Realistic Detection Limits from Confidence Bands - American

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Realistic Detection Limits from Confidence Bands Julia R. Burdge* Department of Chemistry, The University of Akron, Akron, OH 44325-3601 Douglas L. MacTaggart and Sherry O. Farwell Graduate Office, South Dakota School of Mines and Technology, Rapid City, SD 57701-3995

Detection limits are commonly used to quantify the capabilities of analytical instruments and measurement methods. Descriptions of many different statistical methods for determining detection limits have appeared in the analytical chemistry literature (1–10) and in analytical chemistry textbooks (11, 12). The methods most widely used, such as the “3-σ” method and the “2 × S/N ” method, fall under the “classical” approach described by Kaiser (5), Boumans (4), Currie (7, 13), and Long and Winefordner (3). This approach has carried the imprimatur of the ACS (1, 2) and, until recently, of the IUPAC (14, 15). This combination of professional society approval and widespread pedagogy has contributed to the common use of this approach by practicing analytical chemists. Application of the classical approach typically focuses on the standard deviation of blank measurements multiplied by a factor to indicate a statistical confidence level; that is, ksbl. In the common 3-σ method, k = 3 is chosen. This choice represents a theoretical type I (false positive or α) error rate of 0.13%. Taking into account nonnormality and other nonideal conditions, this translates in practice to 5–10% (4, 5, 14, 15). Similar methods substitute a one-sided t-value for k. Some proponents of the classical approach have described methods for dealing with both type I and type II (false negative or β) error rates (1–4, 7 ). This usually involves doubling the value of k. Some workers have incorporated the idea that the standard deviation at the detection limit may be different from the blank standard deviation (7, 16 ). In practice, this is usually ignored except where mandated by the EPA or other regulatory body. Although their use is widespread, these classical methods have a number of shortcomings. Briefly, the most commonly used methods focus on blank measurement distributions and do not address calibration or sample measurement variability. The potential for inaccurate detection limit estimation is significant for results based on calibrations that are derived from inadequate data or fitting models. Also, use of the 3-σ method addresses only α error and neglects β error. The less widely applied regression approach to calculating detection limits, originally described by Hubaux and Vos (6 ), employs response data from samples that contain analyte. Typically, ordinary least squares (OLS) is performed on calibration data obtained from samples containing analyte levels near the blank and detection limit regions. The regression results are then used to construct confidence bands, which lead to detection limits. Use of the regression approach has been promulgated by the Association of Official Analytical Chemists (AOAC) (17) and more recently by the IUPAC (18). A facet of the regression approach that has received minimal recognition in previous articles discussing detection limits is the use of weighted least squares (WLS). Several *Corresponding author. Email: [email protected].

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articles on the application of regression for quantitation purposes have addressed the fact that response variance and standard deviation often increase with increasing analyte level (19–23). Most regression detection limit calculations employ OLS, which assumes constant variance (homoscedasticity) over the analyte level range. Application of OLS to data with nonconstant variance (heteroscedasticity) yields confidence bands and thus detection limits that will not accurately reflect measurement capability. WLS facilitates the determination of realistic detection limits for heteroscedastic data by yielding confidence bands that directly reflect the changing variance. In this work, methods for determining detection limits using OLS and WLS are discussed and applied to selected data sets. The results illustrate the impact of data set size and scedasticity on detection limit results for both well and poorly designed calibration experiments. Results from the most widely used classical methods are included for comparison. Description of Selected Data Sets A calibration data set was generated by analysis of sulfur gas standards with a sulfur chemiluminescence detection (SCD) system. The SCD data acquisition procedure generated 20 sample measurements per second. The resulting data included replicate blank measurements of sulfur-free zero air and replicate measurements at five standard carbonyl sulfide (OCS) concentrations that ranged between 0.3 and 1.5 ppb. The original data set has a large n (2105) and is reasonably, but not perfectly, homoscedastic. These data were manipulated to generate additional data sets to illustrate the following effects on detection limits determined by classical and regression approaches: (i) heteroscedasticity, (ii) large and small n values, and (iii) calibration that does not include blank measurements or sample measurements near the blank. The first manipulation involved amplification of the data’s heteroscedasticity. The residuals from the original regression were multiplied by a linear function to increase the measurement standard deviation with increasing concentration. The new residuals were then added to the predicted y values from the original regression to produce a heteroscedastic data set. These heteroscedastic data were used to determine detection limits by classical, OLS, and WLS methods. The regression lines and two sets of 95% confidence bands (described below) using the heteroscedastic data set for the OLS and WLS cases are shown in Figures 1 and 2, respectively. In these figures, error bars representing 95% confidence intervals and means of the replicate measurements at each level of x were used to summarize this large data set. The next manipulation was the random selection of a data set of small n (35) from the original, nearly homoscedastic data. This smaller data set, shown in Figure 3, was used to illustrate the performance of the classical and OLS

