Precision statistical tools

There is one final statistical distribution, the F-distribution, that is an important addition to the basic statistical tool chest. This distribution is used in the evaluation of two variance estimates to determine whether they are consistent with each other. The QC sample based on 42 estimates (41 degrees of freedom) had a standard deviation of 14.1ng/ml. If we had another set of 31 QC values (30 degrees of freedom), perhaps from a second bioavailability study, with a standard deviation of 19.5ng/ml, we might want to know whether the assay precision values for the two studies were consistent. The variance ratio statistic is sf/s, where is the higher of the two standard deviations, and S2 is the lower of the two. The calculated value of the statistic is compared with tables of critical F-values with - 1 numerator and 1 denomi-... 

System suitability is guaranteed if both apparatus test and validation match their requirements. It is best performed on a routine basis and can be done very easily if the HPLC apparatus is equipped with a computer data system. Then during each run or in well-defined intervals a number of parameters are acquired plate number, resolution, precision, retention time, relative retention time (i.e. k value) and peak asymmetry-also if necessary linearity and limits of quantification. The results are followed by statistical tools, including easy-to-monitor graphical documentation with control charts. ... 

The novice in the field of x-ray spectrometry often feels that analytical precision is largely governed by the easiest-to-calculate error source, the counting error. In practice, however, this is frequently the least important contribution to total error in the system. The next section will discuss sources of error in x-ray spectrometry and some possible solutions. In this section, it will be assumed that many sources of error exist, and an attempt to locate them will be made using a statistical tool called analysis of variance. 

Another important statistical tool is the confidence interval, which is a range of values used to indicate how certain we are that an estimated rate from a sample of dafa is actually the true rate in the population from which the sample is taken. The width of fhe confidence interval serves as a means to indicate how accurate the assessment is, meaning that a narrow confidence interval correlates with a more precise estimate, while a wide confidence interval indicates a less precise one. Another very helpful tool in biostatistics is the p value, which is a measure of the statistical significance of a difference between rates. In other words, we measure how probable it is that the results observed happened by chance. A very small p value means that the estimated differences in rates were very unlikely to have occurred by chance. When analyzing public health data and biostatistics in general, the threshold of p value is less than. 05, which means that there is less than 5% chance that the differences happened by chance alone. 

However, spectra databases are statistical tools to establish the relationships between NMR spectral parameters and chemical environment of individual atoms. So. the results can only be offered with a statistical probability, depending on the quantity and quality of the available database entries. In other words, the accuracy of the predicted data cannot be more precise than the stored data. Usually, C NMR spectra can be calculated for almost any drawn organic structure to an accuracy of 3 ppm or better, apart from stereochemical problems which can not be considered by some databases. Table 9 summarizes the results of the determination of the C NMR chemical shifts for the ten carbon atoms in camphor (2) obtained with different methods. 

Acceptance criteria should take into account method performance attributes and the intended use of the methods. For example, in some instances it may be critical that the method precision and sensitivity (i.e., for impurities) are similar to that obtained by the method development laboratory. In such cases, the samples selected for transfer purposes, the statistical tools applied to demonstrate equivalence and the acceptance criteria should be selected carefully to ensure that the method performance is properly evaluated. On the other hand, in some instances the capabilities (e.g., sensitivity) of the development method may exceed the method performance requirements for commercial release testing. For example, while a gas chromatographic (GC) method may be validated to have sensitivity down to 0.002% for a number of residual solvents monitored during development and, if the specifications are set only on total residual solvents with a limit of 0.5%, it may not be necessary to demonstrate sensitivity for individual solvents to 0.002% to qualify the QC laboratory for routine use. The acceptance criteria should be considered on a case-by-case basis for each method for each product and must be established in advance of the formal testing. 

Control charts were originally developed in the 1920s as a quality assurance tool for the control of manufactured products.Two types of control charts are commonly used in quality assurance a property control chart in which results for single measurements, or the means for several replicate measurements, are plotted sequentially and a precision control chart in which ranges or standard deviations are plotted sequentially. In either case, the control chart consists of a line representing the mean value for the measured property or the precision, and two or more boundary lines whose positions are determined by the precision of the measurement process. The position of the data points about the boundary lines determines whether the system is in statistical control. 

