National Academies Press: OpenBook
« Previous: Appendix C: Commercial Sources of Ambient Ionization Mass Spectrometry Instrumentation
Suggested Citation:"Appendix D: Statistical Calibration." National Research Council. 2012. Assessment of Agent Monitoring Strategies for the Blue Grass and Pueblo Chemical Agent Destruction Pilot Plants. Washington, DC: The National Academies Press. doi: 10.17226/13431.
×

Appendix D
Statistical Calibration

In the following, the ordinary least squares (OLS) and the weighted least squares (WLS) approaches to estimating the calibration function and related interval are reviewed.

OLS ESTIMATION1

As preparation for the following discussion, consider the relationship between response signal y and spiking concentration x in the region of the detection and quantification limits as a linear function of the form

image

where image is a random variable that describes the deviations from the regression line, distributed with mean 0 and constant variance image. The assumption of constant variance is not critical to this approach and will be relaxed in a later section; however, it is useful to simplify the initial exposition. The sample regression coefficient

image

provides an estimate of the population parameter β1 (i.e., the slope of the calibration function). The sample intercept

image

provides an estimate of the population parameter β0 (i.e., the intercept of the calibration function, which describes the mean instrument response or measured concentration when

img

1This section is adapted from Gibbons, 1995.

Suggested Citation:"Appendix D: Statistical Calibration." National Research Council. 2012. Assessment of Agent Monitoring Strategies for the Blue Grass and Pueblo Chemical Agent Destruction Pilot Plants. Washington, DC: The National Academies Press. doi: 10.17226/13431.
×

true concentration x = 0). An unbiased sample estimate of image (i.e., the variance of deviations from the population regression line) is given by

image

where image.

WLS ESTIMATION2

When variance is not constant, as is typically the case in the calibration setting, then the previous OLS solution for constant or “homoscedastic” errors no longer applies. There are several approaches to this problem, but in general the most widely accepted approach is to model the variance as a function of true concentration x and to then use the estimated variances as weights in estimating the calibration parameters, which are now denoted as image and image.

The weighted least squares regression of measured concentration or instrument response (y) on true concentration (x) is denoted by

image

where

image

img

2This section is adapted from Gibbons and Bhaumik, 2001.

Suggested Citation:"Appendix D: Statistical Calibration." National Research Council. 2012. Assessment of Agent Monitoring Strategies for the Blue Grass and Pueblo Chemical Agent Destruction Pilot Plants. Washington, DC: The National Academies Press. doi: 10.17226/13431.
×

image

and the weight image is the variance for sample i, which is computed from those samples with true concentration xi = x. The weighted residual variance is

image

ESTIMATING THE WEIGHTS3

When the number of replicates at each concentration is small, as is typically the case, or there are no replicates, the observed variance at each concentration provides a poor estimate of the true population variance. Two better alternatives are to (1) model the observed variance or standard deviation as a function of true concentration or (2) model the sum of squared residuals as a function of concentration. The latter approach can also be performed iteratively, in which improved estimates of β0 and β1 are obtained from weights computed from the current sum of squared residuals on each iteration. These new estimates of β0 and β1 are in turn used to obtain a new set of estimated weights and so forth until convergence. This algorithm is commonly termed “iteratively reweighted least squares.” An essential element of either approach is to identify a plausible model for the variance function. The following sections consider a few models that are particularly well suited to this problem.

Rocke and Lorenzato Model

To measure the true concentration of an analyte (x), the traditional simple linear calibration model, image with the standard normality assumption on errors, is not appropriate, as it fails to explain increasing measurement variation with increasing analyte concentration, which is commonly observed in analytical data. To overcome this situation, one may propose a log linear model, for example, image, where η is a normal variable with mean 0 and standard deviation image. This model also fails to explain near-constant measurement variation of y for low true concentration level x (Rocke and Lorenzato, 1995). To better model the calibration curve, Rocke and Lorenzato (1995) proposed a combined model that has both types of errors:

img

3This section is adapted from Gibbons and Bhaumik, 2001.

