Factorial-based designs

After the preceding considerations have been taken into account, a test plan is developed to best meet the goals of the program. This might involve one of the standard plans developed by statisticians. Such plans are described in various texts (Table 1) and are considered only briefly here. Sometimes, combinations of plans are encountered, such as a factorial experiment conducted in blocks or a central composite design using a fractional factorial base. 

In this chapter we explore factorial-based experimental designs in more detail. We will show how these designs can be used in their full factorial form how factorial designs can be taken apart into blocks to minimize the effect of (or, if desired, to estimate the effect of) an additional factor and how only a portion of the full factorial design (a fractional replicate) can be used to screen many potentially useful factors in a very small number of experiments. Finally, we will illustrate the use of a Latin square design, a special type of fractionalized design. 

Most screening designs are based on saturated fractional factorial designs. The firactional factorial designs in Section 14.8 are said to be saturated by the first-order factor effects (parameters) in the four-parameter model (Equation 14.27). In other words, the efficiency E = p/f = 4/4 = 1.0. It would be nice if there were 100% efficient fractional factorial designs for any number of factors, but the algebra doesn t work out that way. 

A section has been added to Chapter 1 on the distinction between analytic vs. enumerative studies. A section on mixture designs has been added to Chapter 9. A new chapter on the application of linear models and matrix least squares to observational data has been added (Chapter 10). Chapter 13 attempts to give a geometric feel to concepts such as uncertainty, information, orthogonality, rotatability, extrapolation, and rigidity of the design. Finally, Chapter 14 expands on some aspects of factorial-based designs. 

This section has given an overview of some of the experimental designs that are suitable for collecting data to estimate the coefficients of the first-order and second-order model. Many of these designs are based on factorial and fractional factorial designs. 

Finally it is often useful to be able estimate the experimental error (as discussed in Section 2.2.2), and one method is to perform extra replicates (typically five) in the centre. Obviously other approaches to replication are possible, but it is usual to replicate in the centre and assume that the error is the same throughout the response surface. If there are any overriding reasons to assume that heteroscedasticity of errors has an important role, replication could be performed at the star or factorial points. However, much of experimental design is based on classical statistics where there is no real detailed information about error distributions over an experimental domain, or at least obtaining such information would be unnecessarily laborious. 

Due to the number of variables conventional testing of all the different combinations would require a minimum of 256 experiments eight variables tested at their minimum and maximum values only giving 2 experimental combinations. If apart from the extreme conditions any of the variables were further analysed at intermediate values the total number of experiments would be greatly increased. Thus, in order to reduce the number of experiments required to analyse this number of variables a factorial experimental design of base 2 broken down to 1/16 reduces this to only sixteen experiments. 

One of the most useful symmetrical designs is based on the 4 factorial design (table 2.17), described (using the previous notation) as 4V/4 5 factors at 4 levels are screened in 4 = 16 experiments. We again emphasize that the numbers 0, 1,2, and 3 identify qualitative levels of each variable and have no quantitative significance whatsoever. 

Melzer, G., Esfandabadi, M.E., Franco-Lara, E., and Wittmann, C. (2009) Flux design In silica design of cell factories based on correlation of pathway fluxes to desired properties. BMC Syst. Biol., 3 (1), 120. 

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. 

In this period development efforts in the individual unit operations (sub tasks) leading to the total process are described Where found suitable, fractional factorial design experiments based upon the Taguchi method were conducted to optimize a particular unit operation. No attempt was made at this stage to link various unit operations for the sake of total process optimization. 

An / symmetric B Factory Based on PEP, Conceptual Design Report LBL PUB-5303/SLAC-372/CALT-68-1715, revised June, 1993. 

Factor Interactions - Factor interaction exists when the response does not solely depend on the individual factors but rather on a combination of two or more factors. This interaction effect (which is often important especially in chemical process experiments) can be missed by the use of designs other than a factorial based one. For example, a one factor at a time design will never show the presence of interactions. 

In the early days of the Industrial Revolution, when people first started making parts and systems on a mass scale, it was often enough to rely on the cost and precision advantages afforded by a factory-based production system. In today s world, with an ever increasingly competitive world economy, that is not enough. We need to design parts and products in such a way that we eliminate unnecessary processes. 

