Statistical methods determination limit

Quantitative XRF analysis has developed from specific to universal methods. At the time of poor computational facilities, methods were limited to the determination of few elements in well-defined concentration ranges by statistical treatment of experimental data from reference material (linear or second order curves), or by compensation methods (dilution, internal standards, etc.). Later, semi-empirical influence coefficient methods were introduced. Universality came about by the development of fundamental parameter approaches for the correction of total matrix effects... 

The problem that statisticians have had regarding linearity is the same one that everybody else has had they have not had a good statistic for determining linearity any more than anybody else, so they also have been limited to idiosyncratic empirical methods. But Philip Brown s approach may just form the basis of one. 

The method detection limit is, in reality, a statistical concept that is applicable only in trace analysis of certain types of substances, such as organic pollutants by gas chromatographic methods. The method detection limit measures the minimum detection limit of the method and involves all analytical steps, including sample extraction, concentration, and determination by an analytical instrument. Unlike the instrument detection limit, the method detection limit is not confined only to the detection limit of the instrument. 

The important word in this sentence is predict. It is important, in my opinion, to make a distinction between existence and predictability. Prigogine himself said (much later, in La Fin des Certitudes, LG.7) Every dynamical system must, of course, follow a trajectory, solution of its equations, independently of the fact that we may or may not construct it. Thus, a trajectory exists but cannot be predicted. The impossibility of prediction is therefore related to the impossibility of defining an instantaneous state (in the framework of classical mechanics) as a limit of a finite region of phase space (thus a limit of a result of a set of measurements). For an unstable system, such a region will be deformed and will end up covering almost all of phase space. The necessity of introducing statistical methods appears to me to be due to the practical (rather than theoretical) impossibility of determining a mathematical point as an initial condition. 

The limit of determination is commonly estimated by finding the intercept of extrapolated linear parts of the calibration curve (see point L.D. in fig. 5.1). However, it is often difficult to construct a straight line through the experimental potentials at low concentrations and, moreover, the precision of the potential measurement cannot be taken into consideration. Therefore, it has been recommended that, by analogy with other analytical methods, the determination limit be found statistically, as the value differing with a certain probability from the background [94]. 

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. 

The statistical error determined for K is only a limited measure of the accuracy of the dilatometric measurements. Since the main errors will be similar for each measurement, the accuracy of the method is best determined by estimating the limits of error of each individual measurement.The main source of error and its approximate magnitude should be indicated. 

It should be pointed out that this method for ring analysis and branching analysis is based exclusively on reliable data of n, d, M and a of pure individual hydrocarbons, and holds, within the limits of accuracy of the determination, for widely differing types of branched as well as non-branched saturated hydrocarbon mixtures. It is particularly recommended for the structural analysis of saturated polymers, where other statistical methods (w- -M-method, v-n-d-method, etc.) fail because they have been developed for mineral oils, and are based on correlations of physical data of mineral oil fractions that always show approximately the same small degree of branching 1-2 branchings per molecular weight = 100. 

Slope, standard potential, linear concentration range and limit of detection should be determined using statistic methods, using data obtained for the calibration graph E vs. pSCpt (pSCpt — -log[SCpt]). 

Stability data (not only assay but also degradation products and other attributes as appropriate) should be evaluated using generally accepted statistical methods. The time at which the 95% one-sided confidence limit intersects the acceptable specification limit is usually determined. If statistical tests on the slopes of the regression lines and the zero-time intercepts for the individual batches show that batch-to-batch variability is small (e.g., p values for the level of significance of rejection are more than 0.25), data may be combined into one overall estimate. If the data show very little degradation and variability and it is apparent from visual inspection that the proposed expiration dating eriod will be met, formal statistical analysis may not be necessary. 

In Sections 2 to 4, we review the technology of synthetic oligonucleotide microarrays and describe some of the popular statistical methods that are used to discover genes with differential expression in simple comparative experiments. A novel Bayesian procedure is introduced in Section 5 to analyze differential expression that addresses some of the limitations of current procedures. We proceed, in Section 6, by discussing the issue of sample size and describe two approaches to sample size determination in screening experiments with microarrays. The first approach is based on the concept of reproducibility, and the second approach uses a Bayesian decision-theoretic criterion to trade off information gain and experimental costs. We conclude, in Section 7, with a discussion of some of the open problems in the design and analysis of microarray experiments that need further research. 

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. 

The purpose of statistical evaluation of sample data is to extrapolate from a collection of individual events (e.g., 30 min of process time) to the entire population of events (e.g., 8-h shift). Because microbial monitoring data usually measure the impact of human activity, which is not reproducible exactly from one event to the next, results usually do not fit standard statistical models for normal distributions. In spite of this limitation, it is necessary to summarize the data for comparison to limits. The best statistical methods of evaluation are determined by the nature of the data. Wilson suggests that microbial monitoring data histograms generally resemble Poisson or negative... 

This has been defined (B5) as the smallest single result which, with some assurance, can be distinguished from zero, or, in statistical terms, the smallest single result whose fiducial limits for, say, P = 0.05 do not include zero. This review will be primarily concerned with clinical chemical methods that have acceptable levels of sensitivity, and to which therefore statistical methods of quality control can be applied throughout the range of concentrations which may be encountered in physiological and pathological conditions. To take an extreme example, therefore, the statistical methods of quality control discussed in this review would not be fully applicable to determinations of plasma epinephrine or... 

In the certificate of analysis of a CRM, tables with 95% confidence limits of the certified value are given in relation to the number of capsules and the number of replicates. The certificate does not state the optimal number of capsules or replicates a laboratory should use. An example of a certificate of Bacillus cereus (BCR-CRM 528) is given in Annex 3.3. Statistical methods can be used to determine the required number of capsules and replicates that allow a good judgement of an experiment and that is realisable in practice. 

When a material has been prepared to contain only a few cfp, it is normal that, due to the homogeneity limits of the procedure and the material composition, some capsules contain a few cfp and some others contain no cfp at all. The laboratory classifies the capsules containing cfp as positive. Those containing no cfp are classified as negative. The certificate of analysis of a CRM contains tables with the expected minimum number (with a 95% probability) of positive isolations for a certain number of capsules analysed. The optimal number of capsules that a laboratory should use is not mentioned in the certificate of the CRM. For low level (C)RMs replicates are not possible as the capsule is used as a whole. It is necessary to determine the number of capsules that allows a good evaluation of the performance of the method in the user s laboratory. This number of experiments must remain economically sustainable. This can be done with statistical methods as will be explained with examples below. 

Lubricant condition monitoring is best accomplished by the analysis of numerical data that are associated with the various fluid failure modes [2]. Numerical data can be analysed by statistical methods to determine the relationship between the various test parameters and their respective fluid and machinery failure modes. In addition, the statistical analysis can be used to determine potential data interference sources, the various alarm limits for each parameter and other criteria to be used in the daily evaluation of used oil. Note that it is important to determine all of the causes for variability in parametric data, just as it is necessary to separate changes due to interfering causes from changes with its associated relevant failure modes. 

A stability protocol should describe not only how the stability study is to be designed and carried out but also the statistical method to be used in analyzing the data. This section describes an acceptable statistical approach to the analysis of stability data and the specific features of the stability study that are pertinent to the analysis. In general, an expiration dating or retest period should be determined on the basis of statistical analysis of observed long-term data. Limited extrapolation of the real-time data beyond the observed range to extend the expiration dating or retest period at approval time may be considered if it is supported by the statistical analysis of real-time data, satisfactory accelerated data, and other nonprimary stability data. 

---