Suggested Citation:"Appendix D: Statistical Calibration." National Research Council. 2012. Assessment of Agent Monitoring Strategies for the Blue Grass and Pueblo Chemical Agent Destruction Pilot Plants. Washington, DC: The National Academies Press. doi: 10.17226/13431.
×

image

where y is the r th measurement at the j th concentration level, xjr is the corresponding true concentration, and β0 and β1 are the fixed calibration parameters. In this model, η represents proportional error at higher true concentrations and the ejr ’s are the additive errors that are present primarily at low concentrations. Now assume that η and the ejr ’s are independent and follow normal distributions with means 0 ’s and variances image and image, respectively. Data near zero (i.e., x; 0) determine image, and data for large concentrations determine image. The model specification also indicates that errors at larger concentrations are lognormally distributed and at low concentrations are normally distributed, which agrees with common experience.

In their original paper, Rocke and Lorenzato (1995) derived the maximum likelihood estimators for their model based on maximizing the likelihood function:

image

These computations require complex numerical evaluation of the required integrals. Alternatively, Gibbons et al. (1997) and Rocke and Durbin (1998) have described a WLS solution that involves the following algorithm:

1.Use OLS regression to find initial estimates of β0 and β1 by fitting the linear model:

image

2.Using the sample standard deviation of the lowest concentration as an estimate for image and the standard deviation of the log of the replicates at the highest concentration as an initial estimate for image, refit the model in step 1 using WLS with weights equal to

image

3.Using the new estimates of β0 and β1, compute the predicted response image and standard error of the calibration curve at each concentration x :

image

where m(x) is the number of replicates for concentration x

Suggested Citation:"Appendix D: Statistical Calibration." National Research Council. 2012. Assessment of Agent Monitoring Strategies for the Blue Grass and Pueblo Chemical Agent Destruction Pilot Plants. Washington, DC: The National Academies Press. doi: 10.17226/13431.
×

4.Using WLS, fit the variance function:

image

where

image

and

image

using weights:

image

5.Compute the new estimates of image and

image

6.Iterate until convergence.

In general, this algorithm will converge to positive values of γ and δ. Note that this algorithm uses WLS to compute the parameters of the calibration curve (β0 and β1) as well as the parameters of the variance function (γ and δ). In this way, the lowest concentrations with the smallest variances provide the greatest weight in the estimation. The net result is to not sacrifice precision in estimating the calibration function and corresponding interval estimates at low levels by including higher concentrations in the analysis. This is quite useful if the interest is in low-level detection and quantification.

Exponential Model

An alternative parameterization of the variance function involves modeling the relationship between σ and x as an exponential function of the following form:

image

Although less well theoretically motivated than the Rocke and Lorenzato model, the exponential model provides excellent fit to a wide variety of analytical data (see Gibbons

Suggested Citation:"Appendix D: Statistical Calibration." National Research Council. 2012. Assessment of Agent Monitoring Strategies for the Blue Grass and Pueblo Chemical Agent Destruction Pilot Plants. Washington, DC: The National Academies Press. doi: 10.17226/13431.
×

et al., 1997). The model can be applied either to the observed standard deviations at each concentration or, iteratively, to the sum of squared residuals. For estimating a0 and a1, the traditional approach involves substituting sx for σx and using nonlinear least squares (e.g., Gauss-Newton) or using OLS regression of the natural log transformed observed standard deviation on true concentration (Snedecor and Cochran, 1989). Similarly, WLS can also be used on the regression of image on x using weights

image

Linear Model

The linear model has also been used to model the variance function (Currie, 1995). The linear model is of the form

image

The primary disadvantage of the linear model is that the small sampling fluctuations in the observed sample variance at each concentration can lead to a negative intercept (i.e., a0 < 0) and negative variance estimates. This can lead to improper detection and quantification limit estimates and corresponding interval estimates. As such, the linear model is generally not recommended for routine use. This is not a problem for either of the two preceding models, which can mimic a linear model if required.

