Calibration point

Another way of interpreting absolute risk estimates is through the use of benchmarks or goals. Consider a company that operates 50 chemical process facilities. It is determined (through other, purely qualitative means) that Plant A has exhibited acceptable safety performance over the years. A QRA is performed on Plant A, and the absolute estimates are established as calibration points, or benchmarks, for the rest of the firm s facilities. Over the years, QRAs are performed on other facilities to aid in making decisions about safety maintenance and improvement. As these studies are completed, the results are carefully scrutinized against the benchmark facility. The frequency/consequence estimates are not the only results compared—the lists of major risk contributors, the statistical risk importance of safety systems, and other types of QRA results are also compared. As more and more facility results are accumulated, resources are allocated to any plant areas that are out of line with respect to the benchmark facility. 

From the calibration point of view, manometers can be divided into two groups. The first, fluid manometers, are fundamental instruments, where the indication of the measured quantity is based on a simple physical factor the hydrostatic pressure of a fluid column. In principle, such instruments do not require calibration. In practice they do, due to contamination of the manometer itself or the manometer fluid and different modifications from the basic principle, like the tilting of the manometer tube, which cause errors in the measurement result. The stability of high-quality fluid manometers is very good, and they tend to maintain their metrological properties for a long period. 

Table A2.4 Temperature subranges, deviation functions, and calibration points over the temperature range covered by platinum... 

Table A2.4 is a tabulation of subranges in the temperature region 13.8033 to 1234.93 K, together with the form of the deviation equation that applies to each, and the calibration points from which the coefficients in the deviation equation are to be obtained.

Sres is nearly independent of the number of calibration points and their concentration values x, cf. Figure 2.8. 

There are several ways to test the linearity of a calibration line one can devise theory-based tests, or use common sense. The latter approach is suggested here because if only a few calibration points are available on which to rest one s judgement, a graph of the residuals will reveal a trend, if any is present, while numerical tests need to be adjusted to have the proper sensitivity. It is advisable to add two horizontal lines offset by the measure of repeatability accepted for the method unless the apparent curvature is such that points near the middle, respectively the end of the x-range are clearly outside this reproducibility band, no action need to be taken. 

The horizontal intercepts the lower confidence limit function twice, i.e., if n is small, ires is large, and all calibration points are close together this can be guarded against by accepting Xloq only if it is smaller than... 

Figure 2.15. The limit of detection LOD the minimum signal/noise-ratio necessary according to two models (ordinate) is plotted against log 0(n) under the assumption of evenly spaced calibration points. The three sets of curves are for p = 0.1 (A), 0.05 (B), and 0.02 (C). The correct statistical theory is given by the fine points, while the model presented here is depicted with coarser dots. The widely used S/N = 3. .. 6 models would be represented by horizontals at y = 3. .. 6.

Nonlinear calibration curves are not forbidden, but they do complicate things quite a bit more calibration points are necessary, and interpolation from signal to concentration is often tedious. It would be improper to apply... 

Example 40 Trial calculations are done for Vanaiyt = (0.5) and Vprod = (0.9). .. (1.1) the required t-factors for p = 0.1 turn out to be 1.94. .. 1.66, which is equivalent to demanding n - 1. .. 120 calibration samples. Evidently, the case is critical and needs to be underpinned by experiments. Twenty or 30 calibration points might well be necessary if the calibration scheme is not carefully designed. 

Solution 20-30 calibration points are too many, if only for reasons of expended time. The analyst thus searches for a combination of perhaps n = 8 calibration points and m = 2 replications of the individual samples. This would provide the benefit of a check on every sample measurement without too much additional cost. An inspection of the various contributions in Eq. (2.17) toward the CI(Z) in Table 2.9 reveals the following for n = 8 and m = 2 ... 

Figure 4.39. Variability of back calculated concentrations Concbc- For each concentration range five calibration points were measured, over which a separate regression was run (not shown). Placebo tablets were spiked to the same concentrations and measured in triplicate (short horizontal lines gray trend lines in background). Ten repeat determinations of actual product (vertical bars = Mean + SD) were done. The bold lines pertain to compound A in all concentration ranges, the thin lines to compound B (middle concentration range only).

Determine the limit of detection LOD and limit of quantitation LOQ according to the interpolation at level y = a + CL of the regression line and its lower CL this is sensitive to the calibration-point pattern ... 

Display Table of Detailed Calibration Results) The simulated calibration points Xi, yi), the estimates Y = f(xi), and both the absolute and relative residuals are given. 

Display Calibration Check Graph) The calibration points as obtained under (Accept) above remain as is, but a renewed measurement yes, i of these same samples as unknowns is simulated vertical lines indicate the CL(X) that would be determined for these X = fiyes, mean)- The variability so observed mimics the within-calibration repeatability. Use the button [New Check] to repeat the simulation. 

UV.dat Section 2.2 A set of five calibration points (Absorbance vs. %-of-nominal Concentration) to be used with LINREG, see example used in Chapter 2, starting with Table 2.2. 

In the author s experience, such confirmation is not appropriate when the calibration range is greater than one order of magniffide or calibration points are not chosen carefully. The reason is that lower concentration levels of a calibration graph influence the correlation coefficient to a much smaller extent than higher concentrations. The hypothetical example of calibration results presented in Table 3 demonstrates this very simply. If the amount injected is correlated with the observed peak area in the second column in Table 3, the calibration graph in Figure 2 is obtained. 

TCDD, 2,3,7,8-tetrachlorodibenzo-p-dioxin, TCDF, 2,3,7,8-tetrachlorodibenzofuran. a less than the minimum level at which the analytical system gives recognizable signals and an acceptable calibration point. The MLs for each pollutant are specified in 40 CFR 430. 

Number of calibration points Influence parameters Precision of an analytical result x... 

The number of calibration points, p, their distance and measure at the concentration scale, the number of replicate measurements,... 

The errors are only or essentially in the measured values y as the dependent variable (bsx sy) and in addition, the errors sy are constant in the several calibration points (Homoscedasticity) ... 

If the basic conditions for the use of least squares fitting are not fulfilled (Fig. 6.6), especially if strongly deviating calibration points appear ( outliers or, more exactly, leverage points), the OLS method fails, i.e., the estimated calibration are biased and, therefore, are not representative for the relation between x and y. Whereas normality of the measured values can be frequently obtained by a suitable transformation, especially in the case of outlying calibration points, robust calibration has to be applied (Rousseeuw and Leroy [1987] Danzer [1989] Danzer and Currie [1998]). 

Another way is the robust parameter estimation on the basis of median statistics (see Sect. 4.1.2 Danzer [1989] Danzer and Currie [1998]). For this, all possible slopes between all the calibration points bij = (yj — yi)/(xj — X ) for j > i are calculated. After arranging the b j according to increasing values, the average slope can be estimated as the median by... 

---