Table 14.5 lists the uncoded factor levels, coded factor levels, and responses for a 2 factorial design. Determine the coded and uncoded empirical model for the response surface based on equation 14.10. 

C. Daniel, App/ications of Statistics to lndustria/Experimentation, ]oE Wiley Sons, Inc., New York, 1976. This book is based on the personal experiences and insights of the author, an eminent practitioner of industrial appHcations of experimental design. It provides extensive discussions and concepts, especially in the areas of factorial and fractional factorial designs. "The book should be of use to experimenters who have some knowledge of elementary statistics and to statisticians who want simple explanations, detailed examples, and a documentation of the variety of outcomes that may be encountered." Some of the unusual features are chapters on "Sequences of fractional repHcates" and "Trend-robust plans," and sections entided, "What is the answer (what is the question )," and "Conclusions and apologies."... 

Mathews and Rawlings (1998) successfully applied model-based control using solids hold-up and liquid density measurements to control the filtrability of a photochemical product. Togkalidou etal. (2001) report results of a factorial design approach to investigate relative effects of operating conditions on the filtration resistance of slurry produced in a semi-continuous batch crystallizer using various empirical chemometric methods. This method is proposed as an alternative approach to the development of first principle mathematical models of crystallization for application to non-ideal crystals shapes such as needles found in many pharmaceutical crystals. 

The most recent UK Royal Ordnance Factory (ROF) continuous process for the manuf of TNT is described by Thomas (Ref 90b). It uses a novel method of achieving continuous counter-current contacting between heavy and light phases in the trinitration section and some of the washing stages . The plant is also designed to operate based on the use of 96—100% sulfuric acid in the trinitration stage instead of the oleum used at Radford TNT Purification. 

Gonzalez, A. G., TWo Level Factorial Experimental Designs Based on Multiple Linear Regression Models A Tutorial Digest Illustrated by Case Studies, Analytica Chimica Acta 360, 1998, 227-241. 

A two level full factorial experimental design with three variables, F/P molar ratio, OH/P wt %, and reaction temperature was implemented to analyses the effect of variables on the synthesis reaction of PF resol resin. Based on the composition of 16 components of 10 samples, the effect of three independent variables on the chemical structure was anal3 ed by using 3 way ANOVA of SPSS. The present study provides that experimental design is a very valuable and capable tool for evaluating multiple variables in resin production. 

To extract and evalnate the color pigments from cochineals Dactylopius coccus Costa), a simple method was developed. The procednre is based on the solvent extraction of insect samples nsing methanol and water (65 35, v/v) and a two-level factorial design to optimize the solvent extraction parameters temperature, time, methanol concentration in mixtnre, and yield. For hydrophilic colorants that are more sensitive to temperatnre, water is the solvent of choice. For example, de-aerated water extraction at low temperatnre was applied to separate yellow saffrole and carthamine from saffron (Carthamus tinctorius) florets that contain about 1% yellow saffrole and 0.3% red carthamine. ... 

The simplest and cheapest procedure to obtain standards is based on selective extraction followed by crystallization. A method developed to obtain lycopene from tomato residue using factorial experimental design consisted of a preliminary water removal with ethanol, followed by extraction with EtOAc and two successive crys-talhzation processes using dichloromethane and ethanol (1 4), producing lycopene crystals with 98% purity, measured by HPLC-PDA. Using this approach, bixin was extracted with EtOAc from annatto seeds that were previously washed with... 

Because of the emphasis on experimental design. It Is required that a statistician serve as a member of the design team The assigned tasks and responsibilities for the statistician differ from those for the scientists The primary mechanism for obtaining the experimental design Is to require each scientist on the team to make explicit, documented, numerical predictions for all combinations of the test conditions specified In a factorial table In effect, such predictions require each scientist to quantify the effects of the experimental factors (control variables) on the dependent variable These predictions are based on the scientist s knowledge and assessment of related literature, data, experience, etc Candidate team members who are unable or unwilling to make such predictions are excluded from the team ... 

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