Factorial, two-level

Daniel, C. (1959). Use of half-normal plots in interpreting factorial two-level experiments. Technometrics 1, 311-341. 

THE EXPERIMENTAL CONDITIONS FOR A FULL FACTORIAL TWO LEVEL DESIGN TO TEST THREE HPLC FACTORS... 

Table 4.10 Central composite design for three factors consisting of a full-factorial two-level design and star design.

Two-Level Factorial Design with Three Variables... 

The table of results is laid out in a column, and a second column is constructed in which in the hrst four rows the results would be added sequentially in pahs, e. g. Xi + X2, xj, + X4, x + jcg etc., and the lower four rows are calculated by subuacting the second value from dre preceding value thus, JC2 — JCi, JC4 — JC3 etc., a thh d column is prepared from these results by canying out the same sequence of operations. The process is continued until there are as many columns as the number of variables. Thus in the present tluee-variable, two level-study the process is repeated tluee times (Table 15.1), and in the general -variable, two-level case it is repeated n times. (The general description of uials of this kind where tlrere are n variables and two levels, is 2 factorial uials ). 

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 all the variables that influence the properties of the final product are known, one can use a statistical design (known as a one-half factorial) to optimize the properties of the GPC/SEC gels. Factorial experiments are described in detail by Hafner (10). For example, four variables at two levels can be examined in eight observations. From these observations the significance of each variable as related to the performance of the gel can be determined. An example of a one-half factorial experiment applied to the production of GPC/SEC gel is set up in Table 5.2. The four variables are the type of DVB, amount of dodecane, type of methocel, and rate of stirring. 

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. 

The final structure of resins produced depends on the reaction condition. Formaldehyde to phenol (F/P) and hydroxyl to phenol (OH/P) molar ratios as well as ruction temperahne were the most important parameters in synthesis of resols. In this study, the effect of F/P and OH/P wt%, and reaction temperature on the chemical structure (mono-, di- and trisubstitution of methyrol group, methylene bridge, phenolic hemiformals, etc.) was studied utilizing a two-level full factorial experimental design. The result obtained may be applied to control the physical and chemical properties of pre-polymer. 

A two levels of full factorial experimental design with three independent variables were generated with one center point, which was repeated[3]. In this design, F/P molar ratio, Oh/P wt%, and reaction temperature were defined as independent variables, all receiving two values, a high and a low value. A cube like model was formed, with eight comers. One center point (repeated twice) was added to improve accuracy of the design. Every analysis results were treated as a dependent result in the statistical study. 

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. ... 

Based on the experimental data kinetic parameters (reaction orders, activation energies, and preexponential factors) as well as heats of reaction can be estimated. As the kinetic models might not be strictly related to the true reaction mechanism, an optimum found will probably not be the same as the real optimum. Therefore, an iterative procedure, i.e. optimization-model updating-optimization, is used, which lets us approach the real process optimum reasonably well. To provide the initial set of data, two-level factorial design can be used. 

Table 3. Conversion, selectivity, yield, and reaction rate at various conditions based on two-level factorial design.

This study shows that the optimization of process conditions could be achieved rapidly by a judicious use of statistics and parallel reactors. A two-level factorial method with two center points was used to limit the total number of experiments to ten. Using two identical high-pressure reactors in parallel further shortened the time required to conduct these experiments. For the model reaction of phenol hydrogenation over a commercially available Pd/C, it was experimentally determined that the optimal yield was 73% at 135 °C, 22.5 bar, and 615 ppm w/w NaOH... 

Fig. 5.2. Geometrical representation of a complete two level factorial matrix (three influence factors) with experiments in the centre point (0,0,0)... 

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. 

A replicated two-level factorial experiment is carried out as follows (the dependent variables are yields) ... 

---