Factorial experiments with mixture

In certain practical applications it is advantageous to consider variation of properties not with absolute amounts of components, but with their ratios. If the percentage of each component is not zero, then given upper and lower constraints for the components, ratios of components may be utilized to build conventional factorial designs [22]. The number of ratios in a q-component system is q-1  

using the component ratios as independent factors, the dimensionallity of the problem is reduced by one, and hence the number of experiments is also decreased. 

The points on the line originating from vertex X2 feature a constant ratio of components Xi and X3. In a like manner, the line originating from vertex Xi is the locus of equal ratios of X3 to X2. To meet the orthogonality condition for the design matrix the recourse is made to the linear transformation of Eq. (2.59). 

We seek the functional relationship between the yield of sodium and potassium bicarbonates and the composition of the initial sylvanite solution. The factors controlling the potassium utilization, in the carbonization process are chosen to be the per cent ratios of two of three components making up the system  

To derive the regression equation, we shall use a second-order orthogonal design for k=2, N=9 and the star arm a=l, Fig. 3.31. 

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Finally, the problem was resolved by irradiating standards and mixtures of standards in a factorial experiment. The experiment design was a full factorial experiment with three variables, mercury, selenium, and ytterbium, at two levels with replication and with a center point added to test higher order effects. The pertinent information on treatments and levels of variables are shown in Table VII. 

In the case of constraints on proportions of components the approach is known, simplex-centroid designs are constructed with coded or pseudocomponents [23]. Coded factors in this case are linear functions of real component proportions, and data analysis is not much more complicated in that case. If upper and lower constraints (bounds) are placed on some of the X resulting in a factor space whose shape is different from the simplex, then the formulas for estimating the model coefficients are not easily expressible. In the simplex-centroid x 23 full factorial design or simplex-lattice x 2n design [5], the number of points increases rapidly with increasing numbers of mixture components and/or process factors. In such situations, instead of full factorial we use fractional factorial experiments. The number of experimental trials required for studying the combined effects of the mixture com-... 

Supplementation of Cereals. The next chapter of the story comes from experience in supplementing these breakfast cereal diets with other nutrients. Initially, the breakfast cereal was supplemented in a factorial fashion with protein, a vitamin mixture and several mineral mixtures. Only protein and a trace mineral group containing copper and zinc showed significant effects (6). Next, individual nutrients and pairs of nutrients were tried (7). 

As shown, mixture components are subject to the constraint that they must equal to the sum of one. In this case, standard mixture designs for fitting standard models such as simplex-lattice and simplex-centroid designs are employed. When mixtures are subject to additional constraints, constrained mixture designs (extreme-vertices) are then appropriate. Like the factorial experiments discussed above, mixture experimental errors are independent and identically distributed with zero mean and common variance. In addition, the true response surface is considered continuous over the region being studied. Overall, the measured response is assumed to depend only on the relative proportions of the components in the mixture and not on the amount. 

Mixture designs are applied in cases where the levels of individual components in a formulation require optimization, but where the system is constrained by a maximum value for the overall formulation. In other words, a mixture design is often considered at this stage when the quantities of the factors must add to a fixed total. In a mixture experiment, the factors are proportions of different components of a blend. Mixture designs allow for the specification of constraints on each of the factors, such as a maximum and/or minimum value for each component, as well as for the sum and/or ratio of two or more of the factors. These designs are very specific in nature and are tied to the specific constraints that are unique to the particular formulation. However, as with the discussion of the fractional factorial designs, in order to be most efficient, it is important to provide realistic prior expectations on anticipated effects so the smallest design can be set up to fit the simplest realistic model to the data. 

In calibration it is normal to use several concentration levels to form a model. Indeed, for information on lack-of-fit and so predictive ability, this is essential. Hence two level factorial designs are inadequate and typically four or five concentration levels are required for each compound. However, chemometric techniques are most useful for multicomponent mixtures. Consider an experiment earned out in a mixture of methanol and acetone. What happens if the concentrations of acetone and methanol in a training set are completely correlated If the concentration of acetone increases, so does that of methanol, and similarly with a decrease. Such an experimental arrangement is shown in Figure 2.25. A more satisfactory design is given in Figure 2.26, in which the two... 

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