Statistics calibration and

Statistics is a fundamental and quantitative component of quality assurance. Statistics is also a powerful tool for the forensic chemist, in that it can describe the accuracy and precision of data, assist in answering h)q)ofliesis-based questions, and identify trends and patterns in data. Statistics and the related tools of probability can be applied to direct sampling and provide a reasonable, defensible, and sensible sampling plan. With such a plan in hand, we are ready to move on to multivariate statistics, calibration, and the unification of these concepts under the umbrella of qualify assurance and qualify control. 

Chapter 3 Multivariate Statistics, Calibration, and Quality Control... 

With this brief discussion of multivariate analysis and linear equations completed, we are ready to discuss the calibration of instruments. Often underemphasized, the reliable calibration of equipment and instrumentation is the foundation of data quality and reliability. Accordingly, before leaping into the topic of calibration, we must formally introduce quality assurance and quality control, which in turn will integrate the statistical concepts introduced here and in the previous chapter. The discussion that follows will provide our link between statistics, calibration, and data quality. 

Definition and Uses of Standards. In the context of this paper, the term "standard" denotes a well-characterized material for which a physical parameter or concentration of chemical constituent has been determined with a known precision and accuracy. These standards can be used to check or determine (a) instrumental parameters such as wavelength accuracy, detection-system spectral responsivity, and stability (b) the instrument response to specific fluorescent species and (c) the accuracy of measurements made by specific Instruments or measurement procedures (assess whether the analytical measurement process is in statistical control and whether it exhibits bias). Once the luminescence instrumentation has been calibrated, it can be used to measure the luminescence characteristics of chemical systems, including corrected excitation and emission spectra, quantum yields, decay times, emission anisotropies, energy transfer, and, with appropriate standards, the concentrations of chemical constituents in complex S2unples. 

R. Sundberg, Interplay between chemistry and statistics, with special reference to calibration and the generalized standard addition method. Chemom. Intell. Lab. Syst., 4 (1988) 299-305. 

Frequency domain performance has been analyzed with goodness-of-fit tests such as the Chi-square, Kolmogorov-Smirnov, and Wilcoxon Rank Sum tests. The studies by Young and Alward (14) and Hartigan et. al. (J 3) demonstrate the use of these tests for pesticide runoff and large-scale river basin modeling efforts, respectively, in conjunction with the paired-data tests. James and Burges ( 1 6 ) discuss the use of the above statistics and some additional tests in both the calibration and verification phases of model validation. They also discuss methods of data analysis for detection of errors this last topic needs additional research in order to consider uncertainties in the data which provide both the model input and the output to which model predictions are compared. 

In the following, the stages of the analytical process will be dealt with in some detail, viz. sampling principles, sample preparation, principles of analytical measurement, and analytical evaluation. Because of their significance, the stages signal generation, calibration, statistical evaluation, and data interpretation will be treated in separate chapters. 

From the information-theoretical point of view, calibration corresponds to the coding of the input quantity into the output quantity and, vice versa, the evaluation process corresponds to decoding of output data. From the mathematical viewpoint, qin is the independent quantity in the calibration step and qout the dependent one. In the evaluation step, the situation is reverse qout is the independent, and qin the dependent quantity. From the statistical standpoint, qout is a random variable both in calibration and evaluation whereas qin is a fixed variable in the calibration step and a random variable in the evaluation step. This rather complicated situation has some consequences on which will be returned in Sect. 6.1.2. 

The different statistical character of the three variables becomes most clear in the different uncertainties of the calibration and evaluation lines. Notwithstanding the fundamental differences between xstandard and xsampiey the calculation of the calibration coefficients is carried out by regression calculus. 

The attached worksheet from MathCad ( 1986-2001 MathSoft Engineering Education, Inc., 101 Main Street Cambridge, MA 02142-1521) is used for computing the statistical parameters and graphics discussed in Chapters 58 through 61, in references [b-l-b-4]. It is recommended that the statistics incorporated into this series of Worksheets be used for evaluations of goodness of fit statistics such as the correlation coefficient, the coefficient of determination, the standard error of estimate and the useful range of calibration standards used in method development. If you would like this Worksheet sent to you, please request this by e-mail from the authors. 

Based on previous recommendations [31], a combination of graphical techniques and error index statistics was used for evaluating the goodness-of-fit between the simulated and observed streamflow values, both during the calibration and validation period. The used statistics were the mean error (ME), the percent bias (PBIAS, [32]) and the Nash-Sutcliffe efficiency (NSeff, [33]) ... 

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... 

There are two limitations with the above process 1) the analyst is putting all the eggs in one basket by comparing the sample to just one standard (not very statistically sound), and 2) the calibration constant, K, must truly be constant at the two concentration levels, Cs and Q, (possible, but not guaranteed). Because of these limitations, the concept of the standard curve is used most of the time. 

Recently, introductory books about chemometrics have been published by R. G. Brereton, Chemometrics—Data Analysis for the Laboratory and Chemical Plant (Brereton 2006) and Applied Chemometrics for Scientists (Brereton 2007), and by M. Otto, Chemometrics—Statistics and Computer Application in Analytical Chemistry (Otto 2007). Dedicated to quantitative chemical analysis, especially using infrared spectroscopy data, are A User-Friendly Guide to Multivariate Calibration and Classification (Naes et al. 2004), Chemometric Techniques for Quantitative Analysis (Kramer 1998), Chemometrics A Practical Guide (Beebe et al. 1998), and Statistics and Chemometrics for Analytical Chemistry (Miller and Miller 2000). 

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