ITERATIVELY REWEIGHTED LEAST SQUARES ESTIMATION

An alternative to modeling the observed variance at each concentration is to model the squared residuals as a function of x, and then to use this estimated variance function to obtain weights that are then used in estimating the regression coefficients. This process is iterated until convergence, hence the term “iteratively reweighted least squares”’ (Carroll and Rupert, 1988). As noted by Neter et al. (1990) the methods of maximum likelihood and weighted least squares lead to the same estimators for linear regression models of the form considered here. The previous example of the WLS estimator for the Rocke and Lorenzato model is an example of iteratively reweighted least squares. The general algorithm is as follows:

1. Use OLS estimation to find initial estimates of β0 and β1 by fitting the linear model

image

Suggested Citation:"Appendix D: Statistical Calibration." National Research Council. 2012. Assessment of Agent Monitoring Strategies for the Blue Grass and Pueblo Chemical Agent Destruction Pilot Plants. Washington, DC: The National Academies Press. doi: 10.17226/13431.
×

2. Using the OLS estimates of β00 and β1, compute the predicted response image and the standard error of the calibration curve at each concentration x:

image

where m(x) is the number of replicates for concentration x.

3. Using an appropriate model for the variance function, fit the variance function to the sum of squared residuals:

image

4. Using the provisional weights:

image

recompute β0 and β1 using WLS.

5. Iterate until convergence.

WLS PREDICTION INTERVALS

For WLS estimates of β0 and β1 the estimated variance for a predicted value image is

image

where kj is the estimated variance at concentration xj. An upper (1 -α)100 percent confidence interval for image (i.e., an upper prediction limit for a new measured concentration or instrument response at true concentration xj) is

image

Suggested Citation:"Appendix D: Statistical Calibration." National Research Council. 2012. Assessment of Agent Monitoring Strategies for the Blue Grass and Pueblo Chemical Agent Destruction Pilot Plants. Washington, DC: The National Academies Press. doi: 10.17226/13431.
×

where t is the upper (1 -α)100 percent point of Student’s t -distribution on n - 2 degrees of freedom. For example, at x = 0, the upper prediction limit (UPL) is

image

where image is the variance of the measured concentrations or instrument responses for a sample that does not contain the analyte.

CONFIDENCE REGION FOR AN UNKNOWN TRUE CONCENTRATION

In general practice, measured concentrations are reported as if they are true concentrations, without the benefit of an index of uncertainty. There are two problems with this. First, the measured concentration may provide a biased estimate of the true concentration to the extent that image. Second, even in the absence of bias, the measured concentration is only an estimate of the true concentration, and it has a level of uncertainty that is ignored by simply presenting the measured concentration in the absence of a proper uncertainty interval.

To provide an estimate of true concentration x from measured concentration y for the Rocke and Lorenzato model, compute

image

Bhaumik and Gibbons (2005) derived the asymptotic variance of image as

image

where n0 is the number of calibration measurements at or near zero. As expected, the variance of image depends on x and increases with increasing concentration.

Bhaumik and Gibbons (2005) developed a confidence interval for an unknown true concentration x given a measured concentration y, separately for true concentrations at or near x = 0 and for larger non-zero true concentrations. For a low-level true concentration x0, the (1 -α)100 percent confidence region for x0 is maximage. To construct a confidence interval for an unknown higher level concentration x, they use a lognormal approximation. Let

image

Suggested Citation:"Appendix D: Statistical Calibration." National Research Council. 2012. Assessment of Agent Monitoring Strategies for the Blue Grass and Pueblo Chemical Agent Destruction Pilot Plants. Washington, DC: The National Academies Press. doi: 10.17226/13431.
×

image

where c3 is the approximate variance of

image

The quantity

image

is distributed N(0,1) so that the (1 -α)100 percent confidence region for x0 is obtained by iteratively solving

image

In addition to reporting measured concentrations, the point estimate of x and its 95 percent confidence interval should also be routinely reported; it can be used for the purpose of making both detection decisions and comparisons to regulatory standards. If, for example, the lower 95 percent confidence limit is greater than zero, there is 95 percent confidence that the true concentration is greater than zero. By contrast, if the upper 95 percent confidence limit is less than a regulatory standard, there is 95 percent confidence that the true concentration is less than the regulatory standard, and the corresponding (and potentially less costly) disposal options can be pursued.