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methods for more typical experimental circumstances, that is, where only minimal replication is performed. Next, the small data set was truncated to illustrate the effect of calibration designs that do not include measurement values for the blank or for standards with analyte concentrations close to the blank. This data set is hereafter referred to as “distant from blank”. For this data set, shown in Figure 4, only the three highest concentrations (5 replicates each) from the previous data set were used, giving n = 15. Regression Approach

yU 1 1 y L = y i ± ts r m + n +

9000

9000

8500

8500

SCD Response

SCD Response

Detection of analyte in samples involves both prior calibration using standards and subsequent analysis of the samples. Both measurement procedures contribute to the overall statistical error of the detection process. The magnitude of this error is commonly expressed through the calculation and reporting of confidence bands. What follows is a methodology for estimating detection limits based on confidence

bands generated from calibration experiments. This section will focus on using least squares to calculate these limits for the common analytical situation where external standards are used to calibrate a measurement method. Equation 1 gives the confidence bands for least squares treatment of calibration data based on the proper propagation of errors through independent terms (24–26 ). This expression is for the case of OLS only, that is, constant absolute response variance is assumed. Symbols used in this and following equations are defined in Table 1. Equation 1 gives the confidence bands for the OLS treatment which include error due to the sample analysis step of the measurement process (m replicates per analysis) in addition to error due to the calibration step (n total calibration data points).

8000

7500

7000 0.0

1/2

(1)

S xx

7500

7000 0.5

1.0

1.5

0.0

0.5

8500

8500

SCD Response

9000

8000

7500

1.0

1.5

Figure 2. Heteroscedastic data set (n = 2105) with WLS fit. Outer and inner confidence bands are for m = 1 and m = 100, respectively, with α = β = .05.

9000

0.5

1.0

ppb OCS

Figure 1. Heteroscedastic data set (n = 2105) with OLS fit. Outer and inner confidence bands are for m = 1 (single sample measurement) and m = 100 (reasonable number of sample replicates), respectively, with α = β = .05.

SCD Response

2

8000

ppb OCS

7000 0.0

xi – x

1.5

ppb OCS Figure 3. Reasonably homoscedastic data and OLS fit for small data set (n = 35). Outer and inner confidence bands are for m = 1 and m = 5, respectively, with α = β = .05.

8000

7500

7000 0.0

0.5

1.0

1.5

ppb OCS Figure 4. “Distant from blank” subset (n = 15) of Figure 3 data with OLS fit. Outer and inner confidence bands are for m = 1 and m = 5, respectively, with α = β = .05.

