Proportioning value errors

The method of standard additions can be used to check the validity of an external standardization when matrix matching is not feasible. To do this, a normal calibration curve of Sjtand versus Cs is constructed, and the value of k is determined from its slope. A standard additions calibration curve is then constructed using equation 5.6, plotting the data as shown in Figure 5.7(b). The slope of this standard additions calibration curve gives an independent determination of k. If the two values of k are identical, then any difference between the sample s matrix and that of the external standards can be ignored. When the values of k are different, a proportional determinate error is introduced if the normal calibration curve is used. 

Proportional systematic errors are detected with a Recovery Rate Chart, but not constant systematic errors (e.g. too high blank values). Additionally the spiked analyte might be bound to the matrix differently. This possibly results in a higher recovery rate for the spike than for the originally bound analyte. 

Figure 14-20 Outline of the relation between xl and x2 values measured by two methods subject to proportional random errors. A linear relationship between the target values is assumed.The xl, and x2, values are Gaussian distributed around...

In a situation with proportional random errors, a weighted modification of the correlation coefficient can be computed from sums of squared deviations for xl and x2 values as follows... 

A large proportion of errors in research and engineering as well as in the classroom are due to treating units as if they were not part of the number. In all scientific disciplines, it is essential that a unit be associated with every value in every calculation unless the value is a dimensionless quantity. On September... 

Type I error rates with FOCE-I were consistently near nominal values and were unaffected by number of subjects or number of observations per subject. With large residual variability (42%) and two observations per subject, Type I error rates for FOCE-I were higher than nominal, about 0.075 instead of 0.05. But when the number of observations was increased to four, the Type I error rate decreased to the nominal value and remained there as further increases in the number of observations were examined. Also, when the residual variance was modeled using a proportional residual error model, instead of an exponential residual variance model, the Type I error rate decreased. The major conclusion of this analysis was that FOCE-I should be preferred as an estimation method over FO-approximation and FOCE. 

Proportion Prediction error Sensitivity Specificity PPV NPV 1 -Value... 

Again F a can not be measured since is not known a priori, and in most cases it is assumed that F = F a in Equation [8.66d], thns leading to a potential proportional systematic error in the measured value of since generally Fjj,aa < Fa as a result of occlusion effects. However, an estimate of F a can be obtained by noting from Equation [8.66b] that ... 

Such non-zero intercepts in calibration curves are examples of bias errors. The detection and characterization of bias errors, often appearing together with simultaneous proportional systematic errors (e.g., those associated with values of F < 1), is a major thrust of the approach (Section 8.5.Id) promoted by Youden and Cardone (Youden 1947,1960 Cardone 1983,1986 Ferrus 1987). 

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... 

We will cover later different measures of control performance but the most commonly used is integral over time of absolute error (ITAE). The higher the value of ITAE, the poorer the controller is at eliminating the error. Figure 3.19 shows the impact that switching from proportional-on-PV to proportional-on-error has on ITAE. Both algorithms have been tuned... 

Remember that if a proportional-only controller is configured as proportional-on-PV, it will not respond to changes in SP. This might be considered advantageous since it prevents the operator changing the SP to a value where the offset violates an alarm. However it might create problems with operator acceptance, in which case the proportional-on-error algorithm can be used. 

The modulus shift, Pj, is relatively small and subject to a proportionally large error. However, the shift is required for the superposition. This is in contrast to previous work with EPM and EPDM elastomers where such a shift was not required. The value of Pj decreases with increasing temperature the extent of decrease is similar to what is expected from the theory, PqTq/pT. However, this subject is not pursued any further, since it is somewhat outside the scope of the present study. 

A linear dependence approximately describes the results in a range of extraction times between 1 ps and 50 ps, and this extrapolates to a value of Ws not far from that observed for the 100 ps extractions. However, for the simulations with extraction times, tg > 50 ps, the work decreases more rapidly with l/tg, which indicates that the 100 ps extractions still have a significant frictional contribution. As additional evidence for this, we cite the statistical error in the set of extractions from different starting points (Fig. 2). As was shown by one of us in the context of free energy calculations[12], and more recently again by others specifically for the extraction process [1], the statistical error in the work and the frictional component of the work, Wp are related. For a simple system obeying the Fokker-Planck equation, both friction and mean square deviation are proportional to the rate, and... 

Another way to improve the error in a simulation, at least for properties such as the energy and the heat capacity that depend on the size of the system (the extensive properties), is to increase the number of atoms or molecules in the calculation. The standard deviation of the average of such a property is proportional to l/ /N. Thus, more accurate values can be obtained by running longer simulations on larger systems. In computer simulation it is unfortunately the case that the more effort that is expended the better the results that are obtained. Such is life ... 

Finally, values of sx are directly proportional to transmittance for indeterminate errors due to fluctuations in source intensity and for uncertainty in positioning the sample cell within the spectrometer. The latter is of particular importance since the optical properties of any sample cell are not uniform. As a result, repositioning the sample cell may lead to a change in the intensity of transmitted radiation. As shown by curve C in Figure 10.35, the effect of this source of indeterminate error is only important at low absorbances. This source of indeterminate errors is usually the limiting factor for high-quality UV/Vis spectrophotometers when the absorbance is relatively small. 

A visual inspection of a two-sample chart provides an effective means for qualitatively evaluating the results obtained by each analyst and of the capabilities of a proposed standard method. If no random errors are present, then all points will be found on the 45° line. The length of a perpendicular line from any point to the 45° line, therefore, is proportional to the effect of random error on that analyst s results (Figure 14.18). The distance from the intersection of the lines for the mean values of samples X and Y, to the perpendicular projection of a point on the 45° line, is proportional to the analyst s systematic error (Figure 14.18). An ideal standard method is characterized by small random errors and small systematic errors due to the analysts and should show a compact clustering of points that is more circular than elliptical. 

Some of the inherent advantages of the feedback control strategy are as follows regardless of the source or nature of the disturbance, the manipulated variable(s) adjusts to correct for the deviation from the setpoint when the deviation is detected the proper values of the manipulated variables are continually sought to balance the system by a trial-and-error approach no mathematical model of the process is required and the most often used feedback control algorithm (some form of proportional—integral—derivative control) is both robust and versatile. 

The goal of any statistical analysis is inference concerning whether on the basis of available data, some hypothesis about the natural world is true. The hypothesis may consist of the value of some parameter or parameters, such as a physical constant or the exact proportion of an allelic variant in a human population, or the hypothesis may be a qualitative statement, such as This protein adopts an a/p barrel fold or I am currently in Philadelphia. The parameters or hypothesis can be unobservable or as yet unobserved. How the data arise from the parameters is called the model for the system under study and may include estimates of experimental error as well as our best understanding of the physical process of the system. 

---