Calibration concentrations

Two slopes are compared in a similar manner as are two means the simplest case is obtained when both calibrations are carried out using identical calibration concentrations (as is usual when SOPs are followed) the average variance V u is used in a t-test ... 

Figure 2.8. The slopes and residuals are the same as in Figure 2.4 (50,75,100, 125, and 150% of nominal black squares), but the A -values are more densely clustered 90, 95, 100, 105, and 110% of nominal (gray squares), respectively 96, 98, 100, 102, and 104% of nominal (white squares). The following figures of merit are found for the sequence bottom, middle, top the residual standard deviations +0.00363 in all cases the coefficients of determination 0.9996, 0.9909, 0.9455 the relative confidence intervals of b +3.5%, +17.6%, 44.1%. Obviously the extrapolation penalty increases with decreasing Sx.x, and can be readily influenced by the choice of the calibration concentrations. The difference in Sxx (6250, 250 resp. 40) exerts a very large influence on the estimated confidence limits associated with a, b, Y(x), and X( y ).

Figure 2.10. For n- 5, 10, resp. 20 the estimated CI( j,) and CI(T) (bold) are plotted versus X. The left figure shows the absolute values t j l, while the right one depicts the relative ones, namely 100 t sy/Y in %. At x = 130 one finds Y = 0.553 with a Cl of +0.013 (+2.4%, circles). It is obvious that it would be inopportune to operate in the region below about 90% of nominal if relative precision were an issue (hatched bar). There are two remedies in such a case increase n (and costs) or reduce all calibration concentrations by an appropriate factor, say 10%. The bold lines give the estimates for the regression line (Eq. 2.16), while the thin ones are for individual points (Eq. 2.25).

Figure 2.11. For various combinations of n (5 10 resp. 20) and m (1, 2, resp. 3) the estimated CI(X) is plotted versus absorbance y. The left figure shows the absolute values t ixl, while the right figure depicts the relative ones, namely 100 t Sx/X in %, It is obvious that it would be inopportune to operate in the region below about 90% of nominal (in this particular case below y = 0.36 the absolute error for y = 0.36 is smaller than that for y = 0.6, but the inverse is true for the relative error, see arrows). There are three remedies increase n ox m (and costs), or reduce the calibration concentrations to shift the center of mass (x ean, ymean) below 100/0.42. At y = 0.6 and m - 1 (no replicates ) one finds X = 141.4 with a Cl of 3.39 (+2.4%, circle).

The FDA mandates that of all the calibration concentrations included in the validation plan, the lowest jc for which CV < 15% is the LOD (extrapolation or interpolation is forbidden). This bureaucratic rule results in a waste of effort by making analysts run unnecessary repeat measurements at each of a series of concentrations in the vicinity of the expected LOD in order to not end up having to repeat the whole validation because the initial estimate was off by + or - 20% extrapolation followed by a confirmatory series of determinations would do. The consequences are particularly severe if validation means repeating calibration runs on several days in sequence, at a cost of, say, (6 concentrations) x (8 repeats) x (6 days) = 288 sample work-ups and determinations. 

Example 33 Assume that a simple measurement costs 20 currency units n measurements are performed for calibration and m for replicates of each of five unknown samples. Furthermore, the calibration series of n measurements must be paid for by the unknowns to be analyzed. The slope of the calibration line is > = 1.00 and the residual standard deviation is Sres = 3, cf. Refs. 75, 95. The n calibration concentrations will be evenly spaced between 50 and 150% of nominal, that is for n = 4 x, 50, 83, 117, 150. For an unknown corresponding to 130% of nominal, should be below 3.3 units, respectively < 3.3 = 10.89. What combination of n and m will provide the most economical solution Use Eq. (2.4) for S x and Eq. (2.18) for Vx-Solution since Sxx is a function of the x-values, and thus a function of n (e.g. n = 4 Sxx = 5578), solve the three equations in the given order for various combinations of n and m and tabulate the costs per result, c/5 then select the... 

Of all the requirements that have to be fulfilled by a manufacturer, starting with responsibilities and reporting relationships, warehousing practices, service contract policies, airhandUng equipment, etc., only a few of those will be touched upon here that directly relate to the analytical laboratory. Key phrases are underlined or are in italics Acceptance Criteria, Accuracy, Baseline, Calibration, Concentration range. Control samples. Data Clean-Up, Deviation, Error propagation. Error recovery. Interference, Linearity, Noise, Numerical artifact. Precision, Recovery, Reliability, Repeatability, Reproducibility, Ruggedness, Selectivity, Specifications, System Suitability, Validation. 