JChemEd.chem.wisc.edu • Vol. 76 No. 3 March 1998 • Journal of Chemical Education

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Essentially, eq 1 provides Table 1. Summary of Statistical Terms a pooled standard error by as- Term Description Formula suming, as in the classical apn number of calibration data points n proach, that the measurement m number of post-calibration sample replicates m error inherent in analyzing a Σx i2 – (Σx i)2/n sum of x squared mean deviations Sxx sample is the same as that obΣyi2 – (Σyi )2/n Syy sum of y squared mean deviations served in analyzing the blanks Σx i yi – Σx i Σyi /n sum of xy mean deviations Sxy and other standards. For the reb slope estimate Sxy /Sxx gression approach, however, this assumption can be tested a intercept estimate (Σyi – b Σx i) /n by replicate analysis of the yˆi predicted y value given xi a + bx i standards. residual variance estimate, also often denoted by MSE, sr2 (Syy – bSxy )/(n – 2) Many discussions using exsy x2, sy2 or s 2 pressions similar to eq 1 begin sb2 slope variance estimate sr2/Sxx with a sample measurement x2 s r2 1 sa2 intercept variance estimate size of one, that is, m = 1, with n+S xx introduction of cases for m > g inverse measure of significance of slope estimate (tsb /b)2 1 later. Here, the view is that Σwi xi / Σwi weighted mean x x¯w m = 1 is a special case of m ≥ Σwi xi2 – (Σwi x i )2/ Σwi Swxx weighted sum of x squared mean deviations 1. The two sets of confidence Σwi yi2 – (Σwi yi )2/ Σwi Swyy weighted sum of y squared mean deviations bands shown in each of the Σwi xi yi – Σwi xi Σwi yi / Σwi plots in Figures 1–4 illustrate Swxy weighted sum of x y mean deviations the effect of choosing different bw weighted slope estimate Swxy /Swxx values for m. In each of these aw weighted intercept estimate (Σwi yi – bw Σwi xi )/ Σwi figures, the outer, wider set of weighted residual variance estimate(can be normalized for 2 swr (Swyy – bwSwxy )/(n – 2) bands corresponds to m = 1. comparison with sr2 ) The inner, narrower bands corgw inverse measure of significance of weighted slope estimate (tswr )2/bw2Swxx respond to m significantly larger than one. This narrowfor the OLS case. Assumption of α = β in the derivation of ing effect observed for confidence bands as m increases rethis formula allows the result to be noniterative and exact. sults in decreased detection limits. Constricting the reporting of regression detection limits 1/2 to the case where m = 1 is inconsistent with good laboratory 2t s r 2 gx x2 1 1 xD = +n + – (2) practice (that is, the use of replicates). However, to prevent m S xx 1–g b 1–g the reporting of unrealistically low detection limits via the regression approach, the size of m should be scrupulously For the case of m = 1, eq 2 is algebraically equivalent to representative of how a typical analyte-containing sample is the formula developed by Currie (27) and recently adopted analyzed in practice. by the IUPAC (18). Equation 1 provides a starting point for the derivation Other authors and agencies have adopted formulas that of a detection limit formula through the generation of conare iterative or approximate because of different choices with fidence bands. This approach follows that published by respect to α , β , and the corresponding distributions. The Hubaux and Vos (6 ), which incorporates both α and β errors. AOAC (17) and Coleman (28) choose to allow α and β to Figure 5 demonstrates how confidence bands give rise to a be different from one another. Clayton (9) argues for the use detection limit. The t distributions in this figure should be of noncentral t distributions in place of the usual central t. viewed as rising vertically from the page at x = 0 and x = xD. Gibbons (29) argues for the use of tolerance intervals to Their maxima would then be perpendicular to y0 (the intercept) control the confidence level of detection in multiple samples and yD, respectively. simultaneously. The intersection of the upper confidence band with the Under certain circumstances, one of these more complex y axis at yC , where x = 0, corresponds to the highest signal treatments may be applicable. However, the OLS treatment which could be attributed to a blank 100(1 – α)% of the described here is the most general, in addition to being the time. This is represented by the t distribution with a onemost computationally simple. Also, it has been adopted by tailed α for y at x = 0. the IUPAC (for m = 1), and thus presents one basic formula The intersection of this signal level with the lower conwhich may be used by many analysts to produce comparable fidence band at y L in Figure 5 corresponds to the lowest signal results. that could be attributed to an analyte concentration x D 100(1 – β)% of the time. This is represented by the t distriWeighted Regression bution with a one-tailed β for y at x = x D. The detection limit It is common in instrumental analysis for the response is thus defined as xD. The mean for the distribution of signals variance, estimated by least squares as the residual or regression at x D is y D. variance, sr2, to increase with increasing analyte concentration Equations for y C and y D in Figure 5 can be derived using (19–23). Application of OLS to significantly heteroscedastic eq 1. Transformation into terms of x yields a quadratic data results in the construction of unnecessarily broad confiequation whose root, shown in eq 2, is the detection limit 436



α yD yC

yL

y0

β

0

xD

Figure 5. Least squares fit illustrating the derivation of detection limits using the regression approach. Confidence bands were generated using α = β and one-tailed t distributions at x = 0 and x = x D.

dence bands at the low end of the calibration curve. The detection limit derived from OLS treatment of such data will not be a fair representation of a measurement method’s detection capability.

