Statistical methods calibration

In the case of quantitative analysis, the amount of, or the exact relation between, the constituents of a compound or a mixture have to be estabhshed. The direct relation between the properties of a specimen and the concentration of its constituents could also be the aim of the quantitative investigation. In the latter case so-called calibration models have to be established, the corresponding model parameters have to be estimated, and they have to be confirmed by statistical methods. Calibration models estabhshed this way may then be used to determine, on a statistically verified basis, the concentration of constituents of an analyte within the calibrated range. AU multivariate methods for quantitative analysis mentioned in Fig. 22.2 are employed for the evaluation of spectra. 

Discriminant Analysis (DA) is a multivariate statistical method that generates a set of classification functions that can be used to predict into which of two or more categories an observation is most likely to fall, based on a certain combination of input variables. DA may be more effective than regression for relating groundwater age to major ion hydrochemistry and well construction because it can account for complex, non-continuous relationships between age and each individual variable used in the algorithm while inherently coping with uncertainty in the age values used for calibration, and there is no need to... 

The calibration problem in chromatography and spectroscopy has been resolved over the years with varying success by a wide variety of methods. Calibration graphs have been drawn by hand, by instruments, and by commonly used statistical methods. Each method can be quite accurate when properly used. However, only a few papers, for example ( 1,2,15,16,26 ), show the sophisticated use of a chemometric method that contains high precision regression with total assessment of error. 

We will describe an accurate statistical method that includes a full assessment of error in the overall calibration process, that is, (I) the confidence interval around the graph, (2) an error band around unknown responses, and finally (3) the estimated amount intervals. To properly use the method, data will be adjusted by using general data transformations to achieve constant variance and linearity. It utilizes a six-step process to calculate amounts or concentration values of unknown samples and their estimated intervals from chromatographic response values using calibration graphs that are constructed by regression. 

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. 

What does optimization mean in an analytical chemical laboratory The analyst can optimize responses such as the result of analysis of a standard against its certified value, precision, detection limit, throughput of the analysis, consumption of reagents, time spent by personnel, and overall cost. The factors that influence these potential responses are not always easy to define, and all these factors might not be amenable to the statistical methods described here. However, for precision, the sensitivity of the calibration relation, for example (slope of the calibration curve), would be an obvious candidate, as would the number of replicate measurements needed to achieve a target confidence interval. More examples of factors that have been optimized are given later in this chapter. 

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. 

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]). 

The calibration graph at 510 nm is a straight line and Beer s law is obeyed from 0.5 to 5 [xg/ml of boron in the final measured solution (corresponding to 10-110 xg of boron in the aqueous phase). The molar absorptivity, calculated from the slope of the statistical working calibration graph at 510 nm, was 29051/mol/cm. The Sandell sensitivity was 0.011 xgcm2 of boron. The precision of the method for ten replicate determinations was 0.6%. The absorbance of the reagent blank solution at 510 nm was 0.010 d= 0.003 for ten replicate determinations. Therefore, the detection limit was 0.04 xg/ml of boron in the final measured solution. 

A QA system describes the overall measures that a laboratory uses to ensure the quality of its operation. Typical items include suitable equipment, trained and skilled staff, documented and validated methods, calibration requirements, standards and RMs, traceability, internal QC, PT, nonconformance management, internal audits, and statistical analysis. 

The single most important application of statistical methods in science is the determination and propagation of experimental uncertainties. Quantitative experimental results are never perfectly reproducible. Common sources of error include apparatus imperfections, judgments involved in laboratory technique, and innumerable small fluctuations in the environment. Does the slight breeze in the lab affect a balance When a motor starts in the next building, does the slight power surge affect a voltmeter Was the calibrated volumetric flask perfectly clean ... 

J. N. Miller, Basic statistical methods for analytical chemistry. Part 2. Calibration and regression methods. A review, Analyst, 116 (1991), 3-14. 

A DICE experiment produces large amounts of data, or more exactly, volume values for thousands of gel spots. In order to make correct inferences from these data, statistical methods are quite important. Many of the statistical methods used in DNA microarray studies can be adapted for analyses of gel data. In this section we focus on the calibration and normalization of protein expression data as well as on the detection of differentially expressed proteins resulting from a DICE experiment. 

A problem that arises in many cases is that small raw data samples may contain one or more results which appear to be divergent from the remainder the problem may arise in a calibration experiment or in a set of replicate measurements on a test material. Assuming that the cause of the divergence is not immediately apparent e.g. as a result of a transcription error or instrument failure) it is then necessary to decide whether such an outlier can be rejected entirely from the raw data before data processing e.g. logarithmic transformations) and the usual statistical methods are applied. The treatment of outliers is a major topic in statistics. Three distinct approaches are available. ... 

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... 

An important extension of our large validation studies involves the use of data bases from field studies in the development of improved statistical methods for a variety of problems in quantitative applications of immunoassays. These problems include the preparation and analysis of calibration curves, treatment of "outliers" and values below detection limits, and the optimization of resource allocation in the analytical procedure. This last area is a difficult one because of the multiple level nested designs frequently used in large studies such as ours (22.). We have developed collaborations with David Rocke and Davis Bunch (statisticians and numerical analysts at Davis) in order to address these problems within the context of working assays. Hopefully we also can address the mathematical basis of using multiple immunoassays as biochemical "tasters" to approach multianalyte situations. 

After outliers have been purged from the data and a model has been evaluated visually and/or by, e.g. residual plots, the model fit should also be tested by appropriate statistical methods [2, 6, 9, 10, 14], The fit of unweighted regression models (homoscedastic data) can be tested by the ANOVA lack-of-fit test [6, 9]. A detailed discussion of alternative statistical tests for both unweighted and weighted calibration models can be found in Ref. [16]. The widespread practice to evaluate a calibration model via its coefficients of correlation or determination is not acceptable from a statistical point of view [9]. 

Some definitions are contradictory, meaningless, without benefit or will cause much expenditure of personnel and measurement capacity, e.g. Limit of determination. This is the smallest analyte content for which the method has been vahdated with specific accuracy and precision . Apart from the fact that precision is included in the explanation of accuracy the definition manifests a fundamental inability to give a definition which is fit for practice. A useful definition of the detection and quantification limit is based on a statistical approach to the confidence hyperbola of a methods calibration curve, elaborated by the Deutsche Forschungsgemeinschaft [12]-... 

These differ in the exact way in which variations in the data (responses) are used to predict the concentration. Software for accomplishing multivariate calibration is available from several companies. The use of multivariate statistical methods for quantitative analysis is part of the subdiscipline of chemistry called chemometrics. 

The first three methods are very similar to the methods used for other spectroscopic methods. Statistical calculation methods can be used only in modem x-ray fluorescence instruments that come with the appropriate software. Different manufacturers or companies use different algorithms their instruments. The main purpose of this software is to minimize the influence of measurement errors when computing the results. A wide variety of statistical methods are available. The statistical calculation method saves a lot of experimenting time, because only the analysis of the sample is needed for every analysis. Calibration or analyses of sample with added substance is not required in this case. 

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