Statistical methods error analysis

To improve the statistical precision, replicate samples are processed for each set of conditions. Our error analysis methods have been described previously (28, 54). The cited measurement uncertainties represent single standard deviations at the 68% confidence level. In the case of yield branching ratios these uncertainties follow directly from statistical random error analysis. Speculative estimates of the contributions from possible systematic mechanistic errors have not been included. 

It is hoped that the more advanced reader will also find this book valuable as a review and summary of the literature on the subject. Of necessity, compromises have been made between depth, breadth of coverage, and reasonable size. Many of the subjects such as mathematical fundamentals, statistical and error analysis, and a number of topics on electrochemical kinetics and the method theory have been exceptionally well covered in the previous manuscripts dedicated to the impedance spectroscopy. Similarly the book has not been able to accommodate discussions on many techniques that are useful but not widely practiced. While certainly not nearly covering the whole breadth of the impedance analysis universe, the manuscript attempts to provide both a convenient source of EK theory and applications, as well as illustrations of applications in areas possibly u amiliar to the reader. The approach is first to review the fundamentals of electrochemical and material transport processes as they are related to the material properties analysis by impedance / modulus / dielectric spectroscopy (Chapter 1), discuss the data representation (Chapter 2) and modeling (Chapter 3) with relevant examples (Chapter 4). Chapter 5 discusses separate components of the impedance circuit, and Chapters 6 and 7 present several typical examples of combining these components into practically encountered complex distributed systems. Chapter 8 is dedicated to the EIS equipment and experimental design. Chapters 9 through 12... 

A variety of statistical methods may be used to compare three or more sets of data. The most commonly used method is an analysis of variance (ANOVA). In its simplest form, a one-way ANOVA allows the importance of a single variable, such as the identity of the analyst, to be determined. The importance of this variable is evaluated by comparing its variance with the variance explained by indeterminate sources of error inherent to the analytical method. 

The relative error is the absolute error divided by the true value it is usually expressed in terms of percentage or in parts per thousand. The true or absolute value of a quantity cannot be established experimentally, so that the observed result must be compared with the most probable value. With pure substances the quantity will ultimately depend upon the relative atomic mass of the constituent elements. Determinations of the relative atomic mass have been made with the utmost care, and the accuracy obtained usually far exceeds that attained in ordinary quantitative analysis the analyst must accordingly accept their reliability. With natural or industrial products, we must accept provisionally the results obtained by analysts of repute using carefully tested methods. If several analysts determine the same constituent in the same sample by different methods, the most probable value, which is usually the average, can be deduced from their results. In both cases, the establishment of the most probable value involves the application of statistical methods and the concept of precision. 

The comparison of more than two means is a situation that often arises in analytical chemistry. It may be useful, for example, to compare (a) the mean results obtained from different spectrophotometers all using the same analytical sample (b) the performance of a number of analysts using the same titration method. In the latter example assume that three analysts, using the same solutions, each perform four replicate titrations. In this case there are two possible sources of error (a) the random error associated with replicate measurements and (b) the variation that may arise between the individual analysts. These variations may be calculated and their effects estimated by a statistical method known as the Analysis of Variance (ANOVA), where the... 

Even when the patterns are known to cluster, there remain difficult issues that must be addressed before a kernel-based approach can be used effectively. Two of the more fundamental conceptual issues are the number and size of clusters that should be used to characterize the pattern classes. These are issues for which there are no hard and fast answers. Despite the application of well-developed statistical methods, including squared-error indices and variance analysis, determining the number and size of clusters remains extremely formidable. 

A critical attitude towards the results obtained in analysis is necessary in order to appreciate their meaning and limitations. Precision is dependent on the practical method and beyond a certain degree cannot be improved. Inevitably there must be a compromise between the reliability of the results obtained and the use of the analyst s time. To reach this compromise requires an assessment of the nature and origins of errors in measurements relevant statistical tests may be applied in the appraisal of the results. With the development of microcomputers and their ready availability, access to complex statistical methods has been provided. These complex methods of data handling and analysis have become known collectively as chemometrics. 

Although HTS can process up to a million compounds per day, it has a high possibility of producing both false-negative and false-positive results. Replicate measurements in combination with statistical methods and careful data analysis may help to identify and reduce such errors [69]. 

Laboratory personnel are as intimately involved in TQM as any other employee and aspects of their work touch on all of these ten points. The manner in which TQM principles specifically apply to laboratory personnel, however, is unique to them. They are concerned about analysis methods, choice of laboratory equipment, error analysis, statistics, acquisition of laboratory samples, etc. 

