Proportional-systematic deviation

A constant systematic deviation is given at a significance level, a, if the confidence interval, A q. does not include the value g = 0. In the case that the confidence interval A j does not contain the value = 1, a proportional systematic deviation holds. 

The standard addition method [35] represents a combination of calibration with the aid of both external and internal standards. In ion chromatography, it is used predominantly for the analysis of samples with difficult matrices. Matrix problems may lead to an increase in nonprecision and/or express themselves as constant or proportional systematic deviations of the analytical results. Matrix influence can be identified via calculation of the recovery function. In constant systematic deviation, the error is independent of the analyte component. Such a deviation will cause a parallel shift of the calibration line. A possible origin for this deviation might be a codetection of a matrix component. In proportional systematic deviations, the error depends on the concentration of the analyte component. This type of deviation results in a change of the slope of the calibration line. Deviations of this kind can be caused by individual sample preparation steps such as sample digestion and sample extraction, and also by matrix effects. Systematic deviations can be identified by standard addition and/or calculation of the recovery function. 

Mean value control cards, however, do not indicate any errors related to the sample matrix. It is possible to check the analytical process for matrix influences by determination of the recovery rate, although it only determines matrix-related, proportional systematic deviations. Constant systematic deviations remain undiscovered. Thus, the determination of the recovery rate is only a limited measure for controlling accuracy. If certified reference standards are used, the target value (Xget) is known. Together with the measuring value x eas, the recovery rate can be determined according to ... 

Figure 14-36 Illustration of the systematic difference A< between two methods at a given level Xh according to the regression line.The difference is a result of a constant systematic difference (intercept deviation from zero) and a proportional systematic difference (slope deviation from unity). The dotted line represents the diagonal X2 = XI.

Regarding the influence of the Schlieren effect on analytical procedures, the effect of differences between the temperature of the ambient environment and the sample on the absorbance was originally reported in relation to stopped-flow procedures [84]. The systematic deviation in absorbance was proportional to the temperature difference and was also dependent on the optical properties of the instrument. The effect also manifested itself when identical solutions at different temperatures were mixed [85] a linear dependence of the measurement on the temperature... 

Accuracy refers to the degree of closeness of the determined value to the nominal or known true value under prescribed conditions and is sometimes termed Irueness . A deviation from trueness can be constant (bias or constant systematic error) or vary with the size of the sample and/or the analyte concentration (proportional systematic error. Section 8.1). Precision, in contrast, describes the closeness of agreement (degree of scatter) between a series of measurements obtained from multiple sampling of the same homogenous sample under the prescribed conditions. For a more detailed discussion see Section 8.2. 

The FDA requirement on accuracy states (FDA 2001) Accuracy is determined by replicate analysis of samples containing known amounts of the analyte. Accuracy should be measured using a minimum of five determinations per concentration. A minimum of three concentrations in the range of expected concentrations is recommended. The mean value should be within 15 % of the actual value except at LLOQ, where it should not deviate by more than 20 %. The deviation of the mean from the true value serves as the measure of accuracy . As discussed in Section 8.4.2, such a definition implies that proportional systematic errors... 

The ideal value for W is 100%. With the aid of the recovery rate, the complete process can be assessed. If the true value is found, selectivity, accuracy, and robustness for this concentration level and this matrix under the given experimental conditions are proven. To identify a potential matrix influence, the sample matrix to be analyzed (which does not contain the analyte component) is divided into 10 equal sized portions and spiked with concentrated standard solutions, so that the component concentrations in the spiked samples and in the aqueous cahbration standards are the same. The spiked matrix samples are then analyzed with the corresponding analytical method. Ideally, the recovery function is a straight line with a residual standard deviation corresponding to the process standard deviation of the basic analytical method. In case a proportional systematic or constant systematic deviation is the result of the investigation of the matrix influence, the calibration function obtained with aqueous standards cannot be used for data evaluation the standard addition method has to be applied. 

Experiments, however, showed a systematic deviation from the predictions of Eqs. (7.10) and (7.11) [726, 753]. For example, the speed of thinning, —dh/dt, was observed to be proportional to 1/r - rather than l/r. In addition, usually a faster drainage than predicted by Eqs. (7.10) and (7.11) is observed. There are several reasons for these discrepancies ... 

