Calibration graph errors

In isotope dilution inductively coupled plasma-mass spectrometry (ID-ICP-MS) the spike, the unspiked and a spiked sample are measured by ICP-MS in order to determine the isotope ratio. Using this technique, more precise and accurate results can be obtained than by using a calibration graph or by standard addition. This is due to elimination of various systematic errors. Isotopes behave identically in most chemical and physical processes. Signal suppression and enhancement due to the matrix in ICP-MS affects both isotopes equally. The same holds for most long-term instrumental fluctuations and drift. Accuracy and precision obtained with ID-ICP-QMS are better than with other ICP-QMS calibration... 

Either calibration graphs prepared from standards or the method of standard addition (p. 30) can be used. For the former, the standards should be as similar as possible in overall chemical composition to that of the samples so as to minimize errors caused by the reduction of other species or by variation in diffusion rates. Often, the limiting factor for quantitative work is the level of impurities present in the reagents used. 

The calibration problem in chromatography and spectroscopy has been resolved over the years with varying success by a wide variety of methods. Calibration graphs have been drawn by hand, by instruments, and by commonly used statistical methods. Each method can be quite accurate when properly used. However, only a few papers, for example ( 1,2,15,16,26 ), show the sophisticated use of a chemometric method that contains high precision regression with total assessment of error. 

We will describe an accurate statistical method that includes a full assessment of error in the overall calibration process, that is, (I) the confidence interval around the graph, (2) an error band around unknown responses, and finally (3) the estimated amount intervals. To properly use the method, data will be adjusted by using general data transformations to achieve constant variance and linearity. It utilizes a six-step process to calculate amounts or concentration values of unknown samples and their estimated intervals from chromatographic response values using calibration graphs that are constructed by regression. 

In the calibration problem two related quantities, X and Y, are investigated where Y, the response variable, is relatively easy to measure while X, the amount or concentration variable, is relatively difficult to measure in terms of cost or effort Furthermore, the measurement error for X is small compared with that of Y The experimenter observes a calibration set of N pairs of values (x, y ), i l,...,N, of the quantities X and Y, x being the known standard amount or concentration values and y the chromatographic response from the known standard The calibration graph is determined from this set of calibration samples using regression techniques Additional values of the dependent variable Y, say y., j l,, M, where M is arbitrary, are also observed whose corresponding X values, x. are the unknown quantities of interest The statistical literature on the calibration problem considers the estimation of these unknown values, x, from the observed and the... 

The basic shortcoming of statistically determined calibration graphs as found in the literature has been the omission of confidence bands. When properly constructed, information is available about the error in the calibration process and the resulting uncertainty in the estimated unknown amounts. 

Calibration graphs defined by data with non-negligible error have to be constructed by some kind of smoothing operation. In cases, in which the form of the underlying curve is known a priori, the latter can be approximated by minimizing the squares of deviations. Otherwise a spline function can be used (JJ[, 1 ). The spline function S(x) is constructed to minimize a measure of smoothness defined by... 

One of the opportunities that researchers rarely have is to be in a position of a direct comparison of methods used by several researchers that use the same data. Three of the fenvalerate "unknown" Datasets described and used elsewhere in this volume have been used as primary datasets by 3 research groups in the solution of the calibration problem. Two aspects of the calibration problem, namely, the accuracy of the calibration graph and the description of statistical error as shown by the estimated amount interval are examined here in comparing each of the calibration methods. 

Errors in trace analyses are usually hidden to all except those intimately involved in the sample collection and, later, in the bench analysis. In chromatography, especially, it is too easy to hide behind uncertain work because published research does not concern itself with exactly how the chromatographer makes his quantitative decisions. Today, with the advent of the microprocessor and with the use of black box instruments, the chromatographer knows even less about his calibration graph or line, or the error associated with it. In these instruments, a single point and the origin may determine the calibration graph. Similar problems exist in other modern instrumental analysis techniques. 

Full statistical evaluation of the calibration graph provides useful data about the method s performance characteristics over the applied calibration range such as the standard error of the procedure, sx, or the standard error of estimate, sy. 

Once the analytical parameters have been determined from the method development and the method has proven suitable for routine measurements, internal quality control (IQC) procedures must be established to maintain the validity of the analytical scheme and to better monitor potential sources of errors. The IQC used includes pre- and post-digestion controls, blank determination, half range of the calibration graph checking, and recovery rate of the samples. The stability of the recovery rate with time (Fig. 1.4) shows that the method is robust after using... 

A similar situation occurs when the detection limit is evaluated on the basis of signals obtained for blind samples (in principle not containing the analyte), to which known amounts of the anal3de, for example in the form of a standard solution, are added. The standard deviation of the crossing point of the extrapolated fragment of the calibration graph, y = bC + a, with the concentration (content) of the analyte is the basis for calculation of the detection limit of the procedure [17]. In such a case, error could be caused by the fact that the analyte added to the sample behaves differently to that present in real samples. 

Fig. 3.5 Systematic errors in the standard addition method Cx analyte concentration expected when the calibration dependence is nonlinear in the extrapolation region, c i concentration found when the calibration dependence is reconstructed by a linear calibration graph, c 2 concentration found when analyte in a sample is of different chemical form to that of analyte added to the sample...

For all antimony and bismuth analyses, standard addition as well as a calibration graph were applied to check for possible systematic errors. The results, obtained by the two methods, agreed within 10%. 

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