Calibration line

The scatter of the points around the calibration line or random errors are of importance since the best-fit line will be used to estimate the concentration of test samples by interpolation. The method used to calculate the random errors in the values for the slope and intercept is now considered. We must first calculate the standard deviation Sy/x, which is given by ... 

Equation 19 utilizes the Y-residuals, 1) — Y, where 1) are the points on the calculated best-fit line or the fitted 1) values. The appropriate number of degrees of freedom is A — 2 the minus 2 arises from the fact that linear calibration lines are derived from both a slope and an intercept which leads to a loss of two degrees of freedom. 

We will assume a perfect accelerometer calibration line that reads 3 mA for 1 g of acceleration. 

Hence, by plotting A [or log(l/T)] as ordinate, against concentration as abscissa, a straight line will be obtained and this will pass through the point c = 0, A = 0 (T = 100 per cent). This calibration line may then be used to determine unknown concentrations of solutions of the same material after measurement of absorbances. 

The IR calibration line shown in Fig. 36 may not be valid with the products formed in these series of experiments since in this case the densities and formula weight of the products may not be assumed the same as that of 7, which was used to construct the calibration line. Thus only H1 NMR spectroscopy has been employed to explore the stability of Si-H bonds in the presence of carbenium ions simulating propagating carbenium ions of isobutylene and styrene. 

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. 

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... 

The only recourse is to modify the recovery experiments above in the sense that the sample to be tested itself is used as a kind of blank, to which further analyte is spiked. This results in at least two measurements, namely untreated sample and spiked sample, which can then be used to establish a calibration line from which the amount of analyte in the untreated sample... 

Furthermore, there is the problem that the signal level to whieh one extrapolates need not necessarily be y = 0 if there is any interference by a matrix component, one would have to extrapolate to a level y > 0. This uncertainty can only be cleared if the standard addition line perfectly coincides with the calibration line obtained for the pure analyte in absence of the matrix, i.e. same slope and 100% recovery, see also Figure 3.2. This problem is extensively treated in Refs. 97-101. A modification is presented in Ref. 102. 

Once a mathematical model has been ehosen, there is the option of either fixing certain parameters (see Section 4.10) or fixing certain points, e.g., constraining the calibration line to go through the origin. 3- ,74,ii3... 

The sequence of the innovation, gain vector, variance-covariance matrix and estimated parameters of the calibration lines is shown in Figs. 41.1-41.4. We can clearly see that after four measurements the innovation is stabilized at the measurement error, which is 0.005 absorbance units. The gain vector decreases monotonously and the estimates of the two parameters stabilize after four measurements. It should be remarked that the design of the measurements fully defines the variance-covariance matrix and the gain vector in eqs. (41.3) and (41.4), as is the case in ordinary regression. Thus, once the design of the experiments is chosen... 

Calibration Slope and intercept of the calibration line at time t [1 c] where c is the concentration of the calibration standard measured at time t time constant of the the variations of slope and intercept... 

System equation of a calibration line with drift... 

In this section we derive a system equation which describes a drifting calibration line. Let us suppose that the intercept jC (/ + 1) at a time y + 1 is equal to xfj) at a time j augmented by a value a(j), which is the drift. By adding a non-zero system noise to the drift, we express the fact that the drift itself is also time dependent. This leads to the following equations [5,6] ... 

The model contains four parameters, the slope and intercept of the calibration line and two drift parameters a and p. All four parameters are estimated by applying the algorithm given in Table 41.10. Details of this procedure are given in Ref. [5]. 

Adequate sensitivity should be demonstrated and estimates of the limit of detection (LOD) and the limit of quantitation (LOQ) should be provided. The slope of the calibration line may indicate the ability of the method to distinguish the tme analyte concentration. The LOD of a method is the lowest analyte concentration that produces a reproducible response detectable above the noise level of the system. The LOQ is the lowest level of analyte that can be accurately and precisely measured. For a regulatory method, quantitation is limited by the lowest calibration standard. The techniques for these estimations should be described. 

Table 4 Comparison of the observed signal intensity with calculated response based on the best fit of a linear or a second-order calibration line...

Figure 3 Least squares calibration line for photometric detector. (From Dorschel, C. A., Ekmanis, J. L., Oberholtzer, J. E., Warren, Jr., F. V., and Bidlingmeyer, B. A., LC detectors evaluations and practical implications of linearity, Anal. Chem., 61, 951 A, 1989. Copyright American Chemical Society Publishers. With permission.)...

Houben [256] has compared the determination of flame-retardant elements Br, P, S, K, Cl and F in polycarbonate using commercial (X40 and UniQuant ) software. For the X40 method, a calibration line for each element in PC or PC/ABS blends was mapped for the conversion of intensities to concentrations. With the universal UniQuant method, sensitivity factors (ks) were calibrated with pure standards. The X40 method turned out to be more reliable than UniQuant for the determination of FRs in PC and PC/ABS blends, even in the case of calibration of k values with PC standards. Standard errors of 5 % were achieved for Br, P, S and K, and 20% for Cl and F the latter element could not be determined by means of UniQuant (Table 8.44). GFR PC cannot be quantified with these two methods, because of the heterogeneous nature of the composites. Other difficult matrices for XRF analysis are PBT, PS and PP compounds containing both BFRs and Sb203 (10-30wt %) due to self-absorption of Sb and interelement effects. 

Fig. 6.11. Calibration lines by using several CRM samples (S soil BHB-1, T Tibet soil GBW 8302, R river sediment CRM 320, Ga Gabbro MRG-1, A Andesite AGV-1, Gr Granodiorite GSP-1) and varying the sample weight (Zn 307.6 nm)...

For the basic evaluation of a linear calibration line, several parameters can be used, such as the relative process standard deviation value (Vxc), the Mandel-test, the Xp value [28], the plot of response factor against concentration, the residual plot, or the analysis of variance (ANOVA). The lowest concentration that has been used for the calibration curve should not be less than the value of Xp (see Fig. 4). Vxo (in units of %) and Xp values of the linear regression line Y = a + bX can be calculated using the following equations [28] ... 

The DL and QL for chromatographic analytical methods can be defined in terms of the signal-to-noise ratio, with values of 2 1-3 1 defining the DL and a value of 10 1 defining the QL. Alternatively, in terms of the ratio of the standard deviation of the blank response, the residual standard deviation of the calibration line, or the standard deviation of intercept (s) and slope (5) can be used [40, 42], where ... 

The basis upon which this concept rests is the very fact that not all the data follows the same equation. Another way to express this is to note that an equation describes a line (or more generally, a plane or hyperplane if more than two dimensions are involved. In fact, anywhere in this discussion, when we talk about a calibration line, you should mentally add the phrase ... or plane, or hyperplane... ). Thus any point that fits the equation will fall exactly on the line. On the other hand, since the data points themselves do not fall on the line (recall that, by definition, the line is generated by applying some sort of [at this point undefined] averaging process), any given data point will not fall on the line described by the equation. The difference between these two points, the one on the line described by the equation and the one described by the data, is the error in the estimate of that data point by the equation. For each of the data points there is a corresponding point described by the equation, and therefore a corresponding error. The least square principle states that the sum of the squares of all these errors should have a minimum value and as we stated above, this will also provide the maximum likelihood equation. 

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