DETECTION AND QUANTIFICATION

The previously described WLS prediction limit image corresponds to the concept of a decision limit LC defined by Currie (1968) for the case in which the data arise from a calibration experiment, μ and σ at x = 0 are unknown, and one wishes to make a detection decision for a single future test sample. Measured concentrations (or instrument responses) that exceed the UPL should yield the binary decision of “detected” with (1 - α)100 percent confidence. Note that when the true concentation x = LC, the probability of exceeding the UPL is only 50 percent. As such, Currie defined the

Suggested Citation:"Appendix D: Statistical Calibration." National Research Council. 2012. Assessment of Agent Monitoring Strategies for the Blue Grass and Pueblo Chemical Agent Destruction Pilot Plants. Washington, DC: The National Academies Press. doi: 10.17226/13431.
×

detection limit LD as the 95 percent UPL for a true concentration at LC. The WLS estimate of LC is therefore

image

and the WLS estimate of LD is

image

Note that in order to compute LC and LD, one must have estimates of image and image, which are often unavailable and must be estimated using a model of standard deviation versus concentration, as previously described. The final estimates of LC and LD are obtained from simple repeated substitution beginning from LC =0 and LD = LCuntil convergence (i.e., change of less than 10-4 in estimates of LC and LD on successive iterations).

Finally, Currie (1968) defined the limit of determination LQ as the concentration at which the signal-to-noise ratio is 10 to 1. In the current context, one can estimate LQ by identifying the true concentration at which the estimated standard deviation is one-tenth of its magnitude. Again, a simple iterative approach generally performs quite well (Gibbons and Coleman, 2001; Gibbons et al., 1997).

REFERENCES

Bhaumik, D. and R. Gibbons. 2005. Confidence regions for random-effects calibration curves with heteroscedastic errors. Technometrics 47(2): 223-231.

Carroll, R. and D. Rupert. 1988. Transformation and Weighting in Regression. Boca Raton, FL: CRC Press.

Currie, L. 1968. Limits for qualitative detection and quantitative determination: Application to radiochemistry. Analytical Chemistry 40(3): 586-593.

Currie, L. 1995. Nomenclature in evaluation of analytical methods including detection and quantification capabilities. Pure and Applied Chemistry 67: 1699-1723.

Suggested Citation:"Appendix D: Statistical Calibration." National Research Council. 2012. Assessment of Agent Monitoring Strategies for the Blue Grass and Pueblo Chemical Agent Destruction Pilot Plants. Washington, DC: The National Academies Press. doi: 10.17226/13431.
×

Gibbons, R. 1995. Some statistical and conceptual issues in the detection of low-level environmental pollutants. Environmental and Ecological Statistics 2(2): 125-145.

Gibbons, R. and D. Bhaumik. 2001. Weighted random-effects regression models with applications to interlaboratory calibration. Technometrics 43(2): 192-198.

Gibbons, R. and D. Coleman. 2001. Statistical Methods for Detection and Quantification of Environmental Contamination. New York, N.Y.: John Wiley & Sons, Inc.

Gibbons, R., D. Coleman, and R. Maddalone. 1997. An alternate minimum level definition for analytical quantification. Environmental Science & Technology 31(7): 2071-2077.

Neter, J., W. Wasserman, and M. Kutner. 1990. Applied Linear Regression Models, 2nd Ed. Homewood, IL: McGraw-Hill/Irwin.