Obtaining Weights for Use with WLS Least squares treatment of heteroscedastic data is commonly accomplished through either data transformations or by the application of WLS. For the purpose of calculating detection limits, Clayton (9) and Gibbons (29) show the applicability of square root transformations. Several approaches to the application of WLS to heteroscedastic data exist. As a first order treatment, weights are commonly assigned as the reciprocal of the response variance at each level of x. The weight, wi, for a given xi is 1/ si2, where si is the estimate of the standard deviation in y. The sample size necessary to obtain a precise estimate of a response standard deviation has been variously reported as 8–10 (21, 32) and as at least 20 (4, 5, 14, 15, 17, 20). This level of replication unfortunately is frequently difficult to achieve in practice. Essentially, the weights are scaling factors for the overall residual variance, allowing the width of the resultant confidence bands to vary in a manner that tracks the variation observed in the sample responses over the different concentrations, as illustrated in Figure 2. Thus, when WLS as opposed to OLS is applied to a data set that exhibits increasing variance with increasing concentration, the width of the confidence bands is narrowed at the low end of the curve. This confidence band narrowing, in addition to the narrowing effect of m > 1, results in lower detection limits and provides a more accurate representation of a measurement method’s capability at low analyte concentrations. Another approach to the treatment of heteroscedastic data is to assume or show that the response standard deviation is proportional to either y or x. Equations for slope, intercept, etc., can thus be derived (30, 31). Gibbons (29) describes an approximation method for obtaining detection limits by this approach. In this method, the weighted slope, intercept, etc., are inserted into OLS detection limit formulas. The approach taken in this work for the heteroscedastic data case is to (i) obtain individual standard deviations (s i) at

each level of x via replicate measurement of standards, (ii) model the si as a function of x to obtain weights for any given level of x, (iii) perform WLS using this weighting scheme to obtain estimates of the slope, intercept, and other statistical variables, and (iv) use the WLS results and the modeled weights in a detection limit formula specifically derived for the WLS case. Because the standard deviation for the large n heteroscedastic data set is known to follow a linear trend (i.e., that is how the data set was constructed), the variance model used for illustration here consists of applying OLS to the s i and xi. Other models may be more appropriate, depending on the particular data set (20, 22, 23). In particular, the variance often tends to stop decreasing as the detection limit is approached, thus requiring a more complex model. The model chosen here allows the calculation of wi at any xi for the data sets at hand and facilitates the construction of smooth confidence bands.

Derivation of WLS Detection Limit Formula To derive a detection limit formula for the weighted regression case, eq 1, used to calculate OLS confidence bands, must first be modified as shown in eq 3. xi – xw yU 1 1 y L = y i ± ts wr mw i + Σw i + S wxx

2

1/2

(3)

This equation illustrates how the regression variance is scaled by the weights as discussed above, that is, the scaled regression variance at a given x i will be s wr2/wi for m = 1. For the WLS case, use of Figure 5 and eq 3, followed by transformation into x and solution of the quadratic, yields a more complicated result than the OLS case. The presence of the weighting factors in these equations complicates the derivation of a detection limit formula because wa (the weight at the intercept) and wD (the weight at the detection limit) are not equivalent. These two weighting factors can be obtained from the modeled sample standard deviations. However, the relationship between wD and x D causes their final calculation to require iteration. The result is given in eq 4.

