Selection of the experimental design

To examine the ruggedness of the factors that were selected one could test these factors one variable at a time, i.e. change the level of one factor and keep all other factors at nominal level. The result of this experiment is then compared to the result of experiments with all factors at nominal level. The difference between the two types of experiments gives an idea of the effect of the factor in the interval between the two levels. The disadvantage of this method is that a large number of experiments is required when the number of factors is large. 

For this reason one prefers to apply an experimental design. In the literature a number of different designs are described, such as saturated fractional factorial designs and Plackett-Burman designs, full and fractional factorial designs, central composite designs and Box-Behnken designs [5]. 

In a full factorial design all combinations between the different factors and the different levels are made. Suppose one has three factors (A,B,C) which will be tested at two levels (- and +). The possible combinations of these factor levels are shown in Table 3.5. Eight combinations can be made. In general, the total number of experiments in a two-level full factorial design is equal to 2 with /being the number of factors. The advantage of the full factorial design compared to the one-factor-at-a-time procedure is that not only the effect of the factors A, B and C (main effects) on the response can be calculated but also the interaction effects of the factors. The interaction effects that can be considered here are three two-factor interactions (AB, 

AC and BC) and one three-factor interaction (ABC). From the 2 full factorial design shown in Table 3.5 these seven effects can be calculated. An eighth statistic that can be obtained from this design is the mean result. From a 2 full factorial design therefore 2 statistics can be calculated. The 

NUMBER OF STATISTICS THAT CAN BE CALCULATED FOR DIFFERENT FULL FACTORIAL DESIGNS  

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After selection of the experimental design, the experiments can be defined. For this purpose, the level symbols, —1 and +1, as given in Tables 6 and 7, are replaced by the real factor values, as for instance shown in Tables 2 and 3, respectively, yielding the factor level combinations to be performed. Dummy factors in PB designs can be neglected during the execution of the experimental work. 

In a robustness test the following steps can be identified (a) identification of the variables to be tested, (b) definition of the different levels for the variables, (c) selection of the experimental design, (d) definition of the experimental protocol, (e) definition of the responses to be determined, (f) execution of the experiments and determination of the responses of the method, (g) calculation of effects, (h) statistical and/or graphical analysis of the effects, and (i) drawing chemically relevant conclusions from the statistical analysis and, if necessary, taking measures to improve the performance of the method. A general overview of robustness testing can be found in [35). 

Optimal design theory provides an alternative approach to the selection of an experimental design. For a description of the theory of optimal design see, for example, Atkinson and Donev [22]. 

However, in order to use these criteria in the selection of an experimental design from a set of possible candidate experiments, certain assumptions must be fullfilled (1) The mathematical form of the model to be fitted is perfectly known. (2) The region in which the experiments can be run, i.e. the experimental domain is known, and it is excluded that experiments can be run outside this region. 

With several factors to study, the size of the experimental design becomes large, time consuming, and expensive. Two levels are often selected for an exploratory design. 

One possibility is to control each at a constant level, selected arbitrarily or for economic or technological reasons. The values of these factors must be incorporated into the experimental plan even if they themselves do not vary. They are still part of the experimental design. And then, even if the hypothesis that they do not affect the response is mistaken, they will not affect our conclusions about the influences of the factors studied, but only our estimate of the constant term. 

The choice of the experimental design was the first stage of the optimization of the number of experiments. After the selection of the design, the best way of decreasing the number of tests is to work directly on the factors of the designs. Indeed the dimension of an experiment matrix depends only on the number of factors among and of the selected resolution for the experimentation. 

The selection of the operating principle and the design of the calorimeter depends upon the nature of the process to be studied and on the experimental procedures required. Flowever, the type of calorimeter necessary to study a particular process is not unique and can depend upon subjective factors such as teclmical restrictions, resources, traditions of the laboratory and the inclinations of the researcher. 

Following the movement of airborne pollutants requires a natural or artificial tracer (a species specific to the source of the airborne pollutants) that can be experimentally measured at sites distant from the source. Limitations placed on the tracer, therefore, governed the design of the experimental procedure. These limitations included cost, the need to detect small quantities of the tracer, and the absence of the tracer from other natural sources. In addition, aerosols are emitted from high-temperature combustion sources that produce an abundance of very reactive species. The tracer, therefore, had to be both thermally and chemically stable. On the basis of these criteria, rare earth isotopes, such as those of Nd, were selected as tracers. The choice of tracer, in turn, dictated the analytical method (thermal ionization mass spectrometry, or TIMS) for measuring the isotopic abundances of... 

Reaction and Transport Interactions. The importance of the various design and operating variables largely depends on relative rates of reaction and transport of reactants to the reaction sites. If transport rates to and from reaction sites are substantially greater than the specific reaction rate at meso-scale reactant concentrations, the overall reaction rate is uncoupled from the transport rates and increasing reactor size has no effect on the apparent reaction rate, the macro-scale reaction rate. When these rates are comparable, they are coupled, that is they affect each other. In these situations, increasing reactor size alters mass- and heat-transport rates and changes the apparent reaction rate. Conversions are underestimated in small reactors and selectivity is affected. Selectivity does not exhibit such consistent impacts and any effects of size on selectivity must be deterrnined experimentally. 

In the development of a SE-HPLC method the variables that may be manipulated and optimized are the column (matrix type, particle and pore size, and physical dimension), buffer system (type and ionic strength), pH, and solubility additives (e.g., organic solvents, detergents). Once a column and mobile phase system have been selected the system parameters of protein load (amount of material and volume) and flow rate should also be optimized. A beneficial approach to the development of a SE-HPLC method is to optimize the multiple variables by the use of statistical experimental design. Also, information about the physical and chemical properties such as pH or ionic strength, solubility, and especially conditions that promote aggregation can be applied to the development of a SE-HPLC assay. Typical problems encountered during the development of a SE-HPLC assay are protein insolubility and column stationary phase... 

One goal of our experimental program with the bench-scale unit was to develop the necessary correlations for use in the ultimate design of large commercial plants. Because of the complexity inherent in the three-phase gas-liquid-solid reaction systems, many models can be postulated. In order to provide a background for the final selection of the reaction model, we shall first review briefly the three-phase system. 

The expression x (J)P(j - l)x(j) in eq. (41.4) represents the variance of the predictions, y(j), at the value x(j) of the independent variable, given the uncertainty in the regression parameters P(/). This expression is equivalent to eq. (10.9) for ordinary least squares regression. The term r(j) is the variance of the experimental error in the response y(J). How to select the value of r(j) and its influence on the final result are discussed later. The expression between parentheses is a scalar. Therefore, the recursive least squares method does not require the inversion of a matrix. When inspecting eqs. (41.3) and (41.4), we can see that the variance-covariance matrix only depends on the design of the experiments given by x and on the variance of the experimental error given by r, which is in accordance with the ordinary least-squares procedure. 

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. 

If the structure of the models is more complex and we have more than one independent variable or we have more than two rival models, selection of the best experimental conditions may not be as obvious as in the above example. A straightforward design to obtain the best experimental conditions is based on the divergence criterion. 

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