Linear error propagation

Of all the requirements that have to be fulfilled by a manufacturer, starting with responsibilities and reporting relationships, warehousing practices, service contract policies, airhandUng equipment, etc., only a few of those will be touched upon here that directly relate to the analytical laboratory. Key phrases are underlined or are in italics Acceptance Criteria, Accuracy, Baseline, Calibration, Concentration range. Control samples. Data Clean-Up, Deviation, Error propagation. Error recovery. Interference, Linearity, Noise, Numerical artifact. Precision, Recovery, Reliability, Repeatability, Reproducibility, Ruggedness, Selectivity, Specifications, System Suitability, Validation. 

A related example of linearized error propagation during the isotope dilution measurement of lead in rock samples using the double-spike technique is given by Hamelin et al. (1985). o... 

Each stated uncertainty in this and other tables represents one estimated standard error, propagated to parameters from uncertainties of measurements of wave numbers the uncertainties of the latter measurements were provided by authors of papers [91,93] reporting those data, and the weight of each datum in the non-linear regression was taken as the reciprocal square of those uncertainties. As the reduced standard deviation of the fit was 0.92, so less than unity, the authors... 

The practical performance of a calibration thus is identical in all four cases of ion interference. One determines the isotope ratio mu/m+ "for a number of standards with a known mole ratio X/Y and performs a linear regression on these data. If there is interference by the unlabeled compound, the inverse mole ratio Y/X is used as a variable and if both unlabeled and labeled compound interfere, Y/(X + IY) is used as variable. In the latter case, the introduction of the experimental factor l, however, will be an important source of error propagation, especially if qj p. (Jonckheere, 1982 Jonck-heere et al., 1982). 

The round-off error propagation associated with the use of Shacham and Kehat s direct method for the solution of large sparse systems of linear equations is investigated. A reordering scheme for reducing error propagation is proposed as well as a method for iterative refinement of the solution. Accurate solutions for linear systems, which contain up to 500 equations, have been obtained using the proposed method, in very short computer times. 

Analytic expressions are available for assessing the propagation of errors through linear systems. Such approaches can be used as well when the variances are sufficiently small that the system can be linearized about its expectation value. Numerical techniques are generally needed to assess the propagation of errors through nonlinear systems. 

A multivariate linear regression can successfully be employed for moderately wide range calibration. As an example parabolic regression (17) 31,72) should be mentioned. Higher order polynomials have disadvantages in terms of degree of freedom and error propagation, and the results are less precise. 

The standard addition method is commonly used in quantitative analysis with ion-sensitive electrodes and in atomic absorption spectroscopy. In TLC this method was used by Klaus 92). Linear calibration with R(m=o)=o must also apply for this method. However, there is no advantage compared with the external standard method even worse there is a loss in precision by error propagation. The attainable precision is not satisfactory and only in the order of 3-5 %, compared to 0.3-0.5 % using the internal standard method 93). 

In the following we analyze the influence of errors on our approach and its accuracy and compare the results with those obtained by using linearized kinetics. We consider a nonlinear kinetic example for which a detailed analytical study is possible. We compare that exact solution with the first-order response theory based on appropriate tracer measurements, and also compare it with the response of the linearized kinetic example. An important interest here is in the effects of error propagations in the analysis due to the application to measurements of poor precision. 

An important consideration has been omitted in [3-5], which are devoted to this approach, and that is the usually rather large experimental errors associated with microarray measurements. It is important to know how such errors propagate in the calculations and their effects on the proper identification of the connectivity matrix. A simple example worked out in detail in section 12.5 shows the possible multiplicative effects of such errors in nonlinear kinetic equations. Until this problem is addressed, the approach of linearization must be viewed with caution. 

Figure 8.10 A simple example (linear calibration function with intercept A = 0) demonstrating the effects of both noise (or background) and sensitivity (5, Equation [8.61]) on the error propagation when predicting the content (concentration or amount of analyte) from the noisy signal by projecting the Y-axis onto the content axis (x-axis) through the calibration line. A low noise level combined with a high sensitivity S (slope of calibration line) will thus allow small changes in concentration (or amount) to be detectable, and by extension lead to a lower LOD (Equation [8.61]) this is the origin of the more colloquial use of the term sensitivity to indicate lower LOD (and possibly LLOQ) values.

Here (p) is a fimction of T, partial derivatives can be estimated from finite difference approximations of the measimed data, eg, d p)/dT = [J2p(Ti + i,4>i) - Y,P( ri,4>i)y -N(Ti + i - 7I)].ThevaluesofA< andATare A(p = m L and AT = mrL, where mr and m j, are the slopes of the linear gradients, known from the library preparation procedure. Making these substitutions shows that the error propagation for property p scales as... 

In the calibration routine, peak data of the unknowns are related to those of the calibration standards. From several calibration functions the most suitable for the task can be chosen. Single-level calibration (not to be mistaken for single-standard calibration) is the method of choice when the expected concentration in the unknowns is a fixed value, e.g. in content uniformity tests. It allows the use of a maximum ratio of unknowns to calibration standards with favorable error propagation. Only the results of unknowns within a narrow, user selectable range are accepted for calculations. The width of the range is defined by linearity and slope of the calibration function as established during method validation. 

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