Statistical analysis accuracy

E0so is measured in electrochemistry and is usually known with an accuracy to 0.01 V or better.8 On the other hand 0 is measured with surface physics techniques that have an accuracy of 0.05 eV, rarely better and often worse (because of imperfect surfaces)/9 Thus, Eq. (28) does not ensure an appropriate accuracy for AX, so that the uncertainty may outweigh the value itself. The best way to proceed is to plot E0=q vs. 0 for a number of metals and to derive information about AX from eventually recognizable graphical correlations using statistical analysis. 

Later, Bi(OlT), Bi(2TT), Bi(001), and Bi(101) faces were stud-ied.28,152 253 254,705 The accuracy of the experimental results has been established by statistical analysis. A very slight variation in capacitance (3-6%) with v (from 60 to 21,000 Hz) was observed for electrochemically polished single-crystal Bi. Therefore, to a first approximation, the measured admittance was identified with the differential capacitance C. 

In principle, FCS can also measure very slow processes. In this limit the measurements are constrained by the stability of the system and the patience of the investigator. Because FCS requires the statistical analysis of many fluctuations to yield an accurate estimation of rate parameters, the slower the typical fluctuation, the longer the time required for the measurement. The fractional error of an FCS measurement, expressed as the root mean square of fluorescence fluctuations divided by the mean fluorescence, varies as 1V-1/2, where N is the number of fluctuations that are measured. If the characteristic lifetime of a fluctuation is r, the duration of a measurement to achieve a fractional error of E = N l,/- is T = Nr. Suppose, for example, that r = 1 s. If 1% accuracy is desired, N = 104 and so T = 104 s. 

The primary goal of this series of chapters is to describe the statistical tests required to determine the magnitude of the random (i.e., precision and accuracy) and systematic (i.e., bias) error contributions due to choosing Analytical METHODS A or B, and/or the location/operator where each standard method is performed. The statistical analysis for this series of articles consists of five main parts as ... 

Our statistical analysis reveals a large improvement from cc-pCV(DT)Z to cc-pCV(TQ)Z see Fig. 1.4. In fact, the cc-pCV(TQ)Z calculations are clearly more accurate than their much more expensive cc-pcV6Z counterparts and nearly as accurate as the cc-pcV(56)Z extrapolations.The cc-pCV(TQ)Z extrapolations yield mean and maximum absolute errors of 1.7 and 4.0 kJ/mol, respectively, compared with those of 0.8 and 2.3 kJ/mol at the cc-pcV(56)Z level. Chemical accuracy is thus obtained at the cc-pCV(TQ)Z level, greatly expanding the range of molecules for which ab initio electronic-structure calculations will afford thermochemical data of chemical accuracy. 

Typically extrapolations of many kinds are necessary to complete a risk assessment. The number and type of extrapolations will depend, as we have said, on the differences between condition A and condition B, and on how well these differences are understood. Once we have characterized these differences as well as we can, it becomes necessary to identify, if at all possible, a firm scientific basis for conducting each of the required extrapolations. Some, as just mentioned, might be susceptible to relatively simple statistical analysis, but in most cases we will find that statistical methods are inadequate. Often, we may find that all we can do is to apply an assumption of some sort, and then hope that most rational souls find the assumption likely to be close to the truth. Scientists like to be able to claim that the extrapolation can be described by some type of model. A model is usually a mathematical or verbal description of a natural process, which is developed through research, tested for accuracy with new and more refined research, adjusted as necessary to ensure agreement with the new research results, and then used to predict the behavior of future instances of the natural process. Models are refined as new knowledge is acquired. 

Assuming that/A is 0.01, and we would like to determine this value with the atom-probe to an accuracy of 5% of fA, then the number of ions collected should be at least 39 600. This is already near an impractically large number in the atom-probe analysis. It quickly becomes impractical to determine accurately the concentration of a minority species with the atom-probe at a low counting rate if its concentration is smaller than —0.5%. The following statistical analysis points to a method with which this difficulty can be overcome. 

From the data listed in Tables I-V, we conclude that most authors would probably accept that there is evidence for the existence of a compensation relation when ae < O.le in measurements extending over AE 100 and when isokinetic temperature / , would appear to be the most useful criterion for assessing the excellence of fit of Arrhenius values to Eq. (2). The value of oL, a measure of the scatter of data about the line, must always be considered with reference to the distribution of data about that line and the range AE. As the scatter of results is reduced and the range AE is extended, the values of a dimin i, and for the most satisfactory examples of compensation behavior that we have found ae < 0.03e. There remains, however, the basic requirement for the advancement of the subject that a more rigorous method of statistical analysis must be developed for treatment of kinetic data. In addition, uniform and accepted criteria are required to judge quantitatively the accuracy of obedience of results to Eq. (2) or, indeed, any other relationship. 

Accuracy. The more accurate the sampling method the better. Given the very large environmental variability, however, sampling and analytical imprecision is rardy a significant contribution to overall error, or width of confidence limits, of the final result. Even highly imprecise methods, such as dust count methods, do not add much to overall variability when the variability between workers and overtime is considered. An undetected bias, however, is more serious because such bias is not considered by the statistical analysis and can, therefore, result in gross unknown error. 

We can easily quantify measurement error due to existence of a well-developed approach to analytical methods and laboratory QC protocols. Statistically expressed accuracy and precision of an analytical method are the primary indicators of measurement error. However, no matter how accurate and precise the analysis may be, qualitative factors, such as errors in data interpretation, sample management, and analytical methodology, will increase the overall analytical error or even render results unusable. These qualitative laboratory errors that are usually made due to negligence or lack of information may arise from any of the following actions ... 

Statistics should follow the technical scrutiny, not the other way round. A statistical analysis of data of an interlaboratory study cannot explain deviating results nor can alone give information on the accuracy of the results. Statistics only treat a population of data and provide information on the statistical characteristics of this population. The results of the statistical treatment may give rise to discussions on particular data not belonging to the rest of the population, but outlying data can sometimes be closer to the true value than the bulk of the population (Griepink et al., 1993). If no systematic errors affect the population of data, various statistical tests may be applied to the results, which can be treated either as individual data or as means of laboratory means. When different methods are applied, the statistical treatment is usually based on the mean values of replicate determinations. Examples of statistical tests used for certification purposes are described elsewhere (Horwitz, 1991). Together with the technical evaluation of the results, the statistical evaluation forms the basis for the conclusions to be drawn and the possible actions to be taken. 

We have evaluated three different techniques to generate QSAR models, namely Comparative Molecular Field Analysis (CoMFA), Comprehensive Descriptors for Structural and Statistical Analysis (CODESSA), and Hologram QSAR (HQSAR). More specifically they were evaluated for their utility (predictivity, speed, accuracy, and reproducibility) to predict ER binding activity quantitatively (Tong et al., 1998 Shi et al., 2001). Common to the three QSAR methods is the... 

Salem et al. [48] reported simple and accurate methods for the quantitative determination of flufenamic, mefenamic and tranexamic acids utilizing precipitation reactions with cobalt, cadmium and manganese. The acidic drugs were precipitated from their neutral alcoholic solutions with cobalt sulfate, cadmium nitrate or manganese chloride standard solutions followed by direct determination of the ions in the precipitate or indirect determination of the ions in the filtrate by atomic absorption spectroscopy (AAS). The optimum conditions for precipitation were carefully studied. The molar ratio of the reactants was ascertained. Statistical analysis of the results compared to the results of the official methods revealed equal precision and accuracy. The suggested procedures were applied for determining flufenamic, mefenamic and... 

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