If the analysis of a dynamic NMR spectrum is carried out by an iterative least-squares fitting method, the results are accompanied by estimates of the errors. These are proportional to the square root of the sum of the squares of the deviations of the theoretical spectrum from the experimental one, as well as to the sensitivity of the sum to changes in the value of the parameter considered within the region where the sum attains a minimum. These estimates constitute a measure of the effects, on the resulting parameter values, of random errors. They do not include any effects due to systematic errors such as those involved in the assumed values of certain parameters. Moreover, because of the nonlinearity of the least-squares fitting procedure employed, estimates of the errors have only an approximate statistical significance (Section IV.B.2 and reference 67). 

Systematic error — A kind of -> error that can be ascribed to a definite cause and even predicted if all the aspects of the measurement are known. It is also named determinate error. Systematic errors are usually related to the -> accuracy of a measurement since their deviations are generally of the same magnitude and unidirectional with respect to the true value. There are basically three sources of systematic errors instrumental errors, -> methodic errors, and operative errors [iii]. In addition, systematic errors can be classified as constant errors and - proportional errors [iv]. 

Comparing these results to the experiments, a general tendency can be observed in the case of the Xa results, the relative energies (to the HOMO level, for example) are systematically lower, than the experimental ones. These deviations from the experimental values are approximately proportional to the absolute energy values. The opposite tendency can be observed in the case of the ab initio calculations, where the relative energies are larger than the peak separations in the experiments. The difference in the basis sets and in the description of the electron-electron interaction are the possible sources of these deviations. 

As mentioned earlier, the matrix-related random interferences may not be independent. In this case, simple addition of the components is not correct, because a covariance term should be included. However, we can estimate the combined effect corresponding to the bracket term, which then strictly refers to the CV of the differences (CV b2-rb])- As in the case with constant standard deviations, information on the analytical components is usually available, either from duplicate sets of measurements or from quality control data, and the combined random bias term in the second bracket can then be derived by subtracting the analytical component from CV21. Systematic and random errors can then be determined, and it can be decided whether a new field method can replace an existing one. Figure 14-31 shows an example with proportional random errors around the regression line. 

Splitting of the systematic error into a constant and a proportional component depends on the assumption of linearity, which should be tested. A convenient test is a runs test, which in principle assesses whether the negative and positive deviations from the points to the line are randomly distrib-... 

Samples can be divided into two aliquots and analyzed, and the duplicates used for control purposes. This is a simple quality control procedure that does not require stable control materials and therefore can be used when stable materials are not available or as a supplemental procedure when stable control materials are available. The differences between duplicates are plotted on a range type of control chart that has limits calculated from the standard deviation of the differences. When the duplicates are obtained from the same method, this range chart monitors only random error and thus is not adequate for ensuring the accuracy of the analytical method. When the duplicates are obtained from two different laboratory methods, then the range chart actually monitors both random and systematic errors but cannot separate the two types of errors. The interpretation becomes more difficult, particularly when there are stable systematic differences or biases between the two analytical methods. Multiplicative factors may be necessary to deal with proportional differences, and additive factors may be necessary to allow for constant differences. Interpretation of observed differences becomes more qualitative nevertheless, this procedure still provides a useful way of monitoring the consistency of the data being generated by the laboratory. 

The following is an example of a mathematical/statistical calculation of a calibration curve to test for true slope, residual standard deviation, confidence interval and correlation coefficient of a curve for a fixed or relative bias. A fixed bias means that all measurements are exhibiting an error of constant value. A relative bias means that the systematic error is proportional to the concentration being measured i.e. a constant proportional increase with increasing concentration. 

As we have already seen, the systematic errors in a approach zero as B approaches 90°, and may be eliminated by use of the proper extrapolation function. The magnitude of these errors is proportional to the slope of the extrapolation line and, if these errors are small, the line will be quite flat. In fact, if we purposely increase the systematic errors, say, by using a slightly incorrect value of the camera radius in our calculations, the slope of the line will increase but the extrapolated value of Oq will remain the same. The random errors involved in measuring line positions show up as random errors in a, and are responsible for the deviation of the various points from the extrapolation line. The random errors in a also decrease in magnitude as B increases, due essentially to the slow variation of sin 6 with B at atge ahgtes. 

Figure 3.13 Model parameter estimates as a function of the prior standard deviation for clearance. A 1-compartment model with absorption was fit to the data in Table 3.5 using a proportional error model and the SAAM II software system. Starting values were 5000 mL/h, 110 L, and 1.0 per hour for clearance (CL), volume of distribution (Vd), and absorption rate constant (ka), respectively. The Bayesian prior mean for clearance was fixed at 4500 mL/h while the standard deviation was systematically varied. The error bars represent the standard error of the parameter estimate. The open symbols are the parameter estimates when prior information is not included in the model.

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