Experimental design precision

In the two-sample collaborative test, each analyst performs a single determination on two separate samples. The resulting data are reduced to a set of differences, D, and a set of totals, T, each characterized by a mean value and a standard deviation. Extracting values for random errors affecting precision and systematic differences between analysts is relatively straightforward for this experimental design. 

In reporting the results of such studies, the experimental design should be carefully described so readers are able to assess the precise significance of the reported values. 

Bob is particularly concerned that, although analytical chemistry forms a major part of the UK chemical industry s efforts, it is still not considered by many to be a subject worthy of special consideration. Consequently, experimental design is often not employed when it should be and safeguards to ensure accuracy and precision of analytical measurements are often lacking. He would argue that although the terms accuracy and precision can be defined by rote, their meanings, when applied to analytical measurements, are not appreciated by many members of the scientific community. 

When an analytical method is being developed, the ultimate requirement is to be able to determine the analyte(s) of interest with adequate accuracy and precision at appropriate levels. There are many examples in the literature of methodology that allows this to be achieved being developed without the need to use complex experimental design simply by varying individual factors that are thought to affect the experimental outcome until the best performance has been obtained. This simple approach assumes that the optimum value of any factor remains the same however other factors are varied, i.e. there is no interaction between factors, but the analyst must be aware that this fundamental assumption is not always valid. 

Experimental design A number of formal procedures whereby the effect of experimental variables on the outcome of an experiment may be assessed. These may be used to assess the optimum conditions for an experiment and to maximize the accuracy and precision obtained. 

Bzik, T. J., Henderson, P. B., and Hobbs, J. R, Increasing the Precision and Accuracy of Top-Loading Balances Application of Experimental Design, Anal. Chem. 70, 1998, 58-63. 

The set of selected wavelengths (i.e. the experimental design) affects the variance-covariance matrix, and thus the precision of the results. For example, the set 22, 24 and 26 (Table 41.5) gives a less precise result than the set 22, 32 and 24 (Table 41.7). The best set of wavelengths can be derived in the same way as for multiple linear regression, i.e. the determinant of the dispersion matrix (h h) which contains the absorptivities, should be maximized. 

The measurements of the output vector are taken at distinct points in time, t, with i=d,...,N. The initial condition x0, is also chosen by the experimentalist and it is assumed to be precisely known. It represents a very important variable from an experimental design point of view. 

In Equations 6.61, U denotes the concentration of catalyst present in the reactor (Ck) and u2 the hydrogen pressure (P). As far as the estimation problem is concerned, both these variables are assumed to be known precisely. Actually, as it will be discussed later on experimental design (Chapter 12), the value of such variables is chosen by the experimentalist and can have a paramount effect on the quality of the parameter estimates. Equations 6.61 are rewritten as following... 

SEQUENTIAL EXPERIMENTAL DESIGN FOR PRECISE PARAMETER ESTIMATION... 

The sequential experimental design can be made either for precise parameter estimation or for model discrimination purposes. 

As an example for precise parameter estimation of dynamic systems we consider the simple consecutive chemical reactions in a batch reactor used by Hosten and Emig (1975) and Kalogerakis and Luus (1984) for the evaluation of sequential experimental design procedures of dynamic systems. The reactions are... 

Hosten, L.H. and G. Emig, "Sequential Experimental Design Procedures for Precise Parameter Estimation in Ordinary Differential Equations", Chem. Eng. Sci., 30, 1357 (1975)... 

The reliability of multispecies analysis has to be validated according to the usual criteria selectivity, accuracy (trueness) and precision, confidence and prediction intervals and, calculated from these, multivariate critical values and limits of detection. In multivariate calibration collinearities of variables caused by correlated concentrations in calibration samples should be avoided. Therefore, the composition of the calibration mixtures should not be varied randomly but by principles of experimental design (Deming and Morgan [1993] Morgan [1991]). 

A simple statistical test for the presence of systematic errors can be computed using data collected as in the experimental design shown in Figure 34-2. (This method is demonstrated in the Measuring Precision without Duplicates sections of the MathCad Worksheets Collabor GM and Collabor TV found in Chapter 39.) The results of this test are shown in Tables 34-9 and 34-10. A systematic error is indicated by the test using... 

The first precise or calculable aspect of experimental design encountered is determining sufficient test and control group sizes to allow one to have an adequate level of confidence in the results of a study (that is, in the ability of the study design with the statistical tests used to detect a true difference, or effect, when it is present). The statistical test contributes a level of power to such a detection. Remember that the power of a statistical test is the probability that a test results in rejection of a hypothesis, H0 say, when some other hypothesis, H, say, is valid. This is termed the power of the test with respect to the (alternative) hypothesis H. ... 

The PLS-2 technique is a typical full spectmm method where the data are fitted to many data points, thereby improving the precision and requires a carefully experimental design of the Standard composition of the calibration set order the provide good predictions. In this study training set of 27 representative ternary mixtures was constmcted and the absorption spectra were recorded. In Table 33.1, the compositions of the ternary mixtures employed are summarized. 

Consideration of the effect of experimental design on the elements of the variance-covariance matrix leads naturally to the area of optimal design [Box, Hunter, and Hunter (1978), Evans (1979), and Wolters and Kateman (1990)]. Let us suppose that our purpose in carrying out two experiments is to obtain good estimates of the intercept and slope for the model yj, = Po + Pi i, + r,. We might want to know what levels of the factor x , we should use to obtain the most precise estimates of po and... 

Equation 7.1 is one of the most important relationships in the area of experimental design. As we have seen in this chapter, the precision of estimated parameter values is contained in the variance-covariance matrix V the smaller the elements of V, the more precise will be the parameter estimates. As we shall see in Chapter 11, the precision of estimating the response surface is also directly related to V the smaller the elements of V, the less fuzzy will be our view of the estimated surface. 

Equations 11.11 and 11.14. Which experimental design gives the most precise estimate of PJ ... 

It might also be argued statistically (indeed, it is perhaps one of the major points of this chapter) that the domain of the experimental design represents only a small fraction of the factor space shown a broader design would have given smaller uncertainties and more precise information over the whole factor space (see, for example, Figure 13.8). 

An excellent exposition of split-plot experimental designs can be found in D.R. Cox s book, Planning of Experiments [42]. He states that split-plot designs are particularly useful when one (or more) factors are what he calls classification factors. These factors are included in the experiment to determine whether they modify the effect of the other factors or indicate how the other factors work. The classification factors are included to examine their possible interaction with the other factors. Lower precision is tolerated for comparisons of the classification factors, in order that the precision of the other factors and the interactions can be increased. In the standard terminology associated with split-plot experiments, the classification factors are called whole-plot factors and are applied to the larger experimental units. The smaller experimental units are called subplots. 

W.J. Youden, Experimental design and ASTM committee. Materials Research and Standards, 1 (1961) 862-867. Reprinted in Precision and measurement Calibration (Vol. 1. Special Publication 300), Gaithersburg, MD National Bureau of Standards, 1969, ed. H.H. Ku. 

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