The role of quality in reliability would seem obvious, and yet at times has been rather elusive. While it seems intuitively correct, it is difficult to measure. Since much of the equipment discussed in this book is built as a custom engineered product, the classic statistical methods do not readily apply. Even for the smaller, more standardized rotary units discussed in Chapter 4, the production runs are not high, keeping the sample size too small for a classical statistical analysis. Run adjustments are difficult if the run is complete before the data can be analyzed. However, modified methods have been developed that do provide useful statistical information. These data can be used to determine a machine tool s capability, which must be known for proper machine selection to match the required precision of a part. The information can also be used to test for continuous improvement in the work process. 

One of the main determinants of the number of subjects required to reach the desired statistical power is the precision of the measurement tool utilized. More precise measurements will reduce the number of subjects required. As an example, if a study is being conducted to assess the influence of a dietary supplement on body fat, several measurement tools could be used to assess this outcome. These tools range from low levels of cost and precision (e.g. skinfold measurements) to moderate levels (e.g. bioelectrical impedance) to high levels of cost and precision (dual x-ray absorptiometry - DXA). A study that uses skinfold measurements to measure the outcome will require many more subjects than one which employs DXA. Therefore, it is often less expensive in total to utilize a more expensive measurement tool, because the more precise tool will allow the study to have sufficient power with a smaller number of subjects. 

In general, all observed intemuclear distances are vibrationally averaged parameters. Due to anharmonicity, the average values will change from one vibrational state to the next and, in a molecular ensemble distributed over several states, they are temperature dependent. All these aspects dictate the need to make statistical definitions of various conceivable, different averages, or structure types. In addition, since the two main tools for quantitative structure determination in the vapor phase—gas electron diffraction and microwave spectroscopy—interact with molecular ensembles in different ways, certain operational definitions are also needed for a precise understanding of experimental structures. 

Its precise basis in statistical mechanics makes the virial equation of state a powerful tool for prediction and correlation of thermodynamic properties involving fluids and fluid mixtures. Within the study of mixtures, the interaction second virial coefficient occupies an important position because of its relationship to the interaction potential between unlike molecules. On a more practical basis, this coefficient is useful in developing predictive correlations for mixture properties. 

These techniques include methods that are highly empirical in nature, such as those based on various forms of additivity principles, as well as those that are based on the use of statistical mechanical calculations. Although the latter methods represent precise tools to determine S and Cp from molecular properties, they are of little utility for the prediction of AHj. 

Two aspects are important for IQC (1) the analysis of control materials such as reference materials or spiked samples to monitor trueness and (2) replication of analysis to monitor precision. Of high value in IQC are also blank samples and blind samples. Both IQC aspects form a part of statistical control, a tool for monitoring the accuracy of an analytical system. In a control chart, such as a Shewhart control chart, measured values of repeated analyses of a reference material are plotted against the run number. Based on the data in a control chart, a method is defined either as an analytical system under control or as an analytical system out of control. This interpretation is possible by drawing horizontal lines on the chart x(mean value), x + s (SD) and x - s, x + 2s (upper warning limit) and x-2s (lower warning limit), and x + 3s (upper action or control limit) and x- 3s (lower action or control limit). An analytical system is under control if no more than 5% of the measured values exceed the warning limits [2,6, 85]. 

A statistical index of precision calculated as ([standard deviation x 100] mean). The CV is a measure of the variability in a group of measurements. Since the CV is unitless, it can be used to compare CVs from different experiments . It is also a quality control tool. For example, in the algal microplate toxicity test, algal cell density in control wells at the end of the test exposure period must have a CV not exceeding 20% to meet test acceptability criteria. Volume 1(1,2,3,10). 

Commonly the compromising conditions of routine environmental monitoring lead to restrictions on the accuracy and the precision of sampling and analysis. The purpose of this section is to show that under these conditions multivariate statistical methods are a useful tool for qualitative extraction of new information about the degree of stress of the investigated areas, and for identification of emission sources and their seasonal variations. The results represented from investigation of the impact of particulate emissions can, in principle, be transferred to other environmental analytical problems, as described in the following case studies. 

Because geochemically different clay sources may have been used by potters to produce ceramics for both domestic and trade purposes, neutron activation analysis (NAA) has been used as an independent means of ceramic characterization. Because of the relatively good analytical precision possible with NAA, statistical patterning of NAA data for major, minor, and trace element concentrations may be used as a powerful provenancing tool. 

The procedure applied in the experimental calibration approach consists of acquiring spectral data for the maximum of sample varieties of precisely known values for the compound or indices to be measured, before applying mathematical statistical methods for quantification It is essential to understand that this step is fundamental for the construction of a high-performance analytical tool. Not only must the operator have at his or her disposal the greatest number of representative... 

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