Factorial designs with fractional factorials

Persson Stubberud, K., and Astrom, O. (1998). Separation of ibuprofen, codeine phosphate, their degradation products and impurities by capillary electrophoresis I. Method development and optimization with fractional factorial design. ]. Chromatogr. A 798, 307—314. 

Factorial designs with more than two levels of the factors are quite common, and mixed factorial designs in which the several factors have different numbers of levels might fit certain experimental requirements. The fractional replication of designs of this type is somewhat hazardous, since balanced arrangements are hard to come by ... 

It is impossible to evaluate all possible combinations of substrates, reagents, and solvents by experiments. It is quite cumbersome, even to run a complete factorial design with selected substrates, reagents and solvents, as was described in the examples above. To achieve a more manageable number of test systems, it is possible to use the principles of fractional factorial designs to select test systems by their principal properties To illustrate this, we shall once more make use of the Willgerodt-Kindler reaction. 

Maio, G., von Holst, C., Wenclawiak, B. W., and Darskus, R., Supercritical fluid extraction of some chlorinated benzenes and cyclohexanes from soil optimization with fractional factorial design and simplex. Ana/. Chem., 69, 601-606, 1997. 

If this appears insufficient, the design may be replicated n times. The effects are then estimated with a precision of a/(nN), improved by a factor of V . This also has the advantage that the repeatability o may be estimated. Alternatively N could be increased by employing a larger Plackett-Burman design, or fractional factorial design, possibly carried out in stages, or blocks (1). It may be then be possible to estimate further effects (interactions), as we shall see in chapter 3. 

We take the full factorial design with the number of variables required and select half the experiments. Each half fraction can in turn be divided into quarter fractions. Whereas for a given number of variables at 2 levels there is only one possible full factorial design, there can be many possible fractional factorial designs and they are not at all equivalent to one another. We saw in paragraph II.C.3 that the 2 design consists of 16 experiments representing all possible combinations of... 

With fractional factorial designs the estimates of the effects are always aliased. We need to know how serious is the confounding. If two main effects are confounded then the matrix is not very useful. If a main effect is confounded only with second order interactions then it is very likely that the interaction could be neglected. The... 

After analysing the results, the experimenter may well wish for clarification and he will very likely need to continue the experiment with a second design. This is usually of the same number of experiments as the original, the design remaining fractional factorial overall. 

A 2 fracrional factorial experimental design (7,9) is fonned of k columns corresponding to the k variables, each of which can take two distinct levels, noted (-) and (+) and of Af = 2 " rows corresponding to the 2 experimenis. it represents the fraction 1/2 of the full 2 design 2 " = 2V2. it can be built from a 2" full factorial design, with m = i - r independent variables, the slmcturcs of the columns of the k-m variables left being combinations of the m basic columns. 

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... 

O. L. Davies and co-workers. The Design andAna/ysis of Industria/Experiments, 2nd ed., Hafner, New York, 1956 reprinted by Longman, New York, 1987. This book, which is a sequel to the authors basic text Statistica/Methods in Eesearch and Production, is directed at industrial situations and chemical appHcations. Three chapters are devoted to factorial experiments and one chapter to fractional factorial plans. A lengthy chapter (84 pp.) discusses the deterrnination of optimum conditions and response surface designs, which are associated with the name of G. Box, one of the seven co-authors. Theoretical material is presented in chapter appendices. 

Cropley made general recommendations to develop kinetic models for compUcated rate expressions. His approach includes first formulating a hyperbolic non-linear model in dimensionless form by linear statistical methods. This way, essential terms are identified and others are rejected, to reduce the number of unknown parameters. Only toward the end when model is reduced to the essential parts is non-linear estimation of parameters involved. His ten steps are summarized below. Their basis is a set of rate data measured in a recycle reactor using a sixteen experiment fractional factorial experimental design at two levels in five variables, with additional three repeated centerpoints. To these are added two outlier... 

Because the experimental expenditure increases strongly with the increasing number of influence factors, fractional factorial design FFD (partial factorial design) is applied in such cases. It is not possible to evaluate all the interactions by FFDs but only the main effects. 

Abstract A preconcentration method using Amberlite XAD-16 column for the enrichment of aluminum was proposed. The optimization process was carried out using fractional factorial design. The factors involved were pH, resin amount, reagent/metal mole ratio, elution volume and samphng flow rate. The absorbance was used as analytical response. Using the optimised experimental conditions, the proposed procedure allowed determination of aluminum with a detection limit (3o/s) of 6.1 ig L and a quantification limit (lOa/s) of 20.2 pg L, and a precision which was calculated as relative standard deviation (RSD) of 2.4% for aluminum concentration of 30 pg L . The preconcentration factor of 100 was obtained. These results demonstrated that this procedure could be applied for separation and preconcentration of aluminum in the presence of several matrix. 

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"... 

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