, and not an intrinsic property of the method. Detection limits based on statistical evaluation of replicate measurements (which meet the ACS definition of the detection limit) are more useful to the analytical chemist than limits based on single measurements of the analyte signal (which seem to be required by the IUPAC definition). However, nothing in the confidence-limit-based definition precludes the analyst from making only single determinations on unknown samples if he is unwilling (because of cost) or unable (because of limited sample) to perform replicate determinations. The finite detection limit that would be obtained in this case is readily computed from the con fidence interval formula given by F. J. Linnig and John Mandel ("Which Measure of Precision," Anal. Chem.
1290 A · ANALYTICAL CHEMISTRY, VOL. 55, NO. 13, NOVEMBER 1983
1964, 36, 25 A, Equation 8). The re sulting expression is then closely ap proximated by the formula obtained by Long and Winefordner (Equation 21) using the propagation-of-errors approach. However, the latter equa tion suggests that confidence intervals are symmetrical, whereas in fact, the true confidence intervals are unsymmetrical, with the lower limit always farther away from the expected value of the concentration than the upper limit for all unknowns between the blank and the average value of the concentration standards. Thus, a de tection limit calculated using the IUPAC definition, which requires computing an upper confidence limit for the blank, will always be smaller than the true detection limit defined statistically, which requires comput ing a lower confidence limit for the sample. In practice, however, the dis crepancy may be small, amounting to an error of only a few percent of the true value. I am constantly amazed by the con fusion among analytical chemists con cerning the proper statistical evalua tion of calibration curve data. The sta tistical problems were solved decades ago. Could it be that we simply do not know where to look up the informa tion we need? William R. P o r t e r School of Pharmacy University of Wisconsin 425 N. Charter St. Madison, Wis. 53706
C o r r e c t i o n : There are unfprtunate typographical errors in the R E P O R T
by Long and Winefordner in the June 1983 issue, pp. 712-24 A. Ex pressions appearing in Table I on p. 716 A should read:
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