Constant-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. 

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. 

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 ... 

It is detemrined experimentally an early study was the work of Andrews on carbon dioxide [1], The exact fonn of the equation of state is unknown for most substances except in rather simple cases, e.g. a ID gas of hard rods. However, the ideal gas law P = pkT, where /r is Boltzmaim s constant, is obeyed even by real fluids at high temperature and low densities, and systematic deviations from this are expressed in tenns of the virial series ... 

If regression analysis is used, a warning must be given. The method is purely mechanical and will therefore always yield a result. But if the functions chosen are not appropriate, the curve will show systematic deviations from the straight line of a wavelike character. The procedure must therefore be the following A sufficient number of (x,t) pairs is selected from the experimental material. They are inserted in the chro-nomal and the resulting, usually linear, equations are solved for the constants. 

After the choice of the constants has been made, e.g., by regression analysis, it is absolutely necessary either to draw a large-scale diagram of the relation between the real times and those shown by the clock or better to prepare a table containing the real times and those calculated by means of the chronomal. By this procedure any systematic deviations can at once be detected. If the constants have been found by regression analysis and systematic deviations still exist, this means that the mechanism of the clock is incorrect, i.e., that the functions /< have been incorrectly chosen. 

It is also assumed that, in the case of n-alkane mixtures, the ky-values are independent of the chain length of component j. Dimitrelis and Prausnitz[8] showed that there is a systematic deviation from the Carnahan and Starling[9] repulsive term as the difference in molecular size between two molecules increase. It is thus expected that the value of the interaction parameter ly will be related to the difference in size between the two molecules. It is assumed that the value of ly will approach a constant value when this difference becomes large. The interaction parameters for propane and n-butane were found by fitting this equation of state to the data mentioned above. The parameters are shown in table 2 ... 

Another shortcoming of the Baker theory is the fact that at low alcohol concentrations systematic deviations appear in the [A]//bo vs. [A] plots since the [A]/A o values tend to be constant or increase rather than decrease with decreasing alcohol concentration. 

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. 

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.

Eq. (32) contains no parameters characteristic of the filler. If it successfully represents data on one carbon black, then it should also do so for other blacks, irrespective of particle size or structure. The author s fragmentary data (212) on large particle furnace blacks indicates that this is true. The effects of carbon black structure at constant particle size are shown in Fig. 23 with data on the four carbons of Fig. 11. Although there is some scatter in the experimental points there is no systematic deviation by black structure. The line drawn through the data agrees nearly quantitatively with that of Harwood and Payne. 

For the chosen value of Cp = 1.8 x 10 4, a certain systematic deviation was observed in direct calculations, which the authors attribute either to the dependence of the values of constants on viscosity, or to the incorrectness of the assumption that the probability of chain transfer to the polymer is the same for all the units in the chain. 

Inadequate stoichiometry and poor calibration of the analytical device are interconnected problems. The kinetic model itself follows the stoichiometric rules, but an inadequate calibration of the analytical instrument causes systematic deviations. This can be illustrated with a simple example. Assume diat a bimolecular reaction, A + B P, is carried out in a liquid-phase batch reactor. The density of the reaction mixture is assumed to be constant. The reaction is started with A and B, and no P is present in the initial mixture. The concentrations are related by cp=CoA-Cj=Cob -Cb, i e. produced product, P, equals with consumed reactant. If the concentration of the component B has a calibration error, we get instead of the correct concentration cb an erroneous one, c n ncs, which does not fulfil the stoichiometric relation. If the error is large for a single component, it is easy to recognize, but the situation can be much worse calibration errors are present in several components and all of their effects are spread during nonlinear regression, in the estimation of the model parameters. This is reflected by the fact that the total mass balance is not fulfilled by the experimental data. A way to check the analytical data is to use some fonns of total balances, e.g. atom balances or total molar amounts or concentrations. For example, for the model reaction, A + B P, we have the relation ca+cb+cp -c()a+c0 -constant (again c0p=0). 

In the following section we describe some of these methods and how they may show the different effects of dispersion and systematic error. Then in the remaining two sections of the chapter we will discuss methods for treating heteroscedastic systems. In the first place, we will show how their non-constant standard deviation may be taken into account in estimating models for the kind of treatment we have already described. Then we will describe the detailed study of dispersion within a domain, often employed to reduce variation of a product or process. 

Eighty-nine per cent of the variation in the biological data can be explained with the receptor model. On the other hand, a systematic deviation can be noticed. Some compounds (denoted by bold type in Table 24.1) are described by the model as being too weak by a constant factor of 1.0 or 1.5 orders of magnitude when compared with the experimental affinities. The inspection of the molecular formulae identifies those showing the larger deviation as fluorobenzoyl derivatives and those with the smaller deviation as benzoyl derivatives. In all cases, substances which are not optimally described by the model, possess an electron-deficient aromatic system. This leads to the hypothesis that for binding at the real receptor a type of interaction may be important which is... 

Energy parameters and charge-transfer spectra of complexes of Br2 with several substituted pyridines have now been compared with the force constants k(Br—Br) and k(N- Br2) and with the properties of the donors. The complexes with orr/to-substituted pyridines show systematic deviations from the relations found to be valid for the other donors. The n.m.r. spectrum of the pyridine-Br2 complex in a nematic phase has been obtained and analysed. The results indicate that the donor-acceptor interaction is similar to that found in the solid state for other halogen-pyridine complexes. The equilibrium constant for the formation of the 1 1 complex of Br2 with hexamethyl-phosphortriamide (HMPA) has been determined by n.m.r. spectroscopy. Solid adducts of Br2 with poly-HMPA could also be prepared. 

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