A=

ts wrQ – xw gw bw

B = xw gw xw gw –

2ts wrQ bw

x 2 g C= 1 + w + w Σw i S wxx mw a x 2 Q= 1 + 1 + w mw a Σw i S wxx

1/2

1/2

A ± B + g w S wxx xD =

1 – gw C+ mw D

1 – gw


(4) 437


To date, only the positive root in eq 4 has been found to be useful. Iteration of these equations causes rapid convergence. Typically, only about five iterations are necessary to get a stable result, depending on the precision desired.

for the large n heteroscedastic WLS and small n = 35 data sets. The agreement for the heteroscedastic OLS data set is, of course, not as good, illustrating the effect of using an improper regression model. As discussed, response variability commonly increases with increasing analyte concentration. In such instances, the application of WLS will yield lower estimates of the detection limit than will OLS by narrowing the width of the confidence bands near the origin. This can be explained by noting that WLS reduces the excessive influence of large variabilities in y values distant from the intercept on the standard error of the intercept. For the examples given in Figures 1 and 2, the use of WLS versus OLS results in a reduction of the intercept standard error from 5.1 (OLS) to 1.8 (WLS) for the same heteroscedastic data set. As shown in Table 2, this difference in intercept standard error translates into a reduction in the calculated detection limit from 0.68 ppb (OLS) to 0.29 ppb (WLS) for the m = 1 case, and to a reduction in the calculated detection limit from 0.071 ppb (OLS) to 0.022 ppb (WLS) when m = 100. These results from our rigorous, yet iterative, WLS treatment were compared with results obtained by using the approximation method of Gibbons discussed above in conjunction with our OLS formula. For m = 1, Gibbons’ method yields 0.32 ppb. For m = 100, it yields 0.033 ppb. Thus, Gibbons’ approximation simplifies the calculations for the WLS case and yields results which qualitatively match the more rigorous treatment. The small n = 35 data set presents an example of a reasonably well defined calibration. There are ten replicates of the blank and five replicates at each of five standard concentrations. As shown in Table 2, the classical and regression detection limits generated for this data set are typically slightly higher than the respective classical and regression detection limits for the large n data set. For the small data set with values distant from the blank, however, the classical detection limits are substantially lower and the regression detection limits are substantially higher than for the large n data set. These notable differences are the result of using a calibration curve that is not well defined in the region near the blank and the detection limit. For the classical approach, the detection limits in x will vary with the particular intercept and slope estimates generated from a specific calibration data set. The intercept estimate for the distant from blank data set is higher (7401) than that for the more optimal small n data set (7330), and the slope estimate is correspondingly lower for this nonoptimal data set (692 vs 750, respectively). If the relation between the intercept and slope estimates for the data sets were reversed, as is

Discussion of Results Table 2 presents the results of the application of the classical and regression approaches to the selected data sets. The results for the classical approach were generated by using the distribution of 660 blank responses, which were incorporated in the original large n = 2105 data set. Subsequent conversion from y to x was performed by using the individual calibration curves for each specific data set. The differences among the classical results within the k = 3 and k = 6 columns of Table 2 are thus due to the differences among these individual calibrations. The results for the regression approach were generated using the indicated data sets, their individual calibration curves, and eqs 2 and 4. For the regression approach, α = β = .05 was chosen. Table 2 shows why analysts are reluctant to report detection limits using the regression approach. When confidence bands are generated using an improper regression method, a nonoptimal calibration design, or without including sample replicates, the obtained detection limits can be significantly higher than those calculated using the 3-σ classical approach, which does not account for calibration fitting error and which fixes only α risk. Comparison of the respective values in the m = 1 xD column vs those in the k = 3 column demonstrates the potential for misunderstanding the merit of the regression approach. What the results also show, however, is that with the inclusion of an appropriate number of replicates and an optimized calibration design, the regression approach yields comparable or lower detection limits than the classical approach. This is illustrated by comparing the results in the m > 1 x D column with the results in the k = 3 column for the large n and small n = 35 (i.e., data not distant from blank) data sets. These numbers in the m > 1 x D column possess the enhanced type I and type II error protection that is inherent in the regression approach. An “appropriate” number of replicates can be estimated by consideration of the analytical process. As stated above, the SCD data acquisition procedure used in this work generates a large number of data points very quickly. The use of m = 100, for example, requires a data acquisition period of only five seconds. Obviously, in this case, the confidence bands can be further narrowed by the choice of a larger m value, but the value chosen should be realistic and justifiable within a particular experimental setting. It is interesting to note that for the large n and small Table 2. Comparison of Classical and Regression Detection Limits for Selected Data Sets n = 35 data sets using m = 1, the regression approach Classical Approach Regression Approacha Data Set n gives results very close to 3 Sbl 6 Sbl x D(m = 1) x D(m > 1) those obtained with the claslarge n, heteroscedastic sical approach when both α OLS 2105 0.17 0.3 5 0.68 0.071 (m = 100) and β are taken into acWLS (5 iterations) 2105 0.18 0.36 0.29 0.022 (m = 100) count. This is illustrated by small n, homoscedastic (OLS) 35 0.20 0.39 0.44 0.22 (m = 5) comparing the results in the n , d i s t a n t f r o m b l a n k ( O L S ) 1 5 0 . 1 1 0 . 3 2 0 . 7 4 0 .59 (m = 5) s m a l l m = 1 x D column with the a results in the k = 6 column NOTE: Detection limits are given in ppb. α = β = .05. 438



