Number of experiments needed

In Chapter 15, which was based on reference [1] we began our discussions of factorial designs. If we expand the basic rc-factor two-level experiment by increasing the number of factors, maintaining the restriction of allowing each to assume only two values, then the number of experiments required is 2", where n is the number of factors. Even for experiments that are easy to perform, this number quickly gets out of hand if eight different factors are of interest, the number of experiments needed to determine the effect of all possible combinations is 256, and this number increases exponentially. 

The other obvious way we might want to expand the experiment is to increase the number of levels (values) that some or all of the factors take. In this case, the number of experiments required increases even faster than 2". So, for example, if each factor is at three levels, then the number of experiments needed is 3" (for eight factors, corresponding to our previous calculation, this comes to 6,561 experiments ). In the general case, the number of experiments needed is Tr n , where nt is the number of levels of the ith factor. 

It should be clear at this point that the problem with this scenario is the sheer number of experiments needed, which in the real world translates into time, resources, and expense. Something must be done. 

Because variables in models are often highly correlated, when experimental data are collected, the xrx matrix in Equation 2.9 can be badly conditioned (see Appendix A), and thus the estimates of the values of the coefficients in a model can have considerable associated uncertainty. The method of factorial experimental design forces the data to be orthogonal and avoids this problem. This method allows you to determine the relative importance of each input variable and thus to develop a parsimonious model, one that includes only the most important variables and effects. Factorial experiments also represent efficient experimentation. You systematically plan and conduct experiments in which all of the variables are changed simultaneously rather than one at a time, thus reducing the number of experiments needed. 

A number of experiments need to be performed in order to fully understand the mechanisms of 1,4-dioxane degradation. A thorough analysis of 1,4-dioxane byproducts will be evaluated using various analytical instruments. 

Reductive alkylation is an efficient method to synthesize secondary amines from primary amines. The aim of this study is to optimize sulfur-promoted platinum catalysts for the reductive alkylation of p-aminodiphenylamine (ADPA) with methyl isobutyl ketone (MIBK) to improve the productivity of N-(l,3-dimethylbutyl)-N-phenyl-p-phenylenediamine (6-PPD). In this study, we focus on Pt loading, the amount of sulfur, and the pH as the variables. The reaction was conducted in the liquid phase under kinetically limited conditions in a continuously stirred tank reactor at a constant hydrogen pressure. Use of the two-factorial design minimized the number of experiments needed to arrive at the optimal solution. The activity and selectivity of the reaction was followed using the hydrogen-uptake and chromatographic analysis of products. The most optimal catalyst was identified to be l%Pt-0.1%S/C prepared at a pH of 6. 

Let us first consider a half-fraction factorial design. Only half of the number of experiments needed for a full factorial are performed. For... 

Although application of chemometrics in sample preparation is very uncommon, several optimisation techniques may be used to optimise sample preparation systematically. Those techniques can roughly be divided into simultaneous and sequential methods. The main restrictions of a sequential simplex optimisation [6,7] find their origin in the complexity of the optimisation function needed. This function is a predefined function, often composed of several criteria. Such a composite criterion may lead to ambiguous results [8]. Other important disadvantages of simplex optimisation methods are that not seldom local optima are selected instead of global optima and that the number of experiments needed is not known beforehand. 

Dimensional analysis is a useful tool for examining complex engineering problems by grouping process variables into sets that can be analyzed separately. If appropriate parameters are identified, the number of experiments needed for process design can be reduced and the results can be... 

When there are more than two factors, the possible interactions increase. With three factors there can he three, two-way interactions (1 with 2,1 with 3, and 2 with 3), and now one three-way interaction (a term in Xj X2 X3). The numbers and possible interactions build up like a binomial triangle (table 3.1). However, these higher order interactions are not likely to be significant, and the model can be made much simpler without loss of accuracy by ignoring them. The minimum number of experiments needed to establish the coefficients in equation 3.3 is the number of coefficients, and in this model there is a constant, k main effects and Vzk [k— 1) two-way interaction effects. 

If a complete factorial design has three levels (low, middle and high) for each of three factors (A = temperature B = pH and C = reaction time), it is said to be a 3 X 3 X 3 or 3 CFD and 27 runs are required, each one corresponding to a particular combination of the factor levels. Furthermore, if we replicate each run k times, then the number of experiments needed is k x 3. ... 

Efficient experimental design decreases the number of experiments needed to obtain required information and an estimate of uncertainty in that information. A trade-off is that the fewer experiments we do, the greater the uncertainty in the results. 

For decades researchers have been developing in silico models to minimize the number of experiments needed to identify or map the potential epitopes on the antigen surface. Because of the basic differences in the recognition of B- and T-cell epitopes, researchers have derived separate algorithms and tools for the two types of epitope. This chapter discusses only B-cell epitope prediction models (linear and conformational). Although they are not very different from basic B-cell epitope algorithms, T-cell epitope models have been reviewed in detail elsewhere (7, 8). 

The second method relies on the experimental determination of the kinetic parameters using techniques from biophysics or enzymology. Also in this case problems exist (1) the kinetic parameters are often determined under conditions different from the conditions in the cytoplasm (2) an enormous number of experiments need to be done, even for a network of moderate size, to determine all kinetic parameters experimentally. When the second method is used to parameterize a kinetic model then the resulting model is considered a silicon cell model. A number of silicon cell models exist [25-27, 29, 75-77]. 