Rocke, D. and B. Durbin. 1998. Models and Estimators for Analytical Measurement Methods with Non-constant Variance. Davis, Calif.: University of California at Davis, Center for Image Processing and Integrated Computing.

Rocke, D. and S. Lorenzato. 1995. A two-component model for measurement error in analytical chemistry. Technometrics 37(2): 176-184.

Snedecor, G. and W. Cochran. 1989. Statistical Methods. Ames, IA: Iowa University Press.

Suggested Citation:"Appendix D: Statistical Calibration." National Research Council. 2012. Assessment of Agent Monitoring Strategies for the Blue Grass and Pueblo Chemical Agent Destruction Pilot Plants. Washington, DC: The National Academies Press. doi: 10.17226/13431.
×

This page is blank

Suggested Citation:"Appendix D: Statistical Calibration." National Research Council. 2012. Assessment of Agent Monitoring Strategies for the Blue Grass and Pueblo Chemical Agent Destruction Pilot Plants. Washington, DC: The National Academies Press. doi: 10.17226/13431.
×
Page 151
Suggested Citation:"Appendix D: Statistical Calibration." National Research Council. 2012. Assessment of Agent Monitoring Strategies for the Blue Grass and Pueblo Chemical Agent Destruction Pilot Plants. Washington, DC: The National Academies Press. doi: 10.17226/13431.
×
Page 152
Suggested Citation:"Appendix D: Statistical Calibration." National Research Council. 2012. Assessment of Agent Monitoring Strategies for the Blue Grass and Pueblo Chemical Agent Destruction Pilot Plants. Washington, DC: The National Academies Press. doi: 10.17226/13431.
×
Page 153
Suggested Citation:"Appendix D: Statistical Calibration." National Research Council. 2012. Assessment of Agent Monitoring Strategies for the Blue Grass and Pueblo Chemical Agent Destruction Pilot Plants. Washington, DC: The National Academies Press. doi: 10.17226/13431.
×
Page 154
Suggested Citation:"Appendix D: Statistical Calibration." National Research Council. 2012. Assessment of Agent Monitoring Strategies for the Blue Grass and Pueblo Chemical Agent Destruction Pilot Plants. Washington, DC: The National Academies Press. doi: 10.17226/13431.
×
Page 155
Suggested Citation:"Appendix D: Statistical Calibration." National Research Council. 2012. Assessment of Agent Monitoring Strategies for the Blue Grass and Pueblo Chemical Agent Destruction Pilot Plants. Washington, DC: The National Academies Press. doi: 10.17226/13431.
×
Page 156
Suggested Citation:"Appendix D: Statistical Calibration." National Research Council. 2012. Assessment of Agent Monitoring Strategies for the Blue Grass and Pueblo Chemical Agent Destruction Pilot Plants. Washington, DC: The National Academies Press. doi: 10.17226/13431.
×
Page 157
Suggested Citation:"Appendix D: Statistical Calibration." National Research Council. 2012. Assessment of Agent Monitoring Strategies for the Blue Grass and Pueblo Chemical Agent Destruction Pilot Plants. Washington, DC: The National Academies Press. doi: 10.17226/13431.
×
Page 158
Suggested Citation:"Appendix D: Statistical Calibration." National Research Council. 2012. Assessment of Agent Monitoring Strategies for the Blue Grass and Pueblo Chemical Agent Destruction Pilot Plants. Washington, DC: The National Academies Press. doi: 10.17226/13431.
×
Page 159
Suggested Citation:"Appendix D: Statistical Calibration." National Research Council. 2012. Assessment of Agent Monitoring Strategies for the Blue Grass and Pueblo Chemical Agent Destruction Pilot Plants. Washington, DC: The National Academies Press. doi: 10.17226/13431.
×
Page 160
Suggested Citation:"Appendix D: Statistical Calibration." National Research Council. 2012. Assessment of Agent Monitoring Strategies for the Blue Grass and Pueblo Chemical Agent Destruction Pilot Plants. Washington, DC: The National Academies Press. doi: 10.17226/13431.
×
Page 161
Suggested Citation:"Appendix D: Statistical Calibration." National Research Council. 2012. Assessment of Agent Monitoring Strategies for the Blue Grass and Pueblo Chemical Agent Destruction Pilot Plants. Washington, DC: The National Academies Press. doi: 10.17226/13431.
×
Page 162
Next: Appendix E: Sampling Variability and Uncertainty Analyses »
Assessment of Agent Monitoring Strategies for the Blue Grass and Pueblo Chemical Agent Destruction Pilot Plants Get This Book
×
Buy Paperback | $46.00 Buy Ebook | $36.99
MyNAP members save 10% online.
Login or Register to save!
Download Free PDF

January 2012 saw the completion of the U.S. Army's Chemical Materials Agency's (CMA's) task to destroy 90 percent of the nation's stockpile of chemical weapons. CMA completed destruction of the chemical agents and associated weapons deployed overseas, which were transported to Johnston Atoll, southwest of Hawaii, and demilitarized there. The remaining 10 percent of the nation's chemical weapons stockpile is stored at two continental U.S. depots, in Lexington, Kentucky, and Pueblo, Colorado. Their destruction has been assigned to a separate U.S. Army organization, the Assembled Chemical Weapons Alternatives (ACWA) Element.

ACWA is currently constructing the last two chemical weapons disposal facilities, the Pueblo and Blue Grass Chemical Agent Destruction Pilot Plants (denoted PCAPP and BGCAPP), with weapons destruction activities scheduled to start in 2015 and 2020, respectively. ACWA is charged with destroying the mustard agent stockpile at Pueblo and the nerve and mustard agent stockpile at Blue Grass without using the multiple incinerators and furnaces used at the five CMA demilitarization plants that dealt with assembled chemical weapons - munitions containing both chemical agents and explosive/propulsive components. The two ACWA demilitarization facilities are congressionally mandated to employ noncombustion-based chemical neutralization processes to destroy chemical agents.

In order to safely operate its disposal plants, CMA developed methods and procedures to monitor chemical agent contamination of both secondary waste materials and plant structural components. ACWA currently plans to adopt these methods and procedures for use at these facilities. The Assessment of Agent Monitoring Strategies for the Blue Grass and Pueblo Chemical Agent Destruction Pilot Plants report also develops and describes a half-dozen scenarios involving prospective ACWA secondary waste characterization, process equipment maintenance and changeover activities, and closure agent decontamination challenges, where direct, real-time agent contamination measurements on surfaces or in porous bulk materials might allow more efficient and possibly safer operations if suitable analytical technology is available and affordable.

  1. ×

    Welcome to OpenBook!

    You're looking at OpenBook, NAP.edu's online reading room since 1999. Based on feedback from you, our users, we've made some improvements that make it easier than ever to read thousands of publications on our website.

    Do you want to take a quick tour of the OpenBook's features?

    No Thanks Take a Tour »
  2. ×

    Show this book's table of contents, where you can jump to any chapter by name.

    « Back Next »
  3. ×

    ...or use these buttons to go back to the previous chapter or skip to the next one.

    « Back Next »
  4. ×

    Jump up to the previous page or down to the next one. Also, you can type in a page number and press Enter to go directly to that page in the book.

    « Back Next »
  5. ×

    Switch between the Original Pages, where you can read the report as it appeared in print, and Text Pages for the web version, where you can highlight and search the text.

    « Back Next »
  6. ×

    To search the entire text of this book, type in your search term here and press Enter.

    « Back Next »
  7. ×

    Share a link to this book page on your preferred social network or via email.

    « Back Next »
  8. ×

    View our suggested citation for this chapter.

    « Back Next »
  9. ×

    Ready to take your reading offline? Click here to buy this book in print or download it as a free PDF, if available.

    « Back Next »
Stay Connected!