possible according to their statistical distributions, then the relation between their detection limit estimates would also be reversed. Thus, the classical approach provides an unrealistic estimate of the detection limit in x when the calibration curve used is not well defined in the blank-detection region. In summary, these examples demonstrate that regression detection limits are best estimated using calibration data sets that are as large as is feasible and, more importantly, that provide good coverage of the blank-detection region. In any case, the resultant regression detection limits provide information about the quality of the calibration data used to generate them. When the results appear inordinately high, this is evidence that a better experimental design is necessary. This evidence provides a diagnostic which the classical approach does not consider. Conclusions The classical approach to the calculation of detection limits is an old and familiar tool of the analyst. Its widespread application has received acceptance through popular usage, professional societies, and undergraduate textbooks. With rapidly improving data acquisition capabilities, though, its use should be reexamined. The popular 3- σ method fails to account for sample measurement variability and can give unrealistic results if the calibration curve’s slope and intercept are not well defined in the blank-detection region. It also fails to control type II error. The regression approach has received recent endorsement from the AOAC and the IUPAC. It has yet to make its way into popular usage and textbooks owing in large part to misconceptions about its merits, as well as general unfamiliarity with how to apply it. While proper use of this approach allows reporting of lower detection limits for large data sets, where regression coefficients are precisely estimated, it can also provide a safeguard against the reporting of unrealistically low values, provided a reasonable m is chosen. In situations where the data available are severely limited, use of the regression approach encourages analysts who wish to maximize the probability of obtaining a realistic detection limit estimate to optimize the calibration design to well define the blank-detection region. In conclusion, use of the regression approach for the calculation of detection limits has distinct merit over the classical approach. The regression approach, as described, ensures protection against both α and β errors and accounts for the uncertainty in calibration and sample measurements as well as in blanks. These features result in a realistic evaluation of the practical merits of a particular analytical method. Additionally, in contrast to common perception, the regression approach yields detection limits comparable to those obtained with the widely used 3-σ classical approach, when either approach is properly applied to a well-defined data set. In order for the detection limit calculated by the regression approach to represent an analytical method’s performance realistically, the m value used must be consistent with that method’s specific experimental circumstances, and the regression model selected must be appropriate for the particular data set’s scedasticity. Therefore, detection limits determined by the regression approach should be reported for completely specified analytical methods which indicate the m value, the n value, and other relevant parameters of the method.