The algebraic solution is the classical fitting technique, as exemplified by the linear regression (Chapter 2). The advantage lies in the clear formulation of the numerical algorithm to be used and in the uniqueness of the solution. If one is free to choose the calibration concentrations and the number of... 

Define the calibration concentrations as being spaced linearly, logarithmically, or arbitrarily within the concentration range. 

Choose the endpoints of the calibration range the calibration concentrations are now displayed in the green field. 

The predicted concentrations should fall within the calibration concentration range. 

Patterns in the spectral residuals indicate an inadequacy- in the model and/or relatively large calibration concentration errors. 

Median effective concentration. Result of the 3 pM 2,3,7,8-TCDD calibration concentration prepared by the participants. 

Equation (4.20) was proposed by Hoskuldsson [65] many years ago and has been adopted by the American Society for Testing and Materials (ASTM) [59]. It generalises the univariate expression to the multivariate context and concisely describes the error propagated from three uncertainty sources to the standard error of the predicted concentration calibration concentration errors, errors in calibration instrumental signals and errors in test sample signals. Equations (4.19) and (4.20) assume that calibrations standards are representative of the test or future samples. However, if the test or future (real) sample presents uncalibrated components or spectral artefacts, the residuals will be abnormally large. In this case, the sample should be classified as an outlier and the analyte concentration cannot be predicted by the current model. This constitutes the basis of the excellent outlier detection capabilities of first-order multivariate methodologies. 

To replace the concentration term by the respective signal term one can use Eq. 2-59. From this result both variances for the special points x, = 0 and x, = x follow immediately. The first variance then characterizes the error of the calibration offset. v2 ( ). The second term, in brackets in Eq. 2-61, shows that we can expect minimum errors of the calibration process around the middle of the calibration (concentration) range, x, xc ... 

Principal components are primarily abstract mathematical entities and further details are described in Chapter 4. In multivariate calibration the aim is to convert these to compound concentrations. PCR uses regression (sometimes also called transformation or rotation) to convert PC scores to concentrations. This process is often loosely called factor analysis, although terminology differs according to author and discipline. Note that although the chosen example in this chapter involves calibrating concentrations to spectral absorbances, it is equally possible, for example, to calibrate the property of a material to its structural features, or the activity of a drug to molecular parameters. 

Calibration standards are prepared by dilution of the stock standard. Choose an approximately linear range for Pb on the instrument to be used (normally 0—lOpgml-1). Into four 100 ml volumetric flasks place the following a volume of Pb free iso-octane (see note 3) equal to the volume of gasoline in the diluted sample 0.2ml of the iodine in toluene solution 5 ml of the l%v/v solution of Aliquat 336 and such volumes of the stock solution (after dilution, with MIBK, to lOOpgml-1 Pb if necessary) as are necessary to give the desired calibration concentrations. One standard must contain no Pb. Dilute to the mark with MIBK. Prepare these calibration standards freshly as required do not store. 

TABLE 7-7. Calibration Concentrations (n = 2) Back-Calcnlated from the Ratio of the Peak Areas of ET-743 and ET-729... 

The difference, e.g., 5.0 - 1.4 in the column marked 20 ng/ml, must be attributed to the interpolation error, which in this case is due to the uncertainties associated with the four Rodbard parameters. For this type of analysis, the FDA-accepted quantitation limit is given by the lowest calibration concentration for which CV < 15%, in this case 5 ng/ml the cross indicates... 

As previously indicated, the determination of a drug concentrations in plasma specimens requires the construction of a calibration response curve. This curve is often constructed as a straight line from the measured peak response ratios (y,) plotted against their respective calibrator concentrations (x,). The drug concentration in a specimen or the apparent (back-calculated) calibrator concentration is obtained from a rearrangement of theequation for the calibration line (without error) x,- = (y,- — b)/m. An example of a calibration curve with back-calculated concentrations is given the first four columns of Table 10. 

It is obvious that the lower end of the calibration curve does not provide an accurate representation of the calibrator concentrations, a problem that does not exist at the higher end of the curve. This problem illustrates what can happen when there is violation of the assumption of equal (homogeneous) variance for the y, values. In bioanalytical methods, the largest component of the random error can often be attributed to volume errors. One often finds that the standard deviation of the response (y,) is proportional to the concentration (x,), that is, e,- = kx, where k is a constant. This violates the assumption of homogeneity... 

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