Fractional factorial designs with full factorials

It is possible to selectively choose a subset of 4 of the original 8 factor combinations and use these to fit the reduced model with 100% efficiency. The resulting design is called a fractional factorial design . A full 2 factorial design has two half-replicates as shown in Figures 14.4 and 14.5, or in cube plot form as ... 

Additional constraints in a Simplex method further reduce the area of interest from a full Simplex space. The constraints come from additional information about the possible range of the ingredients. For example, some new information obtained from a fractional factorial design with two ingredients showed that the optional concentration of an ingredient X is 3%. We can restrict the Simplex space with the constraint 0.01 < Xi < O.OS. Since the sum of the ingredients totals 100% of the mixture, a consistent range for X2 would be 0.95 < X2 < 0.99. 

The fractional factorial design is based on an algebraic method of calculating the contributions of factors to the total variance with less than a full factorial number of experiments. Such designs are useful when the numbers of potential factors are relatively large because they reduce the total number of runs required for the overall experiment. However, by reducing the number of runs, a fractional factorial design will not be able to evaluate the impact of some of the factors independently. 

A reduction in the number of necessary test mns, also with the possibility of a larger number of factors (e.g., 11 [30]), can be achieved with fractional factorial experimental plans (fractional factorial designs), which are derived from full... 

A full factorial design contains all possible combinations (L ) between the different factors f and their levels L, with L = 2 for two-level designs. It allows estimating all main and interaction effects between the factors. A FF design will only perform a fraction of the full factorial. A two-level FF design 2 examines factors, each at two levels, in 2 experiments, with 1/2"... 

Note that as the 2 full factorial designs needs 8 runs, it is also possible to select the other half fraction of the experiments, with the complementary signs in column C. In this case the generator would be denoted C = - AB. Furthermore, the product ABC renders a column, denoted I, with all elements showing positive signs. So we call 1 = ABC the "defining relation for this design. 

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-... 

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