Only a limited treatment of weighting schemes for use with the WLS case has been presented here. As discussed, the particular weighting scheme used in any instance depends on the behavior of the response variability for a particular analytical method. This area has received only sparse attention in the literature. Much work remains to be done to provide practicing chemists with information to guide their usage of this statistical methodology. Acknowledgment This work was supported in part by the National Science Foundation under grants ATM 8919513 and ATM 9215351. Literature Cited 1. ACS Committee on Environmental Improvement. Anal. Chem. 1980, 52, 2242–2249. 2. ACS Committee on Environmental Improvement. Anal. Chem. 1983, 55, 2210–2218. 3. Long, G. L.; Winefordner, J. D. Anal. Chem. 1983, 55, 712A–724A. 4. Boumans, P. W. J. M. Spectrochim. Acta 1978, 33B, 625–634. 5. Kaiser, H. Anal. Chem. 1970, 42(4), 26A–59A. 6. Hubaux, A.; Vos, G. Anal. Chem. 1970, 42, 849–855. 7. Currie, L. A. Anal. Chem. 1968, 40, 586–593. 8. Mitchell, D. G.; Garden, J. S. Talanta 1982, 29, 921–929. 9. Clayton, C. A.; Hines, J. W.; Elkins, P. D. Anal. Chem. 1987, 59, 2506–2514. 10. Williams, R. R. Anal. Chem. 1991, 63, 1638–1643. 11. Christian, G. D. Analytical Chemistry, 5th ed.; Wiley: New York, 1994; p 53. 12. Harris, D. C. Quantitative Chemical Analysis, 3rd ed.; Freeman: New York, 1991; p 601. 13. Detection in Analytical Chemistry: Importance, Theory, and Practice; Currie, L. A., Ed.; ACS Symposium Series 361: American Chemical Society: Washington, DC, 1988; Chapter 1. 14. IUPAC Committee on Spectrochemistry and Other Optical Procedures for Analysis. Pure Appl. Chem. 1976, 45, 99–103. 15. IUPAC Committee on Spectrochemistry and Other Optical Procedures for Analysis. Spectrochim. Acta 1978, 33B, 241–245. 16. Glaser, J. A.; Foerst , D. L.; McKee, G. D.; Quave, S. A.; Budde, W. L. Environ. Sci. Technol. 1981, 15, 1426–1435. 17. Wernimont, G. T. In Use of Statistics to Develop and Evaluate Analytical Methods; Spendley, W., Ed.; AOAC: Arlington, VA, 1985; Chapter 3. 18. IUPAC Committee on Analytical Nomenclature. Pure Appl. Chem. 1994, 66, 595–608. 19. Franke, J. P.; de Zeeuw, R. A.; Hakkert, R. Anal. Chem. 1978, 50, 1374–1380. 20. Schwartz, L. M. Anal. Chem. 1979, 51, 723–727. 21. Garden, J. S.; Mitchell, D. G.; Mills, W. N. Anal. Chem. 1980, 52, 2310–2315. 22. Watters, R. L.; Carroll, R. J.; Spiegelman, C. H. Anal. Chem. 1987, 59, 1639–1643. 23. Oppenheimer, L.; Capizzi, T. P.; Weppelman, R. M.; Mehta, H. Anal. Chem. 1983, 55, 638–643. 24. Hunter, J. S. J. AOAC Int. 1981, 64, 574–582. 25. Mandel, J.; Linnig, F. J. Anal. Chem. 1957, 29, 743–749. 26. MacTaggart, D. L.; Farwell, S. O. J. AOAC Int. 1992, 75, 594–608. 27. Currie, L. A. In Trace Residue Analysis: Chemometric Estimations of Sampling, Amount, and Error; Kurtz, D. A., Ed.; ACS Symposium Series 284; American Chemical Society: Washington, DC, 1985; Chapter 5. 28. Coleman, D. E.; Auses, J. P. Alcoa Technical Report No. 16-94-4, 1994. 29. Gibbons, R. D. Statistical Methods for Groundwater Monitoring; Wiley: New York, 1994; Chapter 5. 30. Smith, E. D.; Mathews, D. M. J. Chem. Educ. 1967, 44, 757–759. 31. Anderson, K. P.; Snow, R. L. J. Chem. Educ. 1967, 44, 756–757. 32. Meyers, R. H. Classical and Modern Regression with Applications; Duxbury: North Scituate, MA, 1986; Chapter 6.


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Realistic Detection Limits from Confidence Bands - American

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