As noted in the last section, the correct answer to an analysis is usually not known in advance. So the key question becomes How can a laboratory be absolutely sure that the result it is reporting is accurate First, the bias, if any, of a method must be determined and the method must be validated as mentioned in the last section (see also Section 5.6). Besides periodically checking to be sure that all instruments and measuring devices are calibrated and functioning properly, and besides assuring that the sample on which the work was performed truly represents the entire bulk system (in other words, besides making certain the work performed is free of avoidable error), the analyst relies on the precision of a series of measurements or analysis results to be the indicator of accuracy. If a series of tests all provide the same or nearly the same result, and that result is free of bias or compensated for bias, it is taken to be an accurate answer. Obviously, what degree of precision is required and how to deal with the data in order to have the confidence that is needed or wanted are important questions. The answer lies in the use of statistics. Statistical methods take a look at the series of measurements that are the data, provide some mathematical indication of the precision, and reject or retain outliers, or suspect data values, based on predetermined limits. 

First-order error analysis is a method for propagating uncertainty in the random parameters of a model into the model predictions using a fixed-form equation. This method is not a simulation like Monte Carlo but uses statistical theory to develop an equation that can easily be solved on a calculator. The method works well for linear models, but the accuracy of the method decreases as the model becomes more nonlinear. As a general rule, linear models that can be written down on a piece of paper work well with Ist-order error analysis. Complicated models that consist of a large number of pieced equations (like large exposure models) cannot be evaluated using Ist-order analysis. To use the technique, each partial differential equation of each random parameter with respect to the model must be solvable. 

Davis, J.M., Giddings, J.C. (1985) Statistical method for estimation of number of components from single complex chromatograms theory, computer-based testing, and analysis of errors. Anaf Chem. 57 2168-2177. 

In chemistry, as in many other sciences, statistical methods are unavoidable. Whether it is a calibration curve or the result of a single analysis, interpretation can only be ascertained if the margin of error is known. This section deals with fundamental principles of statistics and describes the treatment of errors involved in commonly used tests in chemistry. When a measurement is repeated, a statistical analysis is compulsory. However, sampling laws and hypothesis tests must be mastered to avoid meaningless conclusions and to ensure the design of meaningful quality assurance tests. Systematic errors (instrumental, user-based, etc.) and gross errors that lead to out-of-limit results will not be considered here. 

Linear regression is undoubtedly the most widely used statistical method in quantitative analysis (Fig. 21.3). This approach is used when the signal y as a function of the concentration x is linear. It stems from the principle that if many samples are used (generally dilutions of a stock solution), it becomes possible to perform variance analysis and estimate calibration error or systematic errors. 

To conclude, the stochastic and error-influenced character of environmental data requires the use of mathematical and statistical methods for further analysis. 

Statistical Method for Estimation of Number of Components from Single Complex Chromatograms Theory, Computer-Based Testing, and Analysis of Errors, J. M. Davis and J. C. Giddings, Anal. Chem., 57, 2168 (1985). 

This chapter has described the various techniques of ceramic powder characterization. These characteristics include particle shape, surface area, pore size distribution, powder density and size distribution. Statistical methods to evaluate sampling and analysis error were presented as well as statistical methods to compare particle size distributions. Chemical analytical characterization although veiy important was not discussed. Surface chemical characterization is discussed separately in a later chapter. With these powder characterization techniques discussed, we can now move to methods of powder preparation, each of which 3uelds different powder characteristics. 

The Fcal ratio is also the result of a statistical method known as the one-way Analysis of Variance (ANOVA), which assumes the model Xij = fij + sij for each observation Xij, where Sij are the independent and normally distributed random errors (sij N(0,a)), and which has as objective to test Ho = i i = fi2 =. .. = jXk. The results of the ANOVA procedure, which include the decomposition of the total sum of squares of the deviation of all observations around the general... 

It is necessary to study stability in solution in the solvent used to prepare sample solutions for injection in order to establish that the sample solution composition, especially the analyte concentration, does not change in the time elapsed between the preparation of the solution and its analysis by HPLC. This is a problem for only a few types of compound (e.g. penicillins in aqueous solution) when the sample solution is analysed immediately after the preparation of the sample solution to be injected. The determination of stability in solution is more of an issue when sample solutions are prepared and then analysed during the course of a long autosampler run. While the acceptance criteria for stabUity in solution may be expressed in rather bland terms by making a statement such as, e.g. the analyte was sufficiently stable in solution in the solvent used for preparing sample solutions for reliable analysis to be carried out , in practice it has to be shown that within the limits of experimental error, the result of the sample solution analysis by the HPLC method is the same for injections at the time for which stability is being validated as for injections immediately subsequent to the sample solution preparation. While this may be done by a subjective assessment of results with confidence limits, strictly speaking a statistical method known as the Student s t-test should be used. 

In the development presented in this article, we have always assumed that the best available statistical methods are to be coupled with the structural analysis. For complex systems, which involve spaces of many dimensions, the error magnification obtained in the transformation from the... 

---