The predictive method of Jandera et al. [628] requires knowledge of the isocratic retention data of all solute components in binary and (preferably) ternary mobile phase mixtures. Once these data are available, the method may be very helpful in obtaining an adequate (but not an optimum) separation with a ternary gradient. Unfortunately, the data required for the application of this predictive method are almost never available, and hence a large number of experiments need to be performed before any predictions can take place. When this is the case the method is of very little practical use. 

The CO + OH CO2 + H reaction has been reported as showing mass independent fractionation [11]. This term, in the strict sense of "mass-independence" used earlier means that the slope of a O/ O fractionation plotted versus the corresponding 0/ 0 fractionation would be about unity, rather than having the conventional mass-dependent value of 0.52. However, what is frequently meant by MIF of oxygen instead is that the quantity A O = S O — 0.525 0 is different from zero. The number of experiments needed to determine is only one, instead of the number needed to determine the slope of a 3-isotope and is commonly used in the literature. Indeed, sometimes the data are such that a three-isotope plot cannot be obtained from the available data. To avoid possible confusion one might term reactions that have a non zero value for A O but for which the vs. slope has not been measured, as being mass-anomalous fractionated, MAF. However, the term MIF is in widespread usage, and includes both the strictly MIF and any mass anomalous fractionation (MAF) reactions. 

The central composite design was often selected because of the limited number of experiments needed to sample the response surfaces. In the separation of As and Se species in tap water, the analysis of isoresponse curves allowed the determination of optimum chromatographic conditions and the robustness of the method [77]. The same design was also used to study the influence of an organic modifier and IPR concentration on retention of biogenic amines in wines. To obtain a compromise between resolution and chromatographic time, optimization through a multi-criteria approach was followed [78]. 

In our study of search problems we have seen that single variable systems can be optimized with ease two-variable systems, with some effort and multivariable systems, only with extreme difficulty if at all. As more variables enter a search problem, the number of experiments needed grows rapidly, and the unimodality assumption becomes less and less plausible. Thus our investigation of search problems leads directly to interaction problems, where the criterion of effectiveness depends on so many factors that it is impractical, or even impossible, to find the optimum by conventional methods. Successful techniques for solving interaction problems involve decomposing a big system into several smaller ones, as we have already done with our lines of search. 

Therefore, we have to analyse the variation of the rate of permeation according to the temperature (zj), the trans-membrane pressure difference (Z2) and the gas molecular weight (Z3). Then, we have 3 factors each of which has two levels. Thus the number of experiments needed for the process investigation is N = 2 = 8. Table 5.13 gives the concrete plan of the experiments. The last column contains the output y values of the process (flow rates of permeation). Figure 5.8 shows a geometric interpretation for a 2 experimental plan where each cube corner defines an experiment with the specified dimensionless values of the factors. So as to process these statistical data with the procedures that use matrix calculations, we have to introduce here a fictive variable Xq, which has a permanent +1 value (see also Section 5.4.4). 

Eor a process with k factors and one response, relation (5.117) can be used to estimate the number of experiments needed by a sequential plan ... 

Candidate drugs being evaluated for development are often one of a series of related compounds that may have similar chemical properties, that is, similar paths of degradation may be deduced. However, this rarely tells us the rate at which they decompose, which is of more importance to pharmaceutical development terms. To elucidate their stability with respect to temperature, pH, light, and oxygen, a number of experiments need to be performed. The major objectives of the preformulation team are, therefore, to identify conditions to which the compound is sensitive, and to identify degradation profiles under these conditions. 

In formulation studies, all the factors affecting the stability of the pharmaceutical product have to be considered. Because the stability of pharmaceuticals is generally affected by numerous and complex factors, quantitative analysis of the role of each would involve a very large and complex series of experiments. The effect of each individual factor would have to be tested under conditions in which all other factors are maintained constant. Factorial analysis attempts to minimize the number of experiments needed to get meaningful results and, therefore, to save time and labor. 

Consider the screening problem of table 2.20. Here we want to know the effect of various diluents, disintegrants, lubricants etc. on the stability of the drug substance. We have not imposed any restraint whatsoever on the numbers of levels for each variable. We could of course test all possible combinations with the full factorial design, which consists of 384 experiments (4 x 2 x 3 x 4 x 2 x 2) Reference to the general additive screening model for different numbers of levels will show that the model contains 12 independent terms. Therefore the minimum number of experiments needed is also 12. 

Even though for this kind of study both quantitative and qualitative factors (especially the latter) may be set at more than 2 levels, the number of coefficients in the model equation, and therefore the number of experiments needed to determine them, both increase sharply with the number of levels once we have decided to study interactions between variables. We will therefore also limit both kinds of variables to 2 levels, in this chapter. This is an artificial limitation and might sometimes prove to be too restrictive, especially in the case of qualitative factors. 

The models most often used to describe the response are first-, second-, and, very occasionally, third-order polynomials. The number of coefficients in a polynomial increases very rapidly with the number of variables and the degree of the model and the number of experiments needed increases at least as rapidly. Also a given model will describe the phenomenon that we are trying to model better over a restricted experimental domain than over a wide one. These three considerations imply certain conditions for using response surface modelling. 

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