Experimental domain

To explore an experimental procedure, the experimenter chooses a range of variation for aU the experimental variables considered (a) for all continuous (quantitative) variables, the upper and lower bounds for their variation are specified (b) for the discrete (qualitative) variables, types of equipment, types of catalysts, nature of solvents etc. are specified. Assume that each experimental variable defines a coordinate axis along which the settings of the variables can be marked. Assume 

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The residual standard deviation is Sr = 5.558, the forecast standard deviation is Sq = 5.63 in the middle of the experimental domain. The coefficient of the linear correlation is r = 0.9966. The table below gives some examples of estimations compared with the extreme experimental values found in literature. The confidence range at 95% of the forecast is added in order to show the quality of the estimate. 

FIGURE 7 Experimental domains when examining (a) two and (b) three factors with either the OVAT or an experimental design approach. ( ) Nominal level Ex, effect of factor X. 

As the definition says, a model is a description of a real phenomenon performed by means of mathematical relationships (Box and Draper, 1987). It follows that a model is not the reality itself it is just a simplified representation of reality. Chemometric models, different from the models developed within other chemical disciplines (such as theoretical chemistry and, more generally, physical chemistry), are characterized by an elevated simplicity grade and, for this reason, their validity is often limited to restricted ranges of the whole experimental domain. 

This model allows us to estimate a response inside the experimental domain defined by the levels of the factors and so we can search for a maximum, a minimum or a zone of interest of the response. There are two main disadvantages of the complete factorial designs. First, when many factors were defined or when each factor has many levels, a large number of experiments is required. Remember the expression number of experiments = replicates x Oevels) " (e.g. with 2 replicates, 3 levels for each factor and 3 factors we would need 2 x 3 = 54 experiments). The second disadvantage is the need to use ANOVA and the least-squares method to analyse the responses, two techniques involving no simple calculi. Of course, this is not a problem if proper statistical software is available, but it may be cumbersome otherwise. 

To develop our calibration data set using an experimental design (see Chapter 2) in order to be, hopefully, reasonably sure that all the experimental domain is represented by the standard solutions to be prepared. 

Adsorbed Phase Entropy. Since Equations 7 and 8 can accurately describe the relationship between q, T, and p, we may use them to calculate the integral molar entropy of the adsorbed phase. At temperatures significantly lower than critical for the adsorbate, the entropy of the adsorbed phase is usually compared with the entropy of the liquid at same temperature in order to compare the freedom of each phase. Because our experimental domain was higher, we shall make this comparison with the gaseous phase compressed to the same density p as determined by Equation 8. 

The presented block diagrams link the factor-fixing accuracy, range of response change and response-surface curvature with the width of factor-variation interval. When selecting a factor variation interval one should, if possible, account for the number of factor variation levels in the experimental domain. Depending on the number of these levels, are the experiment range and optimization efficiency. 

The efficiency of screening experiment designs depends on the form of experimental domain. If this domain suits a total simplex (0design points-trials is recommended. In that case design of experiments includes q-pure components (Xpl.O), of centroid simplex (X = q for all i=l, 2,..., q) and q-internal points with coordinates ... 

Of course, the perceived importance of factors depends on the experimental domain or design region which is the experimental area to be explored and is also called the experimental frame by Zeigler et al. (2000). The clients must supply information about this domain to the simulation analysts, including realistic ranges of the individual factors and limits on the admissible scenarios or combinations of factor levels for example, in some applications the factor values must add up to 100%. 

The simplest experimental domain is a fc-dimensional hypercube where the jtii original factor is coded such that its respective low and high values, lj and hj, correspond to the values —1 and +1 of the corresponding standardised factor. Thus, the level zj of the original factor is transformed to level Xj of the standardised factor, where... 

Finally it is often useful to be able estimate the experimental error (as discussed in Section 2.2.2), and one method is to perform extra replicates (typically five) in the centre. Obviously other approaches to replication are possible, but it is usual to replicate in the centre and assume that the error is the same throughout the response surface. If there are any overriding reasons to assume that heteroscedasticity of errors has an important role, replication could be performed at the star or factorial points. However, much of experimental design is based on classical statistics where there is no real detailed information about error distributions over an experimental domain, or at least obtaining such information would be unnecessarily laborious. 

Table 4. Variables and experimental domain in the D-optimal screening design of enamine synthesis over molecular sieves...

Response surface models are local Taylor expansion models which are valid only in the explored domain. It is often found that the stationary point on the response surface is remote from the explored domain and in the model may not describe any real phenomenon around the stationary point. Mathematically, a stationary point can be a maximum, a minimum, or a saddle point but it sometimes corresponds to unrealistic reponses (e.g. yield > 100%) or unattainable experimental conditions (e.g. negative concentrations of reactants). When the stationary point is outside the explored domain, the response surface is monotonous in the explored experimental domain and zx directions which correspond to small coefficients will describe rising or falling ridges. Exploring such ridges offers a means for optimizing the response even if the response surface should be oddly shaped. 

If we set x = 0 (the centre of the experimental domain) in the above expression we can compute z0 which is the vector z0 = [z10z20z30] of the z, coordinates of the centre point. This gives... 

Sometimes other variables must be investigated such as the pH and/or the ionic strength of the buffer in the mobile phase or the concentration of additives in the mobile phase such as for instance tensio-active substances in micellar chromatography. In such a case the first step in an optimization is to screen these factors and to identify the most important ones for the subsequent optimization. The screening (Section 6.4.2) leads to a definition of the experimental domain in which the optimum is probably situated. This is somewhat similar to the retention optimization step. It is followed by an optimization step (Sections 6.4 and 6.7), in which the most important variables are changed, often according to an experimental design. Similar methods are used in capillary zone electrophoresis. 

Another approach, which is often used in combination with modelling methods, is the so-called threshold method. In many cases, it is possible to define for each response a part of the experimental domain where the response is at least adequate. By combining these areas, one may then find a set of conditions where all responses are adequate. This is for instance the approach followed when the solvent triangle (see Section 6.5) is applied for multicomponent separations. The quality of the separation of each successive pair of substances in the chromatogram is determined. A limit is imposed (e.g. resolution should be at least 1.5) and for each pair the area in the triangle is obtained where this is the case. The part of the triangle where the criterion is not reached is shaded and this is done for each pair (see Fig. 6.4). The triangles for each binary separation are then superimposed and the area, which stays blank after this operation, is the one where all separations yield acceptable results. 

In cases where several variables have to be optimized, one often uses experimental design. An experimented design is a predefined experimental set-up in which a given number of variables are examined with a given number of experiments. The experiments are chosen such that the experimental domain is mapped (covered) in a systematic way. The experimental design selected depends on the goals of the study that is carried out. For instance, some experimental designs make it possible to estimate the effect or the influence of the variables (often also called factors) on the considered response(s). 

A response function (e.g. Eiq. (6.2)) is established for a limited number (< 3) of variables to find the combination of variables for which the optimal response is obtained and this is known to be situated within the experimental domain defined by the levels of the variables. This is done with a so-called response surface design (Section 6.4.3). 

A mistake that is often made is to consider experimental design primarily as a statistical modelling or interpretation method. In the context of chromatographic separation, the most important aspect of experimental design is in our view that it is a way of systematically mapping the experimental domain. In many situations, such as in Fig. 6.8, it is possible to visually interpret the results. In such cases statistical analysis is not needed. 

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