Assessment of Agent Monitoring Strategies for the Blue Grass and Pueblo Chemical Agent Destruction Pilot Plants (2012) / Chapter Skim
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Appendix D: Statistical Calibration
Appendix D: Statistical Calibration
Pages 151-162
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From page 151... ...
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 y = 0 1 x (1)
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From page 152... ...
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 0 w and 1w . The weighted least squares regression of measured concentration or instrument response ( y )
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From page 153... ...
, the traditional simple linear calibration model, y = 0 1 x e 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, y = xe , where is a normal variable with mean 0 and standard deviation .
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From page 154... ...
2. Using the sample standard deviation of the lowest concentration as an estimate for e and the standard deviation of the log of the replicates at the highest concentration as an initial estimate for , refit the model in step 1 using WLS with weights equal to 2 2 w( x)
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From page 155... ...
x = a0 e 1 (21) 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
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From page 156... ...
For estimating a0 and a1 , the traditional approach involves substituting s x for x 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)
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From page 157... ...
An upper (1 ) 100 percent ^ wj (i.e., an upper prediction limit for a new measured confidence interval for y concentration or instrument response at true concentration x j )
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( x x i =1 i i =1 i w 2 ) /k i 2 where s0 is the variance of the measured concentrations or instrument responses for a sample that does not contain the analyte.
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z/2 (38) 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.
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From page 160... ...
( x x i =1 i i =1 i w 2 ) /k i 2 Note that in order to compute LC and LD , one must have estimates of sL and C 2 s LD , which are often unavailable and must be estimated using a model of standard deviation versus concentration, as previously described.
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From page 161... ...
1998. Models and Estimators for Analytical Measurement Methods with Non-constant Variance.
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