Fixed experimental design

In the following two sections we will describe two kinds of interpretive methods. In section 5.5.1 we will discuss simultaneous methods, which involve a fixed experimental design. In the iterative procedures of section 5.5.2, an initial design that consists of a minimum number of experiments is used and the location of the next data point is calculated during the optimization process. 

The model is then used in a calculation step to predict the location of the optimum. This step involves the calculation of the response surface from the retention surfaces using a suitable criterion, the location of the (predicted) optimum on this surface, and a decision about new experiments to be performed. Only this last aspect distinguishes iterative designs from the previously described fixed experimental designs. 

However, there is a risk that the global optimum will not be found if large areas remain unsearched. Therefore, a combination of the two different interpretive methods, a fixed experimental design followed by an iterative procedure to refine the location of the optimum, may be the best possible approach, even though a slightly larger number of experiments may be required than for either of the two methods separately. 

An interpretive method, which combines an initial fixed experimental design with an iterative refinement of the optimum, appears to be the most promising approach. 

In that case, interpretive methods based on fixed experimental designs (window diagrams) may be used. 

The Sentinel method is the outstanding exponent of the group of interpretive methods, as it has already been applied successfully for selectivity optimization in programmed solvent LC. However, other interpretive methods, based either on fixed experimental designs or on iterative procedures, can be applied along the same lines. It was seen in section 6.3.2.3 that the extension of the Sentinel method to incorporate gradient optimization was fairly straightforward. 

Experimental descriptors emerge from a fixed experimental design, and their appearance is subject to the physical or chemical limitations of the measurement technique. The advantage of artificial descriptors is that they can be adjusted and fine-tuned easily to fit to a task due to their pure mathematical nature. The only limitation to this approach is the scientist s imagination. However, there are several constraints to be taken into account when selecting or constructing a molecular descriptor. Todeschini and Consonni pointed these out in their book Handbook of Molecular Descriptors [15]. Let us have a closer look at these constraints. 

Experimental Descriptor is a descriptor that results from an analytical technique, such as spectrometry. Experimental descriptors emerge from a fixed experimental design, and their appearance is subject to the physical or chemical limitations of the measurement technique. 

Finding the End Point Potentiometrically Another method for locating the end point of a redox titration is to use an appropriate electrode to monitor the change in electrochemical potential as titrant is added to a solution of analyte. The end point can then be found from a visual inspection of the titration curve. The simplest experimental design (Figure 9.38) consists of a Pt indicator electrode whose potential is governed by the analyte s or titrant s redox half-reaction, and a reference electrode that has a fixed potential. A further discussion of potentiometry is found in Chapter 11. 

The initial experimental design is shown in Figure 10-14. Water and acetic anhydride are gravity-fed from reservoirs and through a set of rotameters. The water is mixed with the acetic anhydride just before it enters the reactor. Water is also circulated by a centrifugal pump from the temperature bath through coils in the reactor vessel. This maintains the reactor temperature at a fixed value. A temperature controller in the water bath maintains the temperature to within 1°F of the desired temperature. 

Methods of experimental design discussed in most basic statistics books can be applied equally well to minimizing fix) (see Chapter 2). You evaluate a series of points about a reference point selected according to some type of design such as the ones shown in Figure 6.1 (for an objective function of two variables). Next you move to the point that improves the objective function the most, and repeat. 

The estimation of purely experimental uncertainty is essential for testing the adequacy of a model. The material in Chapter 3 and especially in Figure 3.1 suggests one of the important principles of experimental design the purely experimental uncertainty can be obtained only by setting all of the controlled factors at fixed levels and replicating the experiment. 

The art of experimental design is made richer by a knowledge of how the placement of experiments in factor space affects the quality of information in the fitted model. The basic concepts underlying this interaction between experimental design and information quality were introduced in Chapters 7 and 8. Several examples showed the effect of the location of one experiment (in an otherwise fixed design) on the variance and co-variance of parameter estimates in simple single-factor models. 

The above treatment has implicitly assumed that the experimental design was such that the number of trials was fixed at 12 and the observation was the number of heads. However, an alternative design could have been to continue tossing the coin until 3 tails were obtained, and the observation would be n, the number of tosses required to produce the 3 tails. In this case, the statistic for judging the data is just n. But the distribution of n, the number of tosses to produce 3 tails, is given by the negative binomial ... 

If a classic approach were used to estimate the regression coefficients, the process woulSbe to set the variables in the matrix R to predetermined values as dictated by the experimental design. This does not make sense when the measurement tem for R is spectroscopy. One cannot choose to set the different waveld hs to fixed values and collect concentration information on the coiresponSng samples. What can be controlled is the concentration of the components mthe samples (i.c., c). The approach, therefore, is to choose samples with iarying concentrations and measure the spectra on these samples. This is ffe opposite of the classical approach where the independent variable (X) and the dependent variable (y) is measured. 

At this point an attempt has been made to identify all of the important v ari-ables. It has also been decided which variables will be fixed and which will be varied. For the nonfixed variables, the range and number of levels (complexity) have been determined. Classical experimental design tools will be used to specify the design for the controllable variables. The design is selected via a software package or statistical reference materials and it specifies (1) the number of runs, (2) the levels of the variable(s), and (3) tlie order of the runs. 

Amination of i-butanol to diisobutylamine was investigated on vanadium modified granulated Raney nickel catalyst in a fixed bed reactor. The addition of 0.5 wt.% V to Raney nickel improved the yield of amines and the stability of catalyst. Factorial experimental design was used to describe the conversion of alcohol, the yield and the selectivity of secondary amine as a function of strong parameters, i.e. the reaction temperature, space velocity and NHs/i-butanol molar ratio. Diisobutylamine was obtained with 72% yield at 92% conversion and reaction parameters P=13 bar, T=240°C, WHSV=1 g/g h, and molar ratios NH3/iBuOH= 1.7, H2/NH3= 1.9. 

Components with a fixed concentration were Primojel (4%), oxazepam (4%), magnesium stearate (1%) and Aerosil 200R (0.2%). The components with a variable concentration were a-lactose, 6-lactose and rice starch. The concentrations of these components sum up to 90,8% of the total tablet weight, their individual concentrations can be obtained from the experimental design listed in Table 4.3 where the listed values of Xj-Xj are fractions of the total amount of 90.8% for these three components. 

Known Variables — Controllable In any experimental situation, cettain conditions are held constant during the course of the experiment. A batch of copolymer may be made with a measured quantity of catalyst, at a given reaction temperature, and reacted for a certain definite time. The vessel is free of water, die monomers are charged in a definite proportion, and any other conditions that may affect the result are held fixed. The next portion of the experiment may involve one of these conditions controlled at a new level while the others remain fixed. Proper design of experimental programmes presupposes ability to control the important factors so that the variation due to these factors can be calculated. In Chapter VI, experimental designs for various situations are coveted. 

Common practice consists in investigating the influence of one experimental variable (hereafter we will refer to it as a factor while keeping other factors at a fixed value. Then, another factor is selected and modified to perform the next set of experiments, and so forth. This one-factor-at-a-time strategy has been shown to be inefficient and expensive it lacks the ability to detect the joint influence of two or more factors (z.e. it cannot address interactions) and often needs many experiments. An increase in efficiency can be achieved by studying several factors simultaneously and systematically by means of an appropriate type of experimental design. In such a way, the experiments will be able to detect the influence of each factor and also the influence of two or more factors because every observation